Mathematics

""I refuse to prove that I exist," says God, "for proof denies faith, and without faith, I am nothing." "Oh," says man, "but the Babel fish is a dead give-away, isn't it? It proves You exist, and so therefore You don't." "Oh, I hadn't thought of that," says God, who promptly vanishes in a puff of logic. "Ah, that was easy," says man, and for an encore goes on to prove that black is white, and gets killed on the next zebra crossing. Most leading theologians claim that this argument is a load of dingo's kidneys."

"All men are mortal. Socrates was mortal. Therefore, all men are Socrates."

"You can prove anything you want by coldly logical reason—if you pick the proper postulates. We have ours and Cutie [robot QT-1] has his." "Then let’s get at those postulates in a hurry. The storm’s due tomorrow." Powell sighed wearily. "That’s where everything falls down. Postulates are based on assumptions and adhered to by faith. Nothing in the Universe can shake them. ..."

"Aristotle is noted for his writings on logic, physics, biology, psychology, metaphysics, ethics, politics, and literature. These works are marked by sober tone, subtle analysis, and empirical accuracy. In logic, he produced the first textbooks ever written. They deal with some of the problems which Plato had suggested but not considered in detail."

"In the logic of science there is a principle as important as that of parsimony: it is that of sufficient reason. The former directs us to look for simplest causes, the later cautions us not to simplify so far that the explanation is inadequate to the facts to be explained....Parsimony is not itself a simple criterion of a good methodology; we cannot simply count the factors of explanation and say that the theory containing the smallest number is the best. The ideal of parsimony cannot be expressed without the proviso that the conditions for which it is a norm shall themselves be adequate."

"LOGIC, n. The art of thinking and reasoning in strict accordance with the limitations and incapacities of the human misunderstanding. The basic of logic is the syllogism, consisting of a major and a minor premise and a conclusion -- thus:"

"R. A. Fisher, J. Neyman, R. von Mises, W. Feller, and L. J. Savage denied vehemently that probability theory is an extension of logic, and accused Laplace and Jeffreys of committing metaphysical nonsense for thinking that it is."

"No, no, you're not thinking; you're just being logical."

"If the world were a logical place, men would ride side saddle."

"Logic is a large drawer, containing some useful instruments, and many more that are superfluous. A wise man will look into it for two purposes, to avail himself of those instruments that are really useful, and to admire the ingenuity with which those that are not so, are assorted and arranged."

"Contrariwise," continued Tweedledee, "if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic."

"You can only find truth with logic if you have already found truth without it."

"Utility and necessity of logic - It would be a mistake to imagine that, above and beyond what is called the Natural Logic of sound common sense, the study of the Science of Logic is absolutely necessary for right reasoning. Men reasoned rightly before Aristotle ever formulated a canon of logic. It was, in fact, by an analysis of such reasonings that he discovered those canons: they could never have been discovered otherwise. Here as elsewhere the art came before the science; theory followed practice. A man may reason rightly without knowing a single rule of the syllogism; or, conversely, he may know all the details of logic and be an indifferent guide to truth just as a first-rate geometrician may be a failure as an engineer. But still, just as his knowledge of geometry will enable the geometrician to detect the defects in a piece of engineering, so too will an explicit knowledge of the canons of reasoning enable us to discover more readily where the fallacy of a misleading argument lies. Without professing to guard us infallibly from error, logic familiarizes us with the rules and canons to which right reasoning processes must conform, and with the hidden fallacies and pitfalls to which such processes are commonly exposed."

"The real trouble with this world of ours is not that it is an unreasonable world, nor even that it is a reasonable one. The commonest kind of trouble is that it is nearly reasonable, but not quite. Life is not an illogicality; yet it is a trap for logicians. It looks just a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait."

"The morbid logician seeks to make everything lucid, and succeeds in making everything mysterious."

"We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two."

"Pierce wrote as a logician and James as a humanist."

"It is with logic as it is with other sciences. They draw wisdom from the mysterious source of plain experience. Agriculture, e. g., aims to teach the farmer how to cultivate the soil; but fields were tilled long before any agricultural college had begun its lectures. In the same way human beings think without ever having heard of logic. But by practice they improve their innate faculty of thought, they make progress, they gradually learn to make better use of it. Finally, just as the farmer arrives at the science of agriculture, so the thinker arrives at logic, acquires a clear consciousness of his faculty of thought and a professional dexterity in applying it."

"Adherents of formal logic may be compared to a maker of porcelain dishes who would contend that he was simply paying attention to the form of his dishes, pots, and vases, but that he did not have anything to do with the raw material."

"“It’s logical,” Sario said. “Lots of people don’t like coping with logic when it dictates hard decisions. That’s a problem with people, not logic.”"

"These, briefly, are the key elements of the stereotype: logic cripples and constrains; it forces one into narrow and mechanical modes of thought that cut one off from a vast range of superior thoughts, feelings and perceptions; logic is an enemy of wit and humor (Mr. Spock's face was always an impassive mask); logic makes us dull and pedantic (Mr. Spock always spoke in a monotone); logic presupposes a simple-minded, black-and-white, yes-no conception of the world. ... Logic misses the point of half the things we ordinarily say and cannot match the insight of the humblest person's common sense."

"From a drop of water, a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other."

"The first question we should face is: What is the aim of a physical theory? To this question diverse answers have been made, but all of them may be reduced to two main principles: "A physical theory," certain logicians have replied, "has for its object the explanation of a group of laws experimentally established." "A physical theory," other thinkers have said, "is an abstract system whose aim is to summarize and classify logically a group of experimental laws without claiming to explain these laws... Now these two questions — Does there exist a material reality distinct from sensible appearances? and What is the nature of reality? — do not have their source in experimental method, which is acquainted only with sensible appearances and can discover nothing beyond them. The resolution of these questions transcends the methods used by physics; it is the object of metaphysics. Therefore, if the aim of physical theories is to explain experimental laws, theoretical physics is not an autonomous science; it is subordinate to metaphysics... Now, to make physical theories depend on metaphysics is surely not the way to let them enjoy the privilege of universal consent."

"Logic is usually understood nowadays as a study of certain formal systems, though in former times there were philosophers who held that the subject matter of logic was the formal rules of human thought."

"The want of logic annoys. Too much logic bores. Life eludes logic, and everything that logic alone constructs remains artificial and forced."

"To find themselves utterly alone at night where company is desirable and expected makes some people fearful; but a case more trying by far to the nerves is to discover some mysterious companionship when intuition, sensation, memory, analogy, testimony, probability, induction — every kind of evidence in the logician's list — have united to persuade consciousness that it is quite in isolation."

"Logic by definition is that which makes sense: nothing more, nothing less. No miracles. No supernatural hocus-pocus. Everything that takes place, every event, every effect, is logically caused by something, something which preceded it in time, and which provided the physico-chemical causative chain that resulted in the effect. These premises are the foundations of our intellectual existence."

"Logic is a feeble reed, friend. "Logic" proved that airplanes can't fly and that H-bombs won't work and that stones don't fall out of the sky. Logic is a way of saying that anything which didn't happen yesterday won't happen tomorrow."

"To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad."

"Logic is logic. That's all I say."

"Logic is one thing and commonsense another."

"But in this age, logic was a flame that must be frequently starved of fuel."

"I have expos'd myself to the enmity of all metaphysicians, logicians, mathematicians, and even theologians; and can I wonder at the insults I must suffer?"

"Logician: A cat has four paws. Old Gentleman: My dog had four paws. Logician: Then it's a cat. Old Gentleman: So my dog is a cat? Logician: And the contrary is also true."

"Moreover, growing uncertainty surrounded even the one too which the academic philosophers felt they could trust: logic. Two centuries before, Kant had asserted in his Logik (1800): ‘There are but few sciences that can come into a permanent state, which admits of no further alteration. To these belong Logic … We do not require any further discoveries in Logic, since it contains merely the form of thought.’ As late as 1939, a British philosopher asserted: ‘Dictators may be powerful today, but they cannot alter the laws of logic, nor indeed can even God do so.’ Thirteen years later the American philosopher Willard Quine calmly accepted that the definition of logic was undergoing fundamental change: ‘What difference is there in principle between such a shift and the shift whereby Kepler succeeded Ptolemy, or Einstein Newton, or Darwin Aristotle?’ In the decades that followed, many rival systems to classical logic emerged: Bochvar’s many-valued logic, new systems by Birkhoff and Destouches-Février and Reichenbach, minimal logic, deontic logics, tense logics. It became possible to speak of empirical proof or disproof of logic."

"What would be the consequences for the theory of truth, asked one worried logician,’… of the adoption of a non-standard system’? Another, observing systems of modal logic, observed: ‘One gets an uneasy feeling as one discerns and studies more of the systems belonging to this family that it is literally a family, and has the power of reproducing and multiplying, proliferating new systems [of logic] without limit.’ In a world in which even the rules of logic shifted and disintegrated, it is not surprising that modern times did not develop in ways the generation of 1920 would have considered ‘logical’."

"Logic hasn't wholly dispelled the society of witches and prophets and sorcerers and soothsayers."

"Logic is neither a science nor an art, but a dodge."

"Logic is concerned with arguments, good and bad. With the docile and the reasonable, arguments are sometimes useful in settling disputes. With the reasonable, this utility attaches only to good arguments. It is the logician's business to serve the reasonable. Therefore, in the realm of arguments, it is the logician who distinguishes good from bad."

"The book, as it stands, seems to me to be one of the most frightful muddles I have ever read, with scarcely a sound proposition in it beginning with page 45, and yet it remains a book of some interest, which is likely to leave its mark on the mind of the reader. It is an extraordinary example of how, starting with a mistake, a remorseless logician can end up in bedlam."

"Metaphysics may be, after all, only the art of being sure of something that is not so, and logic only the art of going wrong with confidence."

""There is one basis of science," says Descartes, "one test and rule of truth, namely, that whatever is clearly and distinctly conceived is true." A profound psychological mistake. It is true only of formal logic, wherein the mind never quits the sphere of its first assumptions to pass out into the sphere of real existences; no sooner does the mind pass from the internal order to the external order, than the necessity of verifying the strict correspondence between the two becomes absolute. The Ideal Test must be supplemented by the Real Test, to suit the new conditions of the problem."

"Anyone who has heard (Jacques Derrida) lecture in French knows that he is more performance artist than logician. His flamboyant style--using free association, rhymes and near-rhymes, puns, and maddening digressions--is not just a vain pose (though it is surely that). It reflects what he calls a self-conscious "acommunicative strategy" for combating logocentrism."

"The contemporary mathematical and symbolic logic is certainly very different from its classical predecessor, but they share the radical opposition to dialectical logic. In terms of this opposition, the old and the new formal logic express the same mode of thought. it is purged from that “negative” which loomed so large at the origins of logic and of philosophic thought—the experience of the denying, deceptive, falsifying power of the established reality. And with the elimination of this experience, the conceptual effort to sustain the tension between “is” and “ought”, and to subvert the established universe of discourse in the name of its own truth is likewise eliminated from all thought which is to be objective, exact, and scientific. For the scientific subversion of the immediate experience which establishes the truth of science as against that of immediate experience does not develop the concepts which carry in themselves the protest and the refusal. The new scientific truth which they oppose to the accepted one does not contain in itself the judgment that condemns the established reality. ... In contrast, dialectical thought is and remains unscientific to the extent to which it is such judgment."

"This fallacy [appeal to authority] is not in itself an error; it is impossible to learn much in today's world without letting somebody else crunch the numbers and offer us explanations. And teachers are sources of necessary information. But how we choose our "authorities" and place a value on such information, is just another skill rarely taught in our education systems. It's little wonder that to most folk, sound bites and talking heads are enough to count as experts. […] Teaching is reinforcing the appeal to authority, where anybody who seems more intelligent than you must ultimately be right. […] We educators must simply role-model critical thinking. […] Educators themselves have to be prepared to show that “evidence” and “answers” are two separate things by firmly believing that, themselves."

"The pedant and the priest have always been the most expert of logicians—and the most diligent disseminators of nonsense and worse."

"Logic, like whiskey, loses its beneficial effect when taken in too large quantities."

"Logic is a systematic method of coming to the wrong conclusion with confidence."

"Able logicians have a trick of being unutterably wrong when they come to write about life: the instinctive, intuitive sense of human values often fails them quite. Logic is more likely to make a man a fool than the lack of logic is."

"It might ... have been supposed that logicians and psychologists would have devoted special attention to meaning, since it is so vital for all the issues with which they are concerned. But that this is not the case will be evident[1] to anyone who studies the Symposium in Mind (October 1920 and following numbers) on "The Meaning of 'Meaning.'""

"Logicians tell us that a system of ideas containing a contradiction can be used to deduce any statement whatsoever, no matter how absurd."

"Logic and mathematics seem to be the only domains where self-evidence manages to rise above triviality; and this it does, in those domains, by a linking of self-evidence on to self-evidence in the chain reaction known as proof."

"Three conceptions are perpetually turning up at every point in every theory of logic, and in the most rounded systems they occur in connection with one another. They are conceptions so very broad and consequently indefinite that they are hard to seize and may be easily overlooked. I call them the conceptions of First, Second, Third. First is the conception of being or existing independent of anything else. Second is the conception of being relative to, the conception of reaction with, something else. Third is the conception of mediation, whereby a first and second are brought into relation."

"Logical analysis applied to mental phenomenon shows that there is but one law of mind, namely that ideas tend to spread continuously and to affect certain others which stand to them in a peculiar relation of affectibility. In this spreading they lose intensity, and especially the power of affecting others, but gain generality and become welded with other ideas."

"A certain maxim of Logic which I have called Pragmatism has recommended itself to me for diverse reasons and on sundry considerations."

"It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better."

"The utility of a science which enables men to take cognizance of the travellers on the mind's highway, and excludes those disorderly interlopers, verbal fallacies, needs but small attestation. Its searching penetration by definition alone, before which even mathematical precision fails, would especially commend it to those whom the abstruseness of the study does not terrify, and who recognise the valuable results which must attend discipline of mind. Like a medicine, though not a panacea for every ill, it has the health of the mind for its aim, but requires the determination of a powerful will to imbibe its nauseating yet wholesome influence: it is no wonder therefore that puny intellects, like weak stomachs, abhor and reject it."

"Of course I'm inconsistent! Only logicians and cretins are consistent!"

"Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premises which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary."

"The question of "unreality," which confronts us at this point, is a very important one. Misled by grammar, the great majority of those logicians who have dealt with this question have dealt with it on mistaken lines. They have regarded grammatical form as a surer guide in analysis than, in fact, it is. And they have not known what differences in grammatical form are important."

"All traditional logic habitually assumes that precise symbols are being employed. It is therefore not applicable to this terrestial life but only to an imagined celestial existence... logic takes us nearer to heaven than other studies."

"I once received a letter from an eminent logician, Mrs. Christine Ladd Franklin, saying that she was a solipsist, and was surprised that there were no others. Coming from a logician, this surprise surprised me."

"The apparent world goes through developments which are the same as those the logician goes through if he starts from Pure Being and travels on to the Absolute Idea. [...] Why the world should go through this logical evolution is not clear; one is tempted to suppose that the Absolute Idea did not quite understand itself at first, and made mistakes when it tried to embody itself in events. But this, of course, was not what Hegel would have said."

"Pure logic is the ruin of the spirit."

"A crisis in doctrine occurred when they discovered that the square root of two was irrational. That is: the square root of two could not be represented as the ratio of two whole numbers, no matter how big they were. "Irrational" originally meant only that. That you can't express a number as a ratio. But for the Pythagoreans it came to mean something else, something threatening, a hint that their world view might not make sense, the other meaning of "irrational"."

"When emotion brings us ghosts from the past, only logic can root us in the present."

"[My aim is] to design logic as a calculating discipline, especially to give access to the exact handling of relative concepts, and, from then on, by emancipation from the routine claims of natural language, to withdraw any fertile soil from "cliché" in the field of philosophy as well. This should prepare the ground for a scientific universal language that, widely differing from linguistic efforts like Volapük [a universal language like Esperanto, very popular in Germany at the time], looks more like a sign language than like a sound language."

"Conceptual graphs are system of logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. The purpose of the system is to express meaning in a form that is logically precise, humanly readable, and computationally tractable."

"Logic is the beginning of wisdom, not the end."

"Logic is in the eye of the logician."

"An idea starts to be interesting when you get scared of taking it to its logical conclusion."

"It takes extraordinary wisdom and self-control to accept that many things have a logic we do not understand that is smarter than our own."

"Poetry — No definition of poetry is adequate unless it be poetry itself. The most accurate analysis by the rarest wisdom is yet insufficient, and the poet will instantly prove it false by setting aside its requisitions. It is indeed all that we do not know. The poet does not need to see how meadows are something else than earth, grass, and water, but how they are thus much. He does not need discover that potato blows are as beautiful as violets, as the farmer thinks, but only how good potato blows are. The poem is drawn out from under the feet of the poet, his whole weight has rested on this ground. It has a logic more severe than the logician's. You might as well think to go in pursuit of the rainbow, and embrace it on the next hill, as to embrace the whole of poetry even in thought."

"The moment we stop assuming that the ideas of any milieu form static 'propositional systems', and recognize that they constitute historically developing 'conceptual populations', we are free to abandon also the philosophers' traditional assumption that rationality is a sub-species of logicality. ...In non-intellectual contexts... we judge the rationality of a man's conduct, not by how he habitually behaves, but rather how far he modifies his behaviour in new and unfamiliar situations, and it is arguable that the rationality of intellectual performances should be judged, correspondingly, by considering, not the internal consistency of a man's habitual concepts and beliefs, but rather the manner in which he modifies this intellectual position in the face of new and unforeseen experiences."

"Logic tends to reduce everything to identities and genera, to each representation having no more than one single and self-same content in whatever place, time, or relation it may occur to us. And there is nothing that remains the same for two successive moments of its existence. My idea of God is different each time that I conceive it. Identity, which is death, is the goal of the intellect."

"As to the most prudent logicians might venture to deduce from a skein of wool the probable existence of a sheep; so you, from the raw stuff of perception, may venture to deduce a universe which transcends the reproductive powers of your loom."

"Logic is a law which must be obeyed, and man realizes himself only in so far as he is logical. He finds himself in cognition."

"Memory, then, is a necessary part of the logical faculty. ... The proposition A = A must have a psychological relation to time, otherwise it would be At1 = At2."

"Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing."

"[ Fuzzy logic is ] a logic whose distinguishing features are (i) fuzzy truth-values expressed in linguistic terms, e.g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc2.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact."

"Rather like the way the Hubble Space Telescope has made a significant contribution to astronomy in enabling astronomers to discover hidden structures and properties of our distant universe, dynamic geometry software has allowed new worlds to become viewable and tangible in mathematics and particularly in geometry."

"Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means."

"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere."

"In doing mathematics, I express something personal. It is a source of joy to know that, despite this personal aspect, the fruit of my work can be of interest to other mathematicians."

"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances."

"Now comes the Einstein–Podolsky–Rosen entangled state. Now I see faces, people saying, "Oh..?" Don't worry! When you go to the concert, you don't need to be able to read the music, to enjoy the music. ...So here... [are] equations. It's a pleasure for my colleague physicists. If you can't read the equation, listen to me. I'm not going to sing, but... listen to the words... the words are... a way of describing the equations, and you don't need to know the mathematics..."

"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."

"Confused is of course the best state a mathematician can be in; the struggle out of that state is the primary drive for progress."

"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."

"Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a non-mathematical Universe containing living observers."

"We say that the string is 'random' if there is no other representation of the string which is shorter than itself. But we will say that it is 'non-random' if there does exist such an abbreviated representation. ...In general, the shorter the possible representation... the less random... On this view we recognize science to be the search for algorithmic compressions. ...It is simplest to think of mathematics as the catalogue of all possible patterns. ...When viewed in this way, it is inevitable that the world is described by mathematics. ...In many ways the search for a Theory of Everything is a manifestation of a faith that this compression goes all the way down to the bedrock of reality..."

"Mathematics became an experimental subject. Individuals could follow previously intractable problems by simply watching what happened when they were programmed into a personal computer. ...The PC revolution has made science more visual and more immediate ...by creating films of imaginary experiences of mathematical worlds. ...Words are no longer enough."

"Something more than impeccable logic is required in mathematics. An expert logician will not necessarily be a passable mathematician for all his skill in logic, any more than a scholarly prosodist will be a respectable poet for all his mastery of meter."

"A narrative of the decisive epochs in the development of mathematics was wanted. ...Numerous professionals... know from hard experience what mathematical invention means. ...Whoever has himself attempted to advance mathematics is inclined to be more skeptical than the average spectator toward any alleged anticipation of notable progress. ...often what looks like an anticipation ...was not even aimed in the right direction. ...when at length progress started; it proceeded along lines totally different from those which, in retrospect, it 'should' have followed."

"Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed."

"Ruth felt that math was like sex—get all you can, but best not done in public."

"“It’s magic,” the chief cook concluded, in awe. “No, not magic,” the ship’s doctor replied. “It’s much more. It’s mathematics.”"

"Maths is really hard to define. ...Except I like to define maths as this{{center|1=\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11}\cdots}}This formula which links π to the odd numbers... It's true. It's always been there. It's absolutely wonderful. It connects odd numbers to the ratio of a circle, and... if you don't like that, then you have no mathematical soul."

"The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason."

"Over time you will get the wrong answer more times than you get the right answer. That’s not a problem! You’ve learnt what doesn’t work, just try again."

"I would myself say that the purely imaginary objects are the only realities, the ὂντως ὂντα [truest things], in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line."

"You don’t need anybody’s permission to be a great mathematician!"

"This statistical regularity in moral affairs fully establishes their being under the presidency of law. Man is now seen to be an enigma only as an individual; in the mass he is a mathematical problem."

"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic."

"There is probably no other science which presents such different appearances to one who cultivates and one who does not, as mathematics. To [the non-mathematician] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple of bloom of vigorous youth, everywhere stretching out after the "attainable but unattained," and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness."

"Mathematics is the study of anything that obeys the rules of logic, using the rules of logic."

"The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. In Greek, mathematics simply meant learning, and we have adapted this... to define the term as "learning to decide." Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa."

"[F]inding direct measurement so often impossible, we are compelled to devise means of doing it indirectly. Hence arose Mathematics."

"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples..."

"The study of mathematics is the indispensable basis for all intellectual and spiritual progress."

"The progress of mathematics has been most erratic, and... intuition has played a predominant rôle in it. ...It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth it had no part. ...the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied."

"Between the philosopher's attitude towards the issue of reality and that of the mathematician there is this essential difference: for the philosopher the issue is paramount; the mathematician's love for reality is purely platonic."

"The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. ...The conic sections, invented in an attempt to solve the problem of doubling the altar of an oracle, ended by becoming the orbits followed by the planets... The imaginary magnitudes invented by Cardan and Bombelli describe... the characteristic features of alternating currents. The absolute differential calculus, which originated as a fantasy of Reimann, became the mathematical model for the theory of Relativity. And the matrices which were a complete abstraction in the days of Cayley and Sylvester appear admirably adapted to the... quantum of the atom."

"The thing with mathematics is we keep thinking about how things work. And once you figure it out, you then see other things, connections and so on. And two weeks later it's so obvious that you could kick yourself for not having thought of it sooner. The high doesn't last! So you have to enjoy it as long as it does last."

"Underpinning everything... are the laws of physics. These remarkably ingenious laws are able to permit matter to self-organize to the point where consciousness emerges in the cosmos—mind from matter—and the most striking product of the human mind is mathematics. This is the baffling thing. Mathematics is... produced by the human mind. Yet if we ask where mathematics works best, it is in areas like particle physics and astrophysics, areas of fundamental science that are very, very far removed from everyday affairs. ...at the opposite end of spectrum of complexity from the human brain. ...a product of the most complex system we know in nature, the human brain, finds a consonance with the underlying, simplest and most fundamental level, the basic building blocks that make up the world."

"It suggests to me that consciousness and our ability to do mathematics are no mere accident, no trivial detail, no insignificant by-product of evolution that is piggy-backing on some other mundane property. It points to what I like to call the cosmic connection, the existence of a really deep relationship between minds that can do mathematics and the underlying laws of nature that produce them. We have a closed system of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can encode... the very laws of physics that gave rise to it."

"Physicists have been drawn to elegant mathematical relationships that bind the subject together with economy and style, melding disparate qualities in subtle and harmonious ways. But this is to import a new factor into the argument—questions of aesthetics and taste. We are then on shaky ground indeed. It may be that M theory looks beautiful to its creators, but ugly to N theorists, who think that their theory is the most elegant. But then the O theorists disagree with both groups..."

"Mathematics, in an earlier view, is the science of space and quantity; in a later view, it is the science of pattern and deductive structure. Since the Greeks, mathematics is also the science of the infinite."

"Our ignorance of this beautiful subject -- a tree of ideas with ancient roots and modern fruit -- is profound and beset with fear, superstition, and misinformation."

"A marveilous newtrality have these things mathematicall and also a strange participation between things supernaturall, imortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, compounded and divisible."

"The nature of mathematical demonstration is totally different from all other, and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us."

"We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two."

"With a view to summon myself to the search for a science of mathematics in general, I asked myself... what precisely was the meaning of this word mathematics, and why arithmetic and geometry only, and not also astronomy, music, optics, mechanics, and so many other sciences, should be considered as forming a part of it; for it is not enough here to know the etymology of the word. In reality the word mathematics meaning nothing but science, those which I have just named have as much right as geometry to be called mathematics; and nevertheless there is no one, however little instructed, who cannot distinguish at once what belongs to mathematics... from what belongs to the other sciences. But... all the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics..."

"Mathematics in itself, as I say, is independent of experience. It begins with the free choice of symbols, to which are freely assigned properties, and it then proceeds to deduce the necessary rational implications of those properties."

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better."

"Recreational mathematics is a splendid hobby which young and old can equally enjoy. The popularity of Sudoku shows that an aptitude for recreational mathematics is widespread in the population. From Sudoku it is easy to ascend to mathematical pursuits that offer more scope for imagination and originality."

"We tend to think of maths as being an 'exact' discipline, where answers are right or wrong. And it's true that there is a huge part of maths that is about exactness. But in everyday life, numerical answers are sometimes just the start of the debate. If we are trained to believe that every numerical question has a definite, 'right' answer then we miss the fact that numbers in the real world are a lot fuzzier than pure maths might suggest."

"I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission."

"Proof is the idol before whom the pure mathematician tortures himself."

"“There’s more to life than mathematics,” Joan said. “But not much more.”"

"In so far as theories of mathematics speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality."

"It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them, which give us the key to the understanding of the phenomena of Nature."

"There is no royal road to geometry. (μή εἶναι βασιλικήν ἀτραπόν ἐπί γεωμετρίαν, Non est regia [inquit Euclides] ad Geometriam via)"

"If you are interested in the ultimate character of the physical world, or the complete world, and at the present time our only way to understand that is through the mathematical type of reasoning... the great depth of character of the universality of the laws, the relationships of things... I don't know any other way to do it, we don't know any other way to describe it accurately... or to see the interrelationships without it... don't misunderstand me, there are many, many aspects of the world that mathematics is unnecessary for... but we were talking about physics... to not know mathematics is a severe limitation in understanding the world."

"Mathematics offers a common language across borders. It is a real joy."

"Mathematics is a tool which ideally permits mediocre minds to solve complicated problems expeditiously."

"The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things. Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them."

"Numbers exist only in our minds. There is no physical entity that is number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe."

"I united the majority of well-informed persons into a club, which we called by the name of the Junto, and the object of which was to improve our understandings. ... The first members of our club were... Thomas Godfrey, a self-taught mathematician, and afterwards inventor of what is now called Hadley's dial; but he had little knowledge out of his own line, and was insupportable in company, always requiring, like the majority of mathematicians that have fallen in my way, an unusual precision in everything that is said, continually contradicting, or making trifling distinctions—a sure way of defeating all the ends of conversation. He very soon left us."

"Mathematics ought properly to be a model of logical clarity. In actual fact there are perhaps no scientific works where you will find more wrong expressions, and consequently wrong thoughts, than in mathematical ones."

"The forbidding symbols and equations are just another language: code for beautiful ideas that often find surprising uses in the ordinary world we all live in."

"Maths is the language spoken by all the sciences, taking us to the frontiers of knowledge, from the workings of the Universe to the workings of our minds, which enabled us to dream it all up in the first place."

"For many, maths is fundamentally beautiful; indeed many mathematicians won’t be entirely satisfied with their work until it has an elegance, simplicity and grace. Others are drawn by its ‘unreasonable effectiveness’ - its power to explain the world we live in."

"It might have been beans, successful hunts, or victories in battle, but, for millennia, people were using maths to describe things - counting them, measuring them, dividing them up."

"It is a truth universally acknowledged … that ε (pronounced ‘epsilon’) is always a very small number, and usually comes with a δ (pronounced ‘delta’)."

"It's a lonely profession."

"An arguing couple spiraling into negativity and teetering on the brink of divorce is actually mathematically equivalent to the beginning of a nuclear war."

"You can’t think of maths just as this abstract thing that exists only in isolation. I genuinely struggle to find a topic where maths can’t offer you at least some use or insight."

"There’s barely any aspect of our modern lives that hasn’t had a mathematical contribution at some point and yet, if you asked the average person, they might think that maths is just difficult, irrelevant and uninteresting."

"What has philosophy got to do with measuring anything? It's the mathematicians you have to trust, and they measure the skies like we measure a field."

"Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth."

"Humane wis­dom understandeth some propositions so perfectly, and is as abso­lutely certain thereof, as Nature herself; and such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few compre­hended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the neces­sity thereof, than which there can be no greater certainty."

"I've always felt that a teacher can introduce recreational math; and I'm defining it in the very broad sense to include anything that has a spirit of play about it. I don't know of any better way to hook the interests of the students."

"The element of play, which makes recreational mathematics recreational, may take many forms: a puzzle to be solved, a competitive game, a magic trick, paradox, fallacy, or simply mathematics with any sort of curious or amusing fillip."

"There is not much difference between the delight a novice experiences in cracking a clever brain teaser and the delight a mathematician experiences in mastering a more advanced problem. Both look on beauty bare -- that clean, sharply defined, mysterious, entrancing order that underlies all structure."

"Mathematical magic combines the beauty of mathematical structure with the entertainment value of a trick."

"A surprising proportion of mathematicians are accomplished musicians. Is it because music and mathematics share patterns that are beautiful?"

"...[A]ll the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced."

"A set is a unity of which its elements are the constituents. It is a fundamental property of the mind to comprehend multitudes into unities. Sets are multitudes which are also unities. A multitude is the opposite of a unity. How can anything be both a multitude and a unity? Yet a set is just that. It is a seemingly contradictory fact that sets exist. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics. Thinking a plurality together seems like a triviality: and this appears to explain why we have no contradiction. But “many things for one” is far from trivial."

"Even God cannot make two times two not make four."

"From the infinitesimal calculus to the present... the essential progress in mathematics has resulted from successively annexing notions which, for the Greeks or the Renaissance geometers or the predecessors of Riemann, went "outside mathematics" because it was impossible to define them."

"Mathematical methods present... two advantages. Their terminology is precise and concentrated, in a fashion which ordinary language cannot afford to adopt. Further, the symbols which result from their employment have implications which, when brought to light, yield new knowledge. This is deductively reached, but it is none the less new knowledge. With greater precision than is usual, ordinary language may be made to do some, if not a great deal, of this work for which mathematical methods are alone quite appropriate. If ordinary language can do part of it an advantage may be gained. The difficulty that attends mathematical symbolism is the accompanying tendency to take the symbol as exhaustively descriptive of reality. Now it is not so descriptive. It always embodies an abstraction. It accordingly leads to the use of metaphors which are inadequate and generally untrue. It is only qualification by descriptive language of a wider range that can keep this tendency in check."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"A critical step was made sometime before the ninth century AD, when a new partial script was invented, one that could store and process mathematical data with unprecedented efficiency. This partial script was composed of ten signs, representing the numbers from 0 - 9. Confusingly, these signs were known as Arabic numerals even though they were first invented by the Hindus."

"Mere observations, however, are not knowledge. In order to understand the universe, we need to connect observations into comprehensive theories. Earlier traditions usually formulated their theories in terms of stories. Modern science uses mathematics."

"I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.""

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean."

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas."

"Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts. The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers."

"There are men of a certain type of mind who are never wearied with gibing at mathematics, at mathematicians, and at mathematical methods of inquiry. It goes almost without saying that these men have themselves little mathematical bent. I believe this to be a general fact; but, as a fact, it does not explain very well their attitude towards mathematicians. The reason seems to lie deeper. How does it come about, for instance, that whilst they are themselves so transparently ignorant of the real nature, meaning, and effects of mathematical investigation, they yet lay down the law in the most confident and self-satisfied manner, telling the mathematician what the nature of his work is (or rather is not), and of its erroneousness and inutility, and so forth? It is quite as if they knew all about it. It reminds one of the professional paradoxers... They, too, write as if they knew all about it. Plainly, then, the anti-mathematician must belong to the same class as the paradoxer, whose characteristic is to be wise in his ignorance, whereas the really wise man is ignorant in his wisdom. ...What is of greater importance is that the anti-mathematicians sometimes do a deal of mischief. For there are many of a neutral frame of mind, little acquainted themselves with mathematical methods, who are sufficiently impressible to be easily taken in by the gibers and to be prejudiced thereby; and, should they possess some mathematical bent, they may be hindered by their prejudice from giving it fair development. We cannot all be Newtons or Laplace's, but that there is an immense amount of moderate mathematical talent lying latent in the average man I regard as a fact; and even the moderate development implied in a working knowledge of simple algebraical equations can, with common-sense to assist, be not only the means of valuable mental discipline, but even be of commercial importance (which goes a long way with some people), should one's occupation be a branch of engineering for example."

"Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable sub-human who has learned to wear shoes, bathe, and not make messes in the house."

"[E]ven in the most precise part of science, in mathematics, we cannot avoid using concepts that involve contradictions. ...[I]t is well known that the concept of infinity leads to contradictions that have been analyzed, but it would be practically impossible to construct the main parts of mathematics without this concept."

"Dialectical mathematics is a rigorously logical science, where statements are either true or false, and where objects with specified properties either do or do not exist. Algorithmic mathematics is a tool for solving problems. Here we are concerned not only with the existence of a mathematical object, but also with the credentials of its existence. Dialectical mathematics is an intellectual game played according to rules about which there is a high degree of consensus. The rules of the game of algorithmic mathematics may vary according to the urgency of the problem on hand. We never could have put a man on the moon if we had insisted that the trajectories should be computed with dialectic rigor. The rules may also vary according to the computing equipment available. Dialectic mathematics invites contemplation. Algorithmic mathematics invites action. Dialectic mathematics generates insight. Algorithmic mathematics generates results."

"An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us."

"The art of doing mathematics consists in finding that special case which contains all the germs of generality."

"The doctrine of Right and Wrong, is perpetually disputed, both by Pen and the Sword: Whereas the doctrine of Lines, and Figures, is not so; because men care not, in that subject what be truth, as a thing that crosses no mans ambition, profit, or lust. For I doubt not, but if it had been a thing contrary to any mans right of dominion, or to the interest of men that have dominion, That the three Angles of a Triangle, should be equall to two Angles of a Square; that doctrine should have been, if not disputed, yet by the burning of all books of Geometry, suppressed, as far as he whom it concerned was able."

"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination."

"Mathematics is the language of size, shape and order and that it is an essential part of the equipment of an intelligent citizen to understand this language. If the rules of mathematics are the rules of grammar, there is no stupidity involved when we fail to see that a mathematical truth is obvious. The rules of ordinary grammar are not obvious. They have to be learned. They are not eternal truths. They are conveniences without whose aid truths about the sorts of things in the world cannot be communicated from one person to another."

"As soon as a thought or word becomes a tool, one can dispense with actually ‘thinking’ it, that is, with going through the logical acts involved in verbal formulation of it. As has been pointed out, often and correctly, the advantage of mathematics—the model of all neo-positivistic thinking—lies in just this ‘intellectual economy.’ Complicated logical operations are carried out without actual performance of the intellectual acts upon which the mathematical and logical symbols are based. … Reason … becomes a fetish, a magic entity that is accepted rather than intellectually experienced."

"Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat flour from peascods, so pages of formulæ will not get a definite result out of loose data."

"From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician."

"The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit."

"Clearly, if electric action is to be explained in mechanical terms, the mechanism must be supposed to be attached to the electric charges, and to move through space with them. It must extend through the whole of space, because the attraction and repulsion of an electron extend through the whole of space, and it must be the same for all directions in space. Further, as the pattern of events is unaltered by motion, the mechanism must be the same when the electron is in motion as when it is at rest. But experiment shows that an electron in motion exerts additional forces which are not the same for all directions in space; if we picture this electron as moving head-foremost through space, these forces surround it like a belt around its waist. Thus direct experimental evidence shows that the forces exerted by an electron (or... any charged body) can neither be attributed to any mechanism attached to the body, nor through action transmitted through an ether or any medium surrounding the body. We have a perfect specification of the pattern of events written... in the language of mathematics, but this does not admit of interpretation in mechanical terms, or indeed in any terms other than those of mathematics."

"We will always have STEM with us. Some things will drop out of the public eye and will go away, but there will always be science, engineering and technology. And there will always, always be mathematics. Everything is physics and math."

"I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for... science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. ...the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics."

"The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience."

"Those who thought they could distinguish philosophy from mathematics by saying that the former was concerned with quality only, the latter with quantity only, mistook effect for cause. It is owing to the form of mathematical knowledge that it can refer to quanta only, because it is only the concept of quantities that admits of construction, that is, of a priori representation in intuition, while qualities cannot be represented in any but empirical intuition."

"There is a familiar formula—perhaps the most compact and famous of all formulas—developed by Euler from a discovery of De Moivre: eiπ + 1 = 0. ...It appeals equally to the mystic, the scientist, the philosopher, the mathematician."

"My earliest mathematical memory is my father explaining to me the theorem that three angles in a triangle add up to 180 degrees. The idea that something could be proved to be always true was very appealing to me."

"Mathematics in general is fundamentally the science of self-evident things."

"It is impossible, and it has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form."

"There is no doubt... that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgement has caused mathematicians to equate elegance and conciseness at the cost of intelligibility."

"Electromagnetic theory is entirely a mathematical theory illustrated by a few crude physical pictures. These pictures are no more than the clothes that dress up the body of mathematics and make it appear presentable in the society of sciences. ...Though he [James Clerk Maxwell] had tried desperately to build a physical account of electromagnetic phenomena, in his classic Treatise on Electricity and Magnetism he omitted most of this material and emphasized the highly polished, complex mathematical theory. ...Radio waves and light waves operate in a physical darkness illuminated only for those who would carry the torch of mathematics."

"One of the curious things about mathematics that clearly emerges... is that mathematics which is concerned with reasoning nevertheless creates processes which can be applied almost mechanically, that is, without reasoning. The thinking is, so to speak, mechanized and this mechanization enables us to solve complicated problems in no time. We think up processes so that we don't have to think."

"It is impossible to be a mathematician without being a poet in soul."

"There is no mathematical substitute for philosophy."

"It is a well-known experience that the only truly enjoyable and profitable way of studying mathematics is the method of "filling in details" by one's own efforts."

"There is a temptingly simple explanation for the fact that science is mathematical in nature: it is because we give the name of science to those areas of intellectual inquiry that yield to mathematical analysis. ...Science ...deals with precisely those subjects amenable to quantitative analysis, and that is why mathematics is the appropriate language... The puzzle becomes a tautology: mathematics is the language of science because we reserve the name "science" for anything that mathematics can handle. If it's not mathematical, to some degree at least, it isn't really science."

"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world."

"The only thing I am interested in using mathematics for is to have a good time and to help others do the same."

"You should not choose to do mathematics if you want to make money; your salary as a mathematician will never correspond to the amount of time and energy invested in your work."

"You know I always have so many metaphysical enquiries & speculations which intrude themselves, that I never am really satisfied that I understand anything; because, understand it as well as I may, my comprehension can only be an infinitesimal fraction of all I want to understand about the many connexions & relations which occur to me, how the matter in question was first thought of or arrived at, &c., &c."

"I have not in every case been able to avoid the use of the abbreviated and precise terminology of mathematics. To do so would have been to sacrifice matter to form; for the language of everyday life has not yet grown to be sufficiently accurate for the purposes of so exact a science as mechanics."

"All there is in the three worlds, moving or unmoving, all that cannot be described without mathematics."

"Think of it: of the infinity of real numbers, those that are most important to mathematics—0, 1, √2, e and π—are located within less than four units on the number line. A remarkable coincidence? A mere detail in the Creator's grand design? I let the reader decide."

"I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you. I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have even forgotten how to integrate a good many things. But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very well unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in (4), burn (3). This last I do often."

"One of the best ways to sharpen your brain, and to develop intelligence, is to study mathematics. It challenges and strengthens your mind in a way that very few other things do. It’s like going to the gym -- but for your brain!"

"Euclid alone has looked on Beauty bare."

"There are times when I feel like I'm in a big forest and don't know where I'm going. But then somehow I come to the top of a hill and can see everything more clearly. When that happens, it's really exciting."

"I like crossing the imaginary boundaries people set up between different fields—it's very refreshing. There are lots of tools, and you don't know which one would work. It's about being optimistic and trying to connect things."

"One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics."

"Mathematicians may flatter themselves that they possess new ideas which mere human language is yet unable to express. Let them make the effort to express these ideas in appropriate words without the aid of symbols, and if they succeed they will not only lay us laymen under a lasting obligation, but we venture to say, they will find themselves very much enlightened during the process, and will even be doubtful whether the ideas as expressed in symbols had ever quite found their way out of the equations of their minds."

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or "mathematics envy.""

"The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics."

"Since most people find mathematics somewhat forbidding, if not frightening, they find it difficult to understand how it can be regarded as beautiful. ... It is not the visual beauty of a painting or the audio beauty of a musical performance. Nor is it the literary beauty of a great poem; it is entirely intellectual and therefore, while more difficult to perceive, more satisfying when perceived."

"Mathematics is good for the soul, getting things right enlivens a sense of truth, efforts to understand automatically purify desires."

"Mathematics is the source of a wicked intellect that, while making man the lord of the earth, also makes him the slave of the machine."

"Mathematics is the bold luxury of pure reason, one of the few that remain today."

"In their field they [mathematicians] do what we ought to be doing in ours. Therein lies the significant lesson … of their existence. They are an analogy for the intellectual of the future."

"Wer ein mathematisches Buch nicht mit Andacht ergreift, und es wie Gottes-Wort liest, der versteht es nicht."

"Every true mathematician sees mathematics everywhere—in a child's swing or a pendulum, in the outline shape of a tree and that of its leaves, in the clouds"

"When I have needed solace and I have had to depend on my own resources, the mathematics has been there. I am grateful."

"[Mathematics is] the science that draws necessary conclusions."

"You cannot read mathematics superficially; the inescapable abstraction always has an element of self-torture in it, and the one to whom this self-torture is joy is the mathematician."

"Our epoch is the epoch of increasing consciousness; in this field Mathematics has done its bit. It has made us conscious of the limits of its own capabilities."

"More often than not, a piece of mathematics worked out years before—and believed to be totally without practical value—finds a role in the “real” world."

"In the early 1900s, a great mathematician was expected to comprehend the whole of known mathematics. Mathematics was a shallow pool. Today the mathematical waters have grown so deep that a great mathematician can know only about 5% of the entire corpus. What will the future of mathematics be as specialized mathematicians know more and more about less and less until they know everything about nothing?"

"[...] I who do not even dare to say, when one is added to one, whether the one to which the addition was made has become two, or the one which was added, or the one which was added and the one to which it was added became two by the addition of each to the other. I think it is wonderful that when each of them was separate from the other, each was one and they were not then two, and when they were brought near each other this juxtaposition was the cause of their becoming two. And I cannot yet believe that if one is divided, the division causes it to become two; for this is the opposite of the cause which produced two in the former case; for then two arose because one was brought near and added to another one, and now because one is removed and separated from other. And I no longer believe that I know by this method even how one is generated or, in a word, how anything is generated or is destroyed or exists, and I no longer admit this method, but have another confused way of my own."

"Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone."

"What we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all; this common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony then which is the sole objective reality, the only truth we can attain; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better."

"Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts? This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either."

"You're not supposed to have equations in a public lecture, so think of this as a piece of art."

"We should not believe... that commensurability is a quality of every magnitude as of all the numbers; and whoever has not investigated this subject, shows a gross and unseemly ignorance of what the Athenian Stranger says in the seventh treatise of the Book of the Laws, [namely], "And besides there is found in every man an ignorance, shameful in its nature and ludicrous, concerning everything which has the dimensions, length, breadth, and depth; and it is clear that mathematics can free them from this ignorance. For I hold that this [ignorance] is a brutish and not a human state, and I am verily ashamed, not for myself only, but for all Greeks, of the opinion of those men who prefer to believe what this whole generation believes, [namely], that commensurability is necessarily a quality of all magnitudes. For everyone of them says: "We conceive that those things are essentially the same, some of which can measure the others in some way or other. But the fact is that only some of them are measured by common measures, whereas others cannot be measured at all"."

"Number is the ruler of forms and ideas, and the cause of gods and daemons."

"There is geometry in the humming of the strings, there is music in the spacing of the spheres."

"The more advanced the sciences have become, the more they have tended to enter the domain of mathematics, which is a sort of center towards which they converge. We can judge of the perfection to which a science has come by the facility, more or less great, with which it may be approached by calculation."

"An equation means nothing to me unless it expresses a thought of God."

"I used to fear I was not made for mathematics and would look for people to tell me I was on the right track. You need to develop a personal conviction that you are a mathematician, and that what you are doing makes sense."

"Mathematics has the dubious honor of being the least popular subject in the curriculum … Future teachers pass through the elementary schools learning to detest mathematics. They drop it in high school as early as possible. They avoid it in teachers' colleges because it is not required. They return to the elementary school to teach a new generation to detest it."

"Mathematics lays the foundation of all the ex­act sciences. It teaches the art of combining num­bers, of calculating and measuring distances, how to solve problems, to weigh mountains, to fathom the depths of the ocean; but gives no directions how to ascertain the existence of a God."

"Mathematics may be universal in a nonpersonal sense, but whether it can appeal to all people and be understood by all people is another question."

"Sometime, in the future that is knocking at our door, we shall have to retrain ourselves or our children to properly tell the truth. The exercise will be particularly painful in mathematics. The enrapturing discoveries of our field systematically conceal, like footprints erased in sand, the analogical train of thought that is the authentic life of mathematics. Shocking as it may be to the conservative logician, the day will come when currently vague concepts such as motivation and purpose will be made formal and accepted as constituents of a revamped logic, where they will at last be alloted the equal status they deserve, side-by-side with axioms and theorems. Until that day, however, the truths of mathematics will make only fleeting appearances, like shameful confessions whispered to a priest, to a psychiatrist, or to a wife."

"By all accounts mathematics is mankind’s most successful intellectual undertaking. Every problem of mathematics gets solved, sooner or later. Once solved, a mathematical problem is forever finished: no later event will disprove a correct solution. As mathematics progresses, problems that were difficult become easy and can be assigned to schoolchildren.Thus Euclidean geometry is taught in the second year of high school. Similarly, the mathematics learned by my generation in graduate school is now taught at the undergraduate level, and perhaps in the not too distant future, in the high schools. Not only is every mathematical problem solved, but eventually every mathematical problem is proved trivial. The quest for ultimate triviality is characteristic of the mathematical enterprise."

"Mathematics is the study of analogies between analogies. All science is. Scientists want to show that things that don't look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding."

"Extension and abstraction without apparent direction or purpose is fundamental to the discipline. Applicability is not the reason we work, and plenty that is not applicable contributes to the beauty and magnificence of our subject."

"Trying to solve real-world problems, researchers often discover that the tools they need were developed years, decades or even centuries earlier by mathematicians with no prospect of, or care for, applicability."

"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand."

"Recreational mathematics is a type of play which is enjoyable and requires mathematical thinking or skills to engage with; typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts."

"Far from being frivolous, recreational mathematics cuts to the heart of being a mathematician and is deeply linked to development of graduate-level mathematical skills."

"Mathematics is obviously something that women should be able to do very well. It's very intuitive. You don't need a lot of machinery, and you don't need a lot of physical strength. You just need stamina, and women often have a great deal of stamina."

"Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true … If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate."

"It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true."

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry."

"Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the world, but every possible world, must conform."

"Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth."

"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself."

"I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe – because, like Spinoza's God, it won't love us in return."

"Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say."

"Pi's face was masked, and it was understood that none could behold it and live. But piercing eyes looked out from the mask, inexorable, cold and enigmatic."

"In universities, mathematics is taught mainly to men who are going to teach mathematics to men who are going to teach mathematics to.... Sometimes, it is true, there is an escape from this treadmill. Archimedes used mathematics to kill Romans, Galileo to improve the Grand Duke of Tuscany's artillery, modern physicists (grown more ambitious) to exterminate the human race. It is usually on this account that the study of mathematics is commended to the general public as worthy of State support."

"Is Mathematics merely the fire that allows mankind to see the shadows of a more remote ultimate reality?"

"You cannot be cheated, you cannot be lied to. A thing is true or not true, and there is this notion of clarity on which you can base yourself."

"It’s really beautiful to observe, as you progress in your mathematical maturity, how everything is somehow connected."

"10^{50} is a long way from infinity."

"Mathematics is universal. But very little else is."

"There is a wide distinction between the degree of mathematical acquirement necessary for making discoveries, and that which is requisite for understanding what other have done."

"No part of mathematics came to birth in the form that it now appears in a modern textbook: mathematical creativity can be slow, sometimes messy, often frustrating."

"I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes."

"It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. ...it is the great strength of mathematics—what I have elsewhere called its "unreal reality." Mathematics links the abstract world of mental concepts to the real world of physical things without being located in completely in either."

"Math ain't about numbers! If you think math is about numbers, you probably think that Shakespeare is all about words. You probably think that dancing is all about shoes. You probably think that music is all about notes. Math ain't about numbers! Math is about logic, it's about beauty, it's about connections, it's about how you get from one place to another."

"So, nat'ralists observe, a flea Hath smaller fleas that on him prey, And these have smaller still to bite 'em And so proceed ad infinitum."

"Researchers of the brain believe that mathematical truths make little sense to our mind, particularly..the examination of random outcomes."

"Mathematics is principally a tool to meditate, rather than to compute."

"If a mathematician wishes to disparage the work of one of his colleagues, say, A, the most effective method he finds for doing this is to ask where the results can be applied. The hard pressed man, with his back against the wall, finally unearths the researches of another mathematician B as the locus of the application of his own results. If next B is plagued with a similar question, he will refer to another mathematician C. After a few steps of this kind we find ourselves referred back to the researches of A, and in this way the chain closes."

"The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life."

"Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality."

"One occasionally hears the question, is mathematics invented or discovered?—or an answer. As David Wells points out... both answers... are appropriate. Once a game is invented, the consequences are discovered... as it would require a divine intelligence to know just from the rules how a complex game could best be played. When in practice rules are changed, one makes adjustments that will not alter the consequences too dramatically. Analogously, axioms are usually only adjusted and the altered consequences discovered."

"I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be."

"Mathematics is the only true metaphysics."

"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding."

"The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding."

"Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improved clarity."

"Comparatively few engineers are good mathematicians; and... it is fortunate that such is the case; for nature rarely combines high mathematical talent, with that practical tact, and observation of outward things, so essential to a successful engineer. There have been... brilliant exceptions; but they are very rare. But few even of those who have been tolerable mathematicians when young, can, as they advance in years, and become engaged in business, spare the time necessary for retaining such accomplishments."

"For many individuals, as they approach the limit of their abilities, mathematics loses its fun aspect. When a topic is undeveloped, it is recreational to many. As the theory is developed and becomes more abstract, fewer persons find it recreational."

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of... intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. ...We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions."

"The mathematicians know a great deal about very little and the physicists very little about a great deal."

"Mathematics, and perhaps other sciences like physics, have the mission to prepare or improve the human brain, be it the brain of an individual or the collective brain of mankind, for developments yet to come. Just as animals play... in preparation for situations arising later in their lives, it may be that mathematics... is a collection of games. ...may be the only way to change the individual or collective human mind to prepare it for a future that no one can yet imagine. ...[L]ife appears inter alia as a sequence of chemical games... between individuals, or between groups... essentially of a mathematical nature... not... the von Neumann-Morgenstern theory of games, but more general games in the widest sense. This has perhaps... a direct biological role... a book... by... , ...Das Spiel (The Game) ...describes a number of mathematical games or puzzles and discusses the games molecules... play with each other. ... ...we have seen ...starting with a simple pattern and simple recursive rules can lead to unbelievably complicated configurations... that... defy analysis a priori."

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)"

"What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. ...[P]eople know that it deals with numbers and figures, with patterns, relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes. ... that it purports to form the foundations of all rational thought. ...The aesthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is. ...One can ...look conversely at ...the homely side of mathematics ...having to be punctilious ...having to make sure of every step. ...[O]ne cannot stop at drawing with a big, wide brush; all the details have to be filled in ...Mathematicians ...fool themselves ...when they think their main business is to prove theorems without at least indicating why they may be important. If left entirely to the aesthetic criteria, doesn't it compound the mystery? ...[I]n the decades to come there will be more understanding ...of the degree of beauty, though ...the criteria may have shifted ...[to] a super beauty in unanalyzable higher levels. ...It has to appeal to connections with other theories of the external world or to the history of the development of the human brain, or else it is purely aesthetic and very subjective in the sense that music is. ...[E]ven the quality of music will be analyzable ...by mathematizing the idea of analogy."

"Mathematics may be a way of developing physically, that is anatomically, new connections in the brain."

"Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics is at the top of the Vedanga sastras."

"The Koch Curve - Triangles outside triangles outside triangles ad infinitum the Koch curve goes, it's infinitely infinitesimal, this self-similarity shows. A length too long to measure, an area too small to see, what else can this contradiction be, behold fractal geometry."

"It is but right... to apprise you that, diffident of my own decision, the favorable opinion I favor of your performance is founded rather on the explicit and ample testimonies of Gentlemen confessedly possessed of great mathematical knowledge, than on the partial and incompetent attention I have been able to pay to it myself.—But I must be permitted to remark that the subject, in my estimation, holds a higher rank in the literary scale than you are disposed to allow.—The science of figures, to a certain degree, is not only indispensably requisite in every walk of civilised life; but investigation of mathematical truths accustoms the mind to method and correctness in reasoning, and is an employment peculiarly worthy of rational beings. In a clouded state of existence, where so many things appear precarious to the bewildered research, it is here that the rational faculties find a firm foundation to rest upon. From the high ground of mathematical and philosophical demonstration, we are insensibly led to far nobler speculations and sublimer meditations."

"I enjoy being surprised by mathematics and its intrinsic difficulty. The moment I enjoy best is when the pieces of the puzzle fall into one coherent whole."

"[A]ll science as it grows towards perfection becomes mathematical in its ideas."

"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases... mental power... Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility."

"Nothing is more impressive than the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. ...The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact."

"The only reality mathematical concepts have is as cultural elements or artifacts."

"A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and […] he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification!"

"There is no religious denomination in which so much sin has been committed through the misuse of metaphorical expressions as in mathematics."

"I like to find out as much about a mathematical object as possible, just as you might want to understand a person as well as possible."

"Math is perfect (in principle), but mathematicians are not (because they are humans), hence the mathematics that (human) mathematicians do is influenced by the weltanschauung of the people around them."

"All of mathematics is arbitrary. The very language of existence, mathematics, has no reason to be a certain way. But it is."

"Blindly and lawlessly they did all things, Until I taught them how the stars do rise And set in mystery, and devised for them Number, the inducer of philosophies, The synthesis of Letters, and, beside, The artificer of all things, Memory, That sweet Muse-mother."

"The land of easy mathematics where he who works adds up and he who retires subtracts."

"Susan: What is algebra exactly; is it those three-cornered things? Phoebe: It is x minus y equals z plus y and things like that. And all the time you are saying they are equal, you feel in your heart, why should they be."

"I can’t help it, gas escapes from my fundament on the least pretext, it’s hard not to mention it now and then, however great my distaste. One day I counted them. Three hundred and fifteen farts in nineteen hours, or an average of over sixteen farts an hour. After all it’s not excessive. Four farts every fifteen minutes. It’s nothing. Not even one fart every four minutes. It’s unbelievable. Damn it, I hardly fart at all, I should never have mentioned it. Extraordinary how mathematics help you to know yourself."

"They who study mathematiks only to fix their minds, and render them the steadyer to apply to all other things, as there are many who profess to do, are as wise as those who think by rowing boats, to learn to swim."

"Mathematical development in England was at a low ebb in the early decades of the nineteenth century, with Cambridge stagnating in the shadow of Newton, who had produced his mathematics nearly a century and a half earlier. This dead hand of tradition, which stifled much initiative and originality, was in sharp contrast to the situation in France."

"Advanced mathematics opens doors to many fields of study. More importantly, it expands the way you think."

"Yeah, Silver and his math are jokes, because math has a liberal bias. After all, math is the reason Mitt Romney's tax plan doesn't add up."

"I was an atheist, finding no reason to postulate the existence of any truths outside of mathematics, physics and chemistry. But then I went to medical school, and encountered life and death issues at the bedsides of my patients. Challenged by one of those patients, who asked "What do you believe, doctor?", I began searching for answers."

"Magick, mysticism, and the mathematics are triplets."

"In the pure mathematics, we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven."

"A man with all the algebra in the world is often only an ass when he knows nothing else. Perhaps in ten years society may derive advantage from the curves which these visionary algebraists will have laboriously squared. I congratulate posterity beforehand. But to tell you the truth I see nothing but a scientific extravagance in all these calculations. That which is neither useful nor agreeable is worthless. And as for useful things, they have all been discovered; and to those which are agreeable, I hope that good taste will not admit algebra among them."

"As to your Newton, I confess I do not understand his void and his gravity; I admit he has demonstrated the movement of the heavenly bodies with more exactitude than his forerunners; but you will admit it is an absurdity to maintain the existence of Nothing."

"Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!"

"Measuring is one of the more practical uses of mathematics, but the ability and desire to measure aren't always wrapped up with the need to know useful answers."

"Letting numbers take us where we can't go in person — whether that's to the top of a windmill or to the origin and borders of the universe — has been and still is one of humankind's favorite intellectual adventures."

"The Devil: Okay, boys...tonight's homework. Algebra. Xn + Yn = Zn. You're never gonna use that, are you? Imperialism and the First World War. What was done is done. No point thinking about it now. German, French, Spanish. Ja, ja, oui, oui, s, s. It's nonsense. Everyone speaks English anyway. And if they don't, they ought to. So, no homework tonight. But I want you to watch a lot of TV, don't neglect your video games...and I'll see you in the morning. Shall we say 10:00, 10:30? No point in getting up too early."

"Mathematics is really a liberal art if you look at it from a slightly different point of view."

"The mathematical genius can only carry on from the point which mathematical knowledge within his culture has already reached. Thus if Einstein had been born into a primitive tribe which was unable to count beyond three, life-long application to mathematics probably would not have carried him beyond the development of a decimal system based on fingers and toes."

"So far, all of reality seems to be described by exquisite, elegant mathematical equations. We can’t stop now – it’s got to be beautiful all the way down!"

"Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite."

"Halakhic man, well furnished with rules, judgments, and fundamental principles, draws near the world with an a priori relation. His approach begins with an ideal creation and concludes with a real one. To whom may he be compared? To a mathematician who fashions an ideal world and then uses it for the purpose of establishing a relationship between it and the real world. ... The essence of the Halakhah, which was received from God, consists in creating an ideal world and cognizing the relationship between that ideal world and our concrete environment."

"Mathematics is a versatile art; it can be applied to widely different purposes. Math has no morality; it does not care what it counts or what it proves."

"I had been to school most all the time and could spell and read and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics anyway."

"School children and students who love God should never say: “For my part I like mathematics”; “I like French”; “I like Greek.” They should learn to like all these subjects, because all of them develop that faculty of attention which, directed toward God, is the very substance of prayer."

"The number 2 thought of by one man cannot be added to the number 2 thought of by another man so us to make up the number 4."

"The good Christian should beware of mathematicians, and all those who make empty prophesies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell."

"Mathematicians are used to game-playing according to a set of rules they lay down in advance, despite the fact that nature always writes her own. One acquires a great deal of humility by experiencing the real wiliness of nature."

"Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium... cavendi sunt, ne consortio daemoniorum irretiant."

"A mathematician is a machine for turning coffee into theorems."

"I united the majority of well-informed persons into a club, which we called by the name of the Junto, and the object of which was to improve our understandings. ... The first members of our club were... Thomas Godfrey, a self-taught mathematician, and afterwards inventor of what is now called Hadley's dial; but he had little knowledge out of his own line, and was insupportable in company, always requiring, like the majority of mathematicians that have fallen in my way, an unusual precision in everything that is said, continually contradicting, or making trifling distinctions—a sure way of defeating all the ends of conversation. He very soon left us."

"Avec toute l’algèbre du monde on n’est souvent qu’un sot lorsqu’on ne sait pas autre chose. Peut-être dans dix ans la société tirera-t-elle de l’avantage des courbes que des songe-creux d’algébristes auront carrées laborieusement. J’en félicite d’avance la postérité; mais, à vous parler vrai, je ne vois dans tous ces calculs qu’une scientifique extravagance. Tout ce qui n’est ni utile ni agréable ne vaut rien. Quant aux choses utiles, elles sont toutes trouvées; et, pour les agréables, j’espère que le bon goût n’y admettra point d’algèbre."

"Die Mathematiker sind eine Art Franzosen; redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anders."

"To be a mathematician you must love mathematics more than family, religion, money, comfort, pleasure, glory. I do not mean that you must love it to the exclusion of family, religion, and the rest, and I do not mean that if you do love it, you'll never have any doubts, you'll never be discouraged, you'll never be ready to chuck it all and take up gardening instead. Doubts and discouragements are part of life. Great mathematicians have doubts and get discouraged, but usually they can’t stop doing mathematics anyway, and, when they do, they miss it very deeply. [...] Mind you, I am not recommending or insisting that you love mathematics. I am not issuing an order: “If you want to be a mathematician, start loving mathematics forthwith”—that would be absurd. What I am saying is that the love of mathematics is a hypothesis without which the conclusion doesn’t follow. If you want to be a mathematician, look into your soul and ask yourself how much you want to be one. If the wish isn’t very deep and very great, if it is not, in fact, maximal, if you have another desire that takes precedence, or even more than one, then you should not try to be a mathematician. The “should” is not a moral one; it is a pragmatic one. I think that you would probably not succeed in your attempt, and, in any event, you would probably feel frustrated and unhappy."

"The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another."

"Mathematicians seem to have no difficulty in creating new concepts faster than the old ones become well understood."

"I have hardly ever known a mathematician who was capable of reasoning."

"Aristotle, so far as I know, was the first man to proclaim explicitly that man is a rational animal. His reason for this view was one which does not now seem very impressive; it was, that some people can do sums."

"Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs Aristotle to keep her mouth open while he counted."

"These experiences are not 'religious' in the ordinary sense. They are natural, and can be studied naturally. They are not 'ineffable' in the sense the sense of incommunicable by language. Maslow also came to believe that they are far commoner than one might expect, that many people tend to suppress them, to ignore them, and certain people seem actually afraid of them, as if they were somehow feminine, illogical, dangerous. 'One sees such attitudes more often in engineers, in mathematicians, in analytic philosophers, in book keepers and accountants, and generally in obsessional people'. The tends to be a kind of bubbling-over of delight, a moment of pure happiness. 'For instance, a young mother scurrying around her kitchen and getting breakfast for her husband and young children. The sun was streaming in, the children clean and nicely dressed, were chattering as they ate. The husband was casually playing with the children: but as she looked at them she was suddenly so overwhelmed with their beauty and her great love for them, and her feeling of good fortune, that she went into a peak experience . . ."

"New challenges driven by evolving global technology inspire fresh trends and approaches in teaching statistics in business schools of the 21st century."

"In the 1930s English statistical theory was beginning to travel, with contributions from, amongst others, Hotelling and Snedecor in America and Darmois in France, but its home was still in England where there were four important centres: University College London, Rothamsted Experimental Station, Edinburgh University and Cambridge University with University College and Rothamsted far in the lead. Although Cambridge University was slow to adopt modern statistical theory, Cambridge men–Karl Pearson, Edmund Whittaker and Ronald Fisher–had put the other places on the statistical map. University College was the most established centre and its importance went back to 1893 when Karl Pearson, the professor of applied mathematics, first collaborated with Raphael Weldon, the professor of zoology on a subject they called “biometry.” There was a second surge in the “English statistical school” associated with R. A. Fisher who went to work at Rothamsted in 1919."

"I wish that people would be persuaded that psychological experiments, especially those on the complex functions, are not improved (by large studies); the statistical method gives only mediocre results; some recent examples demonstrate that. The American authors, who love to do things big, often publish experiments that have been conducted on hundreds and thousands of people; they instinctively obey the prejudice that the persuasiveness of a work is proportional to the number of observations. This is only an illusion."

"All models are wrong, but some are useful"

"Statisticians, like artists, have the bad habit of falling in love with their models."

"...the statistician knows...that in nature there never was a normal distribution, there never was a straight line, yet with normal and linear assumptions, known to be false, he can often derive results which match, to a useful approximation, those found in the real world."

"Life in financial markets has got no relation to sigmas. I mean, if everybody who had ever operated in financial markets had never had any concept of standard errors, and so on, they would be a lot better off."

"Strange events permit themselves the luxury of occurring."

"If you torture the data enough, nature will always confess."

"A new branch of mathematics was developed over the last 200 years to deal with the more complex aspect of reality: statistics."

"Confucius, Buddha, Jesus and Muhammad would have been bewildered if you'd have told them that in order to understand the human mind and cure its illness you must first study statistics."

"There are no routine statistical questions, only questionable statistical routines."

"Statistics is the science of learning from data, and of measuring, controlling, and communicating uncertainty."

"I had been so terrorized by scientific statistics (if ten million people each leave over three grains of rice from their lunch, how many sacks of rice are wasted in one day; if ten million people each economize one paper handkerchief a day, how much pulp will be saved?) that whenever I left over a single grain of rice, whenever I blew my nose, I imagined that I was wasting mountains of rice, tons of paper, and I fell prey to a mood dark as if I had committed some terrible crime. But these were the lies of science, the lies of statistics and mathematics: you can't collect three grains of rice from everybody."

"In our private life as in our collective life there is no other truth than a statistical one."

"There are three kinds of lies: lies, damned lies, and statistics."

"Statistics has been the most successful information science. Those who ignore Statistics are condemned to reinvent it."

"The rise of biometry in this 20th century, like that of geometry in the 3rd century before Christ, seems to mark out one of the great ages or critical periods in the advance of the human understanding."

"To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of."

"briefly, and in its most concrete form, the object of statistical methods is the reduction of data. A quantity of data, which usually by its mere bulk is incapable of entering the mind, is to be replaced by relatively few quantities which shall adequately represent the whole, or which, in other words, shall contain as much as possible, ideally the whole, of the relevant information contained in the original data."

"It is a statistical commonplace that the interpretation of a body of data requires a knowledge of how it was obtained."

"“Without Statistics and numbers, it's a mere thought of unconvincing opinions.”"

"A well-wrapped statistic is better than Hitler's "big lie"; it misleads, yet it cannot be pinned on you."

"Statistics is both the science of uncertainty and the technology of extracting information from data."

"Statistics are often the last refuge of the antihumanist."

"I think the essential thing if you want to be a good statistician, as opposed to being a mathematician, is to talk to people and find out what they're doing and why they're doing it."

"Most people don’t have a good understanding of just how variable statistics are, even when you are dealing with robots."

"Statistics is the art of stating in precise terms that which one does not know."

"Statistics is the study and informed application of methods for reaching conclusions about the world from fallible observations."

"Each of us has been doing statistics all his life, in the sense that each of us has been busily reaching conclusions based on empirical observations."

"Statistics are like a bikini. What they reveal is suggestive, but what they conceal is vital."

"Politicians use statistics in the same way that a drunk uses lampposts — for support rather than illumination."

"Uncertainty is a personal matter; it is not the uncertainty but your uncertainty."

"Without the aid of statistics nothing like real medicine is possible."

"According to a University College London study (2014), 40% of women surveyed with severe mental illness had suffered rape or attempted rape in adulthood, and 53% of those had attempted suicide as a result. In the general population, 7% of women had been victims of rape or attempted rape, of whom 3% had attempted suicide. If these statistics don't sound accurate to you, your hesitation or disbelief supports another reality about rape research: Because so many individuals who survive a rape may not report a rape (for a multitude of reasons), statistics have limited meaning."

"A statistical procedure is not an automatic, mechanical truth-generating machine."

"While it is easy to lie with statistics, it is even easier to lie without them."

"A statistical analysis, properly conducted, is a delicate dissection of uncertainties, a surgery of suppositions."

"91.7 percent of all statistics are made-up on the spot."

"In God we trust. All others must bring data."

"To understand God's thoughts we must study statistics, for these are the measure of his purpose."

"The true foundation of theology is to ascertain the character of God. It is by the art of statistics that law in the social sphere can be ascertained and codified, and certain aspects of the character of God thereby revealed. The study of statistics is thus a religious service."

"Average a left-hander with a right-hander and what do you get?"

"With four parameters I can fit an elephant, and with five I can make him wiggle his trunk."

"Although its evolution in the United States differed markedly from that of applied mathematics, statistics, too, benefited from the presence of the emigres and from the overall war effort. After a protracted period of professional differentiation from the social scientists and from the social sciences, mathematical statisticians had formed their own society, the Institute of Mathematical Statistics (IMS), in 1935. By 1938, the IMS had also taken over responsibility for the Annals of Mathematical Statistics, a journal that had been founded in 1929 to serve the needs of the more mathematically and theoretically inclined statistical practitioners. Thus, when refugees like Neyman, William Feller, Mark Kac, and Abraham Wald took up positions in the United States at Berkeley, Brown, Cornell, and Columbia, respectively, they were able to participate in a young, but viable, community of mathematical statisticians."

"Numbers and stats bob in a sentimental slop, a swampy slurry of bits of hard data and buckets of mushy manipulation."

"The individual source of the statistics may easily be the weakest link. Harold Cox tells a story of his life as a young man in India. He quoted some statistics to a Judge, an Englishman, and a very good fellow. His friend said, Cox, when you are a bit older, you will not quote Indian statistics with that assurance. The Government are very keen on amassing statistics—they collect them, add them, raise them to the nth power, take the cube root and prepare wonderful diagrams. But what you must never forget is that every one of those figures comes in the first instance from the chowty dar [chowkidar] (village watchman), who just puts down what he damn pleases."

"Thomasina: If there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers? Septimus: We do. Thomasina: Then why do your shapes describe only the shapes of manufacture? Septimus: I do not know. Thomasina: Armed thus, God could only make a cabinet."

"There are two kinds of statistics, the kind you look up and the kind you make up."

"The main problem with statistics is statisticians."

"The best thing about being a statistician is that you get to play in everyone's backyard"

"Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise."

"The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data."

"The death of one man is a tragedy, the death of millions is a statistic."

"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read or write."

"Statistics is a body of methods for making wise decisions in the face of uncertainty"

"While nothing is more uncertain than the duration of a single life, nothing is more certain than the average duration of a thousand lives."

"The bulk of statistical criticism is of the hit-and-run variety – the critic points out some real or fancied flaw and supposes that his job is done. Indeed, some critics appear to labor under the misconception that if some flaw can be found in a study, this automatically in validates the author’s conclusions. Since the critic makes no attempt to develop a tenable counterhypothesis, his performance is on a par with that of a proponent who glances at his data and then jumps to his conclusion."

"How dare we speak of the laws of chance? Is not chance the antithesis of all law?"

"He who believes this (atomism) may as well believe that if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form the Annals of Ennius. I doubt whether fortune could make a single verse of them."

"The generation of random numbers is too important to be left to chance."

"The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He does not throw dice."

"Events may appear to us to be random, but this could be attributed to human ignorance about the details of the processes involved."

"Perhaps randomness is not merely an adequate description for complex causes that we cannot specify. Perhaps the world really works this way, and many events are uncaused in any conventional sense of the word."

"Randomness is a very, very subtle concept with its study properly belonging to statisticians more than mathematicians."

"Consideration of black holes suggests, not only that God does play dice, but that he sometimes confuses us by throwing them where they can't be seen."

"Fretting about a dearth of randomness seems like worrying that humanity might use up its last reserves of ignorance."

"The fact that randomness requires a physical rather than a mathematical source is noted by almost everyone who writes on the subject, and yet the oddity of this situation is not much remarked."

"Random chance was not a sufficient explanation of the Universe — in fact, random chance was not sufficient to explain random chance."

"Random numbers should not be generated with a method chosen at random."

"The sun comes up just about as often as it goes down, in the long run, but this doesn't make its motion random."

"A random sequence is a vague notion in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians and depending somewhat on the uses to which the sequence is to be put."

"Quand une regle est fort composée, ce qui luy est conforme, passe pour irrégulier."

"For I do not believe that it is through the interference of Divine Providence … that the spittle of a certain person moved, fell on a certain gnat in a certain place, and killed it."

"Existence is random. Has no pattern save what we imagine after staring at it for too long. No meaning save what we choose to impose. This rudderless world is not shaped by vague metaphysical forces. It is not God who kills the children. Not fate that butchers them or destiny that feeds them to the dogs. It’s us. Only us."

"Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin."

"One of us recalls producing a 'random' plot with only 11 planes, and being told by his computer center's programming consultant that he had misused the random number generator: 'We guarantee that each number is random individually, but we don’t guarantee that more than one of them is random.' Figure that out."

"The definition of random in terms of a physical operation is notoriously without effect on the mathematical operations of statistical theory because so far as these mathematical operations are concerned random is purely and simply an undefined term."

"While in theory randomness is an intrinsic property, in practice, randomness is incomplete information."

"Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist."

"Dividing by zero...allows you to prove, mathematically, anything in the universe. You can prove that 1+1=42, and from there you can prove that J. Edgar Hoover is a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted? (See appendix A for a proof that Winston Churchill was a carrot.)"

"Zero is even because if you divide it in half, both people get nothing."

"...can't split it out, because you haven't got nowt."

"No one gets owt if it's shared out."

"It is even because the 2 times table it goes 0, 2, 4, 6, 8."

"Um, first I said that um, zero was even but then I guess I revised so that zero, I think, is special because um, I— um, even numbers, like they — they make even numbers; like two, um, two makes four, and four is an even number; and four makes eight; eight is an even number; and um, like that. And, and go on like that and like one plus one and go on adding the same numbers with the same numbers. And so I, I think zero's special. - Benny"

"I didn't think zero was even or odd until yesterday and then someone said it could be even because one below zero and one above zero are both odd, and that made sense. - Sheena"

"I thought zero was an even number, but from the meeting I got sort of mixed up because I heard other ideas I agree with and now I don't know which one I should agree with. … I'm going to listen more to the discussion and find out. - Mei"

"This method, seemingly very clever, actually played into our hands! And so it often happens that an apparently ingenious idea is in fact a weakness which the scientific cryptographer seizes on for his solution."

"Few false ideas have more firmly gripped the minds of so many intelligent men than the one that, if they just tried, they could invent a cipher that no one could break."

"The multiple human needs and desires that demand privacy among two or more people in the midst of social life must inevitably lead to cryptology wherever men thrive and wherever they write."

"The magic words are squeamish ossifrage"

"Feistel and Coppersmith rule. Sixteen rounds and one hell of an avalanche."

"For the computer security community, the moral is obvious: if you are designing a system whose functions include providing evidence, it had better be able to withstand hostile review."

"When a cryptanalyst starts out trying to analyze a new algorithm, his first thought is probably: "Yikes. What a mess. I'll never make sense of this". So there are all sorts of tricks to help you start to probe into the convoluted innards of the cipher. One of these is to attack a weakened version. Later, he may be able to extend the attack to the full strength version; or, if this cannot be done, the reason why it can't at least gives some insight into the strengths and weaknesses of the cipher."

"There is also a side benefit: the difference in strength made by even really subtle changes warns us just how tricky crypto can be..."

"Due to the suspicious nature of crypto users I have a feeling DES will be with us forever, we will just keep adding keys and cycles..."

"The NSA response was, "Well, that was interesting, but there aren't any ciphers like that.""

"The real work in an attack, at least an attack against a well-designed cipher, is modifying the attack technique so that it works. Knudsen's papers are an excellent example of this; he is a master at making an attack work where others have failed. Differentials work where characteristics don't. Truncated differentials work where normal differentials don't. Even this year's exciting find, impossible differentials, are simply another way at looking at a differential attack. A cryptanalyst with a "menu" would have never found any of those attacks, and would have broken far fewer ciphers."

"The obvious mathematical breakthrough would be development of an easy way to factor large prime numbers."

"The point of academic attacks is not exhibiting practical breaks; the point is that only a trained cryptographer can tell whether a given algorithm is secure or not. The author of an algorithm says: "My cipher is secure, and trust me, I am an expert at this. And to prove that I am a real good expert, I challenge other experts to find even the most impractical, academic flaw in my cipher"."

"Just like glue. Commercial ads state that the foobar glue can stick an elephant to the ceiling. Who needs to stick an elephant to the ceiling? But if it can do that, people will trust its sticking strength."

"We didn't do this with just a pencil and some paper. Lots of our notes are in pen. We didn't need to erase much."

"If you think cryptography is the answer to your problem, then you don't know what your problem is."

"This statistical regularity in moral affairs fully establishes their being under the presidency of law. Man is now seen to be an enigma only as an individual; in the mass he is a mathematical problem."

"Probability is the very guide of life."

"It is a truth very certain that, when it is not in our power to determine what is true, we ought to follow what is most probable"

"Al right. I already see you turning off. I can see you say you don't understand me. You can't understand that it could be chance. "I don't like it!" Tough! I don't like it either, but that's the way it is! Ok? I don't understand it either. ..."It must be that Nature knows that it's going to go up or down." No, it must not be that nature knows! We are not to tell Nature what she's gotta be! That's what we found out. Every time we take a guess as how she's got to be, and go and measure... She's clever. She's always got better imagination than we have, and she finds a cleverer way to do it than we have thought of. And in this particular case, the clever way to do it is by probability, by odds. ...[L]ight works by probability."

"My thesis, paradoxically, and a little provocatively, but nonetheless genuinely, is simply this :PROBABILITY DOES NOT EXIST.The abandonment of superstitious beliefs about the existence of Phlogiston, the Cosmic Ether, Absolute Space and Time, ... , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs."

"Probability is too important to be left to the experts. [...] The experts, by their very expert training and practice, often miss the obvious and distort reality seriously. [...] The desire of the experts to publish and gain credit in the eyes of their peers has distorted the development of probability theory from the needs of the average user. The comparatively late rise of the theory of probability shows how hard it is to grasp, and the many paradoxes show clearly that we, as humans, lack a well grounded intuition in the matter. Neither the intuition of the man in the street, nor the sophisticated results of the experts provides a safe basis for important actions in the world we live in."

"How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?"

"There is no Algebraist nor Mathematician so expert in his science, as to place entire confidence in any truth immediately upon his discovery of it, or regard it as any thing, but a mere probability. Every time he runs over his proofs, his confidence increases; but still more by the approbation of his friends; and is raised to its utmost perfection by the universal assent and applauses of the learned world."

"R. A. Fisher, J. Neyman, R. von Mises, W. Feller, and L. J. Savage denied vehemently that probability theory is an extension of logic, and accused Laplace and Jeffreys of committing metaphysical nonsense for thinking that it is."

"They should have known better. The probability of a train derailment was infinitesimal. That meant it was only a matter of time."

"No human being can give an eternal resolution to another or take it from him; If someone objects that then one might just as well be silent if there is no probability of winning others, he thereby has merely shown that although his life very likely thrived and prospered in probability and everyone of his undertakings in the service of probability went forward, he has never really ventured and consequently has never had or given himself the opportunity to consider that probability is an illusion, but to venture the truth is what gives human life and the human situation pith and meaning, to venture is the fountainhead of inspiration, whereas probability is the sworn enemy of enthusiasm, the mirage whereby the sensate person drags out time and keeps the eternal away, whereby he cheats God, himself, and his generation: cheats God of the honor, himself of liberating annihilation, and his generation of the equality of conditions."

"The epistemological value of probability theory is based on the fact that chance phenomena, considered collectively and on a grand scale, create non-random regularity."

"As is known, the question of the objectivity or the subjectivity of probability has divided the world of science into two camps. Some maintain that there exist two types of probability, as above, others, that only the subjective exists, because regardless of what is supposed to take place, we cannot have full knowledge of it. Therefore, some lay the uncertainty of future events at the door of our knowledge of them, whereas others place it within the realm of the events themselves."

"People seem to flinch at the word probability—it has too many syllables. Besides, it sounds mathematical, and it’s become politically correct in our country to be proud of not knowing any mathematics. (We’re already paying the price for that.)"

"The laws of probability are mighty powerful, and they never sleep. If this were more widely understood there’d be a lot less crowing about good luck, and a lot less guilt about bad luck. And we’d have a more civilized world. Some things really do happen by chance, and there is little we can do to change that."

"Feeling threatened is not the same as being threatened, but the difference gets lost. The danger from low levels of radiation is quite low, as expressed as morbidity statistics or probabilities, but there is an unfortunate lack of connection to probability in the average person. Low probabilities are a particular problem of perception. If they were not, then nobody would play the lottery and the gambling industry would collapse."

"In applying dynamical principles to the motion of immense numbers of atoms, the limitation of our faculties forces us to abandon the attempt to express the exact history of each atom, and to be content with estimating the average condition of a group of atoms large enough to be visible. This method... which I may call the statistical method, and which in the present state of our knowledge is the only available method of studying the properties of real bodies, involves an abandonment of strict dynamical principles, and an adoption of the mathematical methods belonging to the theory of probability. … If the actual history of Science had been different, and if the scientific doctrines most familiar to us had been those which must be expressed in this way, it is possible that we might have considered the existence of a certain kind of contingency a self evident truth, and treated the doctrine of philosophical necessity as a mere sophism."

"Einstein's supreme greatness was in transforming physical thinking from that of the culmination of classical physics about 1900 to that of quantum mechanics starting about 1925. ...far more than anyone else, he caused physicists to think in terms of probabilities. He began to do this in his early work in thermodynamics, and he brought such thinking to its first great fruition in 1905 in his work on Brownian movement and in his first work on radiation, in which he introduced the concept of light quanta or photons. Its second, even greater fruition was in his famous paper on the quantum theory of radiation in 1917. That paper illustrated methods that have been in use almost without change ever since, even though the majority of the users have no knowledge that it was Einstein who propounded them. It was in this paper that Einstein postulated the various transition possibilities between two states of a quantized system. ...quantum theory has existed ever since precisely for the purpose of evaluating these probabilities. In this paper of 1917 Einstein postulated in particular the process known as stimulated emission, and inferred the properties of this process. This is the process employed in the... light maser or laser."

"The theory of probability combines commonsense reasoning with calculation. It domesticates luck, making it subservient to reason."

"Why do [people] confuse probability and expectation, that is, probability and probability times payoff? Mainly because much... schooling comes from examples in symmetric environments... the... bell curve... is entirely symmetric."

"At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate."

"Since the end of World War II, I have been working on the many ramifications of the theory of messages. Besides the electrical engineering theory of the transmission of messages, there is a larger field which includes not only the study of language but the study of messages as a means of controlling machinery and society, the development of computing machines and other such automata, certain reflections upon psychology and the nervous system, and a tentative new theory of scientific method. This larger theory of messages is a probabilistic theory, an intrinsic part of the movement that owes its origin to Willard Gibbs and which I have described in the introduction."

"Algorithms are the computational content of proofs."

"The issue with AI, to me, is a very simple one. It's like the term algorithm. We watch companies use algorithms, and now AI, as a means of evading responsibility for their actions… If we endorse the view that AI is all-powerful, we are endorsing the view that it can alleviate people of responsibility for their actions—militarily, socio­economically, whatever. The biggest danger of AI is that we attribute these godlike characteristics to it and therefore let ourselves off the hook. I don't know what the mythological underpinnings of this are, but throughout history there's this tendency of human beings to create false idols, to mold something in our own image and then say we've got godlike powers because we did that."

"Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game...), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms... even logics. ...[I]n arithmetic we find s, which are a whole new 'game'... [T]o identify mathematics with games would be one of those part-for-whole mistakes (like 'all geometry is projective geometry' or 'arithmetic is just logic' from the nineteenth century)... [M]y separation of game analysis from playing games tells... against the analogy of mathematics to the expert play of the game itself."

"Numerical analysis is very much an experimental science."

"Algorithms + Data Structures = Programs"

"Ford, there’s an infinite number of monkeys outside who want to talk to us about this script for Hamlet they’ve worked out."

"Should not a true understanding of life promote care for the future along with the present? This is the immediate duty of every scientist. Until now scientists have dealt with life as finite — is it not now their mission to see life as extending into Infinity? 553."

"In Sorbière's day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisibles and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes's acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and banned the concept.. Even as late as the 1730s... George Berkeley mocked mathematicians for their use of infinitesimals... Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitesimally small. These included, in addition to Wallis: Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton."

"On the one side were ranged the forces of hierarchy and order—Jesuits, Hobbesians, French Royal Courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative "liberalizers" such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come."

"All things were together, infinite both in number and in smallness; for the small too was infinite."

"Empedocles holds that the corporeal elements are four, while all the elements-including those which initiate movement-are six in number; whereas Anaxagoras agrees with Leucippus and Democritus that the elements are infinite."

"Motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuous--that is how it comes about that 'infinite' is often used in definitions of the continuous ('what is infinitely divisible is continuous'). Besides these, place, void, and time are thought to be necessary conditions of motion."

"The science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, e.g. a quality or a point--it is not necessary perhaps that such things should be put under either head. Hence it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, what it is. The appropriateness to the science of this problem is clearly indicated. All who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things."

"Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self-subsistent substance, and not as a mere attribute of some other thing. Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also."

"The Pythagoreans identify the infinite with the even. For this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity. An indication of this is what happens with numbers. If the gnomons are placed round the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same. But Plato has two infinities, the Great and the Small."

"The physicists... always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so-called elements--water or air or what is intermediate between them. Those who make them limited in number never make them infinite in amount. But those who make the elements infinite in number, as Anaxagoras and Democritus do, say that the infinite is continuous by contact-compounded of the homogeneous parts."

"We cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle. Everything is either a source or derived from a source. But there cannot be a source of the infinite or limitless, for that would be a limit of it. Further, as it is a beginning, it is both uncreatable and indestructible. For there must be a point at which what has come to be reaches completion, and also a termination of all passing away. That is why, as we say, there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as Mind or Friendship. Further they identify it with the Divine, for it is 'deathless and imperishable' as Anaximander says, with the majority of the physicists."

"Belief in the existence of the infinite comes mainly from five considerations: 1) From the nature of time--for it is infinite. 2) From the division of magnitudes-for the mathematicians also use the notion of the infinite. 3) If coming to be and passing away do not give out, it is only because that from which things come to be is infinite. 4) Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself. 5) Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody--not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought."

"That what is outside is infinite, leads people to suppose that body also is infinite, and that there is an infinite number of worlds. Why should there be body in one part of the void rather than in another? Grant only that mass is anywhere and it follows that it must be everywhere. Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be."

"The problem of the infinite is difficult: many contradictions result whether we suppose it to exist or not to exist. If it exists, we have still to ask how it exists; as a substance or as the essential attribute of some entity? Or in neither way, yet none the less is there something which is infinite or some things which are infinitely many?"

"The problem... which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite. We must begin by distinguishing the various senses in which the term 'infinite' is used. 1) What is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is 'invisible'). 2) What admits of being gone through, the process however having no termination, or what scarcely admits of being gone through. 3) What naturally admits of being gone through, but is not actually gone through or does not actually reach an end. Further, everything that is infinite may be so in respect of addition or division or both."

"If 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible."

"Anaxagoras gives an absurd account of why the infinite is at rest. He says that the infinite itself is the cause of its being fixed. This because it is in itself, since nothing else contains it--on the assumption that wherever anything is, it is there by its own nature. But this is not true: a thing could be somewhere by compulsion, and not where it is its nature to be."

"The view that there is an infinite body is plainly incompatible with the doctrine that there is necessarily a proper place for each kind of body, if every sensible body has either weight or lightness, and if a body has a natural locomotion towards the centre if it is heavy, and upwards if it is light. This would need to be true of the infinite also. But neither character can belong to it: it cannot be either as a whole, nor can it be half the one and half the other. For how should you divide it? Or how can the infinite have the one part up and the other down, or an extremity and a centre?"

"To suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and an end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. ...clearly there is a sense in which the infinite exists and another in which it does not."

"The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it."

"Our definition then is as follows: A quantity is infinite if it is such that we can always take a part [or piece] outside what has been already taken. On the other hand, what has nothing outside it is complete and whole. For thus we define the whole--that from which nothing is wanting, as a whole man or a whole box. What is true of each particular is true of the whole as such--the whole is that of which nothing is outside. On the other hand that from which something is absent and outside, however small that may be, is not 'all'. 'Whole' and 'complete' are either quite identical or closely akin. Nothing is complete (teleion) which has no end (telos); and the end is a limit."

"Parmenides must be thought to have spoken better than Melissus. The latter says that the whole is infinite, but the former describes it as limited, 'equally balanced from the middle'. ...it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine."

"What is one is indivisible whatever it may be, e.g. a man is one man, not many. Number on the other hand is a plurality of 'ones' and a certain quantity of them. Hence number must stop at the indivisible: for 'two' and 'three' are merely derivative terms, and so with each of the other numbers."

"In the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number. But this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time."

"With magnitudes the contrary holds. What is continuous is divided ad infinitum, but there is no infinite in the direction of increase. For the size which it can potentially be, it can also actually be. Hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens."

"Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that the finite straight line may be produced as far as they wish."

"It is plain that the infinite is a cause in the sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible. All the other thinkers, too, evidently treat the infinite as matter--that is why it is inconsistent in them to make it what contains, and not what is contained."

"It remains to dispose of the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. Some have no cogency; others can be met by fresh objections that are valid. 1) In order that coming to be should not fail, it is not necessary that there should be a sensible body which is actually infinite. The passing away of one thing may be the coming to be of another, the All being limited. 2) There is a difference between touching and being limited. The former is relative to something and is the touching of something (for everything that touches touches something), and latter is an attribute of some one of the things which are limited. On the other hand, what is limited is not limited in relation to anything. Again, contact is not necessarily possible between any two things taken at random. 3) To rely on mere thinking is absurd, for then the excess or defect is not in the thing but in the thought. One might think that one of us is bigger than he is and magnify him ad infinitum. But it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he is the size he is. The thought is an accident. a) Time indeed and movement are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence. b) Magnitude is not infinite either in the way of reduction or of magnification in thought. This concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is."

"If, then, there is some end of the things we do, which we desire for its own sake (everything else being desired for the sake of this), and if we do not choose everything for the sake of something else (for at that rate the process would go on to infinity, so that our desire would be empty and vain), clearly this must be the good and the chief good."

"For if they imagine infinite spaces of time before the world, during which God could not have been idle, in like manner they may conceive outside the world infinite realms of space, in which, if any one says that the Omnipotent cannot hold His hand from working, will it not follow that they must adopt Epicurus’ dream of innumerable worlds? with this difference only, that he asserts that they are formed and destroyed by the fortuitous movements of atoms, while they will hold that they are made by God’s hand, if they maintain that, throughout the boundless immensity of space, stretching interminably in every direction round the world, God cannot rest, and that the worlds which they suppose Him to make cannot be destroyed. ... there is no place beside the world ...no time before the world."

"Every material body has some natural movement, and can change its place. But an infinite body would occupy every place, and every place would be its own place. ...Every mathematical body must be imagined as having some shape. But shape is defined by some term or boundary, and nothing infinite can have a boundary."

"Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity."

"In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered."

"Galileo observed as early as 1638 that there are precisely as many squares 1, 4, 9, 16, 25,... as are positive integers all together. This is evident from the sequences1, 2, 3, 4, 5, 6, ... , n, ... 12, 22, 32, 42, 52, 62, ..., n, ... He thus recognized the fundamental distinction between finite and infinite classes that became current in the late nineteenth century. An infinite class is one in which there is a one-to-one correspondence between the whole class and a subclass of the whole. Or, what is equivalent, there are as many things in one part of an infinite class as there are in the whole class. ...A class whose elements can be put in a one-to-one correspondence with the integers 1, 2, 3, ... is said to be denumerable. All the points in any line segment, finite or infinite in length, form a non-denumerable set. A basic course in calculus starts from the theory of point sets. The distinction between denumerable and non-denumerable classes was not started by Galileo; it was observed about 1840 by Bolzano and in 1878 by Cantor. But Galileo's recognition of the cardinal property of all infinite classes makes him one of the genuine anticipators in the history of calculus. The other was Archimedes."

"If a thing loves, it is infinite."

"If the doors of perception were cleansed everything would appear to man as it is, infinite. For man has closed himself up, till he sees all things thro' narrow chinks of his cavern."

"To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour."

"It is not possible, I think, to rise from the perusal of the arguments of Clark and Spinoza without a deep conviction of the futility of all endeavors to establish, entirely à priori, the existence of an Infinite Being, His attributes, and His relation to the universe. The fundamental principle of all such speculations, viz. that whatever we can clearly conceive, must exist, fails to accomplish its end, even when its truth is admitted. For how shall the finite comprehend the infinite? Yet must the possibility of such conception be granted, and in something more than the sense of a mere withdrawal of the limits of phænomal existence, before any solid ground can be established for the knowledge, à priori, of things infinite and eternal."

"There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite."

"There are no moral or intellectual merits. Homer composed the Odyssey; if we postulate an infinite period of time, with infinite circumstances and changes, the impossible thing is not to compose the Odyssey, at least once."

"Anyone who believes exponential growth can go on forever in a finite world is either a madman or an economist."

"[A]s the great extreme of dimension is sublime, so the last extreme of littleness is in the same measure sublime... when we attend to the infinite divisibility of matter, when we pursue animal life into these excessively small, and yet organized beings... when we push our discoveries yet downward... in tracing which the imagination is lost as well as the sense; we become amazed and confounded at the wonders of minuteness; nor can we distinguish in its effects this extreme of littleness from the vast itself. For division must be infinite as well as addition; because the idea of a perfect unity can no more be arrived at, than that of a complete whole, to which nothing can be added."

"Another source of the sublime is infinity... Infinity has a tendency to fill the mind with that sort of delightful horror, which is the most genuine effect and truest test of the sublime. There are scarce any things which can become the objects of our senses, that are really... infinite. But the eye not being able to perceive the bounds... they seem... infinite, and they produce the same effects... We are deceived in the like manner, if the parts of some large object are so continued to any indefinite number, that the imagination meets no check... Whenever we repeat an idea frequently, the mind... repeats it long after the first cause has ceased... multiplied without end. ...This is the reason of an appearance very frequent in madmen; that they remain... in the constant repetition of some remark... complaint, or song... every repetition reinforces it with new strength... unrestrained by the curb of reason, continues... to the end of their lives."

"There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated."

"Each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite."

"The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great."

"I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures."

"The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds."

"For science, the invention of the differential calculus was a giant step. For the first time in human history the concept of the infinite, which had intrigued philosophers and poets from time immemorial, was given a precise mathematical definition, which opened countless new possibilities for the analysis of natural phenomena. ...According to Zeno, the great athlete Achilles can never catch up with a tortoise... The flaw in Zeno's argument lies in the fact that even though it will take Achilles an infinite number of [procedural] steps to reach the tortoise, this does not take an infinite time. With the tools of Newton's calculus it is easy to show that a moving body will run through an infinite number of infinitely small intervals in a finite time."

"And in that moment, I swear we were infinite."

"The primary Imagination I hold to be the living power and prime agent of all human perception, and as a repetition in the finite mind of the eternal act of creation in the infinite I AM."

"Although velocity was a relative in Newtonian science, yet there did not exist one definite velocity which was assumed to be absolute. This was the infinite velocity. It was assumed that a velocity that was infinite or instantaneous for one observer would remain infinite or instantaneous for all other observers. So far... as velocity was concerned, the sole difference between Einstein and Newton is that with Einstein the absolute or invariant velocity is no longer infinite. ...It is this difference between the invariant velocities of Newton and Einstein which is responsible for all the major differences between classical and Einsteinian science... it is this finiteness of the invariant velocity which precludes us from attaching any absolute value to shape and distance. Einstein's theory proves that molar matter can never move with the absolute speed of light. We are therefore perfectly justified in saying that the velocity of matter remains essentially relative, since it can never attain that critical velocity (i.e., that of light) which is absolute."

"Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum, And the great fleas themselves, in turn, have greater fleas to go on, While these again have greater still, and greater still, and so on."

"Anaximander gave up the idea that water or any other known substance might be the first principle, and held that this is of the nature of the infinite, that is, matter without any determinate property except that of being infinite. All things are developed out of this and return to it again, so that an infinite series of worlds have been generated and have in turn become again resolved into the abstract mass."

"The world sings of an infinite Love: how can we fail to care for it?"

"If I should ask... how many squares there are one might reply truly that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square. ... But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square. This being granted we must say that there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. ... So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former, and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite quantities. When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number. Or if I had replied to him that the points in one line were equal in number to the squares; in another, greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed, have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each. So much for the first difficulty."

"As to the query whether the finite parts of a limited continuum [continuo terminato] are finite infinite in number I will... answer that they are neither finite or infinite."

"I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to eternity there would still remain finite parts which were undivided. ... Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is... getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows that since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri]. ... There is no difficulty in the matter because unity is at once a square, a cube, a square of a square, and all the other powers [dignitā]; nor is there any essential peculiarity in squares or cubes which does not belong to unity; as, for example, the property of two square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional. Consider the two square numbers, 9 and 4; then 3 is the mean proportional between 9 and 1 [\frac{1}{3} = \frac{3}{9}]; while 2 is a mean proportional between 4 and 1 [\frac{1}{2} = \frac{2}{4}]; between 9 and 4 we have 6 as a mean proportional [\frac{4}{6} = \frac{6}{9}]. A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18 [\frac{8}{12} = \frac{18}{27}]; while between 1 and 8 we have 2 and 4 intervening [\frac{1}{2} = \frac{4}{8}]; and between 1 and 27 there lie 3 and 9 [\frac{1}{3} = \frac{9}{27}]. Therefore we conclude that unity is the only infinite number. These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common."

"Neither one nor the other doth follow, for that both the assertions may be true. The Oracle adjudged Socrates the wi­sest of all men, whose knowledg is limited; Socrates acknowledgeth that he knew nothing in relation to absolute wisdome, which is infinite; and because of infinite, much is the same part as is little, and as is nothing (for to arrive... to the infinite number, it is all one to accumulate thousands, tens, or ciphers,) therefore Socrates well perceived his wisdom to be nothing, in comparison of the infinite knowledg which he wanted. But yet, because there is some knowledg found amongst men, and this not equally shared to all, Socrates might have a greater share thereof than others, and therefore verified the answer of the Oracle."

"Nature doth, and in it alone is discovered an infinite wisdom, so that Divine Wisdom may be concluded to be infinitely infinite."

"I must have recourse to a Philosophical distinction, and say that the understanding is to be taken two ways, that is intensivè, or extensivè; and that extensive, that is, as to the multitude of intel­ligibles, which are infinite, the understanding of man is as nothing, though he should understand a thousand propositions; for that a thousand, in respect of infinity is but as a cypher: but taking the understanding intensive, (in as much as that term imports) intensively, that is, perfectly some propositions, I say, that humane wis­dom understandeth some propositions so perfectly, and is as abso­lutely certain thereof, as Nature herself; and such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few compre­hended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the neces­sity thereof, than which there can be no greater certainty."

"Although I might very rationally put it in dispute, whe­ther there be any such centre in nature, or no; being that neither you nor any one else hath ever proved, whether the World be fi­nite and figurate, or else infinite and interminate; yet nevertheless granting you, for the present, that it is finite, and of a terminate Spherical Figure, and that thereupon it hath its centre; it will be requisite to see how credible it is that the Earth, and not rather some other body, doth possesse the said centre."

"Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of s. Of course, models based on the real and complex numbers can be hugely useful, but they are just approximations... There is... no intrinsic value of a theorem that is true for a continuum of agents on a , if it is also true for a finite number of agents of a finite choice space."

"Anaximander... taught that the primary substance was infinite, eternal and ageless and... encompassed the world. This... is transformed into the various substances... Theophrastus quotes from Anaximander: "Into that from which things take their rise they pass away once more... for they make reparation and satisfaction to one another for their injustice according to the ordering of time." In this... the antithesis of Being and Becoming plays the fundamental role. The primary substance, infinite... ageless... undifferentiated Being, degenerates into... forms which lead to endless struggles. ...Becoming is ...a ...debasement of the infinite Being—a disintegration into the struggle ultimately expiated by a return into that ...without shape or character. The struggle ...is the opposition between hot and cold, fire and water, wet and dry, etc. ...[T]emporary victory ...is the injustice for which they ...make reparation in the ordering of time. ...[T]here is "eternal motion," the creation and passing away of worlds from infinity to infinity."

"[E]ven in the most precise part of sience, in mathematics, we cannot avoid using concepts that involve contradictions. ...[I]t is well known that the concept of infinity leads to contradictions that have been analyzed, but it would be practically impossible to construct the main parts of mathematics without this concept."

"Marvin Minsky was a long term friend of Feynman ... He recalls Feynman's suspicions of continuous functions and how he liked the idea that space-time might in fact be discrete. Feynman was fascinated by the question 'How could there possibly be an infinite amount of information in any finite volume?'"

"The Infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite."

"When we turn to the question, what is the essence of the infinite, we must first give ourselves an account as to the meaning the infinite has for reality: let us then see what physics teaches us about it."

"The first naive impression of nature and matter is that of continuity. Be it a piece of metal or a fluid volume, we cannot escape the conviction that it is divisible into infinity, and that any of its parts, however small, will have the properties of the whole. But wherever the method of investigation into the physics of matter has been carried sufficiently far, we have invariably struck a limit of divisibility, and this was not due to a lack of experimental refinement but resided in the very nature of the phenomenon. One can indeed regard this emancipation from the infinite as a tendency of modern science and substitute for the old adage natura non facit saltus its opposite: Nature does make jumps."

"It is well known that matter consists of small particles, the atoms, and that the macroscopic phenomena are but manifestations of combinations and interactions among these atoms. But physics did not stop there: at the end of that last century it discovered atomic electricity of a still stranger behavior. Although up to then it had been held that electricity was a fluid and acted as a kind of continuous eye, it became clear then that electricity too, is built up of positive and negative electrons. Now besides matter and electricity there exists in physics another reality, for which the law of conservation holds; namely energy. But even energy, it is found, does not admit of simple and unlimited divisibility. Planck discovered the energy-quanta. And the verdict is that nowhere in reality does there exist a homogeneous continuum in which unlimited divisibility is possible, in which the infinitely small can be realized. The infinite divisibility of a continuum is an operation which exists in thought only, is just an idea which is refuted by our observations of nature, as well as by physical and chemical experiments."

"The second place in which we encounter the problem of the infinite in nature is when we regard the universe as a whole. Let us then examine the extension of this universe to ascertain whether there exists there an infinitely great. The opinion that the world was infinite was a dominant idea for a long time. Up to Kant and even afterward, few expressed any doubt in the infinitude of the universe. Here too modern science, particularly astronomy, raised the issue anew and endeavored to decide it not by means of inadequate metaphysical speculations, but on grounds which rest on experience and on the application of the laws of nature. There arose weighty objections against the infinitude of the universe. It is Euclidean geometry which leads to infinite space as a necessity. ...Einstein showed that Euclidean geometry must be given up. He considered this cosmological question too from the standpoint of his gravitational theory and demonstrated the possibility of a finite world; and all the results discovered by the astronomers are consistent with this hypothesis of an elliptic universe."

"At the very base of all these different kinds of mathematics, there is an eternal question or eternal mission of mathematics, and that is infinity. Consciously or unconsciously, what mathematicians do is a finitization of infinity. It is impossible to put infinite things into a computer. It doesn’t matter how good a computer may be, how fast it becomes; it cannot compute infinite things. But there the mathematician has a job to do: to formulate a . The model may not match exactly the original phenomenon, but it helps you to understand it. And the model is finite: you can put it in a computer, and the computer computes the exact answer, at least for the model. So mathematicians give infinity a finite shape or a finitely computable and understandable form. This is quite an interesting feature of human nature. To my way of thinking, humans are different from other animals in that humans have a notion of infinity. They never see infinity, they never live infinitely, and even the universe may not last infinitely long. But humans cannot live without the idea of infinity."

"Heisuke Hironaka, as quoted by Allyn Jackson: (quote from p. 1019)"

"All this arguing of infinities is but the ambition of school boys."

"No priestly dogmas, invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common sense more than the doctrine of the infinitive divisibility of extension, with its consequences; as they are pompously displayed by all geometricians and metaphysicians, with a kind of triumph and exultation. A real quantity, infinitely less than any finite quantity, containing quantities infinitely less than itself, and so on in infinitum; this is an edifice so bold and prodigious, that it is too weighty for any pretended demonstration to support, because it shocks the clearest and most natural principles of human reason."

"Even if there were exceedingly few things in a finite space in an infinite time, they would not have to repeat in the same configurations. Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur."

"From the mathematical point of view there are infinitely many... numbers... Thus the first task of "scientific" arithmetic—as contrasted with... "practical" knowledge...— consists in finding such arrangements and orders of the assemblages of monads as will completely comprehend their variety under well-defined properties, so that their unlimited multiplicity may at last be brought within bounds (cf. Nichomachus I, 2). ...When we recall how Plato (Theaetetus 147 C ff.) makes Theaetetus, speaking from a very advanced stage of scientific geometry and arithmetic, describe his procedure... What... appears to Plato so exemplary for Socrates' present inquiry concerning "knowledge", and indeed for every Socratic inquiry of this kind[?]. Theaetetus... divides "the whole realm of number"... into two domains: to one of these belong all those numbers which may arise from a number when it is multiplied by itself... to the other, all those which may arise from the multiplication of one number with another. The first number domain he calls "square," the second "promecic" or "heteromecic" (oblong), designations which recur in all later arithmetical presentations (cf. Diogenes Laertius III, 24). Thus two eide [kinds, forms, or species]... allow us to articulate and delimit a realm of numbers previously incomprehensible because unlimited, especially if we substitute the various eide of polygonal numbers for the one eidos of oblong numbers."

"Greek mathematical thought does, indeed deal first and last with different kinds of numbers. Were it otherwise, how would it be possible to come to terms with the limitlessness of the material with which arithmetic is confronted? Therefore... theoretical arithmetic... attempts to comprehend all possible groupings of monads in general under arrangements which are determinate, i.e., which possess unambiguous characteristics and which may, in turn, be reduced to their own ultimate elements... Now the most comprehensive eide, those which come closest to the rank of ' and are therefore termed "the very first"... are the odd and the even. ...each of these halves nevertheless comprising an unlimited multitude of numbers. But each of these... is now in turn gathered "into one"... by means of certain unambiguous characteristics..."

"In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length. ...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry."

"The Greeks failed to comprehend the infinitely large, the infinitely small, and infinite processes. They "shrank before the silence of the infinite spaces.""

"The Pythagoreans associated good and evil with the limited and unlimited, respectively."

"Aristotle says the infinite is imperfect, unfinished, and therefore unthinkable; it is formless and confused. Only as objects are delimited and distinct do they have a nature."

"To avoid any assertion about the infinitude of the straight line, Euclid says a line segment (he uses the word "line" in this sense) can be extended as far as necessary. Unwillingness to involve the infinitely large is seen also in Euclid's statement of the parallel axiom. Instead of considering two lines that extend to infinity and giving a direct condition or assumption under which parallel lines might exist, his parallel axiom gives a condition under which two lines will meet at some finite point."

"The concept of the infinitely small is involved in the relation of points to a line or the relation of the discrete to the continuous, and Zeno's paradoxes may have caused the Greeks to shy away from this subject."

"The relationship of point to line bothered the Greeks and led Aristotle to separate the two. Though he admits points are on lines, he says that a line is not made up of points and that the continuous cannot be made up of the discrete. This distinction contributed also to the presumed need for separating number from geometry, since to the Greeks numbers were discrete and geometry dealt with continuous magnitudes."

"Because they [the ancient Greeks] feared infinite processes they missed the limit process. In approximating a circle by a polygon they were content to make the difference smaller than any given quantity, but something positive was always left over. Thus the process remained clear to the intuition; the limit process, on the other hand, would have involved the infinitely small."

"The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine. ...Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate."

"Closely related to the problem of the parallel postulate is the problem of whether physical space is infinite. Euclid assumes in Postulate 2 that a straight-line segment can be extended as far as necessary; he uses this fact, but only to find a larger finite length—for example in Book I, Propositions 11, 16, and 20. For these proofs Heron gave new proofs that avoided extending the lines, in order to meet the objection of anyone who would deny that the space was available for the extension."

"Aristotle had considered the question of whether space is infinite and gave six nonmathematical arguments to prove that it is finite; he foresaw that this question would be troublesome."

"In an infinite, eternal universe, the point is that anything is possible, and it's unlikely that we can even begin to scratch the surface of the full range of possibilities."

"As there are an infinity of possible worlds, there are also an infinity of laws, certain ones appropriate to one; others, to another, and each possible individual of any world involves in its concept the laws of its world."

"For the present, such a state of instantaneous transition from inequality to equality, from motion to rest, from convergence to parallelism, or anything of the sort, can be sustained in a rigorous or metaphysical sense, or whether infinite extensions successively greater and greater, or infinitely small ones successively less and less, are legitimate considerations, is a matter that I own to be possibly open to question; but for him who would discuss these matters, it is not necessary to fall back upon metaphysical controversies, such as the composition of the continuum, or to make geometrical matters depend thereon. Of course, there is no doubt that a line may be considered to be unlimited in any manner, and that, if it is unlimited on one side only, there can be added to it something that is limited on both sides. But whether a straight line of this kind is to be considered as one whole that can be referred to computation, or whether it can be allocated among quantities which may be used in reckoning, is quite another question that need not be discussed at this point."

"It will be sufficient if, when we speak of infinitely great (or more strictly unlimited), or of infinitely small quantities (i.e., the very least of those within our knowledge) it is understood that we mean quantities that are indefinitely great or indefinitely small, i.e., as great as you please, or as small as you please, so that the error that any one may assign may be less than a certain assigned quantity. Also, since in general it will appear that, when any small error is assigned, it can be shown that it should be less, it follows that the error is absolutely nothing; an almost exactly similar kind of argument is used in different places by Euclid, Theodosius and others; and this seemed to them to be a wonderful thing, although it could not be denied that it was perfectly true that, from the very thing that was assumed as an error, it could be inferred that the error was non-existent. Thus by infinitely great and infinitely small, we understand something indefinitely great, or something indefinitely small, so that each conducts itself as a sort of class, and not merely as the last thing of a class. If any one wishes to understand these as the ultimate things, or as truly infinite, it can be done, and that too without falling back upon a controversy about the reality of extensions, or of infinite continuums in general, or of the infinitely small, ay, even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of the calculation, just as the algebraists retain imaginary roots with great profit. For they contain a handy means of reckoning, as can manifestly be verified in every case in a rigorous manner by the method already stated. But it seems right to show this a little more clearly, in order that it may be confirmed that the algorithm, as it is called, of our differential calculus, set forth by me in the year 1684, is quite reasonable."

"Dans chaque point réel, qui fait une Monade... il y pourroit lire encor tout le passé, et même tout l'avenir infiniment infini, puisque chaque moment contient une infinité de choses , et qu'il y a une infinité de momens dans chaque partie du temps, et une infinité d'heures, d'années, de siecles, d'eônes, dans toute l'éternité future. Quelle infinité d'infinités infiniment répliquée, quel monde, quel univers dans quelque corpuscule qu'on pourroit assigner."

"The halls rose in a pyramid, becoming even more beautiful as one mounted towards the apex, and representing more beautiful worlds. Finally they reached the highest one which completed the pyramid, and which was the most beautiful of all: for the pyramid had a beginning, but one could not see its end; it had an apex, but no base; it went on increasing to infinity. ...because amongst an endless number of possible worlds there is the best of all, else would God not have determined to create any; but there is not any one which has not also less perfect worlds below it: that is why the pyramid goes on descending to infinity. ...We are in the real true world (said the Goddess) and you are at the source of happiness."

"Outside the ordered universe [is] that amorphous blight of nethermost confusion which blasphemes and bubbles at the center of all infinity—the boundless daemon sultan Azathoth, whose name no lips dare speak aloud, and who gnaws hungrily in inconceivable, unlighted chambers beyond time and space amidst the muffled, maddening beating of vile drums and the thin monotonous whine of accursed flutes."

"He that would enjoy life and act with freedom must have the work of the day continually before his eyes. Not yesterday's work, lest he fall into despair; nor to-morrow's, lest he become a visionary—not that which ends with the day, which is a worldly work; nor yet that only which remains to eternity, for by it he cannot shape his actions. Happy is the man who can recognise in the work of to-day a connected portion of the work of life and an embodiment of the work of Eternity. The foundations of his confidence are unchangeable, for he has been made a partaker of Infinity. He strenuously works out his daily enterprises because the present is given him for a possession. Thus ought Man to be an impersonation of the divine process of nature, and to show forth the union of the infinite with the finite, not slighting his temporal existence, remembering that in it only is individual action possible; nor yet shutting out from his view that which is eternal, knowing that Time is a mystery which man cannot endure to contemplate until eternal Truth enlighten it."

"The concept of infinity came in relatively late, even in Egypt, and... its first fathers were more likely metaphysicians than theologians. In looking backward, as in looking forward, early man was quite unable to imagine endless time. Always he concluded that the animal creation, including his own kind, must have a beginning, and the earth he walked on, with it. Sometimes he ascribed the act of creation to the gods, or to one of them, and sometimes he laid it to a potent being of lesser dignity, usually to a totem animal."

"Swamp Thing: You thought ...that it could not...get worse...you imagined...that things...had reached their limits. Do not...delude yourselves...there are...no limits."

"Doria listened to and depended upon that inner realm of memory and consciousness, her muse of interior music, her beloved companion through whose consolation she escaped on her flight into the infinite."

"It seems to me, that if the matter of our sun and planets and all the matter of the universe, were evenly scattered throughout all the heavens, and every particle had an innate gravity towards all the rest, and the whole of space throughout which this matter was scattered was but finite, the matter on [toward] the outside of this space would, by its gravity, tend towards all the matter on the inside, and, by consequence, fall down into the middle of the whole space, and there compose one great spherical mass. But if the matter was evenly disposed throughout an infinite space it could never convene into one mass; but some of it would convene into one mass and some into another, so as to make an infinite number of great masses, scattered at great distances from one another throughout all that infinite space."

"I fear what I have said of Infinities, will seem obscure to you; but it is enough if you understand, that Infinities when considered absolutely without any Restriction or Limitation, are neither equal nor unequal, nor have any Proportion one to another, and therefore the Principle that all Infinities are equal, is a precarious one."

"Let man then contemplate the whole of nature in her full and grand majesty, and turn his vision from the low objects which surround him. Let him gaze on that brilliant light, set like an eternal lamp to illumine the universe; let the earth appear to him a point in comparison with the vast circle described by the sun; and let him wonder at the fact that this vast circle is itself but a very fine point in comparison with that described by the stars in their revolution round the firmament. But if our view be arrested there, let our imagination pass beyond; it will sooner exhaust the power of conception than nature that of supplying material for conception. The whole visible world is only an imperceptible atom in the ample bosom of nature. No idea approaches it. We may enlarge our conceptions beyond all imaginable space; we only produce atoms in comparison with the reality of things. It is an infinite sphere, the center of which is everywhere, the circumference nowhere. In short it is the greatest sensible mark of the almighty power of God, that imagination loses itself in that thought."

"For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. The ends of things and their beginnings are impregnably concealed from him in an impenetrable secret. He is equally incapable of seeing the nothingness out of which he was drawn and the infinite in which he is engulfed."

"...as nature has graven her image and that of her Author on all things, they almost all partake of her double infinity. Thus we see that all the sciences are infinite in the extent of their researches. For who doubts that geometry, for instance, has an infinite infinity of problems to solve? They are also infinite in the multitude and fineness of their premises; for it is clear that those which are put forward as ultimate are not self-supporting, but are based on others which, again having others for their support, do not permit of finality. ...Of these two Infinites of science, that of greatness is the most palpable, and hence a few persons have pretended to know all things. ...the infinitely little is the least obvious. Philosophers have much oftener claimed to have reached it, and it is here they have all stumbled. ...We need no less capacity for attaining the Nothing than the All. Infinite capacity is required for both, and it seems to me that whoever shall have understood the ultimate principles of being might also attain to the knowledge of the Infinite. The one depends on the other, and one leads to the other. These extremes meet and reunite by force of distance, and find each other in God, and in God alone."

"Excessive qualities are prejudicial to us and not perceptible by the senses; we do not feel but suffer them. Extreme youth and extreme age hinder the mind, as also too much and too little education. In short, extremes are for us as though they were not, and we are not within their notice. They escape us, or we them. This is our true state; this is what makes us incapable of certain knowledge and of absolute ignorance."

"We sail within a vast sphere, ever drifting in uncertainty, driven from end to end. When we think to attach ourselves to any point and to fasten to it, it wavers and leaves us; and if we follow it, it eludes our grasp, slips past us, and vanishes for ever. Nothing stays for us. This is our natural condition, and yet most contrary to our inclination; we burn with desire to find solid ground and an ultimate sure foundation whereon to build a tower reaching to the Infinite. But our whole groundwork cracks, and the earth opens to abysses."

"I hold it equally impossible to know the parts without knowing the whole, and to know the whole without knowing the parts in detail. The eternity of things in itself or in God must also astonish our brief duration."

"Unity joined to infinity adds nothing to it, no more than one foot to an infinite measure. The finite is annihilated in the presence of the infinite, and becomes a pure nothing. So our spirit before God, so our justice before divine justice."

"We know that there is an infinite, and are ignorant of its nature. As we know it to be false that numbers are finite, it is therefore true that there is an infinity in number. But we do not know what it is. It is false that it is even, it is false that it is odd; for the addition of a unit can make no change in its nature. Yet it is a number, and every number is odd or even (this is certainly true of every finite number). So we may well know that there is a God without knowing what He is. Is there not one substantial truth, seeing there are so many things which are not the truth itself?"

"We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite, and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because He has neither extension nor limits."

"When we would pursue virtues to their extremes on either side, vices present themselves insensibly there, in their insensible journeys towards the infinitely little; and vices present themselves in a crowd towards the infinitely great, so that we lose ourselves in them, and no longer see virtues."

"We must relax our minds a little; but this opens the door to debauchery. Let us mark the limits. There are no limits in things. Laws would put them there, and the mind cannot suffer it."

"He who proclaims the existence of the Infinite, and none can avoid it — accumulates in that affirmation more of the supernatural than is to be found in all the miracles of all the religions; for the notion of the Infinite presents that double character that forces itself upon us and yet is incomprehensible. When this notion seizes upon our understanding we can but kneel ... I see everywhere the inevitable expression of the Infinite in the world; through it the supernatural is at the bottom of every heart. The idea of God is a form of the idea of the Infinite. As long as the mystery of the infinite weighs on human thought, temples will be erected for the worship of the Infinite, whether God is called Brahma, Allah, Jehovah, or Jesus; and on the pavement of these temples, men will be seen kneeling, prostrated, annihilated by the thought of the Infinite."

"All that is not thought is pure nothingness...And yet—strange contradiction for those who believe in time—geologic history shows us that life is only a short episode between two eternities of death, and that, even in this episode, conscious thought has lasted and will last only a moment. Thought is only a gleam in the midst of a long night. But it is this gleam which is everything."

"This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely. ...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms. ...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite. This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem... In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general."

"If we have only to classify a finite number of objects, it is easy to preserve these classifications without change. If the number of objects is indefinite, ...[i.e.,] if we are constantly liable to find new and unforeseen objects springing up, it may happen that the appearance of a new object will oblige us to modify the classification, and it is thus that we are exposed to antinomies. There is no actual infinity. The Cantorians forgot this, and so fell into contradiction. It is true that Cantorism has been useful, but that was when it was applied to a real problem, whose terms were clearly defined, and then it was possible to advance without danger. Like the Cantorians, the logicians have forgotten the fact, and they have met with the same difficulties. ...[B]elief in an actual infinity is essential in the Russellian logic, and this... distinguishes it from the Hilbertian logic. Hilbert takes the... view of extension... to avoid the Cantorian antimonies. Russell takes the... view of comprehension... to regard the infinite as actual. And we have not only infinite classes; when we pass from the genus to the species... the number of conditions is still infinite, for they generally express that the object... is in... relation with all the objects of an infinite class. But all this is ."

"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."

"God created infinity, and man, unable to understand infinity, had to invent s."

"This I have tested too frequently to be mistaken by offering to indifferent spectators forms of equal abstract beauty in half tint, relieved, the one against dark sky, the other against a bright distance. The preference is invariably given to the latter... the same preference is unhesitatingly accorded to the same effect in Nature herself. Whatever beauty there may result from effects of light on foreground objects... there is yet a light which the eye invariably seeks with a deeper feeling of the beautiful, the light of the declining or breaking day, and the flakes of scarlet cloud burning like watch-fires in the green sky of the horizon; a deeper feeling... having more of spiritual hope and longing, less of animal and present life... I am willing to let it rest on the determination of every reader, whether the pleasure which he has received from these effects of calm and luminous distance be not the most singular and memorable of which he has been conscious... It is not then by nobler form, it is not by positiveness of hue, it is not by intensity of light... that this strange distant space possesses its attractive power. But there is one thing that it has, or suggests, which no other object of sight suggests in equal degree, and that is—Infinity. It is of all visible things the least material, the least finite, the farthest withdrawn from the earth prison-house, the most typical of the nature of God, the most suggestive of the glory of his dwelling-place. For the sky of night, though we may know it boundless, is dark; it is a studded vault, a roof that seems to shut us in and down; but the bright distance has no limit, we feel its infinity, as we rejoice in its purity of light. ...this expression of infinity in distance... is of that value that no other forms will altogether recompense us for its loss; and... no work of any art, in which this expression of infinity is possible, can be perfect or supremely elevated, without it, and that, in proportion to its presence, it will exalt and render impressive even the most tame and trivial themes. And I think if there be any one grand division, by which it is at all possible to set the productions of painting, so far as their mere plan or system is concerned, on our right and left hands, it is this of light and dark background, of heaven light or of object light."

"Those who assumed innumerable worlds, e.g., Anaximander, Leukippos, Demokritos, and, later on, Epicurus, assumed that they came into being and passed away ad infinitum, some always coming into being and others passing away."

"We know by actual observation only a comparatively small part of the whole universe. I will call this "our neighborhood." Even within the confines of this province our knowledge decreases very rapidly as we get away from our own particular position in space and time. It is only within the solar system that our empirical knowledge extends to the second order of small quantities (and that only for g44 and not for the other gαβ), the first order corresponding to about 10-8. How the gαβ outside our neighborhood are, we do not know, and how they are at infinity of space or time we shall never know. Infinity is not a physical but a mathematical concept, introduced to make our equations more symmetrical and elegant. From the physical point of view everything that is outside our neighborhood is pure extrapolation, and we are entirely free to make this extrapolation as we please to suit our philosophical or aesthetical predilections—or prejudices. It is true that some of these prejudices are so deeply rooted that we can hardly avoid believing them to be above any possible suspicion of doubt, but this belief is not founded on any physical basis. One of these convictions, on which extrapolation is naturally based, is that the particular part of the universe where we happen to be, is in no way exceptional or privileged; in other words, that the universe, when considered on a large enough scale, is isotropic and homogeneous."

"It is only by intuition that the infinite can be apprehended. But why is this? Why cannot the infinite be apprehended by concepts? To see this we must understand that the word "infinite," in the religious sense, has nothing at all to do with that sense of the word in which it is applied to space, time, and the number series. We may call this latter the mathematical infinite to distinguish it from the religious infinite. And it is the confusion between these two which misled us into the false trail of supposing that the infinity of God's mind refers to the amount of His knowledge and that the finitude of man's mind refers to his ignorance. The religious infinite, or in other words the infinity of God, means that than 'which there is no other'. In this sense neither space nor time could be infinite, since space is an "other" to time, and time is an "other" to space."

"What does infinity mean to you? Are you not infinity and yourself?"

"Infinity is the end. End without infinity is but a new beginning."

"Concerning the things needful to be known... we must first take care not to commit ourselves to a search, going back to infinity—that is, in order to discover the best method for finding out the truth, there is no need of another method to discover such method; nor of a third method for discovering the second, and so on to infinity. By such proceedings, we should never arrive at the knowledge of the truth, or indeed at any knowledge at all."

"Things regarded in themselves, and in relation to God, are neither ugly nor beautiful. ...Perfection and imperfection are names which do not differ much from the names beauty and ugliness. ... This I know, that between finite and infinite there is no comparison; so that the difference between God and the greatest and most excellent created thing is no less than the difference between God and the least created thing."

"By God, I mean a being absolutely infinite — that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality. Explanation — I say absolutely infinite, not infinite after its kind: for, of a thing infinite only after its kind, infinite attributes may be denied; but that which is absolutely infinite, contains in its essence whatever expresses reality, and involves no negation."

"For all who have in anywise reflected on the divine nature deny that God has a body. Of this they find excellent proof in the fact that we understand by body a definite quantity... bounded by a certain shape, and it is the height of absurdity to predicate such a thing of God, a being absolutely infinite. But meanwhile... they think corporeal or extended substance wholly apart from the divine nature, and say it was created by God. ...I myself have proved... that no substance can be produced or created by anything other than itself. Further I showed... that besides God, no substance can be granted or conceived. Hence we drew the conclusion that extended substance is one of the infinite attributes of God. However, ...I will refute the arguments of my adversaries, which all start from the following points:— Extended substance, in so far as it is substance, consists... in parts, wherefore they deny that it can be infinite, or, consequently, that it can appertain to God. This they illustrate... If extended substance, they say, is infinite, let it be conceived to be divided into two [equal] parts; each part will then be either finite or infinite. If the former, then infinite substance is composed of two finite parts, which is absurd. If the latter, then one [the original] infinite will be twice as large as another infinite [the part], which is also absurd. Further, if an infinite line be measured out in foot lengths, it will consist of an infinite number of such parts; it would equally consist of an infinite number of parts, if each part measured only an inch: therefore, one infinity would be twelve times as great as the other. Lastly, if from a single point there be conceived to be drawn two diverging lines which at first are at a definite distance apart, but are produced to infinity, it is certain that the distance between the two lines will be continually increased, until at length it changes from definite to indefinable. As these absurdities follow, it is said, from considering quantity as infinite, the conclusion is drawn, that extended substance must necessarily be finite, and consequently, cannot appertain to the nature of God. ... God, it is said, inasmuch as he is a supremely perfect being, cannot be passive; but extended substance, in so far as it is divisible, is passive. It follows, therefore, that extended substance does not appertain to the essence of God. ... I have already answered their propositions; for all their arguments are founded on the hypothesis that extended substance is composed of parts, and such a hypothesis I have shown... to be absurd. ...all these absurdities ...from which it is sought to extract the conclusion that extended substance is finite, do not at all follow from the notion of an infinite quantity, but merely from the notion that an infinite quantity is measureable, and composed of finite parts: therefore ...infinite quantity is not measureable, and cannot be composed of finite parts. This is exactly what we have already proved... Wherefore the weapon which they aimed at us has in reality recoiled upon themselves. ...For ...taking extended substance, which can only be conceived as infinite, one, and indivisible... they assert, in order to prove that it is finite, that it is composed of finite parts, and that it can be multiplied and divided. ... ...If ...we regard quantity as it is represented in our imagination... we shall find that it is finite, divisible, and compounded of parts; but if we regard it as it is represented in our intellect... we shall then, as I have sufficiently proved, find that it is infinite, one, and indivisible. This will be plain enough... if it be remembered, that matter is everywhere the same, that its parts are not distinguishable, except in so far as we conceive matter as diversely modified, whence its parts are distinguished, not really, but modally. For instance... water, in so far as it is water, is produced and corrupted; but in so far as it is substance, it is neither produced nor corrupted. ...inasmuch as besides God... no substance can be granted, wherefrom it could receive its modifications. All things... are in God, and all things... come to pass solely through the laws of the infinite nature of God, and follow... from the necessity of his essence. Wherefore it can in nowise be said, that God is passive in respect to anything other than himself, or that extended substance is unworthy of the Divine nature, even if it be supposed divisible, so long as it is granted to be infinite and eternal."

"From the necessity of the divine nature must follow an infinite number of things in infinite ways—that is, all things which can fall within the sphere of infinite intellect."

"It is properly debated whether irrational numbers are true numbers or fictions. ...where we might try to subject them to numeration [decimal representation] and to make them proportional to rational numbers, we find that they flee perpetually, so that none of them in itself can be freely grasped: a fact that we perceive in the resolving of them... Moreover, it is not possible to call that a true number which is such as to lack precision and which has no known proportion to true numbers. Just as an infinite number is not a number, so an irrational number is not a true number and is hidden under a sort of cloud of infinity. And thus the ratio of an irrational number to a rational number is no less uncertain than that of an infinite to a finite."

"Buzz Lightyear: To infinity and beyond!"

"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means."

"In the Timaeus Plato had expounded a theory that outside the universe, which he regarded as bounded and spherical, there was an infinite empty space."

"During the Middle Ages the universe was regarded as finite, with the earth at its centre. The idea was abandoned during the Scientific Renaissance, and the universe came to be pictured as an indefinitely large number of stars scattered throughout infinite Euclidean space. This conception appeared to be a necessary consequence of the theory of gravitation; for, as Newton pointed out, a finite material universe in infinite space would tend to concentrate in one massive lump."

"The matter which we suppose to be the main constituent of the universe is built out of small self-contained building-blocks, the chemical atoms. It cannot be repeated too often that the word "atom" is nowadays detached from any of the old philosophical speculations: we know precisely that the atoms with which we are dealing are in no sense the simplest conceivable components of the universe. On the contrary, a number of phenomena, especially in the area of spectroscopy, lead to the conclusion that atoms are very complicated structures. So far as modern science is concerned, we have to abandon completely the idea that by going into the realm of the small we shall reach the ultimate foundations of the universe. I believe we can abandon this idea without any regret. The universe is infinite in all directions, not only above us in the large but also below us in the small. If we start from our human scale of existence and explore the content of the universe further and further, we finally arrive, both in the large and in the small, at misty distances where first our senses and then even our concepts fail us."

"A child might be overawed by a great city, but a civil engineer knows that he might demolish it and rebuild it himself. Husserl's philosophy has the same aim: to show us that, although we may have been thrust into this world without a 'by your leave,' we are mistaken to assume that it exists independently of us. It is true that reality exists apart from us; but what we mistake for the world is actually a world constituted by us, selected from an infinitely complex reality."

"When you reach the top of the mountain, keep climbing."

"Adde thou upright, reserving every tenne, And write the Digits downe all with thy pen, The proofe (for truth I say,) Is to cast nine away. From the particular summes, and severall Reject the Nines; likewise from the totall When figures like in both chance to remaine Clear light of working right shal be your gain; Subtract the lesser from the great, noting the rest. Or ten to borrow, you are ever prest, To pay what borrowed was thinks it no paine, But honesty redounding to your gaine."

"The method of arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies."

"Arithmetic and geometry, those wings on which the astronomer soars as high as heaven."

"If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool— so useful that it still exists."

"The chemist smiles at the childish efforts of alchemists but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today."

"A hieratic papyrus, included in the Rhind collection of the British Museum, was deciphered by Eisenlohr in 1877, and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ahmes some time before 1700 B.C., and was founded on an older work believed by Birch to date back as far as 3400 B.C.! This curious papyrus -- the most ancient mathematical handbook known to us -- puts us at once in contact with the mathematical thought in Egypt of three or five thousand years ago. It is entitled "Directions for obtaining the Knowledge of all Dark Things." We see from it that the Egyptians cared but little for theoretical results. Theorems are not found in it at all. It contains "hardly any general rules of procedure, but chiefly mere statements of results intended possibly to be explained by a teacher to his pupils.""

"The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression."

"I couldn't afford to learn it," said the Mock Turtle with a sigh. "I only took the regular course.""What was that?" inquired Alice."Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "and then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

"The late Professor Leslie... [i]n his Philosophy of Arithmetic... entered... into much of its history. ...[O]ne principal, thing to be cautious of is, his almost monomaniac antipathy to every thing Hindoo—a most unfortunate turn... Leslie... generalises... fearfully every now and then. He informs us that it was the practice throughout Europe to reduce the rules of arithmetic to memorial verses, and that [William] Buckley's Arithmetica Memorativa appears at one period to have gained possession of the schools and colleges of England. Now the truth... the verses attributed to Sacrobosco had never... been printed when Leslie wrote; and Buckley... was printed only once... and two or three times as an appendix to a work on logic. Dr. Peacock expresses the truth in saying... before the invention of printing, the practice of writing memorial verses was common, as appears by manuscript libraries. ...[H]ad the practice of using them been common, the presses of the fifteenth and sixteenth centuries would have given them forth in great numbers. But I cannot learn that any metrical work was printed in the fifteenth century, except the Compotus of [Magister] Anianus, and that only once."

"Arithmetic and geometry are much more certain than the other sciences, because the objects of them are in themselves so simple and so clear that they need not suppose anything which experience can call in question, and both proceed by a chain of consequences which reason deduces one from another. They are also the easiest and clearest of all the sciences, and their object is such as we desire; for, except for want of attention, it is hardly supposable that a man should go astray in them. We must not be surprised, however, that many minds apply themselves by preference to other studies, or to philosophy. Indeed everyone allows himself more freely the right to make his guess if the matter be dark than if it be clear, and it is much easier to have on any question some vague ideas than to arrive at the truth itself on the simplest of all."

"No man acquires property without acquiring with it a little arithmetic, also."

"What mathematics, therefore are expected to do for the advanced student at the university, Arithmetic, if taught demonstratively, is capable of doing for the children even of the humblest school. It furnishes training in reasoning, and particularly in deductive reasoning. It is a discipline in closeness and continuity of thought. It reveals the nature of fallacies, and refuses to avail itself of unverified assumptions. It is the one department of school-study in which the sceptical and inquisitive spirit has the most legitimate scope; in which authority goes for nothing. In other departments of instruction you have a right to ask for the scholar’s confidence, and to expect many things to be received on your testimony with the understanding that they will be explained and verified afterwards. But here you are justified in saying to your pupil “Believe nothing which you cannot understand. Take nothing for granted.” In short, the proper office of arithmetic is to serve as elementary 268 training in logic. All through your work as teachers you will bear in mind the fundamental difference between knowing and thinking; and will feel how much more important relatively to the health of the intellectual life the habit of thinking is than the power of knowing, or even facility of achieving visible results. But here this principle has special significance. It is by Arithmetic more than by any other subject in the school course that the art of thinking—consecutively, closely, logically—can be effectually taught."

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

"If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root."

"But in our opinion truths of this kind should be drawn from notions rather than from notations."

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated."

"The higher arithmetic presents us with an inexhaustible store of interesting truths,—of truths too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration 273 till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed."

"Mathematics is the queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

"A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed."

"Statistics began as the systematic study of quantitative facts about the state."

"Arithmetical symbols are written diagrams and geometrical figures are graphic formulas."

"Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions."

"If the potential of every number is in the monad, then the monad would be intelligible number in the strict sense, since it is not yet manifesting anything actual, but everything conceptually together in it."

"God made the integers, all the rest is the work of man. [Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.]"

"An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry."

"I do hate sums, There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Numbers which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different."

"Music is a hidden arithmetic exercise of the soul, which does not know that it is counting. [Musica est exercitium arithmeticae occultum nescientis se numerare animi.]"

"If you find my arithmetic correct, then no amount of vapouring about my psychological condition can be anything but a waste of time. If you find my arithmetic wrong, then it may be relevant to explain psychologically how I came to be so bad at my arithmetic, and the doctrine of the concealed wish will become relevant—but only after you have yourself done the sum and discovered me to be wrong on purely arithmetical grounds. It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error."

"Population, when unchecked, increases in a geometrical ratio, Subsistence, increases only in an arithmetical ratio."

"Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories."

"The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic."

"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory."

"As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious."

"He who refuses to do arithmetic is doomed to talk nonsense."

"On the one side we may say that the purpose of number work is to put a child in possession of the machinery of calculation; on the other side it is to give him a better mastery of the world through a clear (mathematical) insight into the varied physical objects and activities. The whole world, from one point of view, can be definitely interpreted and appreciated by mathematical measurements and estimates. Arithmetic in the common school should give a child this point of view, the ability to see and estimate things with a mathematical eye."

"The “Disquisitiones Arithmeticae” that great book with seven seals."

"That fondness for science, … that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic."

"Good arithmetic contributes powerfully to purposive effort, to concentration, to tenacity of purpose, to generalship, to faith in right, and to the joy of achievement, which are the elements that make up efficient citizenship.... Good arithmetic exalts thinking, furnishes intellectual pleasure, adds appreciably to love of right, and subordinates pure memory"

"Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method."

"The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time."

"Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use."

"1. 0 is a number. 2. The immediate successor of a number is also a number. 3. 0 is not the immediate successor of any number. 4. No two numbers have the same immediate successor. 5. Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers."

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations."

"You see then, my friend, I observed, that our real need of this branch of science [arithmetic] is probably because it seems to compel the soul to use our intelligence in the search after pure truth."

"Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and if visible or tangible objects are obtruding upon the argument, refusing to be satisfied."

"Let us... see the geometer at work and seek to catch his process. The task is not without difficulty; it does not suffice to open a work at random and analyze any demonstration in it. We must first exclude geometry, where the question is complicated by arduous problems relative to the rôle of the postulates, to the nature and the origin of the notion of space. For analogous reasons we can not turn to the infinitesimal analysis. We must seek mathematical thought where it has remained pure, that is in arithmetic."

"Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles (or as we would more naturally say, shot) required to make the shapes in question. ...He presumably thought of the world as atomic, and of bodies as built up of molecules composed of atoms arranged in various shapes. In this way he hoped to make arithmetic the fundamental study in physics as in aesthetics."

"Unfortunately for Pythagoras, his theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. So long as no adequate arithmetical theory on incommensurables existed, the method of Euclid was the best that was possible in geometry. When Descartes introduced co-ordinate geometry, thereby again making arithmetic supreme, he [Descartes] assumed the possibility of a solution of the problem of incommensurables, though in his day no such solution had been found."

"Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for the mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic."

"Not only philosophers were influenced by Plato. Why did the Puritans object to the music and painting and gorgeous ritual of the Catholic Church? You will find the answer in the tenth book of the Republic. Why are children compelled to learn arithmetic? The reasons are given in the seventh book."

"Plato proceeds to an interesting sketch of the education proper to a young man who is to be a guardian. ...The young man chosen for these merits will spend the years from twenty to thirty on the four Pythagorean studies: arithmetic, geometry (plane and solid), astronomy, and harmony. These studies are not to be pursued in any utilitarian spirit, but in order to prepare his mind for the vision of eternal things. In astronomy, for example, he is not to trouble himself too much about the actual heavenly bodies, but rather with the mathematics of motion of ideal heavenly bodies. This may seem absurd to modern ears, but, strange to say, it proved to be a fruitful point of view in connection with empirical astronomy."

"It is noteworthy that modern Platonists, almost without exception, are ignorant of mathematics, in spite of the immense importance that Plato attached to arithmetic and geometry, and the immense influence that they [these studies] had on his philosophy. This is an example of the evils of specialization: a man must not write on Plato unless he has spent so much of his youth on Greek as to have no time for the things that Plato thought important."

"I should agree with Plato that arithmetic, and pure mathematics generally, is not derived from perception. Pure mathematics consists of tautologies, analogous to "men are men," but usually more complicated. To know that a mathematical proposition is correct, we do not have to study the world, but only the meanings of symbols; and the symbols, when we dispense with definitions (of which the purpose is merely abbreviation), are found to be such words as "or" and "not," and "all" and "some," which do not, like "Socrates," denote anything in the actual world. A mathematical equation asserts that two groups of symbols have the same meaning; and so long as we confine ourselves to pure mathematics, this meaning must he one that can be understood without knowing anything about what can be perceived. Mathematical truth, therefore, is, as Plato contends, independent of perception; but it is truth of a very peculiar sort, and is concerned only with symbols."

"The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion."

"Some writers maintain arithmetic to be only the only sure guide in political economy; for my part, I see so many detestable systems built upon arithmetical statements, that I am rather inclined to regard that science as the instrument of national calamity."

""Any graduate of the ___ Business School should be able to beat an index fund over the course of a market cycle." Statements such as these are made with alarming frequency by investment professionals. In some cases, subtle and sophisticated reasoning may be involved. More often (alas), the conclusions can only be justified by assuming that the laws of arithmetic have been suspended for the convenience of those who choose to pursue careers as active managers."

"Time is naturally associated by us with counting, which is the simplest of all rhythms. It is surely no accident that the words 'arithmetic' and 'rhythm' come from the two Greek terms which are derived from a common root meaning 'to flow'."

"The word model is used as a noun, adjective, and verb, and in each instance it has a slightly different connotation. As a noun "model" is a representation in the sense in which an architect constructs a small-scale model of a building or a physicist a large-scale model of an atom. As an adjective "model" implies a degree of perfection or idealization, as in reference to a model home, a model student, or a model husband. As an adjective "model" implies a degree or perfection or idealization, as in reference to a model home, a model student, or a model husband. As a verb "to model" means to demonstrate, to reveal, to show what a thing is like."

"Scientific [mathematical] models have all these connotations. They are representations of states, objects, and events. They are idealized in the sense that they are less complicated than reality and hence easier to use for research purposes. These models are easier to manipulate and "carry" than the real thing. The simplicity of models, compared with reality, lies in the fact that only the relevant properties of reality are represented."

"There is no America. There is no democracy. There is only IBM and ITT and AT&T and DuPont, Dow, Union Carbide, and Exxon. Those are the nations of the world today. What do you think the Russians talk about in their councils of state — Karl Marx? They get out their linear programming charts, statistical decision theories, minimax solutions, and compute the price-cost probabilities of their transactions and investments, just like we do. We no longer live in a world of nations and ideologies, Mr. Beale. The world is a college of corporations, inexorably determined by the immutable bylaws of business. The world is a business, Mr. Beale. It has been since man crawled out of the slime. And our children will live, Mr. Beale, to see that perfect world in which there's no war or famine, oppression or brutality — one vast and ecumenical holding company, for whom all men will work to serve a common profit, in which all men will hold a share of stock, all necessities provided, all anxieties tranquilized, all boredom amused."

"Linear programming is viewed as a revolutionary development giving man the ability to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity. In the real world, planning tends to be ad hoc because of the many special-interest groups with their multiple objectives."

"Linear relationships can be captured with a straight line on a graph. Linear relationships are easy to think about....Linear equations are solvable... Linear systems have an important modular virtue: you can take them apart, and put them together again — the pieces add up."

"There are two types of systems engineering - basis and applied. There is no need to attempt to define the term systems engineering in a manner acceptable to everybody, as Chalmer Jones brings out in his article herein. Systems engineering is, obviously, the engineering of a system. It usually, but not always, includes dynamic analysis, mathematical models, simulation, linear programming, data logging, computing, optimating, etc., etc. It connotes an optimum method, realized by modern engineering techniques. Basic systems engineering includes not only the control system but also all equipments within the system, including all host equipments for the control system. Applications engineering is — and always has been — all the engineering required to apply the hardware of a hardware manufacturer to the needs of the customer. Such applications engineering may include, and always has included where needed, dynamic analysis, mathematical models, simulation, linear programming, data logging, computing, and any technique needed to meet the end purpose - the fitting of an existing line of production hardware to a customer's needs. This is applied systems engineering."

"Linear programming was a new branch of applied mathematics that – in the USA – came into being as a direct consequence of mathematicians’ war work. It was not done under contract with the AMP but by some of the mathematicians employed directly by the armed forces.15 The source was a concrete practical problem within the US Air Forces,16 a logistic problem that eventually led to the mathematical theory of linear programming, and from there to mathematical programming. The person normally associated with the origin of linear programming in the USA is George B. Dantzig. Dantzig was one of the mathematicians hired directly by the armed forces for the war effort. In 1941 he began working at the Combat Analysis Branch of the United States Air Force Headquarters Statistical Control under the leadership of Tex Thornstons. During the war Dantzig worked on what was called “programming planning” methods to calculate Air Force programs. An Air Force program was a kind of activity plan."

"Linear Programming is a generalization of Linear Algebra. It is capable of handling a variety of problems, ranging from finding schedules for airlines or movies in a theater to distributing oil from refineries to markets. The reason for this great versatility is the ease at which constraints can be incorporated into the model."

"Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role. Many industries use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way."

"In another part of the management forest, the mechanistic school was gathering its forces and preparing to outflank the forces of light. First came the numbers men — the linear programmers, the budget experts, and the financial analysts — with their PERT systems and cost-benefit analyses. From another world, unburdened by most of the scientific management ideology and untouched by the human relations school, they began to parcel things out and give some meaning to those truisms, "plan ahead" and "keep records." Armed with emerging systems concepts, they carried the "mechanistic" analogy to its fullest—and it was very productive. Their work still goes on, largely untroubled by organizational theory; the theory, it seems clear, will have to adjust to them, rather than the other way around."

"[The decisionmaking role of the firm has progressed from the neoclassical standpoint of profit maximization to sales maximization, utility maximization, and satisficing. From the Operation Research point of view] ...the ideal picture is that someone, presumable the firm that hires the operations researcher, hands him, on a silver platter, an objective function. By talking to the engineers, or by looking into a few scientific laws, he determines the policy alternatives available and also the model"

"The development (rather than the history) of operations research as a science consists of the development of its methods, concepts, and techniques. Operations research is neither a method nor a technique; it is or is becoming a science and as such is defined by a combination of the phenomena it studies."

"We may recognize the subject of management cybernetics — which is seen as a rich provider of models for doing Operations research."

"The aim of management science is to display the best course of action in a given set of circumstances, and this must include all the circumstances."

"The 19th and first half of the 20th century conceived of the world as chaos. Chaos was the oft-quoted blind play of atoms, which, in mechanistic and positivistic philosophy, appeared to represent ultimate reality, with life as an accidental product of physical processes, and mind as an epi-phenomenon. It was chaos when, in the current theory of evolution, the living world appeared as a product of chance, the outcome of random mutations and survival in the mill of natural selection. In the same sense, human personality, in the theories of behaviorism as well as of psychoanalysis, was considered a chance product of nature and nurture, of a mixture of genes and an accidental sequence of events from early childhood to maturity. Now we are looking for another basic outlook on the world -- the world as organization. Such a conception -- if it can be substantiated -- would indeed change the basic categories upon which scientific thought rests, and profoundly influence practical attitudes. This trend is marked by the emergence of a bundle of new disciplines such as cybernetics, information theory, general system theory, theories of games, of decisions, of queuing and others; in practical applications, systems analysis, systems engineering, operations research, etc. They are different in basic assumptions, mathematical techniques and aims, and they are often unsatisfactory and sometimes contradictory. They agree, however, in being concerned, in one way or another, with "systems," "wholes" or "organizations"; and in their totality, they herald a new approach."

"Linear programming is viewed as a revolutionary development giving man the ability to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity. In the real world, planning tends to be ad hoc because of the many special-interest groups with their multiple objectives."

"The concern of OR with finding an optimum decision, policy, or design is one of its essential characteristics. It does not seek merely to define a better solution to a problem than the one in use; it seeks the best solution... [It] can be characterized as the application of scientific methods, techniques, and tools to problems involving the operations of systems so as to provide those in control of the operations with optimum solutions to the problems."

"OR is the securing of improvement in social systems by means of scientific method"

"By the end of the war the new game theoretic methods that had been developed by von Neumann and Morgenstern were added to the toolkit and mathematical techniques that operations research scientists deployed. These proved very valuable, and game theoretic approaches took on great importance after the war."

"Operations research has many precursors and allied fields, including Taylorism (after Frederick W. Taylor), scientific management and management science, industrial engineering and systems analysis. As one early textbook explained, the roots of OR "are as old as science and the management function. Its name dates back only to 1940" (Churchman et al. 1957: 3). Certainly its practitioners have expended much energy and ink in search of an acceptable definition of OR. Morse tried unsuccessfully to halt the debate by declaring OR to be 'the activity carried on by members of the Operations Research Society' (Morse 1953: 159) But his colleagues were not so easily dissuaded from debate. Much of the concern with definition focused on the sometimes elusive distinctions between OR and neighbouring fields; the attempt to define, or redefine, OR was also born of the desire to allow the subject to evolve beyond the orthodoxy of wartime experience. Crucial considerations included the balance between model and application, and the complexity of the mathematics involved."

"It is hard to say whether increasing complexity is the cause or the effect of man's effort to cope with his expanding environment. In either case a central feature of the trend has been the development of large and very complex systems which tie together modern society. These systems include abstract or non-physical systems, such as government and the economic system. They also include large physical systems like pipe line and power distribution systems, transportation and electrical communication systems. The growth of these systems has increased the need not only for over-all planning, but also for long-range development of the systems. This need has induced increased interest in the methods by which efficient planning and design can be accomplished in complex situations where no one scientific discipline can account for all the factors. Two similar disciplines which emerged about the time of World War II to cope with these problems are called systems engineering and operations research."

"The Mathematical School ; Although mathematical methods can be used by any school of management theory, and have been, I have chosen to group under a school those theorists who see management as a system of mathematical models and processes. Perhaps the most widely known group I arbitrarily so lump are the operations researchers or operations analysts, who have sometimes anointed themselves with the rather pretentious name of "management scientists." The abiding belief of this group is that, if management, or organization, or planning, or decision making is a logical process, it can be expressed in terms of mathematical symbols and relationships. The central approach of this school is the model, for it is through these devices that the problem is expressed in its basic relationships and in terms of selected goals or objectives."

"We should no longer have trouble explaining the scope and methods of operations research to the layman. We already can say: operations research is the activity carried on by members of the Operational Research Society; its methods are those reported in our journal."

"The conclusions of most good operations research studies are obvious."

"It may be said that systems engineering is directed at the design and operating problems of production processes and units, while operations research is applied to problems in other areas of management such as sales, marketing, and external finance."

"Herbert A. Simon's scientific output goes far beyond the disciplines in which he has held professorships: political science, administration, psychology and information sciences. He has made contributions in the fields of science theory, applied mathematics, statistics, operations research, economics and business and public administration (and), in all areas in which he has conducted research, Simon has had something of importance to say.""

"The development of information research has increased considerably the interaction of emerging information science with other disciplines. Librarianship has traditionally had links with education and classification and has drawn ideas from logic and philosophy. But during the last fifty years new insights and methods have been derived from sociology and social psychology, from computer science, from operations research and related quantitative approaches, from communications research, from linguistics, and most recently from the new hybrids: cognitive science and artificial intelligence."

"Operations research is a scientific approach to problem-solving for executive management."

"The Manager and the Instruments of Management Production is a synthesis of functions — always present but not all actually developed — sometimes quite rudimentary — how is synthesis to be effected — influence of strong personality — leadership essential to great results — manager is outside science of management — must use its principles, as chemist is outside the science of chemistry and uses its principles to effect his aims — management science provides the instruments, their successful wielding depends upon capacity of manager."

"In the the past thirty years, what is now known as the science of has grown from a scattered, almost formless beginning into a well-defined, rapidly developing structure. Today this science bids fair to outstrip in public importance and economic usefulness any one of the great sciences that have been applied to serve mankind. For the science of industrial management is the great super-science that takes mechanics, chemistry, physics, electricity, psychology; and Burbank-like, grafts them all into one great, productive, ever-bearing plant."

"Henry R. Towne is unquestionably the pioneer of management science. He began, as early as 1870, the systematic application at the Yale & Towne works, of what are now recognized as efficient management methods. In 1886, his paper "The Engineer as Economist," delivered before the American Society of Mechanical Engineers, probably inspired Frederick W. Taylor, then a young man of twenty, to devote his energies to the labor that formed his life work."

"The last decade has developed new and complex tools for management, in the form of high-speed computational, display, and communication techniques as well as of powerful analytical methods such as information theory, , and . These tools are being applied to continuing industrial problems, such as production and labor force scheduling, inventory control, sales forecast, and data processing (flow of information), as well as to special problems arising in connection with planning, market analysis, and organization. There is a rapidly growing demand for those well trained in the management sciences. Special graduate training in this held may be based on undergraduate study leading to a bachelor's degree in such fields as business administration, engineering, mathematics or the physical sciences, economics, and psychology."

"One cannot help but be struck by the diversity that characterizes efforts to study the management process. If it is true that psychologists like to study personality traits in terms of a person's reactions to objects and events, they could not choose a better stimulus than management science. Some feel it is a technique, some feel it is a branch of mathematics, or of mathematical economics, or of the "behavioral sciences," or of consultation services, or just so much nonsense. Some feel it is for management (vs. labor), some feel it ought to be for the good of mankind — or for the good of underpaid professors."

"There will be no drastic revolution in management functions or organizations in order to encompass systems management. Rather, the adaptation of systems management theory to organizations has been and will continue to be an evolutionary process."

"In management science, it seems most likely to me that important developments will result from the efforts of research scientists working with management engineers and management people in their area. It is also possible that the accumulated, uncorrelated knowledge and experience of management people will have a valuable effort on any general general formulation concerning management.... Management science undoubtedly requires many disciplines, including mathematics, economics, psychology, sociology, engineering, and others. However, we believe that management science can also be defined as a separate science in its own right."

"We have overwhelming evidence that available information plus analysis does not lead to knowledge. The management science team can properly analyse a situation and present recommendations to the manager, but no change occurs. The situation is so familiar to those of us who try to practice management science that I hardly need to describe the cases."

"The purpose of management science is to secure improvement in social systems by means of the scientific method."

"The aim of management science is to display the best course of action in a given set of circumstances, and this must include all the circumstances."

"When discussing management and science, I kept saying to myself over and over again that science could be looked at as a kind of management, or that management could be looked at as a kind of science. Saying that science can become a way of managing didn't imply automation or any other form of mechanical decision making, because none of this is science. Science is the creative discovery of knowledge. Management science is the process of trying to look at science as a management function. Similarly, management can be looked at as a scientific function, that is, as a way of finding out about the world."

"Decision theory can be pursued not only for the purposes of building foundations for political economy, or of understanding and explaining phenomena that are in themselves intrinsically interesting, but also for the purpose of offering direct advice to business and governmental decision makers. For reasons not clear to me, this territory was very sparsely settled prior to World War II. Such inhabitants as it had were mainly industrial engineers, students of public administration, and specialists in business functions, none of whom especially identified themselves with the economic sciences. Prominent pioneers included the mathematician, Charles Babbage, inventor of the digital computer, the engineer, Frederick Taylor and the administrator, Henri Fayol. During World War II, this territory, almost abandoned, was rediscovered by scientists, mathematicians, and statisticians concerned with military management and logistics, and was renamed “operations research” or “operations analysis.” So remote were the operations researchers from the social science community that economists wishing to enter the territory had to establish their own colony, which they called “management science”."

"During and after World War II, a large number of academic economists were exposed directly to business life, and had more or less extensive opportunities to observe how decisions were actually made in business organizations. Moreover, those who became active in the development of the new management science were faced with the necessity of developing decision-making procedures that could actually be applied in practical situations. Surely these trends would be conducive to moving the basic assumptions of economic rationality in the direction of greater realism."

"Now the salient characteristic of the decision tools employed in management science is that they have to be capable of actually making or recommending decisions, taking as their inputs the kinds of empirical data that are available in the real world, and performing only such computations as can reasonably be performed by existing desk calculators or, a little later electronic computers. For these domains, idealized models of optimizing entrepreneurs, equipped with complete certainty about the world - or, a worst, having full probability distributions for uncertain events - are of little use. Models have to be fashioned with an eye to practical computability, no matter how severe the approximations and simplifications that are thereby imposed on them... The first is to retain optimization, but to simplify sufficiently so that the optimum (in the simplified world!) is computable. The second is to construct satisficing models that provide good enough decisions with reasonable costs of computation. By giving up optimization, a richer set of properties of the real world can be retained in the models... Neither approach, in general, dominates the other, and both have continued to co-exist in the world of management science."

"Operations research has many precursors and allied fields, including Taylorism (after Frederick W. Taylor), scientific management and management science, industrial engineering and systems analysis. As one early textbook explained, the roots of OR "are as old as science and the management function. Its name dates back only to 1940" (Churchman et al. 1957: 3). Certainly its practitioners have expended much energy and ink in search of an acceptable definition of OR. Morse tried unsuccessfully to halt the debate by declaring OR to be 'the activity carried on by members of the Operations Research Society' (Morse 1953: 159) But his colleagues were not so easily dissuaded from debate. Much of the concern with definition focused on the sometimes elusive distinctions between OR and neighbouring fields; the attempt to define, or redefine, OR was also born of the desire to allow the subject to evolve beyond the orthodoxy of wartime experience. Crucial considerations included the balance between model and application, and the complexity of the mathematics involved."

"The management science approach to organizational decision making is the analog to the rational approach by individual managers. Management science came into being during World War II. At that time, mathematical and statistical techniques were applied to urgent, large-scale military problems that were beyond the ability of individual decision makers. Mathematicians, physicists, and operations researchers used systems analysis to develop artillery trajectories, antisubmarine strategies, and bombing strategies such as salvoing (discharging multiple shells simultaneously). Consider the problem of a battleship trying to sink an enemy ship several miles away. The calculation for aiming the battleship's guns should consider distance, wind speed, shell size, speed and direction of both ships, pitch and roll of the firing ship, and curvature of the earth. Methods for performing such calculations using trial and error and intuition are not accurate, take far too long, and may never achieve success. This is where management science came in. Analysts were able to identify the relevant variables involved in aiming a ship's guns and could model them with the use of mathematical equations. Distance, speed, pitch, roll, shell size, and so on could be calculated and entered into the equations. The answer was immediate, and the guns could begin firing. Factors such as pitch and roll were soon measured mechanically and fed directly into the targeting mechanism. Today, the human element is completely removed from the targeting process. Radar picks up the target, and the entire sequence is computed automatically."

"The subject of management science has evolved for more than 60 years and is now a mature field within the broad category of applied mathematics. This book will emphasize both the applied and mathematical aspects of management science."

"The key to virtually every management science application is a mathematical model."

""Interactive Decision Theory" would perhaps be a more descriptive name for the discipline usually called Game Theory"

"There are quite a number of novel developments intended to meet the needs of a general theory of systems. We may enumerate them in brief survey:"

"# Cybernetics, based upon the principle of feedback or circular causal trains providing mechanisms for goal-seeking and self-controlling behavior."

"# Information theory, introducing the concept of information as a quantity measurable by an expression isomorphic to negative entropy in physics, and developing the principles of its transmission."

"# Game theory, analyzing in a novel mathematical framework, rational competition between two or more antagonists for maximum gain and minimum loss."

"# , similarly analyzing rational choices, within human organizations, based upon examination of a given situation and its possible outcomes."

"# or relational mathematics, including non-metrical fields such as network and graph theory."

"# Factor analysis, i.e., isolation by way of mathematical analysis, of factors in multivariable phenomena in psychology and other fields"

"# General system theory in the narrower sense (G.S.T.), trying to derive from a general definition of “system” as complex of interacting components, concepts characteristic of organized wholes such as interaction, sum, mechanization, centralization, competition, finality, etc., and to apply them to concrete phenomena."

"Game theory is the study of individual decision-making in the face of competing boundedly rational actors."

"An equilibrium is not always an optimum; it might not even be good. This may be the most important discovery of game theory."

"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution."

"Like all of mathematics, game theory is a tautology whose conclusions are true because they are contained in the premises."

"That strategic rivalry in a long-term relationship may differ from that of a one-shot game is by now quite a familiar idea. Repeated play allows players to respond to each other’s actions, and so each player must consider the reactions of his opponents in making his decision. The fear of retaliation may thus lead to outcomes that otherwise would not occur. The most dramatic expression of this phenomenon is the celebrated "Folk Theorem." An outcome that Pareto dominates the point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a in infinitely repeated games with sufficiently little discounting."

"By the end of the war the new game theoretic methods that had been developed by von Neumann and Morgenstern were added to the toolkit and mathematical techniques that operations research scientists deployed. These proved very valuable, and game theoretic approaches took on great importance after the war."

"Game theory is about how people cooperate as much as how they compete... Game theory is about the emergence, transformation, diffusion and stabilization of forms of behavior."

"Game theory is logically demanding, but on a practical level, it requires surprisingly few mathematical techniques. Algebra, calculus, and basic probability theory suffice. ...the stress placed on game-theoretic rigor in recent years is misplaced. Theorists could worry more about the empirical relevance of their models and take less solace in mathematical elegance. ...[I]f a proposition is proved for a model with a finite number of agents, it is... irrelevant whether it is true for an ifinite number... There are... only a finite number of people, or even bacteria. Similarly, if something is true in games in which payoffs are finitely divisible... it does not matter whether it is true when payoffs are infinitely divisible. There are no payoffs in the universe... infinitely divisible. Even time... continuous in principle, can be measured only by devices with a finite number of s. Of course, models based on the real and complex numbers can be hugely useful, but they are just approximations... There is... no intrinsic value of a theorem that is true for a continuum of agents on a , if it is also true for a finite number of agents of a finite choice space."

"Direct application of the theory of games to the solution of real problems has been rare, and its chief uses have been to offer some insight and understanding into the problems of competition (without actually solving them), and to provide mathematicians with new fields to conquer. Many important real problems involve more than two opponents, are not zero-sum, and exceed the bounds of the most developed versions of game theory."

"Chapter 2 describes the most demanding rational choice theory of all, game theory, which was developed by a genius and assumes that other people are geniuses."

"Game theory can be defined as the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."

"Game theory, however, deals only with the way in which ultrasmart, all knowing people should behave in competitive situations, and has little to say to Mr. X as he confronts the morass of his problem."

"At present game theory has, in my opinion, two important uses, neither of them related to games nor to conflict directly. First, game theory stimulates us to think about conflict in a novel way. Second, game theory leads to some genuine impasses, that is, to situations where its axiomatic base is shown to be insufficient for dealing even theoretically with certain types of conflict situations... Thus, the impact is made on our thinking process themselves, rather than on the actual content of our knowledge."

"(Game theory is) essentially a structural theory. It uncovers the logical structure of a great variety of conflict situations and describes this structure in mathematical terms. Sometimes the logical structure of a conflict situation admits rational decisions; sometimes it does not."

"[G]ame theory has already established itself as an essential tool in the , where it is widely regarded as a unifying language for investigating human behavior. Game theory's prominence in evolutionary biology builds a natural bridge between the life sciences and the behavioral sciences. And connections have been established between game theory and the two most prominent pillars of physics: and quantum theory. ...[M]any physicists, neuroscientists, and social scientists... are... pursuing the dream of a quantitative science of human behavior. Game theory is showing signs of... an increasing important role in that endeavor. It's a story of exploration along the shoreline separating the continent of knowledge from an ocean of ignorance... a story worth telling."

"The last decade has seen a steady increase in the application of concepts from the theory of games to the study of evolution. Fields as diverse as sex ratio theory, animal distribution, contest behaviour and have contributed to what is now emerging as a universal way of thinking about phenotypic evolution... Paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behavior for which it was originally designed"

"The theory of games was first formalised by Von Neumann & Morgenstern (1953) in reference to human economic behaviour. Since that time, the theory has undergone extensive development... Sensibly enough, a central assumption of classical game theory is that the players will behave rationally, and according to some criterion of self-interest. Such an assumption would clearly be out of place in an evolutionary context. Instead, the criterion of rationality is replaced by that of and stability, and the criterion of self-interest by Darwinian fitness."

"Mathematics is what we want to keep for ourselves. When playing games, we stick to the rules (or we are changing the game...), but when doing serious mathematics (not executing algorithms) we make up the rules—definitions, axioms... even logics. ...[I]n arithmetic we find s... a whole new 'game'... [T]o identify mathematics with games would be one of those part-for-whole mistakes (like 'all geometry is projective geometry' or 'arithmetic is just logic' from the nineteenth century)... [M]y separation of game analysis from playing games tells in favour of the analogy of mathematics to analysis of games played by other... agents, and against the analogy of mathematics to the expert play of the game itself."

"One should remember that mathematical logic itself or the study of mathematics as a formal system can be considered a branch of combinatorial analysis. Metamathematics introduces a class of games—"solitaires"—to be played with symbols according to formal rules. One sense of Gödel's theorem is that some properties of these games can be ascertained only by playing them."

"The cybernetics phase of cognitive science produced an amazing array of concrete results, in addition to its long-term (often underground) influence:"

"Game theory brings to the chaos-theory table the idea that generally, societies are not designed, and that most situations don’t come with a rulebook. Instead, people have their own plans and designs on how things should fit together. They want to determine how the game is played, and they see societal designers as myopic busybodies who would imprison them with their theories."

"The implication of game theory, which is also the implication of the third image, is, however, that the freedom of choice of any one state is limited by the actions of the others."

"The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it."

"Those persons who desire to see the doctrine of Proportion treated according to a general method which is plainer than Euclid's, and equally accurate, may consult the geometry of Playfair, [Alexander] Ingram, Leslie, Cresswell, and J. R. Young. Hutton, and other recent writers have adopted the algebraical method in their elements of geometry. Proportion is not properly a geometrical subject."

"Such questions, thus too often rejected by the mathematician as metaphysical, are on the other hand discarded by the metaphysician as mathematical: so that, disowned by both, they fall to the ground without any satisfactory discussion at all."

"On an attentive examination of the methods adopted by modern elementary writers, in laying down the first principles of ratios and proportion, and especially in commenting upon Euclid, I long since experienced a conviction of the extremely unsatisfactory nature of most of their views; and this chiefly, as appearing to me to involve inadequate ideas of Euclid's real principle in treating of proportionals in his 5th book, and of the nature of the quantities which form the subject of investigation."

"The doctrine of ratios and proportion is introduced by Euclid as a part of his system of geometry; and the student seldom fails to remark, that in the treatises on algebra, the same subject is presented under a considerably different form; though he is usually quite unable to determine wherein the essential difference consists; and would probably find but few teachers who could precisely point out the distinction to him."

"Various relations being established in geometry between lines constituted under given conditions, as parts of geometrical figures, if we choose to adopt the idea of expressing these lines by numerical measures, we are then brought to the distinction of such lines being in some cases commensurable in their numerical values, in others not so. Their geometrical relations however are absolutely general, and do not refer to any such distinction."

"Much of the confusion of ideas which has arisen on these subjects has been occasioned by not observing that when we say "two lines are incommensurable," the phrase is in fact elliptical, and we ought always to consider as understood, if not expressed, that "two lines, if referred to numbers, "are incommensurable." The deficiency of exact comparison in such cases is not in the geometrical relation of the quantities, but in the powers and capacities of our numerical system to express them. These observations may be necessary to enable us to appreciate better the opinions which have since been prevalent among mathematicians; more especially as regards the doctrine of proportionals."

"The author's object in this tract is to defend Euclid from the charges of inconsistency which have been brought against him by Sir John Leslie and others, in consequence of the introduction of the doctrine of ratio and proportion as part of his system of geometry. Most of the best writers of geometry (as Legendre) omit this part in their elementary systems, and most teachers in this country pass over the 5th book, and adopting the doctrine of proportionals from algebra, proceed to apply it to the theorems of the 6th book. Professor Powell treats the subject in detail, stating the objections which have been urged against Euclid, and presenting answers to these objections. He begins with a general statement of the question; he then proceeds to the consideration of Euclid's method, or the doctrine of commensurables and incommensurables. He shews that Euclid, in his earlier books, does not even imply the idea of incommensurability. Neither is this introduced in the 5th and 6th books, and it is not till we arrive at the 10th that this edition in geometrical magnitudes, expressed by numerical measures, is broached. In the 11th and 12th books all reference to this distinction is dropped, recurrence being made to the principles of the 5th book. It is again, however, resumed in the 13th book, and is applied to various properties. The author observes, "that much of the confusion of ideas which has arisen on these subjects, has been occasioned by not observing that when we say two lines are incommensurable, the phrase is, in fact, elliptical, and we ought always to consider as understood, if not expressed, that two lines if referred to numbers are incommensurable. The deficiency of exact comparison in such cases is not in the geometrical relation of the quantities, but in the powers and capabilities of our numerical system to express them. Mr. Powell then proceeds to discuss the views of the earlier geometers and of later mathematicians. He points out the misapprehension under which they all labour, from the common mistake of considering that definitions describe the thing defined instead of fixing the meaning of terms. He shews that the mistake must be corrected before reasoning can be admitted on the subject. The nature of abstract quantity is next ably treated of, and the paper concluded in the same philosophic spirit which pervades it throughout."

"The object of the edition now offered to the public, is not so much to give to the writings of Euclid the form which they originally had, as that which may at present render them most useful. One of the alterations made with this view, respects the Doctrine of Proportion, the method of treating which, as it is laid down in the fifth of Euclid, has great advantages, accompanied with considerable defects; of which, however, it must be observed, that the advantages are essential, and the defects only accidental. To explain the nature of the former, requires a more minute examination than is suited to this place, and must, therefore, be reserved for the Notes; but, in the mean time, it may be remarked, that no definition, except that of Euclid, has ever been given, from which the properties of proportionals can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also simple and direct. As to the defects, the prolixness and obscurity, that have so often been complained of in the fifth Book, they seem to arise chiefly from the nature of the language employed, which being no other than that of ordinary discourse, cannot express, without much tediousness and circumlocution, the relations of mathematical quantities, when taken in their utmost generality, and when no assistance can be received from diagrams. As it is plain, that the concise language of Algebra is directly calculated to remedy this inconvenience, I have endeavoured to introduce it here, in a very simple form however, and without changing the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this means, the steps of the reasoning which were before far separated, are brought near to one another, and the force of the whole is so clearly and directly perceived, that I am persuaded no more difficulty will be found in understanding the propositions of the fifth Book, than those of any other of the Elements."

"All who intend to study these pages would do well to read attentively the following directions and observations; for the subject upon which they are written is considered one really difficult. Why it should be considered so, will readily be conceived when such men as Legendre, Leslie, [Thomas] Keith, Bonnycastle, Austin, Brewster, Young, and in fact every one who has attempted to treat the doctrine of Geometrical Proportion on any plan differing from Euclid's, have committed errors, overlooked mistakes, retrenched the generality of Euclid's reasonings, fallen into logical absurdities, or confined the general application of a subject which pervades a whole course of mathematics; while there is not one mistake, oversight, or logical objection in the whole of Euclid's Fifth Book. "In fact, Euclid's Fifth Book is a master-piece of human reasoning.""

"In matters connected with geometry, nothing is to be taken upon trust: mere opinion, unsupported by reasonings which elevate it into proof, must be regarded, in this subject, as of but little worth."

"It would, of course, be a easier task for me to transfer Euclid's fifth book into these pages. I could find little to remark upon in it, as the ancient Geometer has displayed so much sagacity and penetration in this, the most elaborate of all his writings, that he has left to moderns little or no room for improvement: it must be studied just as it is (in Simson's restoration), or else be superseded in instruction by a treatise of equal generality, but greater simplicity. You will understand, therefore, that I do not displace fifth Euclid's book because of its imperfections, or because of its inadequacy to completely its objects; but solely because of its great difficulty to a beginner."

"The subject of Euclid's fifth book is PROPORTION—universal proportion; that is, not numerical proportion merely, but proportion in reference to all magnitudes and quantities whatever, whether numbers, lines, surfaces, solids, or concrete quantities of any kind. With proportion in numbers you are already familiar:—this will be a help. I hope, too, by this time you are also somewhat acquainted with proportion in Algebra: this will be a greater help; for proportion in Geometry really accomplishes no more for things in general than the same doctrine in arithmetic and algebra accomplishes for what the notation of those sciences specially represents; and if this kind of proportion would do for geometry, the fifth book of Euclid would become a very easy matter indeed."

"The obstacle... is that geometrical magnitudes when compared together are in many cases found to be incommensurable;—that is to say, two such magnitudes may be quite incapable of a common measurement—they may be of a nature not to admit of being both measured by one and the same unit of measurement, however minute the measuring unit be taken, and consequently, both cannot be represented by numbers. I have already adverted to an instance of this kind... in the side and diagonal of a square, and to another in the diameter and circumference of a circle. ...That they are so, could not have been found out by such practical or experimental tests as those here adverted to for illustration: they are proved to be so by geometrical reasoning."

"It may be as well to caution you here that you must not speak of a line or quantity, by itself, as being incommensurable; this would be absurd. The diagonal of a square is not itself incommensurable, since it has, of course, its third part, fourth part, hundredth part, &c. and is therefore measurable by each of those parts; but as none of them will also measure the side, the two, considered together, are incommensurable: there exists no measure common to both. In the same way in reference to the circle—the circumference itself is not incommensurable any more than the diameter; for each has its fourth part, sixth part, &c. but it is incommensurable with its diameter: no length whatever can measure both."

"Now although Euclid makes no mention of incommensurable quantities in his fifth book, he was well aware of their existence; and therefore, to render his theorems on proportion general, he had to take care that this class of quantities should be comprehended in his reasonings. But proportion limited to numbers, or to the symbols for numbers, would necessarily exclude incommensurables; he therefore had to proceed quite independently of arithmetic, and to secure to his propositions such a universality that each theorem should rigorously apply, whether the quantities or magnitudes spoken of be measurable, or beyond the powers of numerical representation. He has executed his difficult task with consummate ability; for as Dr. Barrow remarks, "there is nothing, in the whole body of the Elements, of a more subtile invention,—nothing more solidly established, and more accurately handled, than the doctrine of proportionals.""

"I have resolved to replace the fifth book by the following treatise. I cannot promise that you will find the study of it easy; but it will certainly be much less difficult than the corresponding portion of Euclid's work; and yon will enter upon it with considerable advantage, if you postpone the attempt—as I here recommend—till you have read, as far, at least, as page 220 of the Algebra. It will, indeed, facilitate your progress, and agreeably diversify your mathematical labours, if you commence the elementary algebra upon closing the fourth book of Euclid, and read the PRINCIPLES through, before you begin the following treatise."

"The anonymous author of a scholium to Book v., who is perhaps Proclus, tells us that "some say" this Book, containing the general theory of proportion which is equally applicable to geometry, arithmetic, music, and all mathematical science, "is the discovery of Eudoxus, the teacher of Plato." Not that there had been no theory of proportion developed before his time; on the contrary, it is certain that the Pythagoreans had worked out such a theory with regard to numbers, by which must be understood commensurable and even whole numbers (a number being a "multitude made up of units," as defined in Eucl. vii)."

"Death not merely ends life, it also bestows upon it a silent completeness, snatched from the hazardous flux to which all things human are subject."

"A home with a loving and loyal husband and wife is the supreme setting in which children can be reared in love and righteousness and in which the spiritual and physical needs of children can be met. Just as the unique characteristics of both males and females contribute to the completeness of a marriage relationship, so those same characteristics are vital to the rearing, nurturing, and teaching of children."

"Sweet and gentle and sensitive man With an obsessive nature and deep fascination For numbers And a complete infatuation with the calculation Of π."

"Poetry being the sign of that which all men desire, even though the desire be unconscious, intensity of life or completeness of experience, the universality of its appeal is a matter of course."

"I will grow. I will become something new and grand, but no grander than I now am. Just as the sky will be different in a few hours, its present perfection and completeness is not deficient."

"Always aim at complete harmony of thought and word and deed. Always aim at purifying your thoughts and everything will be well."

"Be a life long or short, its completeness depends on what it was lived for."

"It is the consistency of the information that matters for a good story, not its completeness. Indeed, you will often find that knowing little makes it easier to fit everything you know into a coherent [[w:Pattern|pattern."

"Every phase of our life belongs to us. The moon does not, except in appearance, lose her first thin, luminous curve, nor her silvery crescent, in rounding to her full. The woman is still both child and girl, in the completeness of womanly character."

"It is a union with a Higher Good by love, that alone is endless perfection. The only sufficient object for man must be something that adds to and perfects his nature, to which he must be united in love; somewhat higher than himself, yea, the highest of all, the Father of spirits. That alone completes a spirit and blesses it, — to love Him, the spring of spirits."

"One of the great undiscovered joys of life comes from doing everything one attempts to the best of one's ability. There is a special sense of satisfaction, a pride in surveying such a work, a work which is rounded, full, exact, complete in its parts, which the superficial person who leaves his or her work in a slovenly, slipshod, half-finished condition, can never know. It is this conscientious completeness which turns any work into art. The smallest task, well done, becomes a miracle of achievement."

"Because of the Turing completeness theory, everything one Turing-complete language can do can theoretically be done by another Turing-complete language, but at a different cost. You can do everything in assembler, but no one wants to program in assembler anymore."

"Her long hair is loosely tied back, her face very refined and intelligent-looking, with beautiful eyes and a shadowy smile playing over her lips, a smile whose sense of completeness is indescribable."

"The greatest honor that can be paid to the work of art, on its pedestal of ritual display, is to describe it with sensory completeness. We need a science of description. Criticism is ceremonial revivification."

"...a theorem of propositional logic if and only if f(p1, p2 ,..., pn) is a tautology. … He (Emil L.Post) uses the word to discuss the adequacy of a system of functions to express all the possible truth tables (this is nowadays called truth-functional completeness). In this way he shows not only that through the connectives of Principia (∼ and ∨) one can generate all possible truth tables but also that there are only two connectives which can, singly generate all the truth tables."

"She had become such an integral part of my life that, without her light, presence, and solace, I would feel incomplete."

"The first requisite is knowledge, full and complete — which may be made public to the world."

"We have established complete liberty of conscience, and, in consequence, a complete liberty for mental activity. All free and daring souls have before them a well-nigh limitless opening for endeavor of any kind."

"There can be no effective control of corporations while their political activity remains. To put an end to it will be neither a short nor an easy task, but it can be done. We must have complete and effective publicity of corporate affairs, so that the people may know beyond peradventure whether the corporations obey the law and whether their management entitles them to the confidence of the public."

"The sources of poetry are in the spirit seeking completeness."

"Education develops the intellect; and the intellect distinguishes man from other creatures. It is education that enables man to harness nature and utilize her resources for the well-being and improvement of his life. The key for the betterment and completeness of modern living is education. But, 'Man cannot live by bread alone'. Man, after all, is also composed of intellect and soul. Therefore, education in general, and higher education in particular, must aim to provide, beyond the physical, food for the intellect and soul. That education which ignores man's intrinsic nature, and neglects his intellect and reasoning power can not be considered true education."

"Censorship ends in logical completeness when nobody is allowed to read any books except the books that nobody reads."

"A complete flower (manoecious) has both both male and female parts, and only one parent is needed if the plant is self-fruitful or can polinate itself....The complete flower contains four main parts: sepals, petals, staments, and pistils."

"Everything you need you already have. You are complete right now, you are a whole, total person, not an apprentice person on the way to someplace else. Your completeness must be understood by you and experienced in your thoughts as your own personal reality."

"It is amazing how complete is the delusion that beauty is goodness."

"The purpose of relationships is not to have another who might complete you; but to have another with whom you might share your completeness."

"The truth is, I often like women. I like their unconventionality. I like their completeness. I like their anonymity."

"In order theory and related fields such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of some partially ordered set. Notable special usages of the term include the concepts of complete Boolean algebra, complete lattice, and complete partial order (cpo). Furthermore, an ordered field is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound within the field, which should be compared to the (slightly different) order-theoretical notion of bounded completeness. Up to isomorphism there is only one complete ordered field: the field of real numbers (but note that this complete ordered field, which is also a lattice, is not a complete lattice)."

"Though he died in 1946, Keynes still dominated economics at Cambridge in the mid-1960s. It wasn't just that all the leading figures among the faculty, Richard Kahn, Joan Robinson, Brian Reddaway, David Champernowne, Nicholas Kaldor, and James Meade, had been his pupils and/or collaborators. It was the tone of the place. The study of economics, theoretical or empirical, was driven by the desire to improve the conduct of economic policy. This did not mean that pure theory was neglected. Indeed, in those years Robinson was fighting a stirring battle in the realms of high theory with Paul Samuelson and Robert Solow from the Massachusetts Institute of Technology. Nor was Cambridge innocent of the cutting edge of econometric technique. Richard Stone, who, following Keynes's suggestions, had created the first modern national income accounts, was still director of the Department of Applied Economics where much of modern econometrics was pioneered. Nonetheless, we were taught that theory and technique should serve the higher cause of rational economic policies, and to use our newly learned econometric expertise to write essays on unemployment, or inflation, or the balance of trade, or some other topic at the top of the current political agenda."

"Intermediate between mathematics, statistics, and economics, we find a new discipline which, for lack of a better name, may be called econometrics. Econometrics has as its aim to subject abstract laws of theoretical political economy or "pure" economics to experimental and numerical verification, and thus to turn pure economics, as far as possible, into a science in the strict sense of the word."

"(Econometrics is) the unification of economic theory, statistics and mathematics."

"What makes a piece of mathematical economics not only mathematics but also economics is, I believe, this: When we set up a system of theoretical relationships and use economic names for the otherwise purely theoretical variables involved, we have in mind some actual experiment, or some design of an experiment, which we could at least imagine arranging, in order to measure those quantities in real economic life that we think might obey the laws imposed on their theoretical namesakes."

"(Kálmán's) fundamental contributions to modern system theory... provided rigorous mathematical tools for engineering, econometrics, and statistics, and in particular for his invention of the "Kalman filter,"… was critical to achieving the Moon landings and creating the Global Positioning System and which has facilitated the use of computers in control and communications technology."

"In mathematical statistics, as well as econometrics, Ted Anderson is a scholar of immense stature. The scope and diversity of his research in statistical theory is almost a phenomenon in itself. Sometimes it seems that wherever one turns the subject, Ted's mark and influence is already firmly established. His books on multivariate analysis and time series rapidly became accepted as major treatises and are now integral parts of the bookshelves of every statistician."

"Econometrics may be defined as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference."

"As an econ blogger, I get the sense that this is exactly how many Americans still think of economists--as self-appointed defenders of the free market, spinning theories to show that greed is good. Watching those old Milton Friedman videos, I wonder if that picture might have been accurate in the 1960s and 1970s. But some big things have changed in the field of economics, and America should know about them. Three big changes stand out in particular: Econ today is more data-driven, far less politically conservative, and in general much more like engineering than it used to be."

"Econometrics is the name for a field of science in which mathematical-economic and mathematical-statistical research are applied in combination. Econometrics, therefore, forms a borderland between two branches of science, with the advantages and disadvantages thereof; advantages, because new combinations are introduced which often open up new perspectives; disadvantages, because the work in this field requires skill in two domains, which either takes up too much time or leads to insufficient training of its students in one of the two respects."

"The possibility of a stable economic life with full utilization of our resources is still not sufficiently assured, and it is extremely important that it should be so assured, and that the whole world should accept this as a fact. The work that is being done in econometrics is massive, and undaunted by mathematical difficulties, but it appears, at any rate as viewed from outside, to be unclear as to its aim."

"The successes of modern control theory in the design of highly accurate space navigation systems have stimulated its use in the theoretical analyses of economic and biological systems. Similarly, the effectiveness of computer simulation techniques in the macroscopic analyses of physical systems has brought into vogue the use of computer-based econometric models for purposes of forecasting, economic planning, and management."

"One of the most frequently mentioned equations was Euler's equation, e^{i \pi} + 1 = 0. \,\! Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero."

"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence."

"There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre: e^{i \pi} + 1 = 0. \,\! It appeals equally to the mystic, the scientist, the philosopher, the mathematician."

"Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."

"For any real number x, Euler’s formula is"

"This most extraordinary equation first emerged in Leonard Euler’s Introductio in 1748."

"Since high school I have been utterly fascinated by the “mystery” of i = \sqrt{-1} and by the beautiful calculations that flow, seemingly without end, from complex numbers and functions of complex variables."

"-1 = e^iπ, proves that Euler was a sly guy."

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."

"In the entire history of Greek mathematics, all but the incomparable Archimedes and a few of the more heterodox sophists appear to have hated or feared the mathematical infinite. Analysis was thwarted when it might have prospered."

"I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in certain order, or that that order, has been to great extent necessary – being determined by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for the evolution had arrived."

"THEY who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance ; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation."

"The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it.""

"The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions."

"Mathematical Analysis is... the true rational basis of the whole system of our positive knowledge."

"Every attempt to refer chemical questions to mathematical doctrines must be considered, now and always, profoundly irrational, as being contrary to the nature of the phenomena. . . . but if the employment of mathematical analysis should ever become so preponderant in chemistry (an aberration which is happily almost impossible) it would occasion vast and rapid retrogradation...."

"Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression."

"So far we have studies how, for each commodity by itself, the law of demand in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producers. We considered as given and invariable the prices of other commodities and the incomes of other producers; but, in reality the economic system is a whole of which the parts are connected and react on each other. An increase in the incomes of the producers of commodity A will affect the demand for commodities Band C, etc., and the incomes of their producers, and, by its reaction will involve a change in the demand for A. It seems, therefore, as if, for a complete and rigorous solution of the problems relative to some parts of the economic system, it were indispensable to take the entire system into consideration. But this would surpass the powers of mathematical analysis and of our practical methods of calculation, even if the values of all the constants could be assigned to them numerically."

"Machine-held strings of binary digits can simulate a great many kinds of things, of which numbers are just one kind. For example, they can simulate automobiles on a freeway, chess pieces, electrons in a box, musical notes, Russian words, patterns on a paper, human cells, colors, electrical circuits, and so on. To think of a computer as made up essentially of numbers is simply a carryover from the successful use of mathematical analysis in studying models. Most of this series of lectures has been devoted to applications of computers, and this is not the time to give details about their usefulness. I merely wish to point out certain types of things being done with computers today that could not have been done in 1945. Some of these are technological, some are intellectual."

"Mathematical analysis is co-extensive with nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures ; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. Its chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them. If matter escapes us, as that of air and light because of its extreme tenuity, if bodies are placed far from us in the immensity of space, if man wishes to know the aspect of the heavens at successive periods separated by many centuries, if gravity and heat act in the interior of the solid earth at depths which will forever be inaccessible, mathematical analysis is still able to trace the laws of these phenomena. It renders them present and measurable, and appears to be the faculty of the human mind destined to supplement the brevity of life and the imperfection of the senses, and what is even more remarkable, it follows the same course in the study of all phenomena; it explains them in the same language, as if in witness to the unity and simplicity of the plan of the universe, and to make more manifest the unchangeable order which presides over all natural causes."

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe."

"Mathematical Analysis is as extensive as nature herself."

"Perhaps the least inadequate description of the general scope of modern Pure Mathematics I will not call it a definition would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations."

"[D]uring the last half-century, number and measurable quantity have been separated... the idea of number alone has been recognized as the foundation upon which Mathematical Analysis rests, and the theory of extensive magnitude is now regarded as a separate department in which the methods of Analysis are applicable, but as no longer forming part of the foundation upon which Analysis itself rests."

"I discovered that a whole range of problems of the most diverse character relating to the scientific organization of production (questions of the optimum distribution of the work of machines and mechanisms, the minimization of scrap, the best utilization of raw materials and local materials, fuel, transportation, and so on) lead to the formulation of a single group of mathematical problems (extremal problems). These problems are not directly comparable to problems considered in mathematical analysis. It is more correct to say that they are formally similar, and even turn out to be formally very simple, but the process of solving them with which one is faced [i.e., by mathematical analysis] is practically completely unusable, since it requires the solution of tens of thousands or even millions of systems of equations for completion. I have succeeded in finding a comparatively simple general method of solving this group of problems which is applicable to all the problems I have mentioned, and is sufficiently simple and effective for their solution to be made completely achievable under practical conditions."

"Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity."

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"The analyst, who pursues a purely esthetic aim, helps create, just by that, a language more fit to satisfy the physicist."

"L'analyse mathématique, n'est elle donc qu'un vain jeu d'esprit? Elle ne peut pas donner au physicien qu'un langage commode; n'est-ce pa là un médiocre service, dont on aurait pu se passer à la rigueur; et même n'est il pas à craindre que ce langage artificiel ne soit pas un voile interposé entre la réalité at l'oeil du physicien? Loin de là, sans ce langage, la pluspart des anaologies intimes des choses nous seraient demeurées à jamais inconnues; et nous aurions toujours ignoré l'harmonie interne du monde, qui est, nous le verrons, la seule véritable réalité objective."

"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself."

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and to write."

"As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made Mechanics analytical; I mean Euler."

"But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes."

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. ...Fermat applied his method ...and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same,ssumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"An untold amount of intellectual energy has been expended on the quadrature of the circle, yet no conquest has been made by direct assault. The circle-squarers have existed in crowds ever since the period of Archimedes. After innumerable failures to solve the problem at a time, even when investigators possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, while those who still persisted were completely ignorant of its history and generally misunderstood the conditions of the problem. ...But progress was made on this problem by approaching it from a different direction and by newly discovered paths. Lambert proved in 1761 that the ratio of the circumference of a circle to its diameter is incommensurable. Some years ago, Lindemann demonstrated that this ratio is also transcendental and that the quadrature of the circle, by means of the ruler and compass only, is impossible. He thus showed by actual proof that which keen minded mathematicians had long suspected; namely, that the great army of circle-squarers have, for two thousand years, been assaulting a fortification which is as indestructible as the firmament of heaven."

"J.M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Paul Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"The chief object of the present work is, as its title indicates, to furnish to the student examples by which to illustrate the processes of the Differential and Integral Calculus. In this respect it will be seen to agree with Professor Peacock's Collection of Examples ; and indeed if a second edition of that excellent work had been published I should not have undertaken the task of making this compilation. But as Professor Peacock informed me that he had not leisure to superintend the publication of a second edition of his "Examples" which had been long out of print, I thought that I should do a service to students by preparing a work on a similar plan, but with such modifications as seemed called for by the increased cultivation of Analysis in this University."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"It will be sufficient if, when we speak of infinitely great (or more strictly unlimited), or of infinitely small quantities (i.e., the very least of those within our knowledge) it is understood that we mean quantities that are indefinitely great or indefinitely small, i.e., as great as you please, or as small as you please, so that the error that any one may assign may be less than a certain assigned quantity. Also, since in general it will appear that, when any small error is assigned, it can be shown that it should be less, it follows that the error is absolutely nothing; an almost exactly similar kind of argument is used in different places by Euclid, Theodosius and others; and this seemed to them to be a wonderful thing, although it could not be denied that it was perfectly true that, from the very thing that was assumed as an error, it could be inferred that the error was non-existent. Thus by infinitely great and infinitely small, we understand something indefinitely great, or something indefinitely small, so that each conducts itself as a sort of class, and not merely as the last thing of a class. If any one wishes to understand these as the ultimate things, or as truly infinite, it can be done, and that too without falling back upon a controversy about the reality of extensions, or of infinite continuums in general, or of the infinitely small, ay, even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of the calculation, just as the algebraists retain imaginary roots with great profit. For they contain a handy means of reckoning, as can manifestly be verified in every case in a rigorous manner by the method already stated. But it seems right to show this a little more clearly, in order that it may be confirmed that the algorithm, as it is called, of our differential calculus, set forth by me in the year 1684, is quite reasonable."

"To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus."

"In the beginning of the year 1665 I found the method of approximating series and the rule for reducing any dignity [power] of any binomial to such a series [ i.e. the binomial theorem ]. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions [i.e. the elements of differential calculus], and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions [i.e. integral calculus], and in the same year I began to think of gravity extending to the orb of the moon … and having thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly. All this was in the two years of 1665 and 1666, for in those years I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since."

"[S]cientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation..."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"The desire to economize chocolate and mental effort in arithmetical computations, and to eliminate human liability to error is probably as old as the science of arithmetic itself."

"Some, of my unmathematical friends have incautiously urged me to dis-guard a note about the origin of modern calculating machines. This is the proper place to do so, as the Queen of queens has enslaved a few of these infernal things to do some of her more repulsive drudgery. What I shall say about these marvelous aids to the feeble human intelligence will be little indeed, for two reasons: I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly."

"If the variables are continuous, this definition [Ashby’s fundamental concept of machine] corresponds to the description of a dynamic system by a set of ordinary differential equations with time as the independent variable. However, such representation by differential equations is too restricted for a theory to include biological systems and calculating machines where discontinuities are ubiquitous."

"I know that the right kind of leader for the Labour Party is a kind of desiccated calculating-machine who must not in any way permit himself to be swayed by indignation. If he sees suffering, privation or injustice, he must not allow it to move him, for that would be evidence of the lack of proper education or of absence of self-control. He must speak in calm and objective accents and talk about a dying child in the same way as he would about the pieces inside an internal combustion engine."

"Two centuries ago Leibnitz invented a calculating machine which embodied most of the essential features of recent keyboard devices, but it could not then come into use. The economics of the situation were against it: the labor involved in constructing it, before the days of mass production, exceeded the labor to be saved by its use, since all it could accomplish could be duplicated by sufficient use of pencil and paper. Moreover, it would have been subject to frequent breakdown, so that it could not have been depended upon; for at that time and long after, complexity and unreliability were synonymous."

"Babbage, even with remarkably generous support for his time, could not produce his great arithmetical machine. His idea was sound enough, but construction and maintenance costs were then too heavy. Had a Pharaoh been given detailed and explicit designs of an automobile, and had he understood them completely, it would have taxed the resources of his kingdom to have fashioned the thousands of parts for a single car, and that car would have broken down on the first trip to Giza."

"The advanced arithmetical machines of the future will be electrical in nature, and they will perform at 100 times present speeds, or more. Moreover, they will be far more versatile than present commercial machines, so that they may readily be adapted for a wide variety of operations. They will be controlled by a control card or film, they will select their own data and manipulate it in accordance with the instructions thus inserted, they will perform complex arithmetical computations at exceedingly high speeds, and they will record results in such form as to be readily available for distribution or for later further manipulation."

"The needs of business, and the extensive market obviously waiting, assured the advent of mass-produced arithmetical machines just as soon as production methods were sufficiently advanced. With machines for advanced analysis no such situation existed; for there was and is no extensive market; the users of advanced methods of manipulating data are a very small part of the population."

"The law is not a series of calculating machines where answers come tumbling out when the right levers are pushed."

"Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey—having been thus constructed—an intuitive perception of the significance of numbers being out of the question."

"More than a decade will pass before personal computers emerge from the garages of Silicon Valley, and a full thirty years before the Internet explosion of the 1990s. The word computer still has an ominous tone, conjuring up the image of a huge, intimidating device hidden away in an overlit, air-conditioned basement, relentlessly processing punch cards for some large institution: them. Yet, sitting in a non-descript office in McNamara's Pentagon, a quiet forty-seven-year-old civilian is already planning the revolution that will change forever the way computers are perceived. Somehow, the occupant of that office - a former MIT psychologist named J. C. R. Licklider - has seen a future in which computers will empower individuals, instead of forcing them into rigid conformity. He is almost alone in his conviction that computers can become not just superfast calculating machines, but joyful machines: tools that will serve as new media of expression, inspirations to creativity, and gateways to a vast world of online information. And now he is determined to use the Pentagon's money to make that vision a reality...."

"The arithmetical machine produces effects which approach nearer to thought than all the actions of animals. But it does nothing which would enable us to attribute will to it, as to the animals."

"There is then no analogy whatever between the operations of the Chess-Player, and those of the calculating machine of Mr. Babbage, and if we choose to call the former a pure machine we must be prepared to admit that it is, beyond all comparison, the most wonderful of the inventions of mankind."

"In the early days of the computer revolution computer designers and numerical analysts worked closely together and indeed were often the same people. Now there is a regrettable tendency for numerical analysts to opt out of any responsibility for the design of the arithmetic facilities and a failure to influence the more basic features of software. It is often said that the use of computers for scientific work represents a small part of the market and numerical analysts have resigned themselves to accepting facilities "designed" for other purposes and making the best of them. [...] One of the main virtues of an electronic computer from the point of view of the numerical analyst is its ability to "do arithmetic fast." Need the arithmetic be so bad!"

"All that needed for its apprehension, more than the pure intellect, or required the exercise of fancy, imagination, affection, or faith, was distasteful to Cavendish. An intellectual head thinking, a pair of wonderfully acute eyes observing, and a pair of very skilful hands experimenting or recording, are all that I realise in reading his memorials. His brain seems to have been but a calculating engine; his eyes inlets of vision, not fountains of tears; his hands instruments of manipulation which never trembled with emotion, or were clasped together in adoration thanksgiving, or despair; his heart only an anatomical organ, necessary for of the circulation of the blood."

"I am not against migration. It is simply pragmatic to restrict migration, while at the same time encouraging integration and fighting discrimination. I support the idea of the free movement of goods, people, money and jobs in Europe."

"In the sensorimotor area on top of the cortex there are four maps of a little upside down person, distorted I shape with every bit of skin and muscle represented in detail. This upside map is called the sensorimotor homunculus, the little human. The nervous system abounds in such maps, some of which appear to serve as 'self systems', organizing and integrating vast amounts of local bits of information."

"If the word integration means anything, this is what it means: that we, with love, shall force our brothers to see themselves as they are, to cease fleeing from reality and begin to change it."

"...worldview, a new paradigm of perception and explanation, which is manifested in integration, holistic thinking, purpose-seeking, mutual causality, and process-focused inquiry"

"The aims of General Systems Theory: a) There is a general tendency towards integration in the various sciences: natural and social. b) Such integration seems to be centered in a general theory of systems. c) Such theory may be an important means for aiming at exact theory in the non-physical fields of science.(1956, d) Developing unifying principles running “vertically” through the universes of the individual sciences, this theory brings us nearer to the goal of unity of science."

"But no one expected he had ever get very far, because I don’t suppose he could even integrate e to the x. Is such ignorance possible gasped someone?"

"The formation of the differential equations proper to the phenomena is, independent of their integration, a very important acquisition, on account of the approximations which mathematical analysis allows between questions, otherwise heterogeneous."

"Mathematical analysis branch of mathematics that includes the theories of differentiation, integration, measure, limits, infinite series, and analytic functions. These theories areoften studied in the context of real numbers, complex numbers, and real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and technique of analysis."

"Decentralization may bring flexibility and fast response to changing business needs, as well as other benefits, but decentralization also makes systems integration difficult, presents a barrier to standardization, and acts as a disincentive toward achieving economies of scale. As a result, there is a need to balance the decentralization of IT management to business units with some centralized planning for technology, data, and human resources"

"God does not care about our mathematical difficulties. He integrates empirically."

"There is a property in the horizon which no man has but he whose eye can integrate all the parts, that is, the poet. This is the best part of these men's farms, yet to this their land-deeds give them no title."

"In order to consolidate the euro we need to harmonise our economic, fiscal and social policies, hence we are going toward greater integration.We are going to put in place an economic system of governance of the Eurozone. Great Britain is not part of Eurozone; t at the same time the decisions we will take will have great importance to Britain"

"By partnering with Apple, we have the opportunity to add value by integrating the world’s best digital music [iPod music player and iTunes music] offering into HP’s larger digital entertainment"

"He had the charm of all people who believe implicitly in themselves, that of integration."

"Some people pursue wholeness and integration; others get smashed up, and fragments are rescued from the smash of an intensity that the wholeness and integration people do not reach. Then too, the prophetic aspect of the arts is reflected"

"The Internet is ultimately about innovation and integration, but you don't get the innovation unless you integrate Web technology into the processes by which you run your business."

"Secular humanism contrasts with religious humanism, which is an integration of humanist ethical philosophy with religious rituals and beliefs that center on human needs, interests, and abilities."

"...to 'integrate the study of international economics with the study of international politics to deepen our comprehension of the forces at work in the world."

"Surely the mitochondrion that first entered another cell was not thinking about the future benefits of cooperation and integration; it was merely trying to make its own living in a tough Darwinian world."

"An effective information infrastructure must build on enabling technologies to create an integrated and adaptive system that is accessible to all potential users."

"Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane."

"From a fundamental point of view, integration is less demanding than differentiation, as far as the conditions imposed on the class of functions. As a consequence, numerical integration is a lot easier to carry out than numerical differentiation."

"There are a number of standard procedures which can enable a large number of common integrals to be evaluated explicitly. The simplest strategy is integration by substitution which means changing the variable of integration."

"Integration is the annual technocultural fest of Indian Statistical Institute, usually held during the first and second weekend of January each year. It is one of the biggest fests in Kolkata, and attracts participation from all over the world. Integration is relatively a new addition to the set of Calcutta fests, and yet, it has carved out a niche for itself."

"Integrated circuit (IC) is a circuit in which all or some of the circuit elements are inseparably associated and electrically interconnected so that it is considered to be indivisible for the purposes of construction and commerce."

"One does not become enlightened by imagining figures of light, but by making the darkness conscious. The latter procedure, however, is disagreeable and therefore not popular."

"Design is not making beauty, beauty emerges from selection, affinities, integration, love."

"The most highly satisfactory use of the reverse-curve sign of integration used in calculus is for those two S-openings in the top of a violin"

"Now our struggle is for genuine equality, which means economic equality. ... What does it profit a man to be able to eat at an integrated lunch counter if he doesn't earn enough money to buy a hamburger and a cup of coffee?"

"An integrated Approach is a holistic christological approach to exploring and reconsidering the gifts and graces of God for healing the various aspects of human person.An integrated approach will include a biblically based analysis and synthesis of historical Christian traditions pertaining to healing and deliverance as they relate to the practice of counseling."

"We're moving to this integration of biomedicine, information technology, wireless and mobile now - an era of digital medicine. Even my stethoscope is now digital. And of course, there's an app for that."

"The integration of exponentially growing technologies is beginning to empower the patient, enable the doctor, enhance wellness and begin to cure the well before they get sick."

"People who allow others not of their race to live among them will perish, because the inevitable result of a racial integration is racial inter-breeding which destroys the characteristics and existence of a race. Forced integration is deliberate and malicious genocide, particularly for a people like the White Race who are now a small minority in the world."

"I see with much pleasure that you are working on a large work on the integral Calculus [ … ] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics."

"Nature laughs at the difficulties of integration."

"The key distinctive of a truly Christian education is the effective practice of worldview integration that is, an approach to biblical integration that leads to a Christian worldview.”"

"Overcoming the Cold War required courage from the people of Central and Eastern Europe and what was then the German Democratic Republic, but it also required the steadfastness of Western partner over many decades when many had long lost hope of integration of the two Germanys and Europe."

"In a life-long partnership with his wife Jessie, James Grier Miller contributed substantially to the development of behavioural science and to the integration of disciplines through general systems theory, remaining actively engaged in these areas throughout his working life."

"In particular to perform integration — you have to be smart about which integration technique should be used: integration by partial fractions, integration by parts, and so on"

"Common integration is only the memory of differentiation... The artifices by which it [integration] is effected, are changes, not from the unknown to the known, but from the forms in which memory will not serve to those in which it will"

"[Malcolm X] in keeping with the Nation's teachings he espoused black supremacy, advocated the separation of black and white Americans and scoffed at the civil rights movement's emphasis on integration."

"The Generic Enterprise Reference Architecture and Methodology is about those methods, models and tools which are needed to build the integrated enterprise. The architecture is generic because it applies to most, potentially all types of enterprise."

"If you want to make beautiful music, you must play the black and the white notes together.”"

"...way a knowledge-system is knit together is to learn at least a minimum knowledge-of-knowledge, until someday — someday, or some century — an Integrator would come, and things would be fitted together again. So time mattered not at all"

"Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly."

"The powers which tend to preserve, and those which tend to change the condition of the earth's surface, are never in equilibrio; the latter are, in all cases, the most powerful, and, in respect of the former, are like living in comparison of dead forces. Hence the law of decay is one which suffers no exception: The elements of all bodies were once loose and unconnected, and to the same state nature has appointed that they should all return... TIME performs the office of integrating the infinitesimal parts of which this progression is made up; it collects them into one sum, and produces from them an amount greater than any that can be assigned."

"Science itself is badly in need of integration and unification. The tendency is more and more the other way … Only the graduate student, poor beast of burden that he is, can be expected to know a little of each. As the number of physicists increases, each specialty becomes more self-sustaining and self-contained. Such Balkanization carries physics, and indeed, every science further away, from natural philosophy, which, intellectually, is the meaning and goal of science."

"The success and originality of American integration stem precisely from the fact that immigrants' descendants can perpetuate their ancestral cultures while thinking of themselves as Americans in the fullest sense, sharing basic ideals across racial and ethnic barriers. In France, the characteristic attitude of newcomers from North Africa, Turkey, and sub-Saharan Africa is predominantly one of alienation, confrontation, rejection, and hatred."

"It's in the black players and it remains there. The National League was the first to sign black players and remained ahead all these years. And so many outstanding players are black that it's hard to have an outstanding team without your share of black players. It seemed like the National League teams were willing to sign any promising prospect regardless of color, while the American League was only interested in the outstanding, can't-miss black prospect. And if you don't sign the raw talent that needs a few years to develop, you lose out on some outstanding players."

"There is no one-size-fits-all solution to the challenges facing our cities or to the housing crisis, but the two issues need to be considered together. From an urban design and planning point of view, the well-connected open city is a powerful paradigm and an engine for integration and inclusivity."

"This integrative action in virtue of which the nervous system unifies from separate organs an animal possessing solidarity, an individual, is the problem before us."

"It may appear as a superficial inequality, one that can be solved by merely a few reforms, or perhaps by the full integration of women into the labour force. But the reaction of the common man, woman, and child."

"The more specific idea of Evolution now reached is - a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter."

"A true balance between work and life comes with knowing that your life activities are integrated, not separated."

"Globalization (or Globalisation) is the increasing interdependence, integration and interaction among people and corporations in disparate locations around."

"The emergence of a unified cognitive moment relies on the coordination of scattered mosaics of functionally specialized brain regions. Here we review the mechanisms of large-scale integration that counterbalance the distributed anatomical and functional organization of brain activity to enable the emergence of coherent behaviour and cognition."

"Let us therefore see their actual conception, in thought and action, and see how many perspectives it actually takes into account and how many perspectives it attempts to integrate, and thus let us see how deep and how wide runs that bodhisattva vow to refuse rest until all perspectives whatsoever are liberated into their primordial nature."

"It is vision-logic with its centauric/planetary worldview that, in my opinion, holds the only hope for the integration of the biosphere and noosphere, the supranational organization of planetary consciousness, the genuine recognition of ecological balance…"

"It's just like when you've got some coffee that's too black, which means it's too strong. What do you do? You integrate it with cream, you make it weak. But if you pour too much cream in it, you won't even know you ever had coffee. It used to be hot, it becomes cool. It used to be strong, it becomes weak. It used to wake you up, now it puts you to sleep.”"

"I believe in recognising every human being as a human being, neither white, black, brown nor red. When you are dealing with humanity as one family, there's no question of integration or intermarriage. It's just one human being marrying another human being, or one human being living around and with another human being."

"We who follow the Honorable Elijah Muhammad feel that when you try and pass integration laws here in America, forcing white people to pretend that they are accepting black people, what you are doing is making white people act in a hypocritical way."

"Of psychics were truly successful and if their results were not simply the consequence of trickery at worse or good interviewing skills (at best), then why don't law enforcement agencies have psychic detective squads, a real X-Files Unit, or other ways to integrate these paranormal investigative capabilities."

"Molecular orbital (MO) theory describes covalent bond formation as arising from a mathematical combination of atomic orbitals (wave functions) on different atoms to form molecular orbitals, so called because they belong to the entire molecule rather than to an individual atom. Just as an atomic orbital, whether unhybridized or hybridized, describes a region of space around an atom where an electron is likely to be found, so a molecular orbital describes a region of space in a molecule where electrons are most likely to be found."

"How do we construct H2 by using atomic orbitals? An answer to this question was developed by Pauling: Bonds are made by the in-phase overlap of atomic orbitals... The in-phase overlap of the two 1s orbitals results in a new orbital of lower energy called a bonding molecular orbital... On the other hand, out-of-phase overlap between the same two atomic orbitals results in a destabilizing interaction and formation of an antibonding molecular orbital."

"My loyalties will not be bound by national borders, or confined in time by one nation's history, or limited in the spiritual dimension by one language and culture. I pledge my allegiance to the damned human race, and my everlasting love to the green hills of Earth, and my intimations of glory to the singing stars, to the very end of space and time."

"Why should the thirst for knowledge be aroused, only to be disappointed and punished? My volition shrinks from the painful task of recalling my humiliation; yet, like a second Prometheus, I will endure this and worse, if by any means I may arouse in the interiors of Plane and Solid Humanity a spirit of rebellion against the Conceit which would limit our Dimensions to Two or Three or any number short of Infinity."

"According to the big bang model, space, time, and matter simultaneously expanded into physical existence.... Hence, the mathematical of the space=time combines space and time together under a single coordinate system which is specified by length, width, and height while 'time' is considered as the fourth dimension of the co-ordinate."

"Eternity isn't some later time. Eternity isn't a long time. Eternity has nothing to do with time. Eternity is that dimension of here and now which thinking and time cuts out. This is it. And if you don't get it here, you won't get it anywhere. And the experience of eternity right here and now is the function of life. There's a wonderful formula that the Buddhists have for the Bodhisattva, the one whose being (sattva) is illumination (bodhi), who realizes his identity with eternity and at the same time his participation in time. And the attitude is not to withdraw from the world when you realize how horrible it is, but to realize that this horror is simply the foreground of a wonder and to come back and participate in it."

"We have created characters and animated them in the dimension of depth, revealing through them to our perturbed world that the things we have in common far outnumber and outweigh those that divide us."

"It's bigger on the inside!"

"In a sensible theory there are no [dimensionless] numbers whose values are determinable only empirically. I can, of course, not prove that ... dimensionless constants in the laws of nature, which from a purely logical point of view can just as well have other values, should not exist. To me in my 'Gottvertrauen' [faith in God] this seems evident, but there might be few who have the same opinion ..."

"Space has always been the spiritual dimension of architecture. It is not the physical statement of the structure so much as what it contains that moves us."

"Imaginary time is a new dimension, at right angles to ordinary, real time."

"Bailey reached up and shook his arm. "Snap out of it. What the hell are you talking about, four dimensions? Time is the fourth dimension; you can't drive nails into that.""

"Human rights, of course, must include the right to religious freedom, understood as the expression of a dimension that is at once individual and communitarian — a vision that brings out the unity of the person while clearly distinguishing between the dimension of the citizen and that of the believer."

"The supremacy of expediency is being refuted by time and truth. Time is an essential dimension of existence defiant of man's power, and truth reigns in supreme majesty, unrivaled, inimitable, and can never be defeated."

"Faith is sensitiveness to what transcends nature, knowledge and will, awareness of the ultimate, alertness to the holy dimension of all reality. Faith is a force in man, lying deeper than the stratum of reason and its nature cannot be defined in abstract, static terms. To have faith is not to infer the beyond from the wretched here, but to perceive the wonder that is here and to be stirred by the desire to integrate the self into the holy order of living. It is not a deduction but an intuition, not a form of knowledge, of being convinced without proof, but the attitude of mind toward ideas whose scope is wider than its own capacity to grasp. Such alertness grows from the sense for the meaningful, for the marvel of matter, for the core of thoughts. It is begotten in passionate love for the significance of all reality, in devotion to the ultimate meaning which is only God."

"We actually have a candidate for the mind of God. The mind of God we believe is cosmic music, the music of strings resonating through 11 dimensional hyperspace. That is the mind of God."

"How sickness enlarges the dimension of a man's self to himself!"

"Design in art, is a recognition of the relation between various things, various elements in the creative flux. You can't invent a design. You recognize it, in the fourth dimension. That is, with your blood and your bones, as well as with your eyes."

"Science by itself has no moral dimension. But it does seek to establish truth. And upon this truth morality can be built."

"Lois Lane: I can’t describe what Mxyzptlk then became. He had height, width, depth, and a couple of other things, too."

"She was no scholar in geometry or aught else, but she felt intuitively that the bend and slant of the way she went were somehow outside any other angles or bends she had ever known. They led into the unknown and the dark, but it seemed to her obscurely that they led into deeper darkness and mystery than the merely physical, as if, though she could not put it clearly even into thoughts, the peculiar and exact lines of the tunnel had been carefully angled to lead through poly-dimensional space as well as through the underground — perhaps through time, too."

"Time is relative, it can stretch and squeeze, but it can't run backwards. The only thing that can move across dimensions like time is gravity. … Look, Cooper, they're creatures of at least five dimensions, to them the past might be a canyon they can climb into and the future a mountain they can climb up... but to us it's not, okay?"

"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded. "Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child especially a baby might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

"We are not in the Eighth dimension, we are over New Jersey. Hope is not lost."

"There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone."

"You are about to enter another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight Zone!"

"You're traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land of imagination. Next stop, the Twilight zone!"

"You're traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination. That's the signpost up ahead — your next stop, the Twilight Zone!"

"You are traveling through another dimension, a dimension not only of sight and sound but of mind. A journey into a wondrous land whose boundaries are that of imagination. Your next stop, the Twilight Zone!"

"You unlock this door with the key of imagination. Beyond it is another dimension—a dimension of sound, a dimension of sight, a dimension of mind. You're moving into a land of both shadow and substance, of things and ideas. You've just crossed over into the Twilight Zone."

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system."

"He sat back down again in his armchair. “Of course, that name really isn’t accurate. I suppose a pentaract should really be a four-dimensional pentagon, and this is meant to be a picture of a five-dimensional cube.”"

"When you listen to a thought, you are aware not only of the thought but also of yourself as the witness of the thought. A new dimension of consciousness has come in."

"Make it your practice to withdraw attention from past and future whenever they are not needed. Step out of the time dimension as much as possible in everyday life."

"Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing."

"If I am recalling an incident very vividly I go back to the instant of its occurrence; I become absent minded, as you say. I jump back for a moment. Of course we have no means of staying back for any length of time any more than a savage or an animal has of staying six feet above the ground. But a civilized man is better off than the savage in this respect. He can go up against gravitation in a balloon, and why should we not hope that ultimately he may be able to stop or accelerate his drift along the Time Dimension; or even to turn about and travel the other way?"

"There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it."

"Perhaps the first to approach the fourth dimension from the side of physics, was the Frenchman, Nicole Oresme, of the fourteenth century. In a manuscript treatise, he sought a graphic representation of the Aristotelian forms, such as heat, velocity, sweetness, by laying down a line as a basis designated longitudo, and taking one of the forms to be represented by lines (straight or circular) perpendicular to this either as a latitudo or an altitudo. The form was thus represented graphically by a surface. Oresme extended this process by taking a surface as the basis which, together with the latitudo, formed a solid. Proceeding still further, he took a solid as a basis and upon each point of this solid he entered the increment. He saw that this process demanded a fourth dimension which he rejected; he overcame the difficulty by dividing the solid into numberless planes and treating each plane in the same manner as the plane above, thereby obtaining an infinite number of solids which reached over each other. He uses the phrase "fourth dimension" (4am dimensionem)."

"I search for the realness, the real feeling of a subject, all the texture around it... I always want to see the third dimension of something... I want to come alive with the object."

"Motion in three dimensions is not easy to understand. For example, you are probably good at driving a car along a freeway (one-dimensional motion) but would probably have a difficult time in landing an airplane on a runway (three dimensional motion) without a lot of training."

"The mass is a matrix from which all traditional behavior toward works of art issues today in a new form. Quantity has been transmuted into quality. The greatly increased mass of participants has produced a change in the mode of participation."

"Quantity has a quality all its own."

"The demon of quantity, who will soon rule the world, is pressing home his attack and fortifying his positions in literary circles as everywhere else."

"“Aristocracy,” … taken in its etymological sense, means precisely the power of the elect. The elect, by the very definition of the word, can only be the few, and their power, or rather their authority, being due to their intellectual superiority, has nothing in common with the numerical strength on which democracy is based, a strength whose inherent tendency is to sacrifice the minority to the majority, and therefore quality to quantity and the elect to the masses."

"It is the same case with all those pretended syllogistical reasonings, which may be found in every other branch of learning, except the sciences of quantity and number; and these may safely, I think, be pronounced the only proper objects of knowledge and demonstration."

"Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion."

"Our culture, obsessed with numbers, has given us the idea that what we can measure is more important than what we can't measure. Think about that for a minute. It means that we make quantity more important than quality."

"We cannot deny the indictment that we seek solution for practically every problem of life in quantitative terms, and are not fully aware of the limits of this approach."

"The United States and the post-Stalinist Soviet Union … share the same cultural aims. Both issue from the assumption that wealth is a superior and adequate substitute for symbolic impoverishment. Both American and Soviet cultures are essentially variants of the same belief in wealth as the functional equivalent of a high civilization. In both cultures, the controlling symbolism has been stripped down to belief in the efficacy of wealth. Quantity has become quality. The answer to all questions of “what for?” is “more.”"

"If the quality of society could be replaced by quantity, it would be worth while to live in the world at large; but unfortunately a hundred fools in a crowd still do not produce one intelligent man."

"That curious modern hypostatization “service” is often called in to substitute for the now incomprehensible doctrine of vocation. It tries to secure subordination by hypothesizing something larger than the self, which turns out, however, to be only a multitude of selfish selves. The familiar change from quality to quantity may again be noted; one serves not the higher part of the self (this entails hierarchy) … but merely consumer demand. And who admires those at the top of a hierarchy of consumption? Man as a consuming animal is thus seen to be not enough."

"Monotony of evil: never anything new, everything about it is equivalent. ... It is because of this monotony that quantity plays so great a part. A host of women (Don Juan) or of men (Célimène), etc."

"The spirit, overcome by the weight of quantity, has no longer any other criterion than efficiency."

"There are at present fundamental problems in theoretical physics awaiting solution, e.g., the relativistic formulation of quantum mechanics and the nature of atomic nuclei (to be followed by more difficult ones such as the problem of life), the solution of which problems will presumably require a more drastic revision of our fundamental concepts than any that have gone before. Quite likely these changes will be so great that it will be beyond the power of human intelligence to get the necessary new ideas by direct attempts to formulate the experimental data in mathematical terms."

"At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? In my opinion the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

"Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about. They do not even need to know what they are talking about, or, as they themselves say, whether what they say is true. I will explain that. You state the axioms, such-and-such is so, and such-and-such is so. What then? The logic can be carried out without knowing what the such-and-such words mean. If the statements about the axioms are carefully formulated and complete enough, it is not necessary for the man who is doing the reasoning to have any knowledge of the meaning of the words in order to deduce new conclusions in the same language. … But the physicist has meaning to all his phrases. That is a very important thing that a lot of people who come to physics by way of mathematics do not appreciate. Physics is not mathematics, and mathematics is not physics. One helps the other. But in physics you have to have an understanding of the connection of words with the real world."

"Admire the power of the Dirac notation!"

"I do not use Dirac’s bra-ket notation, because for some purposes it is awkward […]"

"... mathematics resembles an iceberg: beneath the surface is the realm of pure mathematics hidden from the public ... Above the water is the tip, the visible part of which we call applied mathematics."

"Proof is the idol before whom the pure mathematician tortures himself."

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature."

"There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand."

"Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true … If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate."

"Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth."

"FORTY + TEN + TEN = SIXTY"

"SEND + MORE = MONEY"

"Perhaps no puzzle of the whole collection caused more jollity or was found more entertaining than that produced by the Host of the "Tabard," who accompanied the party all the way. He called the pilgrims together and spoke as follows: "My merry masters all, now that it be my turn to give your brains a twist, I will show ye a little piece of craft that will try your wits to their full bent. And yet methinks it is but a simple matter when the doing of it is made clear. Here be a cask of fine London ale, and in my hands do I hold two measures—one of five pints, and the other of three pints. Pray show how it is possible for me to put a true pint into each of the measures." Of course, no other vessel or article is to be used, and no marking of the measures is allowed. It is a knotty little problem and a fascinating one. A good many persons to-day will find it by no means an easy task. Yet it can be done."

"A wide variety of economic problems lead to differential, difference, and integral equations. Ordinary differential equations appear in models of economic dynamics. Integral equations appear in dynamic programming problems and asset pricing models. Discrete-time dynamic problems lead to difference equations."

"Mathematical modeling is a mixed blessing for economics. Mathematical modeling provides real advantages in terms of precision of thought, in seeing how assumptions are linked to conclusions, in generating and communicating insights, in generalizing propositions, and in exporting knowledge from one context to another. In my opinion, these advantages are monumental, far outweighing the costs. But the costs are not zero. Mathematical modeling limits what can be tackled and what is considered legitimate inquiry. You may decide, with experience, that the sorts of models in this book do not help you understand the economic phenomena that you want to understand."

"I have not been able to lay my hands on any notes as to Mathematico-economics that would be of any use to you. I have very indistinct memories of what I used to think on the subject. I never read mathematics now: in fact I have even forgotten how to integrate a good many things. But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very well unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in (4), burn (3). This last I do often."

"When I see equations, I see the letters in colors – I don’t know why. As I’m talking, I see vague pictures of Bessel functions from Jahnke and Ernde’s book, with light-tan j’s, slightly violet-bluish n’s, and dark brown x’s flying around. And I wonder what the hell it must look like to the students."

"I remember... at a Board meeting at Cambridge, the subject of Bessel's functions came into the discussion... to include them in an examination syllabus. Their utility in connection with Applied Mathematics having been referred to, a very great Pure Mathematician who was present ejaculated—"Yes, Bessel's functions are very beautiful functions, in spite of their having practical applications." It would have been interesting to have heard what this great man would have said if he had known that Professor Perry would one day propose the desecration of these beautiful functions by recommending them as suitable playthings for young boys."

"[To use spherical harmonies or Bessel functions is] to be able to start in mathematics in Cambridge just about the place where some of the best mathematical men now end their studies forever."

"Two years before the publication of his Dioptrics, viz. in 1609, Kepler had given to the world his great work entitled "The New Astronomy, or Commentaries on the Motions of Mars." The discoveries which this volume records form the basis of physical astronomy. The inquiries by which he was led to them began in that memorable year 1601, when he became the colleague or assistant of Tycho. ...Having tried in vain to represent the motion of Mars by an uniform motion in a circular orbit, and by the cycles and epicycles with which Copernicus had endeavoured to explain the planetary inequalities, Kepler was led, after many fruitless speculations, to suppose the orbit of the planet to be oval; and from his knowledge of the conic sections, he afterward determined it to be an ellipse, with the sun placed in one of its foci. He then ascertained the dimensions of the orbit; and, by a comparison of the times employed by the planet to complete a whole revolution or any part of one, he discovered that the time in which Mars describes any arches of his elliptic orbit, were always to one another as the areas contained by lines drawn from the focus, or the centre of the sun, to the extremities of the respective arches; or, in other words, that the radius vector, or the line joining the Sun and Mars described equal areas in equal times. By examining the inequalities of the other planets he found that they all moved in elliptic orbits, and that the radius vector of each described areas proportional to the times. These two great results are known by the name of the first and second laws of Kepler. The third law, or that which relates to the connexion between the periodic times and the distances of the planets, was not discovered till a later period of his life."

"The figure A D B E [in Figure 9] represents the form of a planetary orbit, which is that of an oval or ellipse. The longest diameter is A B; the shorter diameter D E. The two points F and G are called the foci of the ellipse, around which, as two central points, the ellipse is formed. The sun is not placed in C, the center of the orbit, but at F, one of the foci of the ellipse. When the planet, therefore, is at A it is nearest the sun, and is said to be in its perihelion; its distance from the sun gradually increases until it reaches the point, B, when it is at its greatest distance the sun, and is said to be in its aphelion; when it arrives at the points D and E of its orbit, it said to be at the mean distance. ...F D , the mean distance of the planet from the sun... The orbits of some of the planets are more elliptical than others. ...Most of the orbits, except those of some of the new planets, approach very nearly to the circular form."

"Lagrange, struck with the circumstance that the calculus had never given any inequalities but such as were periodical, applied himself to the investigation of a general question, from which he found by a method peculiar to himself and independent of any approximation, that the inequalities produced by the mutual action of the planets must in effect be all periodical; that the periodical changes are confined within narrow limits; that none of the planets ever has been or ever can be a comet moving in a very eccentric orbit; but that the planetary system oscillates as it were round a medium state from which it never deviates far: that amid all the changes which arise from the mutual actions of the planets, two things remain perpetually the same, viz. the length of the greater axis of the ellipse which the planet describes, and its periodical time round the sun; or, which is the same thing, the mean distance of each planet from the sun and its mean motion remain constant. The plane of the orbit varies, the species of the ellipse and its eccentricity change, but never, by any means whatever, the greater axis of the ellipse, or the time of the entire revolution of the planet. The discovery of this great principle, which we may consider as the bulwark that secures the stability of our system, and excludes all access to confusion and disorder, must render the name of Lagrange for ever memorable in science, and ever revered by those who delight in the contemplation of whatever is excellent and sublime. After Newton's discovery of the elliptic orbits of the planets from gravitation, Langrange's discovery of their periodical inequalities is, without doubt, the noblest truth in physical astronomy, and in respect of the doctrine of final causes, it may truly be regarded as the greatest of all."

"The planetary orbits are not exact circles, but are more or less elliptical, the Sun being situated not at the centre, but at one of the foci of the ellipse. The distance of either focus from the centre indicates the eccentricity of the ellipse. On account of this eccentricity the distance of a planet from the Sun, and its velocity of revolution, vary in different parts of its orbit."

"Kepler's observation of the elliptical rotation of the planets was the first of three laws, quantitatively expressed, which paved the way for Newton's law. Why did the planets move in just this way? Kepler tried to answer this also, but failed. It remained for Newton to supply the answer to this question."

"If the earth attracts the moon, why does not the moon fall to the earth? A glance at the accompanying figure will help to answer this question. We must remember that the moon is not stationary, but travelling at tremendous speed—so much so, that it circles the entire earth every month. Now if the earth were absent the path of the moon would be a straight line, say MB, If, however, the earth exerts attraction, the moon would be pulled inward. Instead of following the line MB it would follow the curved path MB. And again, the moon having arrived at B, is prevented from following the line B'C, but rather B'C. So that the path instead of being a straight line tends to become curved. From Kepler's researches the probabilities were that this curve would assume the shape of an ellipse rather than a circle. ...Kepler's observations of the movements of the planets around the sun was of inestimable value; for from these Newton deduced the hypothesis that attraction varies inversely as the square of the distance. Making use of this hypothesis, Newton calculated what the attractive power possessed by the earth must be in order that the moon may continue in its path. He next compared this force with the force exerted by the earth in pulling the apple to the ground, and found the forces to be identical! "I compared," he writes, "the force necessary to keep the moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly!" One and the same force pulls the moon and pulls the apple—the force of gravity. Further, the hypothesis that the force of gravity varies inversely as the square of the distance had now received experimental confirmation."

"Imagine but a single planet revolving about the sun. According to Newton's law of gravitation, the planet's path would be that of an ellipse—that is, oval—and the planet would travel indefinitely along this path. According to Einstein the path would also be elliptical, but before a revolution would be quite completed, the planet would start along a slightly advanced line, forming a new ellipse slightly in advance of the first. The elliptic orbit slowly turns in the direction in which the planet is moving. After many years—centuries—the orbit will be in a different direction. The rapidity of the orbit's change of direction depends on the velocity of the planet. Mercury moving at the rate of 30 miles a second is the fastest among the planets. It has the further advantage over Venus or the earth in that its orbit... is an ellipse, whereas the orbits of Venus and the earth are nearly circular; and how are you going to tell in which direction a circle is pointing? Observation tells us that the orbit of Mercury is advancing at the rate of 574 seconds (of arc) per century. We can calculate how much of this is due to the gravitational influence of other planets. It amounts to 532 seconds per century. What of the remaining 42 seconds? ... This discrepancy between theory and observation remained one of the great puzzles in astronomy until Einstein cleared up the mystery. According to Einstein's theory the mathematics of the situation is simply this: in one revolution of the planet the orbit will advance by a fraction of a revolution equal to three times the square of the ratio of the velocity of the planet to the velocity of light. When we allow mathematicians to work this out we get the figure 43, which is certainly close enough to 42 to be called identical with it."

"According to Newton's law an isolated planet in its motion around a central sun would describe, period after period, the same elliptical orbit; whereas Einstein's laws lead to the prediction that the successive orbits traversed would not be identically the same. Each revolution would start the planet off on an orbit very approximately elliptical, but with the major axis of the ellipse rotated slightly in the plane of the orbit. When calculations were made for the various planets in our solar system, it was found that the only one which was of interest from the standpoint of verification of Einstein's formulas was Mercury."

"Mercury. The inclination and eccentricity of orbit are greater than those of any of the planets. In obedience to Kepler's law the orbit is an ellipse, one of whose foci is at the sun... Venus. The eccentricity of the orbit is so small that in the plot its centre is scarcely distinguishable from the sun. As a consequence the planet revolves at a nearly uniform velocity. ... Mars. While the eccentricity of Mars' orbit is less than that of Mercury's, its linear eccentricity or actual distance from its center... to the sun, is greater."

"Parabola, ellipse, hyperbola, and circle are called conic sections, or simply conics, because they can all be obtained as plane sections of a circular cone. They were originally studied from that point of view, and nearly all their elementary properties that are known to-day were proved by the Greek geometers more than two thousand years ago. The conic sections are of especial interest because of the fact that the paths of all the heavenly bodies are curves of this kind. This fact was first established by the great German astronomer and mathematician Johannes Kepler, in 1609. He showed that the planet Mars moves in an ellipse; the other planets, including of course the earth, do the same, while many comets move in parabolas or hyperbolas."

"Since the ellipse is the most important of all conic sections in the study of the orbits of the planets, we discuss these curves in more detail, noting that the circle is a special case of an ellipse or, put differently, the ellipse is a symmetrical distortion of a circle."

"Just as the geometric properties of a circle are determined by its radius (a single distance), so are all the geometric properties of an ellipse determined by two distances: its semimajor axis a and its semiminor axis b. We say that the circle has a single geometrical degree of freedom, whereas the ellipse has two such degrees of freedom. ...The larger b is for a given value of a, the rounder the ellipse, and the smaller b is for a given value of a, the flatter the ellipse. If b = a, the ellipse is a circle and if b = 0, the ellipse is a straight line. ...its geometrical properties are a kind of average of the geometrical properties of a circle with radius a and a circle of radius b. ...The arc of the ellipse lies between the circumferences of these two circles; it is inscribed in the larger one and circumscribes the smaller one. ...the ellipse is symmetrical with respect to the y and x axes."

"The Ellipse is the most simple of the Conic Sections, most known, and nearest of Kin to a Circle, and easiest describ'd by the Hand in plano. Though many prefer the Parabola before it, for the Simplicity of the Æquation by which it is express'd. But by this Reason the Parabola ought to be preferr'd before the Circle it self, which it never is. Therefore the reasoning from the Simplicity of the Æquation will not hold. The modern Geometers are too fond of the Speculation of Æquations."

"How can any prediction of the return of a comet be made? and, supposing such a prediction to be possible, How can it thence be shown that this comet of 1835, is the same with those seen in 1682, in 1607, in 1531, and in 1456? Before the first of these questions is answered, the reader must be made acquainted with the nature and properties of a certain curve, called by geometers an ellipse."

"If now the point H be brought nearer to S than it was before, the length of the string remaining the same, and the curve be then described as before, it will be found to have altered its form, so as more nearly to approach that of a circle, as shown in the... figure, and H may be brought so near to S that its form shall scarcely be distinguishable from that of a circle, which figure it would indeed manifestly become, if the point H were made accurately to coincide with S."

"A string, of which both extremities are fixed, passes through a heavy ring. Find the position of equilibrium. Let A and B denote the fixed endpoints of the string and X any position of the ring [on the string]. ...The ring must hang as low as possible. (In fact the ring is heavy; it "wants to come as close to the ground, or to the center of the earth, as possible.) Both parts of the inextensible string, AX and BX, are stretched, and so the ring, sliding along the string, describes and ellipse with foci A and B. Obviously, the position of equilibrium is at the lowest point M of the ellipse where the tangent is horizontal."

"The more man inquires into the laws which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an infinite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws."

"It appears... that the elastic theories of light, if Kelvin's gyrostatic adynamic ether be admitted, have not been wholly routed. Nevertheless the great electromagnetic theory of light propounded by Maxwell (1864, 'Treatise,' 1873) has been singularly apt not only in explaining all the phenomena reached by the older theories and in predicting entirely novel results, but in harmoniously uniting as parts of a unique doctrine, both the electric or photographic light vector of Fresnel and Cauchy and the magnetic vector of Neumann and MacCullagh. Its predictions have, moreover, been astonishingly verified by the work of Hertz (1890), and it is to-day acquiring added power in the convection theories of Lorentz (1895) and others."

"At first the mathematical disciplines were not sharply defined. As knowledge increased, individual subjects split off from the parent mass and became autonomous. Later, some were overtaken and reabsorbed in vaster generalizations of the mass from which they sprang. Thus trigonometry issued from surveying, astronomy, and geometry only to be absorbed, centuries later, in the analysis which had generalized geometry. This recurrent escape and recapture has inspired some to dream of a final, unified mathematics which shall embrace all. Early in the twentieth century it was believed by some for a time that the desired unification had been achieved in mathematical logic. But mathematics, too irrepressibly creative to be restrained by any formalism, escaped."

"Whatever its source, mathematics has come down to the present by the two main streams of number and form. The first carried along arithmetic and algebra, the second, geometry. In the seventeenth century these two united, forming the ever-broadening river of mathematical analysis."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Science is an attempt to represent the known world as a closed system with a perfect formalism. Scientific discovery is a constant maverick process of breaking out at the ends of the system... and then hastily closing it... The act of the imagination is the opening of the system so that it shows new connections. ...every act of imagination is the discovery of likenesses between two things which were thought unlike. ...they introduce new likenesses, whether it is Shakespeare... or Newton saying that the moon in essence is exactly like a thrown apple."

"Up to this point mathematics alone appeared to Descartes worthy of being called a science. ...in order to establish the science or philosophy sought by Descartes, it was sufficient to find a method that should be to philosophy what the method of mathematical deduction is to arithmetic, algebra and geometry. ...How could one pass from these processes, which are especially adapted to particular sciences, to the general method required by general science or philosophy? Descartes would undoubtedly never have conceived such an audacious hope, had not a great discovery of his set him on this track. He had invented analytical geometry... In this way, Descartes substituted for the old methods, which were especially adapted to algebra and geometry as distinct branches, a general method, applicable to what he called the "universal mathematical science," viz., to the study of "the various ratios or proportions to be found between the objects of the mathematical sciences, hitherto regarded as distinct." Not only did this discovery mark a decisive epoch in the history of mathematics, which it provided with an instrument of incomparable simplicity and power, but it furthermore gave Descartes a right to hope for the philosophical method he was seeking. Ought not a last generalization to be possible, by means of which the method he had so happily discovered should become applicable, not only to the "universal mathematical science," but also to the systematic combination of all the truths which our finite minds may permit us to attain?"

"[A]s the great extreme of dimension is sublime, so the last extreme of littleness is in the same measure sublime... when we attend to the infinite divisibility of matter, when we pursue animal life into these excessively small, and yet organized beings... when we push our discoveries yet downward... in tracing which the imagination is lost as well as the sense; we become amazed and confounded at the wonders of minuteness; nor can we distinguish in its effects this extreme of littleness from the vast itself. For division must be infinite as well as addition; because the idea of a perfect unity can no more be arrived at, than that of an complete whole, to which nothing can be added."

"Edmund Burke, ' (1757) p. 81 of the 1898 edition."

"Copernicus had taken one course in treating the earth as virtually a celestial body in the Aristotelian sense—a perfect sphere governed by the laws which operated in the higher reaches of the skies. Galileo complemented this by taking now the opposite course—rather treating the heavenly bodies as terrestrial ones, regarding the planets as subject to the very laws which applied to balls sliding down inclined planes. There was something in all this which tended to the reduction of the whole universe to uniform physical laws, and it is clear that the world was coming to be more ready to admit such a view."

"[T]he attempt to embrace the whole course of things in time and to relate the successive epochs to one another—the transition to the view that time is actually aiming at something, that temporal succession has meaning and that the passage of ages is generative—was greatly influenced by the fact that the survey became wider than that of human history, that the mind gradually came to see geology, pre-history and history in due succession to one another. The new science and the history joined hands and each acquired a new power as a result of their mutual reinforcement. The idea of progress itself gained additional implications when there gradually emerged a wider idea of evolution."

"Let us assert, as our original postulate, that, the multiple (that is, non-being, if taken in the pure state) being the only rational form of a creatable (creabile) nothingness, the creative act is comprehensible only as a gradual process of arrangement and unification, which amounts to accepting that to create is to unite. And, indeed, there is nothing to prevent our holding that union creates. To the objection that union presupposes already existing elements, I shall answer that physics has just shown us (in the case of mass) that experientially (and for all the protests of "common sense") the moving object exists only as the product of its motion."

"The scientific spirit must then lose its present tendency to speciality, and be impelled towards a logical generality; for all the branches of natural philosophy must furnish their contingent to the common doctrine; in order to which they must first be completely condensed and co-ordinated. When the savans have learned that active life requires the habitual and simultaneous use of the various positive ideas that each of them isolates from all the rest, they will perceive that their social ascendency supposes the prior generalization of their common conceptions, and consequently the entire philosophical reformation of their present practice. Even in the most advanced sciences... the scientific character at present fluctuates between the abstract expansion and the partial application, so as to be usually neither thoroughly speculative nor completely active; a consequence of the same defect of generality which rests the ultimate utility of the positive spirit on minor services, which are as special as the corresponding theoretical habits. But this view, which ought to have been long outgrown, is a mere hindrance in the way of the true conception,—that positive philosophy contemplates no other immediate application than the intellectual and moral direction of civilized society; a necessary application, presenting nothing that is incidental or desultory, and imparting the utmost generality, elevation, unity, and consistency, to the speculative character. Under such a homogeneousness of view and identity of aim, the various positive philosophers will naturally and gradually constitute a European body, in which the dissensions that now break up the scientific world into coteries will merge; and with the rivalries of struggling interests will cease the quarrels and coalitions which are the opprobrium of science in our day."

"Prior to Newton, mathematics, chiefly in the form of geometry, had been studied as a fine art without any view to its physical applications other than in very trivial cases. But with Newton all the resources of mathematics were turned to advantage in the solution of physical problems. Thenceforth mathematics appeared as an instrument of discovery, the most powerful one known to man, multiplying the power of thought... It is this application of mathematics to the solution of physical problems, this combination of two separate fields of investigation, which constitutes the essential characteristic of the Newtonian method. Thus problems of physics were metamorphosed into problems of mathematics. ...Newton showed the mark of genius by inventing the integral calculus. As a result... problems which would have baffled Archimedes were solved with ease. ...this new departure in scientific method led to the discovery of the law of gravitation. ...the real significance in Newton's achievement lay ...in his having established the presence of law and order at least in one realm of nature ...the motions of the heavenly bodies. Nature thus exhibited rationality and was not mere blind chaos and uncertainty."

"Newton, in his application of mathematics to physics, had been concerned only with... planetary motions, mechanics, propagation of sound, etc. But when it came to applying the mathematical method to the more intricate physical problems, a considerable advance was necessary... both mathematical and empirical. Thanks to the gradual accumulation of physical data, and... to the efforts of Newton's great successors in the field of pure mathematics (Euler, Lagrange, Laplace), conditions were ripe in the first half of the nineteenth century for a systematic attack on many of nature's secrets. The mathematical theories constructed were known under the general name of theories of mathematical physics. ...they had their prototype in Newton's celestial mechanics. ...they dealt with a wide variety of physical phenomena (electric, hydrostatic, etc.) ...The most celebrated of these theories (such as those of Maxwell, Boltzmann, Lorentz and Planck) were concerned with very special classes of phenomena. But with Einstein's theory of relativity... the scope of our investigations is so widened that we are appreciably nearer than ever before to the ideal of a single mathematical theory embracing all of physical knowledge."

"The equations of gravitation... signify that whenever we recognise the existence of one of these physical magnitudes it is always accompanied by corresponding curvatures of space-time. It is usual to assume that the curvatures are produced by those concrete somethings which we call mass, momentum, energy, pressure. In this way, we must concede a duality to nature; there would exist both matter and space-time, or, better still, matter and the metrical field of space-time. Einstein... attempted to remove this duality by proving that it was possible to attribute the entire existence of the metrical field, hence of space-time, to the presence of matter. This attitude led to a matter-moulding conception of the universe... And... only when this attitude was adhered to could Mach's belief in the relativity of all motion be accepted. Eddington's attitude is just the reverse. He prefers to assume that the equations of gravitation are not equations in the ordinary sense of something being equal to something else. In his opinion they are identities. They merely tell us how our senses will recognize the existence of certain curvatures of space-time by interpreting them as matter, motion, and so on. In other words, there is no matter; there is nothing but a variable curvature of space-time. Matter, momentum, vis viva, are the names we give to those curvatures on account of the varying ways they affect our senses."

"Passing to the laws of motion, we remember that there is but one law: All free bodies (when reduced to point-masses) follow time-geodesics in space-time regardless of whether space-time be flat, as it is (at least approximately) in interstellar space, or whether it be curved by the presence of matter. If space-time is flat, the geodesics are straight and the bodies describe straight courses with constant speeds as referred to a Galilean frame. Thus Newton's law of inertia is seen to express the flatness of space-time. When space-time, and hence its geodesics, are curved by the presence of matter, the courses of free bodies appear to be curved, or else their motion to be accelerated. But whereas, under those conditions, the law of inertia was at fault in classical science, and an additional gravitational influence had to be introduced, in Einstein's theory the general law of geodesic motion still holds good. Inasmuch as the structure of space-time determines the laws of our geometry, the beatings of natural clocks (atoms) and the motion of free bodies, we see that the theory has brought about a fusion between geometry and physics."

"The age-old conflict between our notions of continuity and the scientific concept of number ended in a decisive victory for that latter. This victory was brought about by the necessity of vindicating, of legitimizing... a procedure which ever since the days of Fermat and Descartes had been an indispensable tool of analysis. ...analytic geometry ...this discipline which was born of the endeavors to subject problems of geometry to arithmetical analysis, ended by becoming the vehicle through which the abstract properties of number are transmitted to the mind. It furnished analysis with a rich, picturesque language and directed it into channels of generalization hitherto unthought of. Now, the tacit assumption on which analytic geometry operated was that it was possible to represent the points on a line, and therefore points in a plane and in space, by means of numbers. ...The great success of analytic geometry... gave this assumption an irresistible pragmatic force. ...Under such circumstances mathematics proceeds by fiat. It bridges the chasm between intuition and reason by a convenient postulate. On the one hand, there was the logically consistent concept of real number and its aggregate, the arithmetic continuum; on the other, the vague notions of the point and its aggregate, the linear continuum. All that was necessary was to declare the identity of the two, or, what amounted to the same thing, to assert that: It is possible to assign to any point on a line a unique real number, and, conversely, any real number can be represented in a unique manner by a point on a line. This is the famous Dedekind-Cantor axiom."

"[W]ith a view to summon myself to the search for a science of mathematics in general, I asked myself... what precisely was the meaning of this word mathematics, and why arithmetic and geometry only, and not also astronomy, music, optics, mechanics, and so many other sciences, should be considered as forming a part of it; for it is not enough here to know the etymology of the word. In reality the word mathematics meaning nothing but science, those which I have just named have as much right as geometry to be called mathematics; and nevertheless there is no one, however little instructed, who cannot distinguish at once what belongs to mathematics... from what belongs to the other sciences. But... all the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics... And a proof that it surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides; and consequently, if it contain any difficulties, these exist in the rest, which have themselves the peculiar ones arising from their particular subject-matter, and which do not exist for the general science."

"I think that the correct connection between quantum theory and relativity has not yet been discovered. ...I think that the present methods which theoretical physicists are using are not the correct methods. They use... a renormalization technique which involves handling infinite quantities, and this is not really a mathematically logical process. ...[I]t is just a set of working rules rather than a correct mathematical theory and I don't like this whole development at all. I think that some other important discoveries will have to be made before these questions are put into order. In particular, there is the problem of explaining the , the number 137, which plays an important role in physics, and the question is, why should it be 137 instead of some other number. That has not been explained at all, and I feel that it is necessary to get an explanation of that before one would make an important advance in understanding atomic theory. ...There is quite a different problem with the ratio of the mass of the proton to the mass of the electron, and the question is whether the ratio of these masses remains constant or whether it develops slowly with time."

"You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. And this made me profoundly skeptical. I knew how many great scientists had chased this will-o’-the-wisp of a unified theory. The ground of science was littered with the corpses of dead unified theories. Even Einstein had spent twenty years searching for a unified theory and had found nothing that satisfied him. I admired Dick tremendously, but I did not believe he could beat Einstein at his own game."

"From the present state of theory it looks as if the electromagnetic field, as opposed to the gravitational field, rests upon an entirely new formal motif, as though nature might just as well have endowed the gravitational ether with fields of quite another type, for example, with fields of a scalar potential, instead of fields of the electromagnetic type. Since according to our present conceptions the elementary particles of matter are also, in their essence, nothing else than condensations of the electromagnetic field, our present view of the universe presents two realities which are completely separated from each other conceptually, although connected causally, namely, gravitational ether and electromagnetic field, or — as they might also be called — space and matter. Of course it would be a great advance if we could succeed in comprehending the gravitational field and the electromagnetic field together as one unified conformation. Then, for the first time, the epoch of theoretical physics founded by Faraday and Maxwell would reach a satisfactory conclusion. The contrast between ether and matter would fade away, and, through the general theory of relativity, the whole of physics would become a complete system of thought, like geometry, kinematics, and the theory of gravitation."

"The basic concepts and laws which are not logically further reducible constitute the indispensable and not rationally deducible part of the theory. ...The conception... of the purely fictitious character of the basic principles of theory was in the eighteenth and nineteenth centuries still far from being the prevailing one. But it continues to gain more and more ground because of the ever-widening logical gap between the basic concepts and laws on the one side and the consequences to be correlated with our experiences on the other—a gap which widens progressively with the developing unification of the logical structure, that is with the reduction in the number of the logically independent conceptual elements required for the basis of the whole system."

"Although it is true that it is the goal of science to discover rules which permit the association and foretelling of facts, this is not its only aim. It also seeks to reduce the connections discovered to the smallest possible number of mutually independent conceptual elements. It is in this striving after the rational unification of the manifold that it encounters its greatest successes, even though it is precisely this attempt which causes it to run the greatest risk of falling a prey to illusions."

"We have to realize that a unified theory of the physical world simply does not exist. We have theories that work in restricted regions, we have purely formal attempts to condense them into a single formula, we have lots of unfounded claims (such as the claim that all of chemistry can be reduced to physics), phenomena that do not fit into the accepted framework are suppressed; in physics, which many scientists regard as the one really basic science, we have now at least three different points of view (relativity, dealing with the very large, quantum theory for an intermediate domain and various particle models for the very small) without a promise of conceptual (and not only formal) unification; perceptions are outside of the material universe (the mind-body problem is still unsolved) - from the very beginning the salesman of a universal truth cheated people into admissions instead of clearly arguing for their philosophy. And let us not forget that it was they and not the representatives of the traditions they attacked who introduced argument as the one and only universal arbiter. They praised argument - they constantly violated its principles."

"People are always asking for the latest developments in the unification of this theory with that theory, and they don't give us a chance to tell them anything about what we know pretty well. They always want to know the things we don't know."

"Unlike the chess game... in which the rules become more complicated as you go along, in physics, when you discover new things, it looks more simple. It appears on the whole to be more complicated because we learn about a greater experience—that is, we learn more about more particles and new things—and so the laws look more complicated again. But if you realize all the time what's kind of wonderful—that is, if we expand our experience into wilder and wilder regions of experience—every once in a while we have these integrations when everything's pulled together into a unification, in which it turns out to be simpler than it was before."

"In some ways, science today is less specialized... Consider... physics and chemistry; fifty years ago they were regarded as separate fields. ...Philosophers even gave an "intelligible" reason why physics and chemistry would always be separate... Physics had to do with quantity, chemistry with quality. Then there developed the field of , later the field of . Today it would be difficult to say what the difference is between physics and chemistry... now the laws of chemistry are derived from physics, from thermodynamics, electrodynamics, and from quantum mechanics. ...The same exists between physics and biology, or between economics and anthropology. ...Today we must understand economics as a tribal custom, and tribal customs from the economic point of view. ...The disappearance of the old unity between science and philosophy can hardly be ascribed to the increasing specialization in science."

"The decisive steps toward a clear understanding of non-Euclidean geometry were taken by Riemann, Helmholtz, and Poincaré, who recognized the essential unity of geometry and physics. However, the understanding did not come into its own until Einstein showed that such a combination of geometry and physics was really necessary for the derivation of phenomena which had actually been observed."

"Our understanding of the four basic concepts of Physics—space, time, matter and force—has undergone radical change in the course of work on unification, starting with Maxwell's unification of electricity with magnetism, all the way to present day string theory. What started as four independent concepts, with space and time postulated and the possible forms of matter and force arbitrarily chosen, now appear as different aspects of a rich and novel dynamically determined structure."

"In general the position as regards all such new calculi is this — That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able — without the unconscious inspiration of genius which no one can command — to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"Geoffrey also recognized that the opposite orientations of gut and nervous system posed a problem for his claim that insects and vertebrates represent different versions of the same archetypal animal - and he proposed the first account of the inversion theory to resolve this threat to unification. ...Geoffroy pursued the... aim of establishing a "unity of type" that could generate both s and vertebrates from the same basic blueprint. ...The single grand design includes a gut in the middle and the main nerve cords somewhere on the periphery."

"[A]fter a close study of the experimental work of Michael Faraday,... James Clerk Maxwell succeeded in uniting electricity and magnetism in the framework of the '. ...Beyond uniting... all... electric and magnetic phenomena in one mathematical framework, Maxwell's theory showed—quite unexpectedly, that electromagnetic disturbances travel at a fixed and never-changing speed that turns out to equal that of light. From this, Maxwell realized that visible light itself is nothing but a particular kind of electromagnetic wave... Maxwell's theory also showed that all electromagnetic waves—visible light among them—are the epitome of the peripatetic traveler. They never stop. They never slow down. Light always travels at light speed."

"The notion of a smooth spatial geometry, the central principle of general relativity, is destroyed by the violent fluctuations of the quantum world on short distance scales. ...The equations of general relativity cannot handle the rolling frenzy of the quantum foam. ...There are ...physicists ...who are deeply unsettled by the fact that the two foundational pillars of physics as we know it are at their core fundamentally incompatible, regardless of the ultramicroscopic distances that must be probed to expose the problem. This incompatibility, they argue, points to an essential flaw in our understanding of the physical universe. This opinion rests on an unprovable but profoundly felt view that the universe, if understood at its deepest and most elementary level, can be described by a logically sound theory whose parts are harmoniously united. Physicists have made numerous attempts at modifying either general relativity or quantum mechanics in some manner so as to avoid the conflict, but the attempts... have been met with failure after failure. That is, until the discovery of superstring theory."

"In a paper he sent to Einstein in 1919, Kaluza made an astounding suggestion. He proposed that the spatial fabric of the universe might possess more than the three dimensions... it provided an elegant and compelling framework for weaving together Einstein's general relativity and Maxwell's electromagnetic theory into a single, unified conceptual framework. ...implicit in Kaluza's work and subsequently made explicit and refined by... Oskar Klein in 1926... the spatial fabric of our universe may have both extended and curled-up dimensions. ... Einstein had formulated general relativity in the familiar setting of a universe with three spatial dimensions and one time dimension. The mathematical formalism... however, could be extended fairly directly to write down analogous equations for a universe with additional space dimensions. Under the "modest" assumption of one additional space dimension, Kaluza... derived the new equations. ...Kaluza found extra equations... those Maxwell had written down in the 1880s for deriving the electromagnetic force! ...Kaluza had united Einstein's theory of gravity with Maxwell's theory of light."

"Although the first principles of a science are the first in logical order, they are generally the last in order of discovery. They are arrived at by generalisations of extended experience. They mark the attainment of true scientific inductions, and manifest their correctness by the explanations they are able to afford. They enable us to discern the coherence of large classes of facts, and give us the power to forecast a line of sequences whereby we may direct them to the accomplishment of desired ends, or shape our actions to those coming events which are beyond our control. As an instrument of discovery, first principles are of very little value, and on account of the many chances of error, and of the fascination which the idea of a completed system exercises over the imagination of great minds, the search after them has been fruitful of error. The present undertaking, therefore, is to be regarded not as an attack upon the evolutionism of Lamarck, nor as an attack upon the evolutionism of Lyell or Darwin, nor yet upon the evolutionism of Spencer as regards the development of intelligence, but as an attack upon the theory which attempts to combine all these into one continuous process."

"All knowledge... is unification of the multiple."

"[I]n the nineteenth century, even the could be reduced to mechanics by the assumption that heat really consists of a complicated statistical motion of the smallest parts of matter. By combining the concepts of the mathematical theory of probability with the concepts of Newtonian mechanics Clausius, Gibbs and Boltzmann were able to show that the fundamental laws in the theory of heat could be interpreted as statistical laws following from Newton's mechanics when applied to very complicated mechanical systems."

"Every kind of science, if it has only reached a certain degree of maturity, automatically becomes a part of mathematics."

"Science attempts to confront the possible with the actual. The price to be paid for this outlook, however, turned out to be high. It was... renouncing a unified world view. ...Most other systems of explanation—mythic, magic, or religious—generally encompass everything. They apply to every domain. They answer any possible question. They account for the origin, the present, and the end of the universe. Science proceeds differently. It operates by detailed experimentation... it looks for partial and provisional answers about those phenomena that can be isolated and well defined. ...the beginning of modern science can be dated from the time when such general questions as, "How was the universe created? What is matter made of? What is the essence of life?" were replaced by such limited questions as "How does a stone fall? how does water flow in a tube? How does blood circulate in vessels?" ...While asking general questions led to limited answers, asking limited questions turned out to provide more and more general answers."

"In the history of sciences, important advances often come from... the recognition that two hitherto separate observations can be viewed from a new angle and seen to represent nothing but different facets of one phenomenon. Thus, terrestrial and celestial mechanisms became a single science with Newton's laws. Thermodynamics and mechanics were unified through statistical mechanics, as were optics and electromagnetism through Maxwell's theory of magnetic field, or chemistry and atomic physics through quantum mechanics. Similarly different combinations of the same atoms, obeying the same laws, were shown by biochemists to compose both the inanimate and animate worlds. ... Despite such generalizations, however, large gaps remain... Following the line from physics to sociology, one goes from simpler to the more complex objects... from the poorer to the richer empirical content, as well as from the harder to the softer system of hypotheses and experimentation. ...Because of the hierarchy of objects, the problem is always to explain the more complex in terms and concepts applying to the simpler. This is the old problem of reduction, emergence, whole and parts... an understanding of the simple is necessary to understand the more complex, but whether it is sufficient is questionable. ...the appearance of life and later of thought and language—led to phenomena that previously did not exist... To describe and to interpret these phenomena new concepts, meaningless at the previous level, are required. ...At the limit total reductionism results in absurdity. ...explaining democracy in terms of the structure and properties of elementary particles... is clearly nonsense."

"But even the distant reader must allow that Clifford's mental personality belonged to the highest possible type to say no more. The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal. And if in these modern days we are to look for any prophet or saviour who shall influence our feelings towards the universe as the founders and renewers of past religions have influenced the minds of our fathers, that prophet, if he ever come, must, like Clifford, be no mere sentimental worshipper of science, but an expert in her ways. And he must have what Clifford had in so extraordinary a degree—that lavishly generous confidence in the worthiness of average human nature to be told all truth, the lack of which in Goethe made him an inspiration to the few but a cold riddle to the many."

"Reduced to their most pregnant difference, empiricism means the habit of explaining wholes by parts, and rationalism means the habit of explaining parts by wholes. Rationalism thus preserves affinities with monism, since wholeness goes with union, while empiricism inclines to pluralistic views. No philosophy can ever be anything but a summary sketch, a picture of the world in abridgment, a foreshortened bird's-eye view of the perspective of events."

"Giacomo Rizzolatti... calls these "mirror neurons" and suggests that they provide the first insight into imitation, identification, empathy, and possibly the ability to mime vocalization—the mental processes intrinsic to human interaction. Vilayanur Ramachandran has found evidence of comparable neurons in the premotor cortex of people. ...one can see a whole new area of biology opening up... that can give us a sense of what makes us social, communicating beings. An ambitious undertaking of this sort might... teach us something about the factors that give rise to tribalism, which is so often associated with fear, hatred, and intolerance of outsiders."

"The remarkable insight that characterized Klimpt's later work was contemporaneous with Freud's psychological studies and presaged the inward turn that would pervade all fields of inquiry in Vienna in 1900. This period... was characterized by the attempt to make a sharp break with the past and to explore new forms of expression in art, architecture, psychology, literature, and music. It spawned an ongoing pursuit to link these disciplines. ...Viennese life at the turn of the century provided opportunities in salons and coffeehouses for scientists, writers, and artists to come together in an atmosphere that was at once inspiring, optimistic, and politically engaged. ...science was no longer the narrow and restrictive province of scientists but had become an integral part of Viennese culture. ...a paradigm for how an open dialogue can be achieved."

"Galileo and Newton swept away the last traces of mysticism and superstition that had always been associated with the heavens. The heliocentric theory of Copernicus and Kepler had classed the earth among the other planets, so that there was good reason to believe that the heavens were made of the stuff of earth rather than, as Greek and medieval philosophers had maintained, of some light, perfect, indestructible substance. But the heliocentric theory... was regarded by many as a mathematical contrivance... not physically true. Moreover... the heliocentric theory created difficulties in accounting for the phenomena of motion readily observed here on earth, and hence encountered legitimate objections. The work of Galileo and Newton resolved these difficulties... and incorporated the theory of the heavenly motions in the very same physical theory that treated motions on earth. There could be no doubt now... that the substance of the other planets could be identified with the rock and clay beneath man's feet, for this affirmation is the very essence of the law of gravitation. The identification... wiped out libraries of speculation and dogma..."

"The mathematician and versatile scientist Pierre L. M. de Maupertuis, a keen student if Newton's work on gravitation, made the next decisive step. Like Euler, Maupertuis studied under John Bernoulli. ...After having worked in the theory of light and gravitation, he announced, in 1744, a new minimum principle, the Principle of Least Action, from which he claimed he could deduce the behavior of light and masses in motion. The principle asserts that nature always behaves so as to minimize an integral known technically as action, and amounting to the integral of the product of mass, velocity, and distance traversed by a moving object. From this principle he deduced the Newtonian laws of motion. With sometimes suitable and sometimes questionable interpretation of the quantities involved, Maupertuis managed to show that optical phenomena, too, could be deduced from this principle. Hence, to an extent at least, he succeeded in uniting the optics of the eighteenth century and mechanical phenomena."

"To the scientists of 1850, Hamilton's principle was the realization of a dream. ...from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. ...they made striking progress ...But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics."

"The decisive step leading to the construction of precise and verifiable scientific theories in place of vague and largely speculative accounts was the involvement of mathematics. This step was made by the Pythagoreans. ...In their philosophy of nature the Pythagoreans began with the principle that number is the essence of all substance. ...forms reduced to numbers. Since number is the essence of any object, the explanation of natural phenomena could be achieved only through number. ...Whether by a lucky stroke or by intuitive genius the Pythagoreans did hit upon two doctrines which later proved to be all important. The first is that nature is built in accordance with mathematical principles, and the second that number relationships reveal the order in nature. They underlie and unify the seeming diversity exhibited by nature."

"To define distance in their non-Euclidean geometries, Cayley and Klein proceeded by analogy with a discovery of Laguerre... who had shown that the distances and angles of ordinary Euclidean geometry can be expressed as cross ratios, in other words, that the Euclidean metric geometry is clearly a specialization of projective geometry. The concept of the "absolute" and the definition of distance unified Euclidean and non-Euclidean geometries into a single all-embracing theory."

"Klein showed that the Reimannian species of non-Euclidean geometry can be developed in a fashion completely analogous to the Lobachevskian type by choosing an "imaginary" absolute, that is, an "imaginary" point pair or conic, and an imaginary value of the constant k. Euclidean geometry can also be treated in the same way by choosing a "degenerate" point pair or conic."

"As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics."

"Group theory is the mathematical language of symmetry, and it... seems to play a fundamental role in the very structure of nature. ...In the midst of the fomenting of the new twentieth century physics was the... life of the greatest female mathematician who ever lived, Emmy Noether. ...At Göttingen, Noether achieved fame for her research into the fundamental structure of mathematics. However, she stepped briefly into the realm of theoretical physics... is a profound statement, perhaps running as deeply into the fabric of our psyche as the famous theorem of Pythagoras. Noether's theorem directly connects symmetry to physics, and vice versa. It frames our modern concepts about nature and rules modern scientific methodology. ...For scientists it is the guiding light to unraveling nature's mysteries, as they delve into the innermost fabric of matter ...To this task scientists apply ...the great s ...Emmy Noether's work interweaves our understanding of nature—through physics and mathematics—with the beauty and harmony that surrounds us... Noether's theorem provides a natural centerpiece for any discussion that unifies physics and mathematics, such as in the teaching of these... in a way that enlivens them both."

"I feel that controversies can never be finished, nor silence imposed upon the Sects, unless we give up complicated reasonings in favour of simple calculations, words of vague and uncertain meaning in favour of fixed symbols [characteres]. Thus it will appear that 'every paralogism is nothing but an error of calculation. When controversies arise, there will be no more necessity for disputation between two philosophers than between two accountants. Nothing will be needed but that they should take pen in hand, sit down with their counting-tables and (having summoned a friend, if they like) say to one another: Let us calculate.'"

"Theoretical physicists today have some ideas on how to combine the strong and electroweak forces, and hope eventually to include gravity in a single unified theory of all forces. The very meaning of "unification" in this context is that one harmonizing theoretical structure, mathematical in form, should be made to accommodate formerly distinct theories under one roof. Unification is the theme, the backbone of modern physics, and for most physicists what unification means in practice is the uncovering of tidier, more comprehensive mathematical structures linking separate phenomena."

"During the 1970s Sheldon Glashow, Abdus Salam, and Steven Weinberg... came up with a theory in which there have to be three separate particles carrying the weak force... the three weak carriers are recognized as heavy photons and the weak force is seen as a modified form of electromagnetism. The obvious differences between... interactions are ascribed to the fact that the photon has no mass and the weak carriers have a lot. ... The role of the electroweak theory in the effort to streamline fundamental physics is in some respects debatable. More [three] particles are needed and the mathematical structure... is not entirely beyond reproach. But it ties two separate forces into a single theoretical device... In 1983, physicists... found the W and Z particles. The electroweak theory was thereupon deemed correct, and the number of distinct forces in the world was officially reduced to three—electroweak, strong, and gravitational. Electroweak unification has organized another corner of theoretical physics, as did the quark model before it."

"Functions are the bread and butter of modern scientists, statisticians, and economists. Once many repeated... experiments and observations produce the same functional interrelationships, those may acquire the... status of laws of nature—mathematical descriptions... Descartes' ideas... opened the door for a systematic mathematization of everything—the very essence of the notion that God is a mathematician. ...[B]y establishing the equivalence of two perspectives of mathematics (algebraic and geometric) previously considered disjoint, Descartes expanded the horizons of mathematics and paved the way to the modern era of analysis, which allows [us] to comfortably cross from one mathematical discipline to another."

"While Descartes' theory of vortices was spectacularly wrong (as Newton ruthlessly pointed out later), it was still interesting, being the first serious attempt to formulate a theory of the universe as a whole based upon the same laws that apply on the Earth's surface. In other words, to Descartes there was no difference between terrestrial and celestial phenomena—the Earth was part of a universe that obeyed uniform physical laws."

"In the late 1960s... Steven Weinberg, Sheldon Glashow, and Abdus Salam developed a theory that treats the electromagnetic force and the weak nuclear force in a unified manner. ...the electroweak theory, predicted the existence of three particles (...W+, W-, and Z bosons) that had never before been observed. The particles were unambiguously detected in 1983 in accelerator experiments... led by... Carlo Rubbia and Simon van der Meer."

"The more sluggish positive charges are at first of less interest,—but the behaviour of electrons cannot be fully and properly understood without a knowledge of the nature and properties of the positive constituent too. According to Larmor, positive charge must be the mirror-image of negative charge, in essential constitution. The positive electron has not, as far as I know, been as yet observed free. Some think it cannot exist in a free state, that it is in fact the rest of the atom of matter from which a negative unit charge has been removed; or, to put it crudely, that "electricity" repels "electricity," and "matter" repels "matter," but that Electricity and Matter in combination form a neutral substance which is the atom of matter as we know it. Such a statement is an extraordinary and striking return to the views expressed by that great genius, Benjamin Franklin."

"My purpose is merely to illustrate the issue involved in our question about the unification of science. A complete unification... would be a unification downward, finding its ultimate and universal laws in mechanics; and it would include in its scope all the movements of human bodies. Those who assert the possibility of a rigorously complete unification thus imply a denial of all physical efficacy to thoughts and feelings as such. Those, on the contrary, who assert such efficacy deny by implication the possibility of a complete unification of even the laws of the motion of matter. They tacitly or explicitly introduce a real discontinuity into the fabric of science."

"The significance of the contention that the laws of the several sciences are discontinuous appears chiefly when you thus regard the sciences as corresponding to stages in the process of evolution. ...that means that at certain points in the evolutionary sequence matter begins to behave in essentially new ways, develops novel properties and methods of action which were in no true sense contained in or implied by its earlier characteristics and performances. If, on the other hand, all the laws of biology and chemistry are ultimately reducible to, and deducible from, the laws of some fundamental branch of physics, that means that, in a very thorough-going sense, the first morning of creation wrote what the last dawn of reckoning shall read."

"Heraclitus. ...change and incessant movement is the basis, and the only basis, of all things and that what is illusory is the idea of a central, or indeed of any other, unity: the Universe is a stream of incessant and infinitely minute changes. The Atomists. From this springs naturally the atomistic theory of Leucippus and Democritus. This theory is an endeavour to give a sort of solidity and reality to the mutability of Heraclitus, whilst retaining his controversial advantages in the denial of an all-embracing One. The veritable original of things is taken by these Atomists to be, not one, but innumerable, indefinitely minute, homogeneous atoms, the mere mechanical combination of which makes up the variety of nature."

"Nature does not begin with elements, as we are obliged to begin with them. It is certainly fortunate... that we can... turn aside our eyes from the over powering unity of the All, and allow them to rest on individual details. But we should not omit, ultimately, to complete and correct our views by a thorough consideration of the things which for the time being we left out of account."

"Velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of 'MM'. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

"The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws."

"The object of the book is philosophical, in the sense now accepted by many and by divergent schools—i.e., it desires to contribute something towards a unification of thought."

"In both Newton's theory and Maxwell's theory the unification consists, partly, in showing that two different processes or phenomena can be identified, in some way, with each other—that they belong to the same class or are the same kind of thing. Celestial and terrestrial objects are both subject to the same gravitational-force law, and optical and electromagnetic processes are one and the same. Because each of these theories unifies such a diverse range of phenomena, they have traditionally been thought to possess a great deal of explanatory power."

"In whatever they focus on, physicists seek the simplicity in complexity and the unity in diversity. Like philosophers, their intellectual siblings, they are driven by the conviction that the universe is within the human power to understand and that if you look beneath its variety and intricacy, you will find comprehensible rules."

"Since the ancients made great account of the science of Mechanics in the investigation of natural things; and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phænomena of nature to the laws of mathematics; I have in this treatise cultivated Mathematics... The ancients considered Mechanics in a twofold respect; as rational, which proceeds accurately by demonstration, and practical. To practical Mechanics all the manual arts belong, from which Mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that Mechanics is so distinguished from Geometry, that what is perfectly accurate is called Geometrical, what is less so is called Mechanical. But the errors are not in the art, but in the artificers. ...the description of right lines and circles, upon which Geometry is founded, belongs to Mechanics. ...To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from Mechanics; and by Geometry the use of them, when so solved, is shewn. And it is the glory of Geometry that from those few principles, fetched from without, it is able to produce so many things. Therefore Geometry is founded in mechanical practice, and is nothing but that part of universal Mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that Geometry is commonly referred to their magnitudes, and Mechanics to their motion. In this sense Rational Mechanics will be the science of motions resulting from any forces whatsoever and of the forces required to produce any motions, accurately proposed and demonstrated. ...we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces whether attractive or impulsive. And therefore we offer this work as mathematical principles of philosophy. For all the difficulty of philosophy seems to consist in this, from the phenomena of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena."

"The most immediate result of this unbalanced specialisation has been that to-day, when there are more "scientists" than ever, there are much less "cultured" men than, for example, about 1750. And the worst is that with these turnspits of science not even the real progress of science itself is assured. For science needs from time to time, as a necessary regulator of its own advance, a labour of reconstitution, and, as I have said, this demands an effort towards unification, which grows more and more difficult, involving, as it does, ever-vaster regions of the world of knowledge. Newton was able to found his system of physics without knowing much philosophy, but Einstein needed to saturate himself with Kant and Mach before he could reach his own keen synthesis."

"In early physical systems we have optics dealing with phenomena perceived by the eye; acoustics treating of auditory percepts, and so on. The subjective concepts of "tone" and "colour" have now been replaced by the objectified concepts of frequency of vibration; and wave-length. The object of this process of elimination is, according to Planck, the striving towards a unification of the whole theoretical system, so that it shall be equally significant for all intelligent beings."

"The unification of knowledge is the natural consequence of the intellectual and moral development of the race."

"The Pythagoreans were the first who attempted a complete classification of the facts of the universe. Their effort, though feeble, was in the right direction; for the first principle of perception is analysis, or classification; and knowledge can never be unified until an ultimate or complete analysis has been performed."

"All the early thinkers sought with wonderful perseverance the knowledge of the First Cause. The of Aristotle, though they had been separately recognized, had not all been proclaimed necessary. Aristotle... gave his chief attention to the solution of the problem of First Causes. He maintained that there were four, as follows: First, the Material Cause, or Essence; second, the Substantial [or Formal] Cause; third, the Efficient Cause, or the principle of motion; fourth, the Final Cause, or the Purpose and End."

"It is the opinion of all competent authorities that the fundamental principles of Kant's philosophy declare against the possibility of a unification of knowledge."

"Ever since Hermann Minkowski's now infamous comments in 1908 concerning the proper way to view space-time, the debate has raged as to whether or not the universe should be viewed as a four-dimensional, unified whole wherein the past, present, and future are regarded as equally real or whether the views espoused by the possibilists, historicists, and presentests regarding the unreality of the future (and, for presentests, the past) are more accurate. Now, a century after Minkowski's proposed block universe first sparked debate, we present a new, more conclusive argument in favor of eternalism."

"A third stage appears between 7 and 8, which we shall call the stage of incipient cooperation. Each player now tries to win, and all, therefore, begin to concern themselves with the question of mutual control and of unification of the rules. But while a certain agreement may be reached in the course of one game, ideas about the rules in general are still rather vague. In other words, children of 7-8, who belong to the same class at school and are therefore constantly playing with each other, give, when they are questioned separately, disparate and often entirely contradictory accounts of the rules observed in playing marbles."

"Now what is science? ...it is before all a classification, a manner of bringing together facts which appearances separate, though they are bound together by some natural and hidden kinship."

"Maxwell got a huge bonus for understanding the unification of electricity and magnetism. He understood the nature of light! When I first heard about this in high school I thought this was the coolest thing, and I still do. It's what we're all trying to do."

"Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premises which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary."

"Science itself is badly in need of integration and unification. The tendency is more and more the other way … Only the graduate student, poor beast of burden that he is, can be expected to know a little of each. As the number of physicists increases, each specialty becomes more self-sustaining and self-contained. Such Balkanization carries physics, and indeed, every science further away, from natural philosophy, which, intellectually, is the meaning and goal of science."

"There is obviously only one alternative, namely the unification of minds or consciousnesses. Their multiplicity is only apparent, in truth there is only one mind."

"In Newton's system of mechanics... there is an absolute space and an absolute time. In Einstein's theory time and space are interwoven, and the way in which they are interwoven depends on the observer. Instead of three plus one we have four dimensions."

"Einstein’s dissent from quantum mechanics and immersion in the search for a unified field theory were not failures but anticipations. After all, even if many string theorists would disagree with Einstein about the incompleteness of quantum mechanics, much of what goes on in string theory these days looks a lot like what Einstein was doing in his Princeton years, which was trying to find new mathematics that might extend general relativity to a unification of all the forces and particles in nature."

"In both quantum theory and general relativity, we encounter predictions of physically sensible quantities becoming infinite. This is likely the way that nature punishes impudent theorists who dare to break her unity. ...If infinities are signs of missing unification, a unified theory will have none. It will be what we call a finite theory."

"A scientific hypothesis may be defined in general terms as a provisional or tentative explanation of physical phenomena. But what is an explanation in the true scientific sense? The answers to this question which are given by logicians and men of science, though differing in their phraseology, are essentially of the same import. Phenomena are explained by an exhibition of their partial or total identity with other phenomena. Science is knowledge; and all knowledge, in the language of Sir William Hamilton is a "unification of the multiple." "The basis of all scientific explanation," says Bain, "consists in assimilating a fact to some other fact or facts. It is identical with the generalizing process." And "generalization is only the apprehension of the One in the Many." Similarly Jevons: "Science arises from the discovery of identity amid diversity," and "every great advance in science consists in a great generalization pointing out deep and subtle resemblances." ...the author just quoted in another place: "Every act of explanation consists in detecting and pointing out a resemblance between facts, or in showing that a greater or less degree of identity exists between apparently diverse phenomena." All this may be expressed in familiar language thus: When a new phenomenon presents itself to the man of science or to the ordinary observer, the question arises in the mind of either: What is it?—and this question simply means: Of what known, familiar fact is this apparently strange, hitherto unknown fact a new presentation—of what known, familiar fact or facts is it a disguise or complication? Or, inasmuch as the partial or total identity of several phenomena is the basis of classification (a class being a number of objects having one or more properties in common), it may also be said that all explanation, including explanation by hypothesis, is in its nature classification. Such being the essential nature of a scientific explanation of which an hypothesis is a probatory form, it follows that no hypothesis can be valid which does not identify the whole or a part of the phenomenon, for the explanation of which it is advanced, with some other phenomenon or phenomena previously observed. This first and fundamental canon of all hypothetical reasoning in science is formally resolvable into two propositions, the first of which is that every valid hypothesis must be an identification of two terms—the fact to be explained and a fact by which it is explained; and the second that the latter fact must be known to experience."

"The theories that we describe here provide the basis of progress toward a unification of macroeconomics and microeconomics."

"Lagrange's "" is perhaps his most valuable work and still amply repays careful study. ...the full power of the newly developed analysis was applied to the mechanics of points and rigid bodies. The results of Euler, of D'Alembert, and of the other mathematicians of the Eighteenth Century were assimilated and further developed from a consistent point of view. Full use of Lagrange's own made the unification of the varied principles of statistics and dynamics possible..."

"In the year 1749 he first suggested his idea of explaining the phenomena of thunder-gusts and of the aurora borealis, upon electrical principles. He points out many particulars in which lightning and electricity agree; and he adduces many facts, and reasonings from facts, in support of his positions. In the same year he conceived the astonishingly bold and grand idea of ascertaining the truth of his doctrine, by actually drawing down the lightning, by means of sharp pointed iron rods raised into the region of the clouds. Even in this uncertain state, his passion to be useful to mankind displays itself in a powerful manner. Admitting the identity of electricity and lightning, and knowing the power of points in repelling bodies charged with electricity, and in conducting their fire silently and imperceptibly, he suggested the idea of securing houses, ships, &c. from being damaged by lightning, by erecting pointed rods, that should rise some feet above the most elevated part, and descend some feet into the ground or the water."

"We only call an elephant or a bacterium an 'organism' because, by analogy we attribute to those beings a similar unification of sensation and of consciousness to that we are conscious of in ourselves; but in human societies and in humanity this essential indication is lacking, and therefore, however many other indications we may detect that are common to humanity and to an organism, in the absence of that essential indication, the acknowledgement of humanity as an organism is incorrect."

"Plato wove together separate threads from three earlier philosophers: the mathematics of Pythagoras, the atomism of Demokritos, and the four elements of Empedokles. As happens with the best scientific syntheses, the resulting theory transformed the components from which it started, and was intellectually more powerful than any of them. For these geometrical atoms differed from those of Demokritos in having a limited number of definite shapes, governed by precise mathematical theorems; and furthermore, they were no longer immutable, but could change into one another in ways that could be related back to their geometrical compositions. As a result, Plato could envisage transmutations of a kind that Demokritos did not allow for, and so introduced a new, quantitative element into the analysis of material change. ...For the regular solids can all be built up from two simple triangles... the fundamental elements of his theory."

"In matter-theory, as in astronomy, the Church's commitment to Aristotle was in due course to prove an embarassment. In both branches of science his speculative distinction between terrestrial and celestial matter was insecure from the very beginning. His own most loyal commentator, ... had already dreamt of a theory unifying all things, and John Philoponos... had rejected the distinction between terrestrial and celestial matter outright. Nevertheless, it was still an axiom of almost a thousand years later."

"Descartes' importance for science does not lie in the details of his cosmology. His decipherment of Nature might be crude, yet he had the courage to insist that mechanical sense could be made of the workings of Nature, throughout the realms of physics, chemistry, and even physiology. By reasserting the unity and rationality of Nature, he did as much as any man to put seventeenth-century scientists back on the intellectual road first trodden by the Greeks."

"From Pappus it appears, however, that the early Mathematicians had at first some reluctance in admitting either the Conic Sections or superior curves in the solution of problems, considering them as not strictly geometrical; but afterwards these lines became objects of much curious investigation, even among the ancients; and in modern times ultimately were of the most extensive utility, both in abstract and in physical science."

"More than a hundred years have elapsed since Benjamin Franklin, employing a phraseology now superseded, put forth a theory of matter. It was pronounced "a delusion" by the physicists of the nineteenth century, but the scientists of the twentieth century, according to Sir Oliver Lodge, may be forced to rehabilitate it as the only means of issue from the labyrinth in which all physical study is now involved. ...the Franklin theory is that electricity and matter in combination form a neutral substance, which is the atom of matter as we know it. The most interesting part of the problem for ourselves, says Sir Oliver, is the explanation of matter in terms of electricity, the view that electricity is, as Franklin seems to have supposed, the fundamental "substance." What we men of to-day have been accustomed to regard as an indivisible atom of matter is thus built up out of electricity. All atoms—atoms of all sorts of "substances"—are built up of the same thing. In our day... the theoretical and proximate achievement of what philosophers from Franklin's day to ours have always sought—a unification of matter—is offering itself to physical inquiry."

"To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought. ...[W]e ...endeavour to imagine the world as one connected set of things which underlies all the perceptions of all people."

"We can describe general relativity using either of two mathematically equivalent ideas: curved space-time or metric field. Mathematicians, mystics and specialists in general relativity tend to like the geometric view because of its elegance. Physicists trained in the more empirical tradition of high-energy physics and quantum field theory tend to prefer the field view, because it corresponds better to how we (or our computers) do concrete calculations. ...the field view makes Einstein's theory of gravity look more like the other successful theories of fundamental physics, and so makes it easier to work toward a fully integrated, unified description of all the laws. ...I'm a field man."

"I feel that we are so close with string theory that—in my moments of greatest optimism—I imagine that any day, the final form of the theory may drop out of the sky and land in someone's lap. But more realistically, I feel that we are now in the process of constructing a much deeper theory than anything we have had before and that well into the twenty-first century, when I am too old to have any useful thoughts on the subject, younger physicists will have to decide whether we have in fact found the final theory."

"Because of white racism's ability to bludgeon us into believing that we are inferior beings and therefore incapable of learning math and the sciences, we must spend a significant amount of our learning and teaching time unlearning and unteaching. This is to say, for example, that when a brother or sister reaches the freshman college level, he or she has already been subjected to at least 17 years of conditioning that dictates: "You are too black and too ignorant to understand such lily-white and intelligent things as math, chemistry, physics, etc." Thus, a major part of the teacher's initial instructional time must be spent dealing with the psychological block against learning math—or any of the sciences."

"Histories make men wise; poets, witty; the mathematics, subtile; natural philosophy, deep; morals, grave; logic and rhetoric, able to contend."

"By the beginning of the seventeenth century we may say that the fundamental principles of arithmetic, algebra, theory of equations, and trigonometry had been laid down, and the outlines of the subjects as we know them had been traced. It must be, however, remembered that there were no good elementary text-books on these subjects; and a knowledge of them was therefore confined to those who could extract it from the ponderous treatises in which it lay buried. Though much of the modern algebraical and trigonometrical notation had been inroduced, it was not familiar to mathematicians, nor was it even universally accepted; and it was not until the end of the seventeenth century that the language of the subjects was definitely fixed. Considering the absence of good text-books, I am inclined... to admire the rapidity with which it came into universal use, than to cavil at the hesitation to trust to it alone which many writers showed."

"Those intending to continue in mathematics or science or technology... believe that a survey of the main directions along which living mathematics has developed would enable them to decide more intelligently in what particular field of mathematics, if any, they would find a lasting satisfaction. ...It is astonishing how few students entering serious work in mathematics or its applications have even the vaguest idea of the highways, the pitfalls, and the blind alleys ahead of them. Consequently, it is the easiest thing in the world for an enthusiastic teacher... to sell his misguided pupils a subject that has been dead for forty or a hundred years, under the sincere delusion that he is disciplining their minds. With only the briefest glimpse of what mathematics in this twentieth century—not in 2100 B.C.—is about, any student of normal intelligence should be able to distinguish between live teaching and dead mathematics. He will be less likely to drown in the ditch or perish in the wilderness."

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"The student can actually carry out the mathematical tasks in an authentically historical setting. He can do long division like the ancient Egyptians, solve quadratic equations like the Babylonians, and study geometry just as the student in Euclid's day. To get involved in the same processes and problems as the ancient mathematicians and to effect solutions in the face of the same difficulties they faced is the best way to gain appreciation of the intelligence and ingenuity of the scholars of early times."

"Students enjoy... and gain in their understanding of today's mathematics through analyzing older and alternative approaches."

"1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves. 2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra. 3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry. Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics."

"The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression."

"In mathematics the art of asking questions is more valuable than solving problems."

"The author holds that our school curricula, by stripping mathematics of its cultural content and leaving a bare skeleton of technicalities, have repelled many a fine mind. It is the aim of this book to restore this cultural content and present the evolution of number as the profoundly human story which it is. ...the historical method has been freely used to bring out the rôle intuition has played in the evolution of mathematical concepts. And so the story of number is here unfolded as a historical pageant of ideas, linked with the men who created those ideas and with the epochs which produced the men."

"The systematic exposition of a textbook in mathematics is based on logical continuity and not on historical sequence; but the standard high school course in mathematics fails to mention this fact, and therefore leaves the student under the impression that the historical evolution of number proceeded in the order in which the chapters of the textbook were written. This impression is largely responsible for the widespread opinion that mathematics has no human element. For here, it seems, is a structure that was erected without a scaffold: it simply rose in its frozen majesty, layer by layer! Its structure is faultless because it is founded on pure reason, and its walls are impregnable because they were reared without blunder, error or even hesitancy, for here human intuition had no part! In short the structure of mathematics appears to the layman as erected not by the erring mind of man but by the infallible spirit of God. The history of mathematics reveals the fallacy of such a notion."

"Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise."

"A finished or even a competent reasoner is not the work of nature alone... education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one... in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not. ..Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:— 1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general. 3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion. 4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if... reason is not to be the instructor, but the pupil. 5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded. ...These are the principal grounds on which... the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life."

"I have tried to say to students of mathematics that they should read the classics and beware of secondary sources. This is a point which Eric Temple Bell makes repeatedly... in '... that the men of whom he writes learned their mathematics not by studying in school or by reading textbooks, but by going straight to the sources and reading the best works of the masters who preceded them. It is a point which in most fields of scholarship at most times in history would have gone without saying. ...The purpose of a secondary source is to make the primary sources accessible to you."

"Should I teach them from the point of view of the history of science, from the applications? My theory is that the best way to teach is to have no philosophy, [it] is to be chaotic and [to] confuse it in the sense that you use every possible way of doing it. ...so as to catch this guy or that guy on different hooks as you go along, [so] that during the time when the fellow who's interested in history's being bored by the abstract mathematics... the fellow who likes the abstractions is being bored another time by the history—if you can do it so you don't bore them all, all the time, perhaps you're better off. ...I don't know how to answer the question of different kinds of minds with different interests... after many, many years of trying to teach and trying all kinds of different methods, I really don't know how to do it."

"More often than we are aware, it is the jargon which is the hurdle that a student cannot overcome rather than the mathematical concepts being introduced."

"Some of the ancient methods of calculation are particularly suited to mental arithmetic. ...Multiplication by a power of two is easily performed by successive doubling—a method fundamental to Egyptian multiplication (and division). Many 'tricks'... have been known for centuries. ...think of multiplication by a number close to a power of 10. How many children have been asked laboriously to multiply by 97 instead of multiplying by 100, [multiplying the original number again] by 3, and subtracting? 'Trick' methods... very often... can introduce principles...(100 - 3)n = 100n - 3n...is to make use of the distributive property... though... we do not have to put this in such technical language in the classroom."

"One of the great mistakes in the teaching of algebra is to present it as if it were a subject unrelated to arithmetic. ...Many schoolchildren... arrive at the conclusion that algebra and arithmetic are different subjects. ...the history of mathematics teaches us that even cubic and quartic equations were successfully solved without the benefit of modern notation. ...The use of words instead of symbols can illustrate how close the formulations are to the original... problems."

"The contemporary decline in interest in geometry and its gradual disappearance from school curricula... should be deplored... Geometry is the most visual of the mathematical disciplines. It is not in principle divorced from numbers, and hence neither is it divorced from algebra. Many a pupil's understanding of algebraic proofs would be considerably reinforced by... visual geometrical proofs which were the hallmark of Greek mathematics and to some extent of Arab mathematics also. ...where a geometrical proof is clear and immediate, as... with... many algebraic identities such as (a \pm b)^2 = a^2 \pm 2ab + b^2\!, the geometry should not be forgotten. The Greeks were some of the greatest teachers of all time... [and] geometric algebra was in many ways [their] greatest achievement ..."

"Reflect on how many real problems of life require solutions which are meaningless unless they are whole numbers. The absence of Diophantine analysis from school curricula is difficult to justify. The methods... are not beyond the capabilities of schoolchildren."

"We cannot expect to present at pre-university level the more abstract approaches to the definition[s]... of the late nineteenth and early twentieth century. We... accept... that the natural numbers are 'given'... we do not need to define them. Our pupils will discover ways in which numbers relate to the real world for themselves—all we have to do is to provide the environment in which this can happen easily and effectively."

"The games which can be built up from the simple idea of dots and lines... can be a productive source of teaching material. After all, s provided the Pythagoreans and neo-Pythagoreans with important theorems... It is well worth while... looking at the games played by the undeveloped peoples of the world. ...These ...are much closer to reality than... sophisticated and expensive forms of entertainment... which... choke the natural initiative... Simple mathematical games ...stimulate initiative and suggest other games which children can invent for themselves... For children, life is naturally simple. Let us not complicate it for them sooner than is necessary, least of all in our teaching of mathematics... which already has enough complications of its own."

"It is a student's understanding of mathematics, above all other subjects, which suffers most from unenlightened teaching methods. ...the troubles may well stem mainly from the first year or two of the child's encounter with numbers... if children come to fear them or to be bored with them, they will eventually join the ranks of the present majority... If, on the other hand, numbers are made a... source of adventure and exploration from the beginning, there is a good chance that the level of numeracy in society can be raised... There is a real role here for the history of mathematics—and the history of number in particular—for history emphasizes the diversity of approaches and methods which are possible and frees us from the straitjacket of contemporary fashions in mathematics education. It is, at the same time, interesting and stimulating in its own right."

"Geometry is grasping space. And since it is about the education of children, it is grasping that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it."

"The history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. ...If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys."

"Using the history of algebra, teachers of the subject, either at the school or at the college level, can increase students' overall understanding of the material. The "logical" development so prevalent in our textbooks is often sterile because it explains neither why people were interested in a particular algebraic topic in the first place nor why our students should be interested in that topic today. History, on the other hand, often demonstrates the reasons for both. With the understanding of the historical development of algebra, moreover, teachers can better impart to their students an appreciation that algebra is not arbitrary, that it is not created "full-blown" by fiat. Rather, it develops at the hands of people who need to solve vital problems, problems the solutions of which merit understanding. Algebra has been and is being created in many areas of the world, with the same solution often appearing in disparate times and places. ...professors can stimulate their students to master often complex notions by motivating the material through the historical questions that prompted its development. In absorbing the idea, moreover, that people struggled with many important mathematical ideas before finding their solutions, that they frequently could not solve problems entirely, and that they consciously left them for their successors to explore, students can better appreciate the mathematical endeavor and its shared purpose."

"Algebra is a machine, or more accurately, a collection of machines. There is a machinery to factor expressions, to decompose complicated fractions into simpler ones, and so on. The object of the conversion in every case is to obtain a form more useful for the problem in hand. ...Elementary algebra as a whole is a huge machine to mechanize thinking. ...The mechanization of processes that have to be used repeatedly is... a great gain, since one does not have to think about them. They become habitual like washing and dressing. ...in itself elementary algebra is of no great interest. ...generally speaking, a machine is valuable because it turns out a useful product. ... The individual techniques of algebra are like single notes selected at random from large and magnificent musical compositions. ...these notes ...employed in the investigation of more significant undertakings, help to form beautiful patterns of reasoning. ...Unfortunately, the usefulness of the techniques of algebra has caused many people to mistake the means for the end and to emphasize these menial techniques to the exclusion of the larger ideas and goals of mathematics. The students who are bored by the process of algebra are more perceptive than those who have mistakenly identified algebraic processes with mathematics."

"The usual courses present segments of mathematics that seem to have little relationship to each other. The history may give perspective on the entire subject and relate the subject matter of the courses not only to each other but also to the main body of mathematical thought."

"The usual courses in mathematics are... deceptive in a basic respect. They give an organized logical presentation which leaves the impression that mathematicians go from theorem to theorem almost naturally, that mathematicians can master almost any difficulty, and that subjects are completely thrashed out and settled. The succession of theorems overwhelms the student... The history, by contrast, teaches us the development of a subject is made bit by bit with results coming from various directions. We learn, too, that often decades and even hundreds of years of effort were required before significant steps could be made. In place of the impression that subjects are completely thrashed out one finds that what is attained is often but a start, that many gaps have to be filled, or that the really important extensions remain to be created."

"There is a danger to the humanities in the present educational crash programs designed to produce a large number of mathematicians, physical scientists, engineers, and technical workers. ...Part of the... objective of the present work is to supply material which can serve as a cultural background or supplement for all those who are receiving rapid, concentrated exposure to recent advanced mathematical concepts, without any opportunity to examine the origins or gradual historical development of such ideas. Hence, although designed for the layman, this book would be helpful in courses in the history, philosophy, or fundamental concepts of mathematics."

"I am well aware of the negative attitude of so many students toward the subject. There are many reasons for this, one of them no doubt being the esoteric, dry way in which we teach the subject. We tend to overwhelm our students with formulas, definitions, theorems, and proofs, but we seldom mention the historical evolution of these facts, leaving the impression that these facts were handed to us, like the Ten Commandments, by some divine authority. The history of mathematics is a good way to correct these impressions."

"The principal aim of mathematical education is to develop certain faculties of the mind, and among these intuition is not the least precious. It is through it that the mathematical world remains in touch with the real world, and even if pure mathematics could do without it, we should still have to have recourse to it to fill up the gulf that separates the symbol from reality."

"Zoologists maintain that the embryonic development of an animal recapitulates in brief the whole history of its ancestors throughout geologic time. It seems it is the same in the development of minds. The teacher should make the child go over the path his fathers trod; more rapidly, but without skipping stations. For this reason the history of science should be our first guide."

"Euclid's manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument."

"Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery."

"The cookbook gives a detailed description of ingredients and procedures but no proofs for its prescriptions or reasons for its recipes; the proof of the pudding is in the eating. … Mathematics cannot be tested in exactly the same manner as a pudding; if all sorts of reasoning are debarred, a course of calculus may easily become an incoherent inventory of indigestible information."

"Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning, but I contend also that there is no other subject in the usual curricula of the schools that affords a comparable opportunity to learn plausible reasoning. ...let us learn proving, but also let us learn guessing."

"I shall often discuss mathematical discoveries... I shall try to make up a likely story how the discovery could have happened. I shall try to emphasize the motives underlying the discovery, the plausible inferences that led to it... everything that deserves imitation. ...I... present also examples of historic interest, examples of real mathematical beauty, and examples illustrating the parallelism of the procedures in other sciences, or in everyday life."

""Groping" and "muddling through" is usually described as a solution by trial and error. ...a series of trials, each of which attempts to correct the error committed by the preceding and, on the whole, the errors diminished as we proceed and the successive trials come closer and closer to the desired final result. ...we may wish a better characterization ..."successive trials" or "successive corrections" or "successive approximations." ...You use successive approximations when ...looking for a word in the dictionary ...A mathematician may apply the term ...to a highly sophisticated procedure ...to treat some very advanced problem ...that he cannot treat otherwise. The term even applies to science as a whole; the scientific theories which succeed each other, each claiming a better explanation ...may appear as successive approximations to the truth. Therefore, the teacher should not discourage his students from using trial and error—on the contrary, he should encourage the intelligent use of the fundamental method of successive approximations. Yet he should convincingly show that for ...many ... situations, straightforward algebra is more efficient than successive approximations."

"At each stage of its development the human race has had a certain climate of opinion, a way of looking, conceptually, at the world. The next glimmer of fresh understanding had to grow out of what was already understood. The next move forward, halting shuffle, faltering step, or stride with some confidence, was developed upon how well the [human] race could then walk. As for the human race, so for the human child. But this is not to say that to teach science we must repeat the thousand and one errors of the past, each ill-directed shuffle. It is to say that the sequence in which the major strides forward were made is a good sequence in which to teach them. The genetic method is a guide to, not a substitute for, judgement."

"Why should the typical student be interested in those wretched triangles? ...Without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science. We may question the propriety of allowing a learner, at the commencement of his Geometrical studies, to exhibit Geometrical demonstrations in Algebraical symbols. Surely it is not too much to apprehend that such a practice may occasion serious confusion of thought."

"Since formal mathematics is no more than a culturally-dependent system of aesthetics, while it may continue to be taught like Western music, there is no need to impose its consequences on K-12 children. What we need to teach children is practical mathematics... In particular, it may be worth re-examining whether one might want to teach as entirely separate subjects, from the outset, the two mathematical streams: practical mathematics and formal mathematics, with their distinct notions of number and proof."

"The education of the child must accord both in mode and arrangement with the education of mankind, considered historically. In other words, the genesis of knowledge in the individual, must follow the same course as the genesis of knowledge in the race. In strictness, this principle may be considered as already expressed by implication; since both being processes of evolution, must conform to those same general laws of evolution... and must therefore agree with each other. Nevertheless this particular parallelism is of value for the specific guidance it affords. To M. Comte we believe society owes the enunciation of it; and we may accept this item of his philosophy without at all committing ourselves to the rest."

"Nowhere in all ancient mathematics do we find any attempt at what we call demonstration. No argumentation was presented, but only the prescription of certain rules: "Do such and so." We are ignorant of the way the theorems were found... To those who have been educated in Euclid's strict argumentation, the entire Oriental way of reasoning seems at first strange and highly unsatisfactory. But this strangeness wears off when we realize that most of the mathematics we teach our present-day engineers and technicians is still of the "do such, do so" type, without much attempt at rigorous demonstration. Algebra is still being taught in many high schools as a set of rules rather than as a science of deduction. Oriental mathematics never seems to have been emancipated from the millenial influence of the problems of technology and administration, for the use of which it had been invented."

"One makes sense of , whether fictional or factual, by a mental construction that is sometimes called the world of story ...[T]he imaginative effort is a standard way of understanding what people say... In order to understand connected speech about concrete things, one imagines them. ...The capacity to do this... encourages empathy, but it also allows one to do mathematics. ...This is often fun, and it is a form of playing with ideas. ...This ludic aspect of mathematics is emphasized by in his semiotic analysis of mathematics and acknowledged by David Wells in his comparison of mathematics and games. ...The ludic aspect is something that undergraduates, many of whom have decided that mathematics is either a guessing game... or the execution of rigidly defined procedures, need to be encouraged to do when they are learning new ideas. They need to fool around with them to become familiar... Changing the s and seeing what a function looks like... to learn how the function behaves. ...Mathematical research involves a good deal of fooling around, which is part of why it is a pleasurable activity... This is not competitive..."

"A liberal Education ought to include both Permanent Studies which connect men with the culture of past generations, and Progressive Studies which make them feel their community with the present generation, its businesses, interests and prospects. ...When the Student is once well disciplined in geometrical mathematics, he may pursue analysis safely and surely to any extent. But though modern mathematics may thus be very fitly studied as a sequel to the older forms of mathematical science... these modern methods cannot supply the place of the ancient subjects as the Permanent Studies in our Educational course."

"The use of mathematical study, with which we have to do, is not to produce a school of eminent mathematicians, but to contribute to a Liberal Education of the highest kind."

"So far as civilisation is connected with the advance and diffusion of human knowledge, civilisation flourishes when the prevalent education is mathematical, and fades when philosophy is the subject most preferred."

"The study of mathematics is apt to commence in disappointment... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it."

"Mathematics as an Element in the History of Thought."

"Few contemporaries were as profoundly read in the history of mathematics as was De Morgan. No subject was too insignificant to receive his attention. ...In [his] article "Induction (Mathematics)," first printed in 1838, occurs, apparently for the first time, the name "mathematical induction"; it was adopted and popularized by I. Todhunter in his Algebra. The term "induction" had been used by John Wallis in 1656, in his Arithmetica infinitorum; he used the induction known to natural science. In 1686 Jacob Bernoulli criticised him for using a process which was not binding logically and then advanced in place of it the proof from n to n + 1. This is one of the several origins of the process of mathematical induction. From Wallis to De Morgan, the term "induction" was used occasionally in mathematics, and in a double sense, (1) to indicate incomplete inductions of the kind known in natural science, (2) for the proof from n to n + 1. De Morgan's "mathematical induction" assigns a distinct name for the latter process. The Germans employ more commonly the name "vollständige Induktion," which became current among them after the use of it by R. Dedekind in his Was sind und was sollen die Zahlen, 1887."

"One who extended the theory of equations somewhat further than Vieta was Albert Girard... Like Vieta this ingenious author applied algebra to geometry, and was the first who understood the use of negative roots in the solution of geometric problems. He spoke of imaginary quantities; inferred by induction that every equation has as many roots as there are units in the number expressing its degree; and first showed how to express the sums of their powers in terms of the coefficients."

"A more modern attempt to explain the fruitfulness of mathematical reasoning is that of Poincaré, who finds it all due to the principle of mathematical induction. This principle of mathematical induction is undoubtedly of wide application, though there are many regions even in arithmetic where it is difficult to see its application, e.g., the science of prime numbers, a science dealing entirely with non-recurring individuals. But the important thing to observe is that this principle of mathematical induction is entirely different from the induction that prevails in the physical sciences."

"It is absolutely certain that if a proposition is established by mathematical induction, it will never be disproved, i.e., if a general proposition is true of n + 1 whenever it is true of n, and also of 1, then no possible number can arise of which this proposition is not true, for the principle of mathematical induction is used in defining all finite integers. Whether, therefore, we agree with Russell and call the principle of mathematical induction a definition, or concede to Poincaré that it is a special axiom, a synthetic proposition a priori, the fact remains that reasoning from it is a purely deductive procedure."

"The propositions of arithmetic, the... operations, for instance, which play such a fundamental rôle even in the most simple calculations, must be demonstrated by deductive methods. What is the principle involved? Well, this principle has been variously called mathematical induction, and complete induction, and that of reasoning by recurrence. The latter is the only acceptable name, the others being misnomers. The term induction conveys an entirely erroneous idea of the method, for it does not imply systematic trials."

"It is significant that we owe the first explicit formulation of the principle of recurrence to the genius of Blaise Pascal... Pascal stated the principle in a tract called The Arithmetic Triangle which appeared in 1654. Yet... the gist of the tract was contained in the correspondence between Pascal and Fermat regarding a problem in gambling, the same correspondence which is now regarded as the nucleus from which developed the theory of probabilities. It surely is a fitting subject for mystic contemplation that the principle of reasoning by recurrence, which is so basic in pure mathematics, and the theory of probabilities, which is the basis of all deductive sciences, were both conceived while devising a scheme for the division of stakes in an unfinished match of two gamblers."

"Despite the age-long tyranny exercised by the Aristotelian logic... Of all argument forms, there is one which, viewed as the figure of the way in which the mind gains certainty that a specified property belonging, but not immediately by definition, to each element of a denumerable assemblage of elements does so belong, enjoys the distinction of being at once perhaps the most fascinating, and, in its mathematical bearings, doubtless the most important single form in modern logic. This form is that variously known as reasoning by recurrence, induction by connection (De Morgan), mathematical induction, complete induction, and Fermatian induction—so called by C. S. Peirce, according to whom this mode of proof was first employed by Fermat. Whether or not such priority is thus properly ascribed, it is certain that the argument form in question is unknown to the Aristotelian system, for this system allows apodictic certainty in case of deduction only, while it is the distinguishing mark of mathematical induction that it yields such certainty by the reverse process, a movement from the particular to the general, from the finite to the infinite. Of the various designations of this mode argument, "mathematical induction" is undoubtedly the most appropriate, for though one not be able to agree with Poincaré that the mode in question is characteristic of mathematics, it is peculiar to science, being indeed, as he has called it, "mathematical reasoning par excellence.""

"This procedure is the demonstration by recurrence. We first establish a theorem for n = 1; then we show that if it is true of n - 1, it is true of n, and thence conclude that it is true for all the whole numbers. ..Here then we have the mathematical reasoning par excellence, and we must examine it more closely. ...The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms. ...to arrive at the smallest theorem [we] can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite. This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem... In this domain of arithmetic,.. the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general."

"We can not... escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. ...Neither can this rule come to us from experience... This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetic a priori judgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry. ...it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it."

"But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided by induction? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily, because it is only the affirmation of a property of the mind itself."

"We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name."

"Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable."

"Mathematical induction, which is purely ordinal... may be stated as follows: A series generated by a one-one relation, and having a first term, is such that any property, belonging to the first term and to the successor of any possessor of the property, belongs to every term of the series."

"The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle."

"Many years ago I published in the Formulaire de Mathématique of Professor Peano an account of the first discovery of mathematical induction as due to the Italian Maurolycus. But this paper seems to have had only a small diffusion. ...the most original of his works is the treatise on arithmetic "Arithmeticorum libri duo" written in the year 1557 and printed in Venice in the year 1575 in the collection "D. Francisci Maurolyci Opuscula mathematica." In the Prolegomena to this work he points out that neither in Euclid nor in any other Greek or Latin writer (among them he enumerates Iamblichus, Nicomachus, Boetius) is there, to his knowledge, a treatment of the polygonal and polyhedral numbers, and he reproaches Jordanus for having been content with a useless repetition of what was written by Euclid. "Nos igitur [he says] conabimur ea, quae super hisce numerariis formis nobis occurrunt, exponere: multa interim faciliori via demonstrantes, et ab aliis authoribus aut neglecta, aut non animadversa supplentes." This new and easy way is nothing else than the principle of mathematical induction. This principle is used at the beginning of the work only in the demonstration of very simple propositions, but in the course of the treatise is applied to the more complicated theorems in a systematic way. ...Was Pascal unaware of the book of Maurolycus? In his Traité du triangle arithmétique printed perhaps in the year 1657, he never mentions Maurolycus, notwithstanding that, in my opinion, this treatise is only an application of the method discovered by Maurolycus. But Pascal, shortly after, being engaged in the polemic concerning the cycloid, in the well known letter, "Lettre de Dettonville à Carcavi" had to demonstrate a proposition concerning the triangular and pyramidal numbers. He says then:"Cela Est Aisé Par Maurolic."It is strange to point out that not even the name of Maurolycus has been included in the Table analytique of the old edition of the works of Pascal, and more strange that the editors of the new edition of the "Oeuvres" of Pascal in a very incomplete historical note before the reimpression of the Traité du triangle arithmétique never mention the name of one of the greatest European mathematicians of the sixteenth century."

"In his address to the Mathematical Section at the British Association Meeting of 1869 at Exeter, Professor J. J. Sylvester laid much stress upon the employment of inductive philosophy in mathematics. He said that he was aware that many who had not gone deeply into the principles of mathematical science believed that inductive philosophy, or the method of evolving new truths by induction, was reserved for the experimental sciences, and that the methods of investigation in mathematical science might all be classified as deductive. He went on to say that this opinion is not a correct one, and that many valuable results are obtained in mathematical science by induction, or reasoning from particulars to generals, which could not otherwise be obtained so easily. Although making a distinction between mathematical induction and the induction used in natural philosophy, De Morgan, in his article in the 'Penny Cyclopædia' on this subject, states that an instance of mathematical induction occurs in every equation of differences and in every recurring series. Taking the definition of induction as given by Dr. Whateley, namely, "a kind of argument which infers respecting a whole class what has been ascertained respecting one or more individuals of that class," it will be evident to any experimenter in chemical or physical science who is also acquainted with the use of induction in mathematical science, that mathematical induction is of a higher and more perfect kind than the induction used in the physical sciences, especially when it assumes the form of successive induction as De Morgan calls it, and as it is employed in recurring series. It is this high class of reasoning which is involved in the construction of series that recur according to a given law, that makes the use of recurring series so valuable in unitation."

"M. Poincaré finds the answer to these questions in the so-called 'mathematical induction' which proceeds from the particular to the more general, but at the same time does so by steps of the highest degree of certitude. In this process he sees the creative force of mathematics, which leads to real proofs and not mere sterile verifications. The illustrations used to make the thought clear are taken from the beginnings of arithmetic, where mathematical thought has remained least elaborated and uncomplicated by the difficult questions related to the notion of space. In successive instances it is shown how more general results are obtained from fundamental definitions and from previous results by means of mathematical induction. In each case the advance is made by virtue of that "power of the mind which knows that it can conceive of the indefinite repetition of the same act as soon as this act is at all possible. The mind has a direct intuition of this power and experience gives only the opportunity to use it and to become conscious of it." The conviction that the method of mathematical induction is valid our author regards as truly an à priori synthetic judgment; the mind can not tolerate nor conceive its contradictory and could not even draw any theoretic consequences from the assumption of the contradictory. No arithmetic could be built up, rejecting the axiom of mathematical induction, as the non-Euclidean geometries have been built up, rejecting the postulate of Euclid."

"Fourier's analytical theory of heat (final form, 1822), devised in the Galileo-Newton tradition of controlled observation plus mathematics, is the ultimate source of much modern work in the theory of functions of a real variable and in the critical examination of the foundation of mathematics."

"These doubts did not halt mathematical creation. Technicians working on the superstructure did not drop their tools and scurry down to the basement because some of their underpinning needed reinforcing. Continuing their own highly specialized labors, they left the necessary task to experts who understood what they were about. The building had not collapsed as late as 1940; and while those engaged in elaborating the superstructure but seldom concerned themselves with what the consolidators of the foundations were doing, they had at least come to tolerate their presence in the building. The misconceptions and recriminations of the 1900's gave way in the 1930's to a first crude approximation to harmony. It was as if the American Federation of Labor and the Committee for Industrial Organization had at last decided to bury the hatchet elsewhere than in either's skull, and get on with the job."

"I shall now address you on the subject of the present situation in research in the foundations of mathematics. Since there remain open questions in this field, I am not in a position to paint a definitive picture of it for you. But it must be pointed out that the situation is not so critical as one could think from listening to those who speak of a foundational crisis. From certain points of view, this expression can be justified; but it could give rise to the opinion that mathematical science is shaken at its roots."

"The truth is that the mathematical sciences are growing in complete security and harmony. The ideas of Dedekind, Poincare, and Hilbert have been systematically developed with great success, without any conflict in the results. It is only from the philosophical point of view that objections have been raised. They bear on certain ways of reasoning peculiar to analysis and set theory. These modes of reasoning were first systematically applied in giving a rigorous form to the methods of the calculus. [According to them,] the objects of a theory are viewed as elements of a totality such that one can reason as follows: For each property expressible using the notions of the theory, it is [an] objectively determinate [fact] whether there is or there is not an element of the totality which possesses this property. Similarly, it follows from this point of view that either all the elements of a set possess a given property, or there is at least one element which does not possess it."

"As soon as I have put it into order I intend to write and if possible to publish a work on parallels. At this moment, it is not yet finished, but the way which I have followed promises me with certainty the attainment of my aim, if it is at all attainable. It is not yet attained, but I have discovered such magnificent things that I am myself astonished at the result. It would forever be a pity, if they were lost. When you see them, my father, you yourself will concede it. Now I cannot say more, only so much that from nothing I have created another wholly new world. All that I have hitherto sent you compares to it as a house of cards to a castle."

"I should regard it as a great misfortune if you were to allow yourself to be deterred by the 'clamors of the Bœotians' from explaining your views of geometry. From what Lambert has said and [Ferdinand Karl] Schweikart orally communicated, it has become clear to me that our geometry is incomplete and stands in need of a correction which is hypothetical and which vanishes wher. the sum of the angles of a plane triangle is equal to 180°. This would be the true geometry and the Euclidean the practical, at least for figures on the earth."

"Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an , which allows a certain amount of self-reference. And by this time everyone was pretty bored."

"The world is totally connected. Whatever explanation we invent at any moment is a partial connection, and its richness derives from the richness of such connections as we are able to make. ...mathematics suffer from the same partiality. Gödel, Turing, and Tarski all proved this. Gödel proved that you cannot have a complete axiomatization of the whole of mathematics, that every system which you devise is partial and suffers from one great shortcoming. If it is consistent, there are theorems which are true that cannot be proved in it. And Turing showed that every machine that we can devise is like a formal system, and that therefore no machine can do all of mathematics. And Tarski put it even more boldly when he said that no universal language for all of science can exist in all cases without paradox."

"Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real."

"In the summer of 1914 I attended Frege's course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. As an example he quoted Weyerstrass's definition: "A number is a series of things of the same kind"... On this he commented with an impish smile: "According to this definition, a railroad train is also a number; this number may then travel from Berlin, pass through Jena... He criticized in particular the lack of attention to certain fundamental distinctions, e.g., ...between the symbol and the symbolized, ...between a logical concept and a mental image or act, and that between a function and the value of a function. Unfortunately, his admonitions go unheeded even today."

"It is in set theory that we encounter the greatest diversity of foundational opinions. This is because even the most devoted advocates of the various new axioms would not argue that these axioms are justified by any basic ‘intuition’ about sets. ... One may vary the rank of sets allowed. Conventional mathematics rarely needs to consider more than four or five iterations of the power set axiom applied to the set of integers. More iterations diminish our sense of the reality of the objects involved."

"An article by Henri Poincaré entitled The Nature of Mathematical Reasoning... appeared in 1894 as the first of a series of investigations into the foundations of the exact sciences. It was a signal for a throng of other mathematicians to inaugurate a movement for the revision of the classical concepts, a movement which culminated in the nearly complete absorption of logic into the body of mathematics."

"The two great conceptual revolutions of twentieth-century science, the overturning of classical physics by Werner Heisenberg and the overturning of the foundations of mathematics by Kurt Gödel, occurred within six years of each other within the narrow boundaries of German-speaking Europe. ...A study of the historical background of German intellectual life in the 1920s reveals strong links between them. Physicists and mathematicians were exposed simultaneously to external influences that pushed them along parallel paths. ...Two people who came early and strongly under the influence of Spengler's philosophy were the mathematician Hermann Weyl and the physicist Erwin Schrödinger. ...Weyl and Schrödinger agreed with Spengler that the coming revolution would sweep away the principle of physical causality. The erstwhile revolutionaries David Hilbert and Albert Einstein found themselves in the unaccustomed role of defenders of the status quo, Hilbert defending the primacy of formal logic in the foundations of mathematics, Einstein defending the primacy of causality in physics. In the short run, Hilbert and Einstein were defeated and the Spenglerian ideology of revolution triumphed, both in physics and in mathematics. Heisenberg discovered the true limits of causality in atomic processes, and Gödel discovered the limits of formal deduction and proof in mathematics. And, as often happens in the history of intellectual revolutions, the achievement of revolutionary goals destroyed the revolutionary ideology that gave them birth. The visions of Spengler, having served their purpose, rapidly became irrelevant."

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

"Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic."

"When we examine the classical set-theoretic foundations of mathematics, we see that the only sets that play a role are sets of restricted type; at the risk of understatement, only sets of rank < ω + ω. Further examination reveals four fundamental principles about sets used: the existence of an infinite set; the existence of the power set of any set; every property determines a subset of any set; and the axiom of choice."

"That the sum of the angles cannot be smaller than 180&deg;; this is the real difficulty, the rock on which all endeavors are wrecked. I surmise that you have not employed yourself long with this subject. I have pondered it for more than thirty years, and I do not believe that any one could have concerned himself more exhaustively with this... than I, although I have not published anything on this subject. The assumption that the sum of the three angles is smaller than 180&deg; leads to a new geometry entirely different from our Euclidean,—a geometry which is throughout consistent with itself, and which I have elaborated in a manner entirely satisfactory to myself, so that I can solve every problem in it with the exception of the determining of a constant, which is not a priori obtainable. The larger this constant is taken, the nearer we approach the Euclidean geometry, and an infinitely large value will make the two coincident. The propositions of this geometry appear partly paradoxical and absurd to the uninitiated, but on closer and calmer consideration it will be found that they contain in them absolutely nothing that is impossible. Thus the three angles of a triangle... can be made as small as we will, provided the sides can be taken large enough; whilst the area of a triangle, however great the sides may be taken, can never exceed a definite limit, nay, can never once reach it. All my endeavors to discover contradictions or inconsistencies in this non-Euclidean geometry have been in vain, and the only thing in it that conflicts with our reason is the fact that if it were true there would necessarily exist in space a linear magnitude quite determinate in itself, yet unknown to us. But I opine that, despite the empty word-wisdom of the metaphysicians, in reality we know little or nothing of the true nature of space, so much so that we are not at liberty to characterize as absolutely impossible things that strike us as unnatural. If the non-Euclidean geometry were the true geometry, and the constant in a certain ratio to such magnitudes as lie within the reach of our measurements on the earth and in the heavens, it could be determined a posteriori. I have, therefore, in jest frequently expressed the desire that the Euclidean geometry should not be the true geometry, because in that event we should have an absolute measure a priori."

"I have also in my leisure hours frequently reflected upon another problem, now of nearly forty years' standing. I refer to the foundations of geometry. I do not know whether I have ever mentioned to you my views on this matter. My meditations here also have taken more definite shape, and my conviction that we cannot thoroughly demonstrate geometry a priori is, if possible, more strongly confirmed than ever. But it will take a long time for me to bring myself to the point of working out and making public my very extensive investigations on this subject, and possibly this will not be done during my life, inasmuch as I stand in dread of the clamors of the Bœotians, which would be certain to arise, if I should ever give full expression to my views. It is curious that in addition to the celebrated flaw in Euclid's Geometry, which mathematicians have hitherto endeavored in vain to patch and never will succeed, there is still another blotch in its fabric to which, so far as I know, attention has never yet been called and which it will by no means be easy, if at all possible, to remove. This is the definition of a plane as a surface in which a straight line joining any two points lies wholly in that plane. This definition contains more than is requisite to the determination of a surface, and tacitly involves a theorem which is in need of prior proof."

"The ease with which you have assimilated my notions of geometry has been a source of genuine delight to me, especially as so few possess a natural bent for them. I am profoundly convinced that the theory of space occupies an entirely different position with regard to our knowledge a priori from that of the theory of numbers (Grössenlehre); that perfect conviction of the necessity and therefore the absolute truth which is characteristic of the latter is totally wanting to our knowledge of the former. We must confess in all humility that a number is solely a product of our mind. Space, on the other hand, possesses also a reality outside of our mind, the laws of which we cannot fully prescribe a priori."

"Leibniz believed not only that it was a metaphysical fact that all truths are reducible to primary logical truths, but also that, given an appropriate formal language, all truths should be capable of a priori proof. The means of carrying out such proofs was the subject of one of Leibniz's earliest works, his dissertation De Arte Combinatoria (On the Art of Combinations) written in 1666... In it Leibniz reveals his vision of a Characteristica Universalis, or universal characteristic, that would operate as a formal logic through which all true propositions would be demonstrable, merely through adherence to syntactical rules..."

"Euclid could inscribe regular polygons of 3, 4, 5, 15 sides or numbers obtained by doubling these. Those of 7, 9, 11, 13, 14 sides no man ever could or ever will geometrically inscribe. When on the evening of March 30th, 1796, Gauss showed to his student friend, the Hungarian, Wolfgang Bolyai, the formula which gave the geometric inscription of the regular polygon of 17 sides, it was with the remark that this alone could be his epitaph, if it were not a pity to omit so much that went with it. Was it this break beyond Euclid's enchanted bounds that started these two young men in that re-sifting of the very foundations of geometry which led to those new conceptions of the whole subject just now, after another hundred years, beginning to be taught in America's foremost universities?"

"For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring."

"It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning."

"I feel that controversies can never be finished, nor silence imposed upon the Sects, unless we give up complicated reasonings in favour of simple calculations, words of vague and uncertain meaning in favour of fixed symbols [characteres]. Thus it will appear that 'every paralogism is nothing but an error of calculation. When controversies arise, there will be no more necessity for disputation between two philosophers than between two accountants. Nothing will be needed but that they should take pen in hand, sit down with their counting-tables and (having summoned a friend, if they like) say to one another: Let us calculate.'"

"These primitive propositions … suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. …All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions"

"The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A! ...Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? ...If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism."

"I come now to the capital work of Hilbert which he communicated to the Congress of Mathematicians at Heidelberg... of which...an English translation due to Halsted appeared in The Monist. ...the author's aim is analogous to that of Russell, but on many points he diverges from his predecessor. "But," he says, "on attentive consideration we become aware that in the usual exposition of the laws of logic certain fundamental concepts of arithmetic are already employed; for example, the concept of the aggregate, in part also the concept of number. "We fall thus into a vicious circle and therefore to avoid paradoxes a partly simultaneous development of the laws of logic and arithmetic is requisite." ...what Hilbert says of the principles of logic in the usual exposition applies likewise to the logic of Russell. So for Russell logic is prior to arithmetic; for Hilbert they are 'simultaneous.' We shall find... other differences still greater... I prefer to follow step by step the development of Hubert's thought... "Let us take as the basis of our consideration first of all a thought-thing 1 (one)." Notice that in so doing we in no wise imply the notion of number, because it is understood that 1 is here only a symbol and that we do not at all seek to know its meaning. "The taking of this thing together with itself respectively two, three or more times ..." Ah! this time it is no longer the same; if we introduce the words 'two,' 'three,' and above all 'more,' 'several,' we introduce the notion of number; and then the definition of finite whole number which we shall presently find, will come too late. Our author was too circumspect not to perceive this begging of the question. So at the end of his work he tries to proceed to a truly patching-up process. Hilbert then introduces two simple objects 1 and =, and and considers all the combinations of these two objects, all the combinations of their combinations, etc. It goes without saying that we must forget the ordinary meaning of these two signs and not attribute any to them. Afterwards he separates these combinations into two classes, the class of the existent and the class of the non-existent... entirely arbitrary. Every affirmative statement tells us that a certain combination belongs to the class of the existent; every negative statement tells us that a certain combination belongs to the class of the non-existent. Note now a difference of the highest importance. For Russell any object whatsoever, which he designates by x, is an object absolutely undetermined and about which he supposes nothing; for Hilbert it is one of the combinations formed with the symbols 1 and =; he could not conceive of the introduction of anything other than combinations of objects already defined."

"It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor, a priori, whether it is possible. From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space—the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are—like all matters of fact—not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small."

"Hardy... in vain, tried to convince him to learn classical foundations of mathematics and, in particular, the rigorous expositive method of mathematical demonstrations. Every time Hardy introduced a problem, Ramanujan considered it ex novo [new] applying unconventional reasoning which was sometimes incomprehensible to his fellow colleagues."

"The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles... and will be established by strict symbolic reasoning... The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken... to defend... against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established. The other object of this work... is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables—which forms the chief part of philosophical logic—is the endeavour to see clearly... the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them..."

"Dedekind proves mathematical induction, while Peano regards it as an axiom. This gives Dedekind an apparent superiority, which must be examined. ...not because of any logical superiority, it seems simpler to begin with mathematical induction. And it should be observed that, in Peano's method, it is only when theorems are to be proved concerning any number that mathematical induction is required. The elementary Arithmetic of our childhood, which discusses only particular numbers, is wholly independent of mathematical induction; though to prove that this is so for every particular number would itself require mathematical induction. In Dedekind's method, on the other hand, propositions concerning particular numbers, like general propositions, demand the consideration of chains. Thus there is, in Peano's method, a distinct advantage of simplicity, and a clearer separation between the particular and the general propositions of Arithmetic. But from a purely logical point of view, the two methods seem equally sound; and it is to be remembered that, with the logical theory of cardinals, both Peano's and Dedekind's axioms become demonstrable."

"Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has been connected with science, logic with Greek. But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one. They differ as boy and man: logic is the youth of mathematics and mathematics is the manhood of logic. This view is resented by logicians who, having spent their time in the study of classical texts, are incapable of following a piece of symbolic reasoning, and by mathematicians who have learnt a technique without troubling to inquire into its meaning or justification. Both types are now fortunately growing rarer. So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premises which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary."

"The philosophical tradition that goes from Descartes to Husserl, and indeed a large part of the philosophical tradition that goes back to Plato, involves a search for foundations: metaphysically certain foundations of knowledge, foundations of language and meaning, foundations of mathematics, foundations of morality, etc. […] Now, in the twentieth century, mostly under the influence of Wittgenstein and Heidegger, we have come to believe that this general search for these sorts of foundations is misguided."

"Much has been written on the history of calculus... However, historians tend to harp on the question of logical justification and to spend a disproportionate amount of time on the way it was handled in the nineteenth century. This not only obscures the boldness and vigor of early calculus, but it is overly dogmatic about the way in which calculus should be justified. ...the sheer diversity of foundations for calculus suggests that we have not yet got to the bottom of it."

"As a science mathematics has been adapted to the description of natural phenomena, and the great practitioners in this field... have never concerned themselves with the logical foundations of mathematics, but have boldly taken a pragmatic view of mathematics as an intellectual machine which works successfully. Description has been verified by further observation, still more strikingly be prediction, and sometimes, more ominously, by control of natural forces. Happily, unresolved problems... still remain as challenges."

"In conversations, some quite recent, on the present status of the foundations of mathematics, von Neumann seemed to imply that in his view, the story is far from having been told. Gödel's discovery should lead to a new approach to the understanding of the role of formalism in mathematics, rather than be considered as closing the subject."

"The general truths concerning relations of space which depend upon the axioms and definitions contained in Euclid's Elements, and which involve only properties of straight lines and circles, are termed Elementary Geometry: all beyond this belongs to the Higher Geometry. To this latter province appertain... all propositions respecting the lengths of any portions of curve lines; for these cannot be obtained by means of the principles of the Elements alone. Here then we must ask to what other principles the geometer has recourse, and from what source these are drawn. Is there any origin of geometrical truth which we have not yet explored? The Idea of a Limit supplies a new mode of establishing mathematical truths. ...a curve is not made up of straight lines, and therefore we cannot by means of any of the doctrines of elementary geometry measure the length of any curve. But we may make up a figure nearly resembling any curve by putting together many short straight lines, just as a polygonal building of very many sides may nearly resemble a circular room. And in order to approach nearer and nearer to the curve we may make the sides more and more small more and more numerous. ...by multiplying the sides we may approach more and more closely to the curve till no appreciable difference remains. The curve line is the Limit of the polygon; and in this process we proceed on the Axiom, that "What is true up to the limit is true at the limit." ... thus the relations of the elementary figures enable us to advance to the properties of the most complex cases. A Limit is a peculiar and fundamental conception, the use of which in proving the propositions of the Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. ...The ancients did not expressly introduce this conception of a Limit into their mathematical reasonings, although in the application of what is termed the Method of Exhaustions they were in fact proceeding upon an obscure apprehension of principles equivalent to those of the Method of Limits."

"From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2."

"The abstract formulation of mathematics seems to date back to the German mathematician Moritz Pasch. At any rate, he was the first to study in detail the axioms concerning the order of points on a straight line and to state clearly the assumptions involved in the idea of "betweenness." ...But to the Italian Giuseppe Peano belongs the credit of developing this point of view systematically. His idea, which he began to elaborate about 1880, is to put the whole of mathematics on a purely formal basis, and for this purpose he invented a symbolism of his own. In 1893 he began the publication of a "Formulario di matematica," which is a synopsis of the most important propositions of the different branches of mathematical science, with their demonstrations, expressed entirely in terms of symbolic logic. ...An immense change in the point of view toward the foundations has been brought about since this abstract formulation was put forward. ...Vailati has suggested that this change is very similar to that which a nation undergoes when it changes from a monarchic or aristocratic form of government to a democracy. The point of view fifty years ago was very largely that the foundations of mathematics were axioms; and by axioms were meant self-evident truths, that is, ideas imposed upon our minds a priori, with which we must necessarily begin any rational development of the subject. So the axioms dominated over mathematical science, as it were, by the divine right of the alleged inconceivability of the opposite. And now, what is the new point of view? The self-evident truth is entirely banished. There is no such thing. What has taken the place of it? Simply a set of assumptions concerning the science which is to be developed, in the choice of which we have considerable freedom. The choice of a set of assumptions is very much like the election of men to office. There is no logical reason why we should not choose the more complex propositions; but as a matter of fact we usually choose the simpler, because it is easier to work with them. Not all propositions reach the high position of assumptions; they are elected for their fitness to serve, and their fitness is very largely determined by their simplicity, by the ease with which the other propositions may be derived from them."

"Mathematics is a most conservative science. Its system is so rigid and all the details of geometrical demonstration are so complete, that the science was commonly regarded as a model of perfection. Thus the philosophy of mathematics remained undeveloped almost two thousand years."

"It would be wrong... to assume that the mathematicians of former ages were not conscious of the difficulty. They always felt that there was a flaw in the Euclidean foundation of geometry, but they were satisfied to supply any need of basic principles in the shape of axioms, and it has become quite customary (I might almost say orthodox) to say that mathematics is based upon axioms. In fact, people enjoyed the idea that mathematics, the most lucid of all the sciences, was at bottom as mysterious as the most mystical dogmas of religious faith."

"Metageometry has always proved attractive to erratic minds. Among the professional mathematicians, however, those who were averse to philosophical speculation looked upon it with deep distrust, and therefore either avoided it altogether or rewarded its labors with bitter sarcasm. Prominent mathematicians did not care to risk their reputation, and consequently many valuable thoughts remained unpublished. Even Gauss did not care to speak out boldly, but communicated his thoughts to his most intimate friends under the seal of secrecy, not unlike a religious teacher who fears the odor of heresy. He did not mean to suppress his thoughts, but he did not want to bring them before the public unless in mature shape."

"The labors of Lobatchevsky and Bolyai are significant in so far as they prove beyond the shadow of a doubt that a construction of geometries other than Euclidean is possible and that it involves us in no absurdities or contradictions. This upset the traditional trust in Euclidean geometry as absolute truth, and it opened at the same time a vista of new problems foremost among which was the question as to the mutual relation of these three different geometries. It was Cayley who proposed an answer which was further elaborated by Felix Klein. These two ingenious mathematicians succeeded in deriving by projection all three systems from one common aboriginal form called by Klein Grundgebild or the Absolute. In addition to the three geometries hitherto known to mathematicians, Klein added a fourth one which he calls elliptic. Thus we may now regard all the different geometries as three species of one and the same genus and we have at least the satisfaction of knowing that there is terra firma at the bottom of our mathematics, though it lies deeper than was formerly supposed."

"A store of information may be derived from Bertrand A. W. Russell's essay on the Foundations of Geometry. He divides the history of metageometry into three periods: The synthetic, consisting of suggestions made by Legendre and Gauss; the metrical, inaugurated by Riemann and characterized by Lobatchevsky and Bolyai; and the projective, represented by Cayley and Klein, who reduce metrical properties to projection and thus show that Euclidean and non-Euclidean systems may result from "the absolute.""

"Prof. B. J. Delbœuf and Prof. H. Poincaré have expressed their conceptions as to the nature of the bases of mathematics, in articles contributed to The Monist. The latter treats the subject from a purely mathematical standpoint, while Dr. Ernst Mach in his little book Space and Geometry, in the chapter "On Physiological, as Distinguished from Geometrical, Space" attacks the problem in a very original manner and takes into consideration mainly the natural growth of space conception. His exposition might be called "the physics of geometry.""

"Hermann Grassmann's Lineare Ausdehnungslehre is the best work on the philosophical foundation of mathematics from the standpoint of a mathematician. ...Victor Schlegel called attention to the similarity of Hamilton's theory of vectors to Grassmann's concept of Strecke, both being limited straight lines of definite direction. Suddenly a demand for Grassmann's book was created in the market; but alas! no copy could be had, and the publishers deemed it advisable to reprint the destroyed edition of 1844."

"The problem of the philosophical foundation of mathematics is closely connected with the topics of Kant's Critique of Pure Reason. It is the old quarrel between Empiricism and Transcendentalism. Hence our method of dealing with it will naturally be philosophical, not typically mathematical."

"The data of mathematics are not without their premises; they are not, as the Germans say, voraussetzungslos; and though mathematics is built up from nothing; the mathematician does not start with nothing. He uses mental implements, and it is they that give character to his science."

"At the bottom of the difficulty there lurks the old problem of apriority, proposed by Kant and decided by him in a way which promised to give to mathematics a solid foundation in the realm of transcendental thought. And yet the transcendental method finally sent geometry away from home in search of a new domicile in the wide domain of empiricism."

"Arithmetic automorphic representation theory is one of the most active areas in current mathematical research, centering around what is known as the Langlands program."

"The Langlands Program was initiated by Robert Langlands in the late 1960s in order to connect number theory and harmonic analysis [L1]. In the last 40 years a lot of progress has been made in proving the Langlands conjectures, but much more remains to be done. We still do not know the underlying reasons for the deep and mysterious connections suggested by these conjectures. But in the meantime, these ideas have propagated to other areas of mathematics, such as geometry and representation theory of infinite-dimensional Lie algebras, and even to quantum physics, bringing a host of new ideas and insights. There is hope that expanding the scope of the Langlands Program will eventually help us get the answers to the big questions about the Langlands duality. ... [L1] R. Langlands, Problems in the theory of automorphic forms, in Lect. Notes in Math. 170, pp. 18–61, Springer Verlag, 1970."

"At the heart of Langlands' program is the general notion of an "automorphic representation" π and its L-function L(s, π ) ... both defined via group theory and the theory of harmonic analysis on so-called adele groups ... The conjectures of Langlands ... amount (roughly) to the assertion that the other zeta-functions arising in number theory are but special realizations of these L(s, π ). Herein lies the agony as well as the ecstasy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and p-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding."

"The functoriality conjecture is at the heart of the Langlands program and will undoubtedly remain as a challenge to number theorists for many decades to come. Shortly after formulating his program, however, Langlands proposed to test it in two interdependent settings. The first was the framework of Shimura varieties, already understood by Shimura as a natural setting for a non-abelian generalization of the Shimura-Taniyama theory of complex multiplication. The second was the phenomenon of endoscopy, which can be seen alternatively as a classification of the obstacles to the stabilization of the trace formula or as an opportunity to prove the functoriality conjecture in some of the most interesting cases. After three decades of research, much of it by Langlands and his associates, these two closely related experiments are coming to a successful close, at least for classical groups, thanks in large part to the recent proof of the so-called Fundamental Lemma by Waldspurger, Laumon, and especially Ngô."

"... it was Langlands who, in 1966 at the age of 30, amalgamated and vastly generalized the essential ideas from his contemporaries as well as the recent past: • the ideas of Selberg, Harish-Chandra, and Gelfand which are rooted in Eisenstein series, harmonic analysis, and representation theory of certain classes of noncompact groups; • the usage of adelic structure on groups championed by Godement as well as Tamagawa and Satake; and • Artin’s legacy on class field theory and his quest for a nonabelian class field theory. Among what Langlands had created was a series of interlocking conjectures which later became the foundation of the Langlands program. Those conjectures seek to connect deep arithmetic questions with the highly structured theory of infinite-dimensional representations of Lie groups. The latter is part of harmonic analysis. Those visionary conjectures have exposed, quite unexpectedly, the deeply entwined nature of several seemingly unrelated branches of mathematics."

"He liked putting different pieces of mathematics together: geometry, analysis, topology… so automorphic forms should have appealed to him. But for some reason he didn’t get interested in that at the time. I think the junction between Grothendieck and Langlands was realized only in 1972 at Antwerp. Serre had given a course on Weil’s theorem in 1967–1968. But after 1968 Grothendieck had other interests. And before 1967 things were not ripe. I’m not sure."

"Automorphic forms can be thought of as fundamental particles in harmonic analysis, which deals in part with waves and frequencies. In a symphony orchestra, automorphic forms issue instructions and work with eigenvalues — the different speeds a violin string moves when struck, for instance — to produce the notes played."

"Computational fluid dynamics (CFD) techniques are used to simulate a number of phenomena. Obviously, crashing waves and oceans can leverage CFD, but explosions, fireballs, and smoke effects all make use of CFD nowadays as well."

"CFD merely provides tools for solving the equations of fluid motion; it does not change the conceptual landscape in any fundamental way. Still, it is so powerful that it has become indispensable to the practice of aeronautical engineering."

"One morning early in my (Hersh's) years as a thesis student of Peter Lax, I entered my mentor's office to find him glowing in smiles. “Louis is back!” he cried out to me. Louis, I wondered? Oh yes, Louis Nirenberg, also one of the partial differential specialists on the faculty of NYU's Courant Institute. He had been on leave in England; now he was back home! At the time. I didn't get it. Louis Nirenberg and Peter Lax were grad students together at NYU. Then they both stayed on to become famous faculty members there—Louis, a world master at elliptic partial differential equations, and Peter, a world master of hyperbolic PDEs. They hardly ever collaborated or produced joint publications. But their conversations and their intellectual and emotional interactions were a vital part of their creativity and success."

"Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods."

"The first systematic attack on a problem involving a partial differential equation was carried out in a sequence of 1746 papers by Jean Le Rond d'Alembert (1717-1783), who sought the fundamental modes of vibration of a vibrating string."

"The development of the theory of P.D.E. is closely linked with advances in complex analysis; in fact, Riemann’s approach to the study of conformal mapping via the Dirichlet principle led to the systematic development of the theory of elliptic P.D.E. and associated variational problems. The application of these methods to the theory of several complex variables was initiated by Hodge in his theory of harmonic integrals on compact manifolds. It is this work that led H. Weyl to prove the fundamental hypoellipticity theorem, known as Weyl’s lemma, which in turn led to the development of the general theory of elliptic P.D.E."

"If the original work of Calderón and Zygmund was related to elliptic PDE's, later developments allowed applications of their theory to parabolic equations and to general hypoelliptic operators, and the more recent explosion of interest in the theory of oscillatory integrals and in problems involving curvature has much to do with hyperbolic equations."

"The study of waves is important to virtually every branch of science and engineering. Indeed, waves are also important to everyday life. Sound waves allow us to hear, and electromagnetic waves allow us to see."

"The mass gap is the reason, if you will, that we do not see classical nonlinear Yang-Mills waves. They are a good approximation only under inaccessible conditions. I have spent most of my career wishing that we had a really good way to quantitatively understand the mass gap in four-dimensional gauge theory. I hope that this problem will be solved one day."

"Ergodic theory is a mathematical subject that studies the statistical properties of deterministic dynamical systems. It is a combination of several branches of pure mathematics, such as measure theory, functional analysis, topology, and geometry, and it also has applications in a variety of fields in science and engineering, as a branch of applied mathematics."

"Combinatorics and Algebraic Geometry have enjoyed a fruitful interplay since the nineteenth century. Classical interactions include invariant theory, theta functions, and enumerative geometry."

"The result of a counting process is independent of the way in which it is done. This is really an assumption about what counting is. It would be senseless to try to prove it, because it is so basic; either you see it or you don't—but in the latter case, a proof won't help you a bit."

"The mathematical theory of critical phenomena is currently undergoing intense development. Intertwined with the science of phase transitions, it draws on ideas from probability theory and statistical physics."

"[A]s a liquid changes into a gas at the critical temperature T_c, the heat capacity diverges as c \sim \frac{1}{\left|T-T_c\right|^{0.11008 \ldots}}. The exponent is not known precisely. It is thought not to be a rational number, but should instead be viewed as a universal mathematical constant, similar to \pi"

"Branching processes provide perhaps the simplest example of a phase transition. They occur naturally as a model of the random evolution of a population that changes in time as a result of births and deaths."

"The end of the fifties marked somewhat of a watershed for continuous time Markov chains, with two branches emerging a theoretical school following Doob and Chung, attacking the problems of continuous-time chains through their sample paths, and using measure theory, martingales, and stopping times as their main tools; and an applications oriented school following Kendall, Reuter and Karlin, studying continuous chains through the transition function, enriching the field over the past thirty years with concepts such as reversibility, ergodicity, and stochastic monotonicity inspired by real applications of continuous-time chains to queueing theory, demography, and epidemiology."

"Modular forms have long played a key role in the theory of numbers, including most famously the proof of Fermat's Last Theorem."

"Homological algebra, conceived as a general tool reaching beyond all special cases, was invented by Cartan and Eilenberg (their book “Homological Algebra” appeared in 1956). This book is a very precise exposition, but limited to the theory of modules over rings and the associated functors “Ext” and “Tor”."

"Gaudí was drawn not just to the aesthetics of the catenary but also to what it represented mathematically. His use of catenaries made the structural mechanics of a building a principal feature of its design. Gaudí realized that the entire architecture of a building could be drafted using a model of hanging chains... when he was commissioned to design a church the Colònia Güell... he made an upside-down skeleton of the project. Instead of using metal chains, he used string weighed down by hundreds of sachets containing lead shot. The weight of each sachet on the string created a mesh of 'transformed' catenary curves. The arches of these transformed catenaries were the most stable curves to withstand a corresponding weight at the same position (such as the roof, or building materials)."

"In 1690... Jacob Bernoulli brought up the problem of the catenary in a memoir... in the '...Huygens' solution represents the past... a complex, though skillful, geometrical method. Leibniz, using his new [infinitesimal calculus] reaches a correct analytical formula...y/a = (b^\frac{x}{a} + b^\frac{-x}{a})/2 where a is [a] segment... and b... corresponds to... e... ...supplied two correct constructions ...presents valid statistical arguments and... new and important... equations of equilibrium in differential form. ...In 1697-1698, Jacob Bernoulli was the first to derive the general equations that not only solved the problem, but also permitted the treatment of the more general theme of the equilibrium of a flexible rope, subject to any distribution of tangential (f_t) and normal (f_n) forces. Bernoulli's equations are...\frac{dT}{ds} + f_t = 0, \qquad \frac{T}{r} + f_n= 0where T is the tension, s the curvilinear abscissa, and r the radius of curvature."

"The largest catenary structure of masonry is the Great Hall of the Palace of Taq Kisra, at Qesiphon, then the capital of Persia."

"The trolley-wire must... be suspended with only a very small sag, and to obtain this result without excessive tension in the wire the span must be relatively short, i.e. of the order of 10 ft. to 15 ft. It is obvious that for these short spans the method of construction adopted in tramway practice would be unsuitable, both from the mechanical as well as the electrical standpoint. However, by adopting the catenary system—that is, supporting the trolley-wire from another wire, suspended with considerable sag between supports of moderate span—we are able to obtain a level trolley-wire with a relatively small number of supporting structures. The wire from which the trolley-wire is supported is called the "catenary" or "messenger" wire, and by insulating this wire from the supporting structures there is no necessity for insulated hangers on the trolley-wire."

"Within a shed erected on the construction site of the church of the Sagrada family... Gaudí... made an upside-down model using lightweight cables to represent the structural lines of the future church—a model based on the structural notion of the inverted catenary. ...Analogically represented by little pouches filled with lead pellets the action of the stresses has been done ...The resulting chain configurations are used to determine the geometrical shapes and structural profiles of columns, pillars, arches, and vaults. ...Vicens Vilarrubias i Valls took photos of the model ...Gaudí used these photos upside-down to draw over them the external and internal elevations, studies of details and sections of the building."

"Corol. 6.—In a vertical plane, but in an inverted situation, the chain will preserve its figure without falling, and therefore will constitute a very thin arch or fornix: that is, infinitely small, rigid, polished spheres, disposed in an inverted curve of a catenaria, will form an arch no part of which will be thrust outwards or inwards by other parts, but, the lowest parts remaining firm, it will support itself by means of its figure... none but the catenaria is the figure of the true and legitimate arch or fornix. And when the arches of other figures is supported, it is because in their thickness some catenaria is included. ...From Corol. 5... it may be collected, by what force an arch or buttress presses a wall outwardly, to which it is applied. For this is the same with that part of the force sustaining the chain, which draws according to a horizontal direction. For the force which in the chain draws inwards, in an arch equal to the chain drives outwards. All other circumstances, concerning the strength of walls to which arches are applied, may be geometrically determined from this theory, which are the chief things in the construction of edifices."

"What has been objected by an anonymous author, in the Leipsic Acts of Feb. 1699, in his animadversions on my demonstrations concerning the catenary, is this: that I have undertaken to demonstrate, after my manner, a matter found out and published by others seven years ago. This is true, and I cannot find any thing in it that is blame worthy. Those great men Huygens, Leibnitz, and Bernouilli, have discovered and communicated many properties of the catenaria, but without demonstration. I have contrived demonstrations, which was the thing I undertook to do. But was this matter that is the nature and primary properties of the catenaria all found out and published by others? ...From all ages architects have made use of arches in public buildings, as well for strength as beauty. Yet what was the true geometrical figure of an arch was not known before my demonstrations came out."

"It was comparatively late that the theory of arches attracted the notice of mathematicians. Dr. Hooke gave the hint, that the figure of a perfectly flexible cord or chain, suspended from two points, was the proper form for an arch. Galileo considered the catenary as a parabolic curve, and John Bernouilli appears to have been the first who discovered its nature. Dr. Gregory (Phil. Trans. 1697) published an investigation of its properties, and observes that the inverted catenary is the best form for an arch on account of its lightness. This is true so long as it is not pressed by an extraneous weight. It is not, however, capable of bearing a load on any part, much less of being filled up on the spandrels, which must be the case in practice. Other considerations must be involved before it can be fitted to receive a roadway or other weight, either upon its crown or haunches."

"The flexible chain, hanging under the action of applied force, will assume a certain shape, namely the catenary if the chain is subjected only to its own weight, or a if the load is uniformly distributed horizontally. Whatever the load, there will be a corresponding shape, and the structural action in all cases is the same; purely tensile forces are transmitted along the centre line of the chain."

"As Hooke saw in 1675 with his ut pendet continuum flexile, sic stabit contiguum rigidum inversum, ...a hanging chain may be inverted to give a satisfactory arch to carry the same loads, but working in compression rather than tension. The compressive arch, however, if of vanishingly small thickness, would be in unstable equilibrium, and stability is conferred in practice by making the arch ring of finite depth. Now if purely compressive forces, without bending, are to be transmitted from one portion of the arch to the next (as purely tensile forces are transmitted in the chain), then the arch centre line can accept only a single type of loading. Thus a parabolic arch can carry only a uniformly distributed horizontal load (although the magnitude of the load is arbitrary). It is the depth in a real arch which enables the arch to carry wider ranges of loading; a large number of different idealized centre-line arches can be contained within a given practical profile. ...[T]his must be so, or no mediaeval bridge would have survived its decentering."

"The true Mathematical and Mechanical Form of all manner of Arches for building with the true butment necessary to each of them, a Problem which no Architectonick Writer hath ever yet attempted, much less perform'd. ...Ut pendet continaum flexile, sic stabit contiguum rigidum, which is the Linea Catenaria."

"I will begin with the subject of your bridge... and it is with great pleasure that I learn... that the execution of the arch of experiment exceeds your expectations. ...You hesitate between the catenary and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium. ...I would propose that you make your middle rail an exact catenary, and the interior and exterior rails parallels to that. It is true, they will not be exact catenaries, but they will depart very little from it; much less than portions of circles will."

"The Catenary has its name from Catena, a chain; being the curve which a regular and very flexible chain will assume, if suspended loosely from both ends. It seems to have been first noticed by the famous Galileo, who proposed it as the figure of an arch of equilibration, but unfortunately mistook it for a . In fact, the Catenary, near its vertex, differs insensibly from that curve, but afterwards deviates more considerably. ...the Parabola diverges faster from its axis than the Catenary. The error of Galileo in confounding those two curves was not perceived till , in 1669, ascertained, by actual experiment, that the Catenary is neither a Parabola nor an . It was in 1691, that the penetrating genius of James Bernoulli discovered the true nature of the catenarian curve. A similar investigation was soon produced by John Bernoulli, by Huygens, and by Leibnitz. This latter philosopher, whose powers of invention and stores of learning were alike transcendant, discovered the fine relation of the Catenary to the Logarithmic Curve."

"The most difficult properties of the Catenary were revealed before the close of the seventeenth century. This curve is entitled to particular attention, not only because it throws light on the theory of arches, but because it applies directly to the construction of suspended bridges, which are now deservedly coming into repute."

"A non-catenary curve might be perfectly doable, but it takes more material, it has bigger beam sections, and overall it is much more complicated to construct... Even if the cladding falls out, the interiors and everything else falls away and the whole thing turns to dust and rubble and sand, [the catenary] should still stand."

"I love the catenary because it tells the story of holding up the roof."

"The arch is one of those brilliant innovations... Spanning... with horizontal beams is a losing game. ...By converting all the stress that fractures the middle of... stone beams—technically tension—into compression on stone piers larger... spaces could be spanned. ...But shift the pressure even slightly off center, and the pillar is likely to collapse. ...In their early incarnations, the limitations of both arch and dome was the ability of craftsmen to shape the stones carefully enough to create blocks precisely in the wedge shapes needed for a particular arch. Despite their mathematical sophistication in most other respects, the architects of antiquity lacked a proper geometric solution to the ideal form of the arch. (It was not until 1675 that the English polymath Robert Hooke described mathematically the shape of an arch loaded in pure compression, that is, with no tension, by showing how it describes an upside-down version of the catenary curve of a hanging chain.) As a result, the only way they could design an arch, and its component stones, was completely by eye, and... such tolerances commanded high prices. Rome overcame this drawback with typical ingenuity, first replacing stones and mortar... and expensive stonecutters with relatively cheap bricklayers. Even more ingeniously, some anonymous Roman builder found how to combine the mortar—in Latin pulvis puteoli—with lime, sand, and gravel to make the first concrete. ...The concrete domes of Rome were not surpassed until the age of steel."

"Certainly the most striking contemporary example of a similar form [ arch] is to be found in St. Louis' ... In its incredible scale and construction out of metal plates this structure also serves as a convenient reminder of the important developments of the production of iron and steel that took place during the Industrial Revolution and that have so significantly affected arches as well as all other types of structural forms for the past 150 years."

"Concrete being such a fluid and dynamic material... finds its identity once it is contained. ...A few... who used the forming materials at hand [were]... Antoni Gaudi... ... ... Felix Candela... ... ... Miguel Fisac... Many of these early innovators pushed the computational envelope... Some, like Antoni Gaudi, looked to nature for inspiration. The question... Do we need to "reinvent forming" or just draw from nature, i.e., gravity—catenary action? as Gaudi did. Alan Chandler in fabric framework notes "...for Felix Candela and Christopher Alexander fabric acted as a permanent shutter (framework)..." Chandler speaks of the family of fabric construction that includes... s... Pneumatic structures... Hydrostatic structures and... Shell structures derived from membrane form-finding. When faced with extremely complicated and complex shapes Heinz Isler and Antoni Gaudi used fabric as a modeling tool. These visionaries recognized that hanging chains and fabrics, forming catenaries, are in pure tension and when inverted are in pure compression and very stable. Gaudi, whose ing preceded the works of Candela... looked to nature and natural forms—an approach today called biomimicry..."

"One can roll noncircular wheels over appropriate road surfaces. The most striking example of this is the fact that a can roll on a road that consists of linked catenaries (the catenaries are defined by y = - cosh x) ...(the ride is not smooth in the sense that the center does not move forward at a constant rate of rotation, but... the actual ride... feels quite smooth). This animation was inspired by an exhibit at San Francisco's ..."

"If 99.99% of the universe is in the plasma state then for the remaining 0.01% must we learn a whole discipline of Fluid Mechanics or Hydrodynamics? The answer is No! More often than not, hydrodynamics provides a first level description of an astrophysical fluid."

"Fluid mechanics is a part of applied mathematics, of physics, of many branches of engineering, certainly civil, mechanical, chemical, and aeronautical engineering, and of naval architecture and geophysics, with astrophysics and biological and physiological fluid dynamics to be added."

"Curiosity about at least two of the branches of fluid mechanics and their applications has a long and distinguished history, for in the Proverbs of Solomon the son of David, king of Israel, it was stated in the words of Agur the son of Jakeh that "There be three things which are too wonderful for me, yea four which I know not," of which two were "The way of an eagle in the air" and "The way of a ship in the midst of the sea," which I take to be questions of aerodynamics and naval architecture, questions that concern us still."

"One wonders if the foundations of fluid mechanics or gas dynamics would ever have been secured if Euler or Boltzmann had justified their work to themselves in terms of its contribution to the design of jet aircraft."

"Does set theory, once we get beyond the integers, refer to an existing reality, or must it be regarded, as formalists would regard it, as an interesting formal game? ... A typical argument for the objective reality of set theory is that it is obtained by extrapolation from our intuitions of finite objects, and people see no reason why this has less validity. Moreover, set theory has been studied for a long time with no hint of a contradiction. It is suggested that this cannot be an accident, and thus set theory reflects an existing reality. In particular, the Continuum Hypothesis and related statements are true or false, and our task is to resolve them."

"Henri Poincaré thought the theory of infinite sets a grave malady and pathologic. "Later generations," he said in 1908, "will regard set theory as a disease from which one has recovered."

"Modern descriptive set theory builds on both the classical descriptive set theory and on recursion theory. It has become clear in the 1950’s that the topological approach of classical descriptive set theory and the recursion theoretic techniques of logical definability describe the same phenomena. Modern descriptive set theory unified both approaches, as well as the notation."

"Frege's development began with... "an epoch-making little book" called ', translated as Concept Script or Conceptual Notation... Just as Aristotle's ' is the foundation of traditional or syllogistic logic—the logic of the categorical three-term syllogism, Frege's Begriffsschrift is the keystone of modern or mathematical logic."

"In 1920 logic was mostly a philosopher's garden. There were also a few mathematicians there, cultivating the logical roots of the mathematical tree. Today, Recursion Theory, Set Theory, Model Theory and Proof Theory, logic's major subdisciplines, have become full-fledged branches of mathematics."

"Now this ratio of the single to the double arises, no doubt, from the ternary number, since one added to two makes three; but the whole which these make reaches to the senary, for one and two and three make six. And this number is on that account called perfect, because it is completed in its own parts: for it has these three, sixth, third, and half; nor is there any other part found in it, which we can call an aliquot part. The sixth part of it, then, is one; the third part, two; the half, three. But one and two and three complete the same six. And Holy Scripture commends to us the perfection of this number, especially in this, that God finished His works in six days, and on the sixth day man was made in the image of God. And the Son of God came and was made the Son of man, that He might re-create us after the image of God, in the sixth age of the human race."

"Of the perfection of the number six, which it the first of the numbers which is composed of its aliquot parts. These works are recorded to have been completed in six days... because six is a ,—not because God required a protracted time... but because the perfection of the works was signified by the number six."

"What English mathematicians were most intrigued by and, at times, embarrassed about was the explanation Ramanujan offered regarding his methodology for arriving at solutions. A devout Hindu, Ramanujan claimed he derived most of his solutions with the help of the goddess Namgiri. He claimed that "an equation for me has no meaning unless it expresses a thought of God," and the quantity 2n-1 stood for "the primordial God and several divinities." ...Ramanujan was unable to provide any explanation of his "methodology" in formal mathematical terms."

"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."

"Throughout history philosophers and mystics have sought a compact key to universal wisdom, a finite formula or text which, when known and understood, would provide the answer to every question. The Bible, the Koran, the mythical secret books of Hermes Trismegistus, and the medieval Jewish Cabala have been so regarded. Sources of universal wisdom are traditionally protected from casual use by being hard to find, hard to understand when found, and dangerous to use, tending to answer more and deeper questions than the user wishes to ask. Like God the esoteric book is simple yet undescribable, omniscient, and transforms all who know It. The use of classical texts to foretell mundane events is considered superstitious nowadays, yet, in another sense, science is in quest of its own Cabala, a concise set of natural laws which would explain all phenomena. In mathematics, where no set of axioms can hope to prove all true statements, the goal might be a concise axiomatization of all “interesting” true statements."

"The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction."

"Number mysticism was not original with the Pythagoreans. The number seven, for example, had been singled out for special awe, presumably on account of the seven wandering stars or planets from which the week (hence our names for the seven days of the week) is derived. The Pythagoreans were not the only people who fancied that the odd numbers had male attributes and the even female... Many early civilizations shared various aspects of numerology, but the Pythagoreans carried number worship to its extreme..."

"When a man counts one, two, three, he is not only doing mathematics, he is on the path to the mysticism of numbers in Pythagoras and Vitruvius and Kepler, to the Trinity and the signs of the Zodiac."

"Figuration can be one of the values or qualities that the Divine Artificer uses to decorate or adorn the invisible first principles of the creation (the word kosmos for 'universe' means 'adornment'). Thus arithmetic and geometric figuration, with the resultant harmonic (musical) figurations, are the most essential tools that Plato posits the Divine Craftsman uses to adorn the 'likeness' of the perfect cosmos, which he is attempting to delineate is the psycho-cosmogony of the Timeaus."

"At Babylon... we see a practical polytheism... combined with the application of the exact sciences, and the gods of heaven subjected to the laws of mathematics. This strange association is to us almost incomprehensible, but it must be remembered that at Babylon a number was a very different thing from a figure. Just as in ancient times and, above all, in Egypt, the name had a magic power, and ceremonial words formed an irresistible incantation, so here the number possesses an active force, the number is a symbol, and its properties are sacred attributes. Astrology is only a branch of mathematics, which the heavens have revealed to mankind by their periodic movements."

"A number of aspects of mathematics are not much talked about in contemporary histories of mathematics. We have in mind business and commerce, war, number mysticism, astrology, and religion. In some instances, writers, hoping to assert for mathematics a noble parentage and a pure scientific experience, have turned away their eyes. Histories have been eager to put the case for science, but the Handmaiden of the Sciences has lived a far more raffish and interesting life than her historians allow."

"To what extent does mathematics itself function as a religion. Insofar as the "laws of mathematics" are properties possessed by certain shared concepts, they resemble doctrines of an established church. An intelligent observer seeing mathematicians at work and listening to them talk, if he himself does not study or learn mathematics, might conclude that they are devotees of exotic sects, pursuers of esoteric keys to the universe. Nonetheless... theologians notoriously differ in their assumptions about God... mathematics seems to be a totally coherent unity with complete agreement on all important questions: especially with the notion of proof, a procedure by which a proposition about an unseen reality can be established... different mathematicians, using different methods, working in different centuries, will find the same answers. Can we conclude that mathematics is a form of religion, and in fact the true religion?"

"All thinges which are, & have beyng, are found under a triple diversitie generall. For, either, they are demed Supernaturall, Naturall, or of a third being. Thinges Supernaturall are immateriall, simple, indivisible, incorruptible, & unchangeable. Things Naturall are materiall, compounded, divisible, corruptible, and changeable. Thinges Supernaturall, are, of the minde onely, comprehended: Things Naturall, of the sense exterior, are able to be perceived. In thinges Naturall, probabilitie and conjecture hath place: But in things Supernaturall, chief demonstration, & most sure Science is to be had. By which properties & comparasons of these two, more easily may be described the state, condition, nature and property of those thinges, which, we before termed of a third being: which, by a peculier name also, are called Thynges Mathematicall. For, these, beyng (in a maner) middle, betwene thinges supernaturall and naturall, are not so absolute and excellent, as thinges supernatural: Nor yet so base and grosse, as things naturall: But are thinges immateriall: and neverthelesse, by materiall things hable somewhat to be signified. And though their particular Images, by Art, are aggregable and divisible, yet the generall Formes, notwithstandyng, are constant, unchangeable, untransformable, and incorruptible. Neither of the sense can they, at any tyme, be perceived or judged. Nor yet, for all that, in the royall mynde of man, first conceived. But, surmountyng the imperfection of conjecture, weenyng and opinion, and commyng short of high intellectuall conception, are the Mercurial fruite of Dianœticall discourse, in perfect imagination subsistyng. A mervaylous neutralitie have these thinges Mathematicall, and also a strange participation betwene thinges supernaturall, immortall, intellectual, simple and indivisible: and thynges naturall, mortall, sensible, compounded and divisible. Probabilitie and sensible prose, may well serve in thinges naturall: and is commendable: In Mathematicall reasoninges, a probable Argument, is nothyng regarded: nor yet the testimony of sense, any whit credited: But onely a perfect demonstration, of truthes certaine, necessary, and invincible: universally and necessaryly concluded: is allowed as sufficient for an Argument exactly and purely Mathematical. ... Neither Number, nor Magnitude, have any Materialitie. First, we will consider of Number, and of the Science Mathematicall, to it appropriate, called Arithmetike: and afterward of Magnitude, and his Science, called Geometrie. ...How Immateriall and free from all matter, Number is, who doth not perceave? yea, who doth not wonderfully wonder at it? For, neither pure Element, nor Aristotele's Quinta Essentia, is hable to serve for Number, as his propre matter. Nor yet the puritie and simplenes of Substance Spirituall or Angelicall, will be found propre enough thereto. And therefore the great & godly Philosopher Anitius Boetius, sayd... All thinges (which from the very first originall being of thinges, have bene framed and made) do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator."

"Newton's goal was incomparably more vast than the discovery of the "mathematical principles of natural philosophy." Newton wished to penetrate the divine principles beyond the veil of nature, and beyond the veils of human record and received revelation as well. His goal was the knowledge of God, and for achieving that goal he marshaled the evidence from every source available to him: mathematics, experiment, observation, reason, revelation, historical method, myth, the tattered remnants of ancient wisdom. ...one result of the restricted interests of modernity has been to look askance at Newton's biblical, chronological, and alchemical studies: to consider his pursuit of prisca sapientia [ancient wisdom] as irrelevant. None of those was irrelevant to Newton, for his goal was considerably more ambitious than a knowledge of nature. His goal was Truth, and for that he utilized every possible resource."

"The Pythagoreans... were fascinated by certain specific ratios, and especially by those which relate what we call today the arithmetic [A = \frac{a + b}{2}], geometric [G = \sqrt{ab}], and harmonic [H = \frac{2ab}{a + b}] means. ... The particular ratios between these means in which the Pythagoreans were interested were...A:G = G:HThe other relationship is expressed bya:A = H:bThe Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the ns by Pythagoras himself after having been taken prisoner in Egypt. s lay at the heart of the Pythagorean theory of music. If a string is divided into 12 parts, the ratio 12:6, or 2:1, gives us the octave. If the arithmetic and harmonic means of 12 and 6 are now taken, we haveA = \frac{6 + 12}{2} = 9andH = \frac{2 \times 6 \times 12}{6 + 12} = 8The [perfect proportions] 9:6 and 12:8 both equal to 3:2, correspond to the fifth in the theory of music. Similarly... 8:6 and 12:9, both equal to 4:3, corresponding to the fourth. In this way, certain intervals, crucial in the theory of music, were all obtained by ratios involving the numbers 1, 2, 3, and 4, which came to be of mystical significance for they also represented the perfect triangle, yielding the 10, the sum of 1, 2, 3, and 4. These ratios of the musical fifth and fourth were used by the Pythagoreans to obtain the whole tone of the ,\frac{3}{2} : \frac{4}{3} = 9:8and the semi-tone,\frac{4}{3} : (\frac{9}{8})^2 = 256:243."

"Throughout man's history there has been a close connection between the natural numbers and the activities of the divine Creator. The natural numbers have always been seen by some at least in every civilization to be symbolic of deep esoteric religious beliefs. The Babylonians had a hierarchy of sixty gods each of which was associated with one of the first sixty natural numbers. ...In ancient India, we find religious significance assigned to each of the first 101 numbers, and the Mayan civilization the first thirteen numbers all represented gods. There is something of a parallel in other sciences. Astronomy grew from astrology and chemistry from alchemy. It is therefore not surprising that number theory, first developed by the Pythagoreans, was associated with their religious beliefs and practices."

"In 1882 Mittag-Leffler suggested that Cantor's work should be translated into French, but he was more positive toward Cantor's "mathematics" than his "philosophy." Hermite and Poincaré agreed that French readers of Cantor might object to "research which is at the same time philosophical and mathematical and where arbitrariness has an excessive place." Hermite thought that Cantor's work was more German metaphysics than mathematics. Perhaps for that reason he suggested as translator a Jewish priest at San-Sulpice, observing that "[Cantor's] philosophical turn of mind will not be an obstacle for a translator who knows Kant." During the translation work, Hermite and Poincaré made suggestions for cuts in the philosophical parts. And even some of the mathematics seemed too abstract to them; they observed, as Poincaré remarked, that "higher infinities" seemed to "have a whiff of form without matter, which is repugnant to the French spirit.""

"We left him after the battle of Prague in 1620. He remained in Prague until December, and then took up his winter quarters with a portion of the Duke of Bavaria's troops left in the extreme south of Bohemia. At this time a new and strange influence had come within his life. In that wonderfully productive winter spent with the Bavarian troops, while active operations were impossible, Descartes heard much of what was going on in the literary and scientific world. Amongst other things, he heard of that strange brotherhood of which we have so often read and yet of which we know so little—the Order of the Rosicrucians. They, it was said, taught a new wisdom, the hitherto undiscovered science. This was enough to excite Descartes' interest: Germany was thoroughly aroused; something had been discovered which was to be kept to the few initiated ones. The same Descartes who was in the habit of disdaining the work of others, began to think he had been precipitate in his judgments. Here was he searching for the Truth patiently and with difficulty, and there were men who declared the way had been opened to them. If these were simple imposters, then it was the duty of any honest man to expose their imposition; but if on the issue which to him was all-important, they had found any light, then, as he told his friend, how despicable would it be in him to disdain to be taught anything out of which he might obtain new knowledge. He made it his duty to discover a member of this learned body, in order to discuss the matter with him and subsequently settle his own conclusions. [footnote] The treatise which Descartes specially dedicates to the Order, is that which was written in 1619-20 and never published, the Polybii cosmopolitani Thesaurus mathematicus, which "sets forth the true means of solving this science, and in which it is demonstrated that nothing further can be supplied by the human mind"; "it is specially designed to relieve the pains of those who, entangled in the Gordian knot of the sciences, uselessly waste the oil of their genius." It is dedicated to all learned men, and more especially to the illustrious Brethren of the Rosicrucian Order in Germany. This may have been the treatise that in his journal he promises, if he can obtain sufficient books, and if it seems worthy of publication, on 23rd September 1620, though why the date should be thus fixed, we do not know. Probably when the time came he did not consider it "worthy," and now all is lost excepting the simple title."

"The word witr literally means "odd number," and tradition says: "God is odd, He loves the odd." Musalmāns pay the greatest respect to an odd number. It is considered unlucky to begin any work, or to commence a journey on a day, the date of which is an odd number."

"The work of Copernicus and Kepler is the work of men searching the universe for the harmony their commingled religious and scientific beliefs assured them must exist, and in aesthetically satisfactory mathematical form. To men who were convinced that an omnipotent being, designing a mathematical universe, would certainly prefer the superior features of the new theory, there could only be one conclusion—the heliocentric theory was true. ... There was a mystic element in their thinking that now seems anomalous in great scientists. ... But despite the presence of mystic, poetic, and religious influences, Copernicus and Kepler were thoroughly rational in rejecting any speculations or conjectures that did not agree with observations. What distinguishes their work from medieval vaporizings is not only the mathematical framework of their theories but their insistence of making the mathematics fit reality. In addition, the preference they showed for a simpler mathematical theory is a thoroughly modern scientific attitude. ...their willingness to think in terms of, and indeed their preference for, a heliocentric theory, despite the arguments ...against the reasonableness of such a view in their time, marks them as heroes in the intellectual battles man has fought to attain a secure dwelling place for science."

"When one compares the pre-Greek and Greek understanding of the concepts of mathematics... [one] notes the sharp transition from the concrete to the abstract... One advantage of treating abstraction is the gain in generality. ...Another advantage ...abstracting ..frees the mind from burdensome and irrelevant details ...The emphasis ...was part and parcel of their outlook on the entire universe. ...Pythagoreans and Platonists maintained that truths could be established only about abstractions. ...impressions received by the senses are inexact, transitory, and constantly changing ...truth, by its very meaning, must consist of permanent, unchanging, definite entities and relationships. ...man may rise to the contemplation of ideas. These are eternal realities and the true goal of thought, whereas mere "things are the shadows..." Thus Plato would say that... reality is in the universal type or idea... Beauty, Justice, Intelligence, Goodness, Perfection, and the State, are independent of the superficial appearances... of the flux of life... of... biases and warped desires... they are ...constant and invariable, and knowledge concerning them is firm and indestructible. ...physical or sensible objects suggest the ideas just as diagrams of geometry suggest abstract geometrical concepts... but one must not lose himself in trivial and confusing minutiae. The abstractions of mathematics possessed a special importance for the Greeks. ...to pass from a knowledge of the world of matter to the world of ideas, man must train ...These highest realities blind the person ...The study of mathematics helps make the transition from darkness to light. ...man learns to pass from concrete figures to abstract forms ...this study purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas. ...to lift the mind above mundane considerations and enable it to apprehend the final aim of philosophy, the idea of the Good."

"Plus one, minus one, plus one, minus one, etc. By adding the first two terms, the next two, and so forth, the result is converted into another for which each term is zero. Grandi, the Italian Jesuit, had concluded the possibility of creation from this series; because the result is always equal to &frac12;, he saw the unborn fraction of infinitely many zeros, or nothingness. It was thus that Leibnitz saw in his binary arithmetic the image of creation. He imagined that Unity represented God, and Zero the void; and that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in this system of numeration. This conception was so pleasing to Leibnitz that he communicated it to the Jesuit, [Claudio Filippo] Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, since he was very fond of the sciences, to Christianity. I mention this merely to show how childhood prejudices may lead astray even the greatest men."

"A fifteenth century trend toward Platonic and Pythagorean mysticism, accompanying the growth of humanism over... earlier Scholastic philosophy, encouraged the previously denied use of the infinite and infinitesimal in geometry. As a result... mathematics was viewed as independent of the senses, not bound by empirical investigations, and thus free to use the infinitite and infinitesimal, provided that no inconsistencies resulted. By the early seventeenth century this view opened up bold new approaches."

"No one has attempted a language or characteristic which includes at once both the arts of discovery and of judgement, that is, one whose signs and characters serve that same purpose that arithmetical signs serve for numbers, and algebraic quantities for quantities taken abstractly. ...since God has bestowed these two sciences on mankind, he has sought to notify us that a far greater secret lies hidden in our understanding, of which these are but the shadows. ...When I... took up logic and philosophy... I once raised a doubt concerning the categories. I said that just as we have categories or classes of simple concepts, we ought also to have a new class of categories in which propositions or complex terms themselves may be arranged in their natural order. For I had not even dreamed of demonstrations at that time and did not know that the geometricians do exactly what I was seeking when they arrange propositions in an order such that one is demonstrated from the other. ...I necessarily arrived at this remarkable thought... that a kind of alphabet of human thoughts can be worked out and that everything can be discovered and judged by a comparison of the letters of this alphabet and an analysis of the words made from them. ...I wrote a Dissertation of the Art of Combinations...published... in 1666, and in which I laid my remarkable discovery before the public. This dissertation was... such as might be written by a youth just out of the schools... not yet conversant with the real sciences. For mathematics was not cultivated in those parts; if I had spent my childhood in Paris, as did Pascal, I might have advanced these sciences earlier. ... Why... no mortal has ever essayed so great a thing—this has often been an object of wonder to me. ...these considerations should have occurred from the very first, just as they occurred to me as a boy interested in logic, before I even touched on ethics, mathematics, or physics, solely because I always looked for first principles. The true reason for straying from the portal of knowledge is... that principles usually seem dry and not very attractive... Yet I am most surprised at the failure of three men to undertake so important a thing—Aristotle, Joachim Jung, and René Descartes. For when Aristotle wrote the Organon and the Metaphysics he laid open the inner nature of concepts with great skill. Joachim Jung... is a man... of such rare judgement and breadth of mind that I cannot think of anyone, not even excepting Descartes himself, from whom a great revival of science might better have been expected, if only he had been known and supported. ...As for Descartes ...since he had aimed at his own excessive applause, he seems to have broken off the thread of his investigation and to have been content with metaphysical meditations and geometrical studies by which he could draw attention to himself. ...he did not adequately think through the full reason and force of the thing. For had he seen a method of setting up a reasonable philosophy with the same unanswerable clarity as arithmetic, he would hardly have used any way other than this to establish a sect of followers, a thing which he so earnestly wanted. For by applying this method of philosophizing, a school would from its very beginning, and by the very nature of things, assert its supremacy in the realm of reason in a geometrical manner and could never perish nor be shaken until the sciences themselves die through the rise of a new barbarism among mankind. As for me, I kept this line of thought. ...For this is what I finally discovered ...Nothing more is necessary to establish the characteristic which I an attempting, at least to the point sufficient to build the grammar of this wonderful language and a dictionary for the most frequent cases, or what amounts to the same thing, nothing more is necessary to set up the characteristic numbers for all ideas than to develop a philosophical and mathematical 'course of studies'... based on a certain new method which I can set forth... a few selected men could finish the matter in five years. It would take them only two, however, to work out by an infallible calculus the doctrines most useful for life, that is, those of morality and metaphysics."

"As soon as his fevered imagination had cooled, he determined at once to produce his invention, and notes on the 23rd February (1620) he was thinking of finding a publisher; but presently he changed his intention; and this treatise... is by Baillet suspected to have been possibly the Olympica... Descartes had heard much of the Rosicrucians,—a hidden confraternity who were believed to have attained some mysterious key to natural knowledge apart from theology, and who were supposed to be spread all through society. A considerable literature of attack and of apology as regards this sect then occupied public interest. Baillet tells us all that was then known about them. In a MS. called Cartesii liber de studio bonœ mentis ad Musœum, Descartes confessed that he had done all he could to find out a member of the brotherhood and learn what he might of their magic secrets, but was completely and permanently unsuccessful. Nevertheless, he had talked so much about it at the time, that he found himself set down as a Rosicrucian, and had some difficulty in clearing himself of the imputation. But in the winter of 1619-20 he had not yet given up hopes of finding out this mystery, and the title of a book found among Leibnitz's transcripts gives us the clue to the lost treatise of this date. It was Polybii cosmopolitani Thesaurus mathematicus (I translate the sequel), 'in which are set forth the true means of solving all the difficulties of this science, and there is demonstrated that, as regards it, nothing further can be supplied by the human mind; with the intention of challenging the delay, and exposing the rashness of those who promise to show new marvels in all the sciences, as well as to relieve the torture (Iabores cruciabiles) of many, who, entangled in some of the s of this science, night and day spend uselessly the oil of their genius,—now offered to the learned of all the world, and especially celeberrimis in Germania Fratribus Roseœcrucis.' ... This interesting though confused title shows clearly what Descartes' inventum mirabile was... simply the solution of all geometrical problems by algebraical symbols. What agitated his mind so greatly was that the discovery would not cease there, but that by means of this new and improved calculus he could apply mathematical demonstration to all the realm of nature. ... But at this time he had only simplified his mathematics so as to make it a general method of investigation. It remained for him to likewise so simplify nature as to make it capable of submitting to his analysis. ...He ...soon turned aside to , to make trial of his new method of solving problems on Faulhaber and other mathematicians of distinction. The story of is repeated, mutatis mutandis, in the case of Faulhaber. He first despised, and then sarcastically challenged, the young inquirer, who on this occasion, however, showed considerable self-confidence, and not only solved the problems proposed, but showed general methods of doing so, and even of determining the solubility of various new problems, or the reverse. He also solved the problems proposed by Peter Roten [Roth] in reply to a challenge of Faulhaber in his algebra. These successes must have made Descartes feel assured of his inventum mirabile as far as mathematics went. But he presently suspended further study..."

"Now Faulhaber was a real Rosicrucian and a very ardent one, and one is justified in assuming, in spite of Baillet's denials, that Descartes found in him the man he was seeking, and that through him Descartes entered into direct contact with the intellectual atmosphere of the Rosecrucians. Might not such contact, however fleeting, have a declining influence upon the moral lines and aims of the philosopher's life? May we not even ask ourselves whether at its origin. Descartes' great idea did not permit the supposition that he intended—an intention that became hazier as time went on—fearlessly to transpose to the plane of everyday reason and of the most widespread common sense the design followed on the plane of alchemic mysteries by the naive Rosicrucians, and in doing so render it much less "uplifted," but much more efficacious—mathematics replacing the Cabala leading to universal knowledge, the hermetic sciences and their occult qualities giving place henceforth to geometric physics and the art of mechanics, as the elixir of life to the laws of rational medicine?"

"As nothing on earth is more conservative than religion, we have still a world of symbolism existing amongst us which is far older than our sects and books, our creeds and articles, a relic of a forgotten pre-historic past. Untold ages before writing was invented, it is believed that men attempted to express their ideas in visible forms. Yet how can a savage, who is unable to count his fingers up to five, and has no idea of abstract number, apart from things, whose habits and thoughts are of the earth, earthy, form a conception of the high and holy One who inhabiteth eternity? Even under the highest forms of ancient civilisation, abundant proofs exist that the imagination of men, brooding over the idea of the Unseen and the Infinite, were bounded by the things which were presented in their daily experience, and which most moved their passions, hopes and fears. Through these, then, they attempted to embody such religious ideas as they felt. They could not teach others without visible symbols to assist their conceptions; and emblems were rather crutches for the halting, than wings to help the healthy to soar. Mankind in all ages has clung to the visible and tangible. The people care little for the abstract and unseen."

"Hardy had a] profound conviction that the truths of mathematics described a bright and clear universe, exquisite and beautiful in its structure, in comparison with which the physical world was turbid and confused. It was this that made his friends... think that in his attitude to mathematics there was something which, being essentially spiritual, was near to religion."

"In our modern era, God and mathematics are usually placed in totally separate arenas of human thought. But... this has not always been the case, and even today many mathematicians find the exploration of mathematics akin to a spiritual journey. The line between religion and mathematics becomes indistinct. In the past, the intertwining of religion and mathematics has produced useful results and spurred new areas of scientific thought. ...In many ways, the mathematical quest to understand infinity parallels mystical attempts to understand God. Both religion and mathematics attempt to express relationships between humans, the universe, and infinity. Both have arcane symbols and rituals, and impenetrable language. Both exercise the deep recesses of our minds and stimulate our imagination. Mathematicians, like priests, seek "ideal," immutable truths and then often try to apply these truths to the real world. Some atheists claim another similarity: mathematics and religion are the most powerful evidence of the inventive genius of the human race. Of course, there are also many differences between mathematics and religion."

"Only a fool would try to compress several millennia's blending of mathematics and religion. We proceed."

"For such purposes as these a very slender knowledge of geometry, and a small portion of arithmetic would suffice;—but as for any considerable amount thereof, and great progress in it, we must inquire how far they tend to this,—namely, to make us apprehend more easily the idea of the good:—and we say that all things contribute thereto, which compel the soul to turn itself to that region in which is the happiest portion of true being, which it must by all means perceive. ...this science has an entirely opposite nature to the words employed in it by those who practise it. ...They speak somehow most absurdly, and necessarily so, since all the terms they use seem to be with a view to operation and practice,—such as squaring, producing, adding, and such like sounds; whereas on the other hand, the whole science should be studied for the sake of real knowledge. ...Is this then further to be agreed on? ...That [it be studied] with a view to the knowledge of eternal being, and not of that which is subject to generation and destruction? ...It would have a tendency, therefore... to draw the soul to truth, and to cause a philosophic intelligence to direct upwards [the thoughts] which we now improperly cast downwards. ...We must give special orders, that the inhabitants of that fine state of yours should by no means omit the study of geometry, since even its by-works are not inconsiderable. ...it is not altogether a trifle, but rather difficult to persuade that by these branches of study some organ of the soul in each individual, is purified and rekindled like fire, after having been destroyed and blinded by other kinds of study,—an organ, indeed, better worth saving than ten thousand eyes, since by that alone can truth be seen."

"As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space."

"It arouses our innate knowledge. ..takes away the forgetfulness and ignorance [of our former existence] that we have from birth,. . . fills everything with divine reason, moves our souls towards Nous,. . . and through the discovery of pure Nous leads us to the blessed life."

"[T]he soul has its essence in mathematical ideas, and it has a prior knowledge of them. .. and brings of them to light when it is set free of the hindrances that arise from sensation. For our sense-perceptions engage the mind with divisible things... and...every divisible thing is an obstacle to our returning upon ourselves. ... Consequently when we remove these hindrances. . we become knowers in actuality..."

"An equation for me has no meaning unless it expresses a thought of God."

"It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosophers, except in France. Philosophical works on the subject, such as Cohen's Princip der Infinitesimalmethode und seine Geschichte, are vitiated, as regards the constructive theory, by an undue mysticism inherited from Kant, and leading to such results as the identification of intensive magnitude with the extensive infinitesimal."

"It seems necessary to say something at the outset in justification of the scientific as against the mystical attitude. Metaphysics, from the first, has been developed by the union or the conflict of these two attitudes. Among the earliest Greek philosophers, the Ionians were more scientific and the Sicilians more mystical. But among the latter, Pythagoras, for example, was in himself a curious mixture of the two tendencies: the scientific attitude led him to his proposition on right-angled triangles, while his mystic insight showed him that it is wicked to eat beans. Naturally enough, his followers divided into two sects, the lovers of right-angled triangles and the abhorrers of beans; but the former sect died out, leaving, however, a haunting flavour of mysticism over much Greek mathematical speculation, and in particular over Plato's views on mathematics. Plato, of course, embodies both the scientific and the mystical attitudes in a higher form than his predecessors, but the mystical attitude is distinctly the stronger of the two, and secures ultimate victory whenever the conflict is sharp. Plato, moreover, adopted from the the device of using logic to defeat common sense, and thus to leave the field clear for mysticism—a device still employed in our own day by the adherents of the classical tradition. The logic used in defence of mysticism seems to me faulty as logic..."

"Mathematics has ceased to seem to me non-human in its subject matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal. I think that the timelessness of mathematics has none of the sublimity that it once seemed to me to have, but consists merely in the fact that the pure mathematician is not talking about time. I can no longer find any mystical satisfaction in the contemplation of mathematical truth. ... One effect of the War was to make it impossible to go on living in a world of abstraction. ... I have no longer the feeling that intellect is superior to sense, and that only Plato's world of ideas gives access to the 'real' world. I used to think of sense, and of thought which is built on sense, as a prison... I now think of sense, and of thoughts built on sense, as windows... I think that we can, however imperfectly, mirror the world, like Leibniz's monads; and I think it is the duty of the philosopher to make himself as undistorting a mirror as he can. ...to recognize such distortions ...Of these, the most fundamental is that we view the world from the here and now, not with that large impersonality which theists attribute to the Deity. To achieve such impartiality is impossible for us, but we can travel a certain distance towards it. To show the road to this end is the supreme duty of the philosopher."

": Prithee, no more prattling; go; I'll hold. This is the third time; I hope good luck lies in odd numbers. Away, go! They say there is divinity in odd numbers, either in nativity, chance, or death. Away!"

"One thing that particularly interested Plato was the mysticism of numbers. In his Republic (Book VIII) he speaks in an obscure fashion of a certain mystic number, but he does not make clear what this number is. He calls it "the lord of better and worse births"... One theory is that 60, or 12,960,000 is the Platonic number. This number played an important part in the mysticism of the Hindus and the Babylonians, and it is possible that Pythagoras found it on the banks of the Euphrates, if he really studied there, and that he took it with him to Crotona, passing it on to his disciples, who, in turn, told it to Plato and his followers. ...More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch..."

"Num. xxxiii. 1. 'Build me here seven alters and prepare me here seven oxen and seven rams.' The ancients were very superstitious about certain numbers, supposing that God delighted in odd numbers.Around his waxen image first I wind Three woollen fillets, of three colours joined; Thrice bind about his thrice devoted head, Which round the sacred alter thrice is led. Unequal numbers please the gods."

"In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the s: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the ... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the ... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology."

"Leibniz discovered his future with the help of a Nuremberg society of alchemists. ...In private ...he exhibited an avid interest in the subject throughout his life—as did many of his contemporaries, such as Isaac Newton. Indeed, he was so certain that he would one day soon discover the means to turn lead into gold that at one point he fretted that the resulting oversupply of the yellow metal might drive down the price and thus deprive him of hard-earned profits."

"Among the first of those who bade adieu to the Scholastic creed was the Cardinal Nicolas Cusanus, a man of rare sagacity and an able mathematician; who arranged and republished the Pythagorean Ideas, to which he was much inclined, in a very original manner, by the aid of his Mathematical knowledge. He considered God as the unconditional Maximum, which at the same time, as Absolute Unity, is also the unconditional Minimum, and begets of Himself and out of Himself, Equality and the combination of Equality with Unity (Son and Holy Ghost). According to him, it is impossible to know directly and immediately this Absolute Unity (the Divinity); because we can make approaches to the knowledge of Him only by the means of Number or Plurality. Consequently he allows us only the possession of very imperfect notions of God, and those by mathematical symbols. It must be admitted that the Cardinal did not pursue this thought very consequently, and that his view of the universe which he connected with it, and which represented it as the Maximum condensed, and thus become finite, was very obscure. Nor was he more successful in his view of the one-ness of the Creator and of Creation, or in his attempt to explain the mysteries of the Trinity and Incarnation, by means of this Pantheistic Theism. Nevertheless, numerous profound though undeveloped observations on the faculty of cognition, are found in his writings, interspersed with his prevailing Mysticism. For instance, he observes, that the principles of knowledge possible to us are contained in our ideas of Number (ratio explicata) and their several relations; that absolute knowledge is unattainable to us (precisio veritatis inattingibilis, which he styled docta ignorantia), and that all which is attainable to us is a probable knowledge (conjectura). With such opinions he expressed a sovereign contempt for the Dogmatism of the Schools."

"Neo-Pythagoreans, like Nicomachus of Gerasa... and Iamblichus... revel in... number mysticism. ...The purpose of the layman's Introduction to Arithmetic... is to explain in a manner intelligible to every one, the wonderful and divine properties of numbers. In a very entertaining way he tells about s... square, rectangular and s... gnomic numbers and spatial numbers, about ratios and the distinction between multiple and epimoric."

"The development of the theory of numbers had its root in the number mysticism of Pythagoras, attained a strictly scientific character in the theory of the even and the odd, and led finally to the systematic establishment of the theory of the divisibility and the proportion of numbers as found in Book VII of the Elements."

"Quite aside from the fact that mathematics is the necessary instrument of natural science, purely mathematical inquiry in itself... by its special character, its certainty and stringency, lifts the human mind into closer proximity with the divine than is attainable through any other medium. Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means. ...The connection between mathematics of the infinite and the perception of God was pursued most fervently by Nicholas of Cusa, the thinker who... intoned the new melody of thought which with Leonardo, Bruno, Kepler, and Descartes gradually swells into a triumphant symphony. He recognizes that the scholastic form of thinking, Aristotelian logic, which is based on... the excluded third, cannot... think the absolute, the infinite... every kind of "rational" theology is rejected, and "mystic" theology takes its place. ...The true love of God is amor de intellectualis [intellectual love]. ...Cusanus does not refer to the mystic form of contemplation, but rather to mathematics and its symbolic method. ...This urge finds its simplest expression in the sequence of numbers, which can be driven beyond any place by repeated addition of one."

"In certain of their aspects, as well as in the attitudes of their practitioners, there is a close analogy between religion and mathematics. To some of its practitioners, religion represents an ideal essentially superpersonal, existing independently of its devotees; to others, it affords rules by which to live, having no intellectual content whatever. And like religion, mathematics traces its beginnings back to prehistoric primitive cultures, as a result of which the numbers with which we count, the so-called "natural numbers"... are frequently regarded in a mystical fashion not ordinarily associated... Also like religion, mathematics furnishes a fascinating study from an anthropological viewpoint, only the meagre beginnings of which are to be found in antropological literature—a gap which I am hoping to help fill..."

"Socrates, in the Phædrus, delivers to us three characters who are elevated from sense, because they fill up and accomplish the primary life of the soul, i.e. the philosopher, the lover, and the musician. But the beginning and path of elevation to the lover, is a progression from apparent beauty, using as excitations the middle forms of beautiful objects. But to the musician, who is allotted the third seat, the way consists in a transition from sensible to invisible harmonies, and to the reasons existing in these. So that to the one, sight is the instrument of reminiscence, and to the other, hearing. But to him who is by nature a philosopher, from whence and by what means is reminiscence the prelude of intellectual knowledge, and an excitation to that which truly is, and to truth itself. For this character also, on account of its imperfection, requires a proper principle: for it is allotted a natural virtue, an imperfect eye, and a degraded manner. It must therefore be excited from itself; and he who is of such a nature, rejoices in that which is. But to the philosopher, says Plotinus, the mathematical disciplines must be exhibited, that they may accustom him to an incorporeal nature, and that afterwards using these as figures, he may be led to dialectic reasons, and to the contemplation of all the things which are. And thus it is manifest, from hence, that the mathematics are of the greatest utility to philosophy."

"Whatever to imperfect natures appears difficult and arduous in obtaining the true knowledge of the gods, the mathematical reasons render, by their images, credible, manifest, and certain. Thus, in numbers, they indicate the significations of super-essential properties, but they evince the powers of intellectual figures, in those figures which fall under cogitation. Hence it is, that Plato, by mathematical forms teaches us many and admirable sentences concerning the gods, and the philosophy of the Pythagoreans; using these as veils, conceals from vulgar inspection the discipline of divine sentences. For such is the whole of the Sacred and Divine Discourse, that of Philolaus in his Bacchics, and the universal method of the Pythagoric narration concerning the Gods. But it especially refers to the contemplation of nature, since it discloses the order of those reasons by which the universe is fabricated, and that proportion which binds, as Timæus says, whatever the world contains, in union and consent; besides it conciliates in amity things mutually opposing each other, and gives convenience and consent to things mutually disagreeing, and exhibits to our view simple and primary elements, from which the universe is composed, on every side comprehended by commensurability and equality, because it receives convenient figures in its proportions, and numbers proper to every production, and finds out their revolutions and renovations, by which we are enabled to reason concerning the best origin, and the contrary dissolution of particulars. In consequence of this, as it appears to me, Timaeus discloses the contemplation concerning the nature of the universe, by mathematical names, adorns the origin of the elements with numbers and figures, referring to these their powers, passions, and energies; and esteeming as well the acuteness as the obtruseness of angles, the levity of sides, or contrary powers, and their multitude and paucity to be the cause of the all-various mutation of the elements."

"When we consider... peculiarities of mental constitution, reflect upon the nature of the intellectual matrix which brought forth mathematics as a science, and call to mind that universally, and at all times, curious superstitions and extravagant notions appear to have been associated with ideas of number, the hypothesis not unnaturally suggests itself that even now the philosophy of mathematics may not wholly have freed itself from mystical implication. ...it may be that all ratiocinative processes, no matter what the subject, in which the current and continual substitution of symbols (of any kind) for concepts is a prime condition of the effective conduct of the process, are provocative of that attitude of mind. ...and I incline to think that not infrequently the more gifted is the individual for the prosecution of purely symbolic trains of thought the greater is the provocation to this mental attitude."

"Discussion of the principles which underlie... development of mathematical expression has been confined almost exclusively within the inner circle of mathematicians interested in the philosophy of their science. But well-nigh up to the close of last century the explanations of this development—especially with regard to Imaginary Quantity—were almost purely mystical: they really explained nothing, threw no light on the motives and processes of thought which obscurely prompted it, and thus left it as much of an enigma as ever. In recent years, however, a real endeavour has been made to unravel this long-standing difficulty in the development of algebraic symbolism; it is even averred that we have now at last a completely adequate explanation of it. The claim is... that the doctrine has been purged of its mystical implications. But... the claim outruns the performance: the purgation is not thorough..."

"The necessity of ratiocination by symbols does carry in its train a tendency to mysticism from which even the keenest intellects cannot always free themselves. ...[footnote:] Some thinkers have spoken of, or alluded to, a certain 'reaction of language upon thought' which results in error or confusion... It is to this kind of illusion or, to... the mental attitude which makes it possible, that I refer when I speak of mysticism, the mystical tendency or bias."

"Even in pure mathematics, conspicuously the domain of definite and stable conceptions, careful investigation might be found to reveal evidence of an infiltration of the mystical. ...this fact, if so it be, in some measure accounts for the almost insuperable difficulty which many strenuous minds find in mastering even the first principles of the subject, as well as for the purely intellectual repugnance with which they regard it."

"Things must be reduced again to what they seem; it is vain and terrible to take them for what we find they are. M. Bergsen is at bottom an apologist for very old human prejudices, an apologist for animal illusion. His whole labour is a plea for some vague but comfortable faith which he dreads to have stolen from him by the progress of art and knowledge. ...Mr. Bergsen is afraid of space, of mathematics, of necessity, and of eternity; he is afraid of the intellect and the possible discoveries of science; he is afraid of nothingness and death. These fears may prevent him from being a philosopher in the old and noble sense of the word; but they sharpen his sense for many a psychological problem, and make him the spokesman of many an inarticulate soul. Animal timidity and animal illusion are deep in the heart of all of us. Practice may compel us to bow to the conventions of the intellect, as to those of polite society; but secretly, in our moments of immersion in ourselves, we may find them a great nuisance, even a vain nightmare. Could we only listen undisturbed to the beat of protoplasm in our hearts, would not that oracle solve all the riddles of the universe, or at least avoid them. ...it is necessary for the mystic to sally forth and attack his enemy on his own ground. If he refuted physics and mathematics simply out of his own faith, he might be accused of ignorance of the subject. He will therefore study it conscientiously, yet with a certain irritation and haste to be done with it, somewhat as a Jesuit might study Protestant theology. ...in retracing a free inquiry in his servile spirit, he remains deeply ignorant, not indeed of its form, but of its nature and value. ...physical science never solicited of anybody that he should be wholly absorbed in the contemplation of atoms, and worship them; that we must worship and lose ourselves in reality, whatever reality may be, is a mystic aberration, which physical science does nothing to foster. Nor does any critical physicist suppose that what he describes is the whole of the object; he merely notes the occasions on which its sensible qualities appear, and calculates events. Because the calculable side of nature is his province, he does not deny that events have other aspects. ...If he chances to call the calculable elements of nature her substance, as it is proper to do, that name is given without passion; he may perfectly proclaim with Goethe that it is accidents, in the farbiger Abglanz [colorful reflection in Faust], that we have our life. ... The horror of mechanical physics arises, then, from attributing to that science pretensions and extensions which it does not have; it arises from the habits of theology and metaphysics being imported inopportunely into science. Similarly, when M. Bergsen mentions mathematics, he seems to be thinking of the supposed authority it exercises—one of Kant's confusions—over the empirical world, and trying to limit and subordinate that authority, lest movement should somehow be removed from nature, and vagueness from human thought. But nature and human thought are what they are; they have enough affinity to mathematics, as it happens, to suggest that study to our minds, and to give those who go deep into it a great, though partial, mastery over things."

"Even the cautious and patient investigation of truth by science, which seems the very antithesis of the mystic's swift certainty, may be fostered and nourished by that very spirit of reverence in which mysticism lives and moves. ...Instinct, intuition, or insight is what first leads to the beliefs which subsequent reason confirms or confutes; but the confirmation, where it is possible, consists, in the last analysis, of agreement with other beliefs no less instinctive. Reason is a harmonising, controlling force rather than a creative one. Even in the most purely logical realm, it is insight that first arrives at what is new. ...both intuition and intellect have been developed because they are useful and ...they are useful when they give truth and become harmful when they give falsehood. ...intuition ...seems on the whole to diminish as civilisation increases. It is greater, as a rule, in children than in adults, in the uneducated than in the educated. In advocating the scientific restraint and balance, as against the self-assertion of a confident reliance upon intuition, we are only urging, in the sphere of knowledge, that largeness of contemplation, that impersonal disinterestedness, and that freedom from practical preoccupations which have been inculcated by all the great religions of the world. Thus our conclusion, however it may conflict with the explicit beliefs of many mystics, is, in essence, not contrary to the spirit which inspires those beliefs, but rather the outcome of this very spirit as applied in the realm of thought."

"Logic has been pursued by those of the great philosophers who were mystics. But since they usually took for granted the supposed insight of the mystic emotion, their logical doctrines were presented with a certain dryness, and were believed by their disciples to be quite independent of the sudden illumination from which they sprang. Nevertheless their origin clung to them, and they remained—to borrow a useful word from Mr. Santayana—"malicious" in regard to the world of science and common sense. It is only so, that we can account for the complacency with which philosophers have accepted the inconsistency of their doctrines with all the common and scientific facts which seem best established and most worthy of belief."

"Everyone knows that to read an author simply in order to refute him is not the way to understand him; and to read the book of Nature with a conviction that it is all illusion is just as unlikely to lead to understanding. If our logic is to find the common world intelligible, it must not be hostile, but must be inspired by a genuine acceptance such as is not usually to be found among metaphysicians."

"Among these friends of Andreae's were many whom we find later in other humanistic societies: Wilhelm von der Wense and Tobias Adami, pupils of Campanella, Johann Kepler, discoverer of the laws of planetary motion, , , Theodor Haak, Samuel Hartlib, , and Comenius. That these various societies had deeper motives than those generally ascribed to them is certain. The Italian academies, after the pattern of which the "Order of the Palm" was founded, must have been, to a certain extent at least, secret societies, since neither their organization, their symbolism, their forms, nor the list of membership was communicated to outsiders, and their real aims were concealed while publicity was given to purposes of a genuinely innocent and popular nature. That the German organizations were not the mere language societies they were generally considered is apparent when we look at the activities of their members. They emphasized the study of the mother-tongue, it is true, but there was hardly a writer among them who was not also interested in the study of natural philosophy, in religion, in mathematics or astronomy, so much so, in fact, that to most of them clung the suspicion of heresy—that they were Rosicrucians and as such members of a religious sect highly dangerous to the church and liable of course to persecution. Members of the seventeenth century academies were natural philosophers, reformers, theologians, educators, statesmen, poets, noblemen; such members there were, as Bacon, Giordano Bruno, Comenius, Robert Boyle, J. B. van Helmont, Campanella, Hugo Grotius, Leibniz, Oxenstierna, Valentin Andreae, Spanheim, Pufendorf, Opitz. Throughout the whole list of membership there runs a line of spiritual relationship in the fact of their tolerance for the beliefs of others, a tolerance remarkable for the seventeenth century. With this they united strict opposition to the scholastic method. They were seriously religious, even to the extent of being mystics, but they understood the essence of Christianity differently from the ruling dogma. They treat not only of the relation of man to God, but of man to nature and of men to each other. For them a knowledge incapable of helping mankind had no value; a science shut off from the people in its language is useless; hence their emphasis of the vernacular. To make all knowledge fruitful for the education of the human race and thus lead the race on to a higher stage of development was one of their great ideals. Their turn for the practical led them on in their striving for a general reformation of the whole world. With their keen sense of the significance of fraternal organization, they formed unions which were intended to benefit the whole man and his whole mode of thinking, to influence his whole life. Their activities were in no way directed, as has been claimed, toward "childish play with symbols and signs but toward inclusive spiritual, religious, philosophical, and scientific aims," the carrying out of which, in those times, could be accomplished only under secret organization. The difficulties under which they labored compelled them to proceed with extreme caution, concealing their real interests and exhibiting to the world only what they considered secondary. When the time and place is more propitious for a franker carrying out of their plans and purposes of reform, we shall find them in a country of larger opportunity; we shall find them in England."

"Besides the early Christian ideal, which recognized and encouraged the connection between the teachings of Christ and the ancient wisdom of platonism, there was in early times another which emphasized and endeavored to develop the antithesis more than the connection. From the time when the new Christian state church came to life, and sacrificial religion and the belief in devils and the priesthood were restored, a struggle of life and death developed between the church and the so called philosophic schools. "The fraternity saw that it had to draw down the mask still further over its face than formerly, and the 'House of the Eternal,' the 'Basilika,' the 'Academies,' and the 'Museums' became workshops of stone cutters, latomies, and or innocent guilds, unions, and companies of every variety. But all later greater religious movements and tendencies which maintained the old beliefs, whether they appeared under the names of mysticism, alchemy, natural philosophy, humanism, or special names and disguises, as workshops or societies, have preserved more or less truly the doctrine of the "sacred numbers" and the number symbolism, and found the keys of wisdom and knowledge in the rightly understood doctrine of the eternal harmony of the spheres.""

"Several interesting peculiarities should not be omitted, as for instance, that Leibniz, about 1667, was secretary of an alchemist's society (of so called rosicrucians) in Nuremberg. Leibniz describes alchemy as an "introduction to mystic theology" and identifies the concepts of "Arcana Naturae" and "Chymica.""

"Presentday topology consists of two distinct parts: point set topology and algebraic topology. The first has mainly been the prerogative of Poland plus a strong American component: the school of R. L. Moore (of Austin, Texas)."

"If geometry is dressed in a suit coat, topology dons jeans and a T-shirt."

"Topologists are interested not only in finite-dimensional spaces (for example, subspaces of Rn), but also in infinite-dimensional ones, such as the spaces occurring in quantum field theory."

"In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."

"In Euler’s investigation of the bridges of Königsberg, he discovered that it was the general arrangement of features that was important, not their exact locations. This observation led to the creation of graph theory, one of the earliest incarnations of topology."

"Category theory plays somewhat the same role in algebra and topology that set theory plays in analysis."

"The cornerstone of Category Theory is the Yoneda lemma. It asserts that a category C may be embedded in the category C^\wedge of all contravariant functors from this category to the category Set of sets, the morphisms in Set being the usual maps. This allows us, in some sense, to reduce Category Theory to Set Theory. The Yoneda lemma naturally leads to the notion of representable functor, and in particular to that of adjoint functor."

"The purpose of sheaf theory is quite general: it is to obtain global information from local information, or else to define “obstructions” which characterize the fact that a local property does not hold globally any more: for example a manifold is not always orientable, or a differential equation can be locally solvable, but not globally."

"Today, ring theory is a fertile meeting ground for group theory (group rings), representation theory (modules), functional analysis (operator algebras), Lie theory (enveloping algebras), algebraic geometry (finitely generated algebras, differential operators, invariant theory), arithmetic (orders, Brauer groups), universal algebra (varieties of rings), and homological algebra (cohomology of rings, projective modules, Grothendieck and higher K-groups)."

"I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for... science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. ...the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics."

"My decision to leave applied mathematics for economics was in part tied to the widely-held popular belief in the 1960s that macroeconomics had made fundamental inroads into controlling business cycles and stopping dysfunctional unemployment and inflation."

"Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate."

"From Pythagoras to Boethius, when pure mathematics consisted of arithmetic and geometry while applied mathematics consisted of music and astronomy, mathematics could be characterized as the deductive study of 'such abstractions as quantities and their consequences, namely figures and so forth' (Acquinas ca. 1260). But since the emergence of abstract algebra it has become increasingly difficult to formulate a definition to cover the whole of the rich, complex and expanding domain of mathematics."

"Pure mathematics is much more than an armoury of tools and techniques for the applied mathematician. On the other hand, the pure mathematician has ever been grateful to applied mathematics for stimulus and inspiration. From the vibrations of the violin string they have drawn enchanting harmonies of Fourier Series, and to study the triode valve they have invented a whole theory of non-linear oscillations."

"In the winter of 2008, Jenifer and I visited Chennai Mathematical Institute. This remarkable Institute is the creation of Seshadri. It is a unique blend of an American style liberal arts college with traditional Indian guru one-on-one teaching, adding physics, computer science, history and music to its maths curriculum. Only in India could an intellectual with no business or management experience, who spends all his spare time singing classical south Indian music, have been the catalyst for such a unique educational experiment."

"Theta functions pervade all of mathematics ranging from the theory of partial differential equations, mathematical physics, to algebraic geometry, number theory and more recently to representation theory."

"The theory of invariants came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head."

"Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics."

"It may be observed of mathematicians that they only meddle with such things as are certain, passing by those that are doubtful and unknown. They profess not to know all things, neither do they affect to speak of all things. What they know to be true, and can make good by invincible arguments, that they publish and insert among their theorems. Of other things they are silent and pass no judgment at all, choosing rather to acknowledge their ignorance, than affirm anything rashly. They affirm nothing among their arguments or assertions which is not most manifestly known and examined with utmost rigour, rejecting all probable conjectures and little witticisms. They submit nothing to authority, indulge no affection, detest subterfuges of words, and declare their sentiments, as in a court of justice, without passion, without apology; knowing that their reasons, as Seneca testifies of them, are not brought to persuade, but to compel."

"There is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect."

"Bell's theorem is the most profound discovery of science."

"Dynamical systems theory began in the late nineteenth century with the work of Henri Poincaré (1854–1912). He solved an important problem in celestial mechanics by using techniques of dynamical systems. In modern times, Stephen Smale, Adrien Douady, John Milnor, Lennart Carleson, and many other notable mathematicians have helped to develop this theory into a powerful mathematical tool. The subject is exciting in that it integrates analysis, geometry, and computer graphics into a whole that is greater than the sum of its parts."

"The only general of revolution which can degenerate into a cylinder, a cone, or a plane is the hyperboloid."

"Rotation of a ... forms a hyperboloid in which the hyperbola becomes a meridian of this surface..."

"All screws of a given pitch belonging to a system of the third order are the generators of a certain hyperboloid. There is... a different hyperboloid for each pitch. ...all these hyperboloids are concentric."

"[F]or the hyperboloid of one sheet and the hyperbolic paraboloid the curvature is negative at every point."

"Upon the hyperboloid of one sheet, and likewise upon the hyperbolic paraboloid, the two lines of striction coincide."

"When an hyperboloid of revolution of one sheet is deformed into another ruled surface, the circle of gorge becomes a Bertrand curve and the generators are parallel to the corresponding binormals of the conjugate Bertrand curve."

"The hyperboloid of revolution (generated as a surface of revolution) is a special case of the elliptical hyperboloid... The shape of a small portion of the hyperboloid of revolution around the equator is similar to that of the , but larger portions are quite different, as are the s. The image of the Gauss map of the whole hyperboloid of revolution omits disks around the north and south poles, whereas the image of the Gauss map of the whole catenoid omits only the north and south poles."

"One way to deal with the matters... is to work with a set of numerical values. I can do this with the help of Ball's Cartesian equation for the hyperboloids of reguli of the 3-system (1900). This equation is quoted and proved by Hunt (1978)... I shall choose the three principle screws of a 3-system of motion... I have, (a) accorded with convention, (b) ensured that the pitch quadric will be real... Ball's equation... clearly represents a series of concentric quadric surfaces... This circumstance of there being none of the concentric hyperboloids coaxial with one of the principal axes is a characteristic of the 3-system. ...within a certain, central zone of the system, the intersections among the hyperboloids are complicated and not easy to understand. Outside that zone, however... the hyperboloids appear in relation to one another... Each successive hyperboloid is wholly 'outside' its predecessor (or wholly 'inside' as the case may be), and no intersections are apparent. ...outside a certain, central zone, only one real screw can be found to pass through a generall chosen point..."

"The equation of the tangent plane at the point (a, 0, 0) of the conicoid \frac{x^2}{a^2} \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 1 is x = a; this meets the surface in straight lines whose projection on the plane x = 0 are given by the equation \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 0. These lines are clearly real when the surface is an hyperboloid of one sheet, and imaginary when the surface is an , or an hyperboloid of two sheets. Hence the hyperboloid of one sheet is a . The hyperbolic paraboloid is a particular case of the hyperboloid of one sheet; hence the hyperbolic paraboloid is also a ruled surface. This can be proved at once from the equation of the paraboloid. For, the tangent plane at the origin is z = 0, and this meets the paraboloid ax^2 + by^2 + 2z = 0 in the straight lines given by the equations ax^2 = by^2 = 0, z = 0; the lines are clearly real when a and b have different signs, and are imaginary when a and b have the same sign. Hence an hyperboloid of one sheet (including an hyperbolic paraboloid as a particular case) is the only ruled conicoid in addition to a cone, a cylinder, and a pair of planes."

"Suppose we start with a disk radiator and we place a mirror in the form of a hyperboloid of revolution coincident with a set of flow lines... We truncate the mirror at some distance so the open end is a circle... Considering the inside as a mirror, this forms a nonimaging concentrator with unusual properties. The foci of the hyperbolas in this section are at... the ends of the diameter of the original disk. Then all rays entering the aperture... and pointing somewhere inside the disk will be reflected by the mirror so as to strike, eventually, the inner disk... Thus, the concentrator takes all rays from the virtual source... which can pass the entry aperture... and concentrate them into an exit aperture. This result is easily proved for rays in the meridional section... the extreme angle rays emerge from the exit aperture but only after an infinite number of reflections. ...rays at angles inside the extreme angles all emerge. Thus, in the meridional plane this is a concentrator of maximal theoretical concentration. This property holds for skew rays, although this is not quite so obvious. ...When used in reverse, the same design produces a virtual ring that fills the space between a Lambertian source and the larger diameter... [disk]. The visual effect produced is striking."

"[F]or the bare hyperboloid, both real and Lambertian sources are in the same plane. This is a limitation in certain applications; however, we can manipulate these positions with lenses to place the sources at more convenient locations. ...this is an ideal concentrator. ...we have postulated an ideal lens while the hyperboloid is an ideal concentrator, albeit operating on a virtual source. The design considerations readily follow from the geometry of the hyperbola."

"A surface which may be generated by a moving straight line is called a '. The plane, the cone, and the cylinder are simple examples... the hyperbolic paraboloid is a . ...[T]he unparted hyperboloid \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{ c^2} = 1 is a ruled surface having two sets of rectilinear generators, i.e., ...through every point of it two straight lines may be drawn, each of which shall lie entirely on the surface."

"If we write the equation of the hyperboloid in the form \frac{x^2}{ a^2} - \frac{z^2}{c^2} = 1 - \frac{y^2}{b^2}, \qquad (1)It is evident that (1) is the product of two equations\begin{array}{lcl} \frac{x}{a} - \frac{z}{c} = k_1(1 - \frac{y}{b}), \\ \frac{x}{a} + \frac{z}{c} = \frac{1}{k_1}(1 + \frac{y}{b}), \qquad (2) \end{array}for any value of k_1. But (2) are the equations of a straight line... Moreover this straight line lies entirely on the surface, since the coordinates of every point of it satisfy (2) and hence (1). As different values are assigned to k_1, we obtain a series of straight lines lying entirely on the surface. Conversely if P_1( x_1, y_1, z_1) is any point of (1), \frac{\frac{x_1}{a} - \frac{z_1}{c}}{1 - \frac{y_1}{b}} = \frac{1 + \frac{y_1}{b}}{\frac{x_1}{a} + \frac{z_1}{c}}Therefore P_1 determines the same value of k_1 from both equations (2). Hence every point of (1) lies in one and only one line (2). We may also regard (1) as the product of the two equations\begin{array}{lcl} \frac{x}{a} - \frac{z}{c} = k_2(1 + \frac{y}{b}), \\ \frac{x}{a} + \frac{z}{c} = \frac{1}{k_2}(1 - \frac{y}{b}), \qquad (3) \end{array}whence it is evident that there is a second set of straight lines lying entirely on the surface, one and only one of which may be drawn through any point of the surface. Equations (2) and (3) are the equations of the rectilinear generators, and every point of the surface may be regarded as the point of intersection of one line from each set."

"In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in the physical order of ideas. Perhaps the most striking example is to be found in the development of abstract Dynamics. The greatest treatise which the world has seen, on this subject, is Lagrange's Mécanique Analytique, published in 1788. ...conceived in the purely abstract Mathematical spirit ...Lagrange's idea of reducing the investigation of the motion of a dynamical system to a form dependent upon a single function of the of the system was further developed by Hamilton and Jacobi into forms in which the equations of motion of a system represent the conditions for a stationary value of an integral of a single function. The extension by Routh and Helmholtz to the case in which "ignored co-ordinates" are taken into account, was a long step in the direction of the desirable unification which would be obtained if the notion of potential energy were removed by means of its interpretation as dependent upon the kinetic energy of concealed motions included in the dynamical system. The whole scheme of abstract Dynamics thus developed upon the basis of Lagrange's work has been of immense value in theoretical Physics, and particularly in statistical Mechanics... But the most striking use of Lagrange's conception of generalized co-ordinates was made by Clerk Maxwell, who in this order of ideas, and inspired on the physical side by... Faraday, conceived and developed his dynamical theory of the Electromagnetic field, and obtained his celebrated equations. The form of Maxwell's equations enabled him to perceive that oscillations could be propagated in the electromagnetic field with the velocity of light, and suggested to him the Electromagnetic theory of light. Heinrich Herz, under the direct inspiration of Maxwell's ideas, demonstrated the possibility of setting up electromagnetic waves differing from those of light only in respect of their enormously greater length. We thus see that Lagrange's work... was an essential link in a chain of investigation of which one result... gladdens the heart of the practical man, viz. wireless telegraphy."

"Full use of Lagrange's own made the unification of the varied principles of statistics and dynamics possible—in statistics by the use of the principle of virtual velocities, in dynamics by the use of . This led... to generalized coordinates and to the equation of motion in their "Lagrangian" form... Newton's geometrical approach was now fully discarded; Lagrange's book was a triumph of pure analysis."

"The treatment of the kinetics of a material system by the method of generalised coordinates was first introduced by Lagrange, and has since his time been greatly developed by the investigations of different mathematicians. Independently of the highly interesting, although purely abstract science of theoretical dynamics which has resulted from these investigations, they have proved of great and continually increasing value in the application of mechanics to thermal, electrical and chemical theories, and the whole range of ."

"When the position of every point of a material system can be determined in terms of any independent variables n in number, the system is said to possess n degrees of freedom, and the n independent variables are called the generalised coordinates. The choice of the particular independent variables is perfectly arbitrary, and may be varied indefinitely, but the number of degrees of freedom cannot be either increased or diminished."

"In a rigid body free to move in any manner there are six degrees of freedom, and the generalised coordinates most frequently chosen in this case are the three rectangular coordinates of some point in the body and three angular coordinates determining the orientation of the body about that point, generally the angles &theta;, &phi;, &psi; of ordinary occurrence in rigid dynamical problems."

"When the body degenerates into a material straight line the number of degrees of freedom is reduced to five; and when this straight line is constrained to move parallel to some fixed plane the number of degrees of freedom is still further reduced to four."

"[M]ental Abstraction... is not [only the] Property of Mathematics, but is common to all Sciences. For every Science considers the Nature of its own Subject abstracted from all others; forms its own general Precepts and Theorems; and separates its own Properties from the Properties of others... For example, Physics considers the constitutive Principles, Matter, Form, &c. of Body in general; then the Affections common to all Bodies, viz. Quantity, Place, Motion, Rest, and the like; from whence it descends to the next lower Species, investigating their particular Natures and Properties; but meddles not with particular Bodies or Individuals, as well because they are innumerable and distinguished from one another by innumerable Differences... The same way Geometry proposes Magnitude for the Subject of its Enquiry, not the peculiar Magnitude of this or that Body, but the Magnitude taken universally; together with its general Affections, viz. Divisibility, Congruence, Proportionality, a Capacity of different Situation and Position, Mobility &c. declaring these to be inherent to it, and after what manner they are so: Next it defines the various Species of Magnitude, (viz. a Line, Superficies, and a Body or Solid) and particularly draws forth and demonstrates their distinct Properties; continually dividing these Species into others more contract, and searching and proving their Affections by universal Propositions, Rules and Theorems lawfully demonstrated, till it has wholly exhausted its Subject, and descended to the very lowest Species. And these Theorems however more or less general as to their Matter, may be truly and properly accommodated to Subjects particular to themselves. True Mathematical Abstraction then, is such as agrees with all other Sciences and Disciplines, nothing else being meant (whatsoever some do strangely say of it) than an Abstraction from particular Subjects, or a distinct Consideration of certain things more universal, others less universal being ommitted and as it were neglected."

"They who are acquainted with the present state of the theory of Symbolical Algebra, are aware that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. ... It has happened in every known form of analysis, that the elements to be determined have been conceived as measurable by comparison with some fixed standard. The predominant idea has been that of magnitude, or more strictly of numerical ratio. The expression of magnitude, or of operations upon magnitude, has been the express object for which the symbols of Analysis have been invented, and for which their laws have been investigated. Thus the abstractions of the modern Analysis, not less than the ostensive diagrams of the ancient Geometry, have encouraged the notion that Mathematics are essentially, as well as actually, the Science of Magnitude. ... [T]his conclusion is by no means necessary. If every existing interpretation is shewn to involve the idea of magnitude, it is only by induction that we can assert that no other interpretation is possible. ...The history of pure Analysis is, it may be said, too recent to permit us to set limits to the extent of its applications. ...That to the existing forms of Analysis a quantitative interpretation is assigned, is the result of the circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis. It is upon the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place among the acknowledged forms of Mathematical Analysis, regardless that in its object and in its instruments it must at present stand alone."

"The chemist, in describing some mineral, may present all its attributes, color mass, density, volume, molecular construction, and its properties exhibited in relation to heat, light, sound, and electric waves, and the resulting conception will not be mathematical. On the other hand, in describing its form as cubical, he relies upon the purely mathematical abstraction involved in the definition of a cube, which includes that of the mathematical plane, which involves the geometric line, which, in turn, resolves into an assemblage of points. Not one of these conceptions finds its realization in physical phenomena, but at best an approximation. The types conceived by the mind remain, however, definite and unchangeable. The idea of the ultimate element, the "point," best illustrates the nature of these abstractions, and involves the first difficulty that lies in the way of the understanding. The point is defined as having but one positive attribute, and that is, position in space. ...The mathematical point cannot be realized by this process, for the resulting principle is this: all material things are indefinitely divisible, and, no matter how small, possess magnitude and occupy space. The negative of this thought is the point, which does not have magnitude, but position only. It is a negative thought concept, a mathematical abstraction..."

"Change of state involves what is meant by the word "time," which, like space, is a necessary condition for thought, or the formation of concepts. These occur in succession, as do all phenomena involving a change, and whenever motion or change is involved, time is required, and the universal knowledge of this fact is an elementary abstraction. Unity apart from any concrete thing... is the primary abstraction in arithmetic, and number and the theory of numbers grow from this concept to one of great importance. Since time and space are necessary to the realization of sensible objects and the phenomena of motion, the attributes of time and space adhere of necessity to the theory of these subjects, and hence the principles demonstrated in the abstract concerning time and space relations, are applicable to such phenomena. Herein lies the secret of the universal application of pure mathematical deductions to the many sciences that sum up human knowledge of the universe."

"Geometry can in no way be viewed... as a branch of mathematics, instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry."

"Abstraction is the immediate ulterior result of analysis. We may speak of the analysis of the mathematical whole, and so of the abstraction of any of its parts. Wherever analysis may take place, abstraction likewise, is possible. ...The reason on account of which the analysis and abstraction of the mind are directed to the parts of the metaphysical whole as such, lies in the fact that the mental division of an object into its mathematical, or separable, parts, is not sufficient even for the ends of ordinary thought. We cannot, from such a division, adequately understand and express the nature of things. This purpose requires that we should consider and designate inseparable parts, such as powers, shapes, magnitudes, and attributes generally."

"The word element is a term which frequently occurs in philosophy. It signifies any of those parts of an object into which it is or may be separated by analysis; and which, therefore, may be separately considered by abstraction. ...A notion of a thing may be formed by the composition of mathematical parts, and such a composition in its relation to parts, and such a composition in relation to the object might be spoken of as mathematical conception."

"It was only in the nineteenth century that the winds of change started to blow. First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of "quantity" and of "measurement" beyond recoginition. Second, the rapidly multiplying of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and "existence" into the abstractions themselves."

"Berkeley Bishop of Cloyne was a man of first-rate talents, distinguished as a metaphysician, a philosopher, and a divine. His geometrical knowledge, however, which, for an attack on the , was more essential than all his other accomplishments, seems to have been little more than elementary. The motive which induced him to enter on discussions so remotely connected with his usual pursuits has been variously represented; but whatever it was, it gave rise to , in which the author professes to demonstrate, that the new analysis is inaccurate in its principles, and that, if it ever lead to true conclusions, it is from an accidental compensation of errors that cannot be supposed always to take place. The argument is ingeniously and plausibly conducted, and the author sometimes attempts ridicule with better success than could be expected from the subject; thus when he calls ultimate ratios the ghosts of departed quantities, it is not easy to conceive a witty saying more happily fastened on a mere mathematical abstraction."

"Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say."

"[W]hile we put number into objects, on the other hand we derive our idea of number only from the presence of the world external to the mind. We see a group of people, and we begin by making an abstraction ("people"), and we say, "Here are ten people"—thus calling them all by the one abstract name, even though the individuals be very different. "A careful observation shows us, however, that there are no objects exactly alike; but by a mental operation of which we are quite unconscious, although it holds within itself the entire secret of mathematical abstraction, we take in objects which seem to be alike, rejecting for the time being their differences. Here is to be found the source of calculation." So the idea of number is generated in the mind by the sense perception of a group of things supposed to be alike."

"In solving a problem, be it one in the calculus, in algebra, or in the second year of arithmetic, we begin by substituting for the actual things certain abstractions represented by symbols; we think in terms of these abstractions, aided by symbols, and finally from our result we pass back to the concrete and say that we have solved the problem. It is all a matter of "one to one correspondence," it being easier for us to work with the abstract numbers and their corresponding figures than to work with the actual objects. Fundamentally the process is something like this: 1. By abstraction we pass to numbers. 2. Thence we pass to symbols, and we make an equation, either openly, as in algebra, or concealed, as in many forms of arithmetic. This equation we solve, the result being a symbol. 3. We find the number corresponding to this symbol, and say that the problem is solved. All this does not mean that primary number is to be merely a matter of symbols. It means that in mathematics we find it more convenient to work purely with symbols, translating back to the corresponding concrete form as may be desired. And so those teachers who fear lest the child shall drift into thinking in symbols instead of in number, are really fearing that the child shall drift into mathematics."

"I don’t call myself a “Platonist,” since I don’t even know what it even means to say that mathematical truths “exist in a Platonic realm.” Indeed, that concrete way of putting things even strikes me as a step backwards—as if astronomers could one day send probes to Plato’s realm to find out what’s in it, or prove that it never existed. The concepts of “existence” and “reality” that we use when discussing physical objects are just flat-out irrelevant here. I instead call myself an “anti-anti-Platonist”: someone who’s not literally committed to a “Platonic realm” of mathematical truth, but who given the choice, would much rather that people believe in such a realm than that they spout the self-evident garbage that goes with denying math’s intellectual autonomy."

"What’s the greatest discovery in the history of thought? Of course, it’s a silly question – but it won’t stop me from suggesting an answer. It’s Plato’s discovery of abstract objects. Most scientists, and indeed most philosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum. Mathematicians, on the other hand, are at least slightly sympathetic. Working day-in and day-out with primes, polynomials and principal fibre bundles, they have come to think of these entities as having a life of their own. Could this be only a visceral reaction to an illusion? Perhaps, but I doubt it."

"Perhaps nobody should be particularly surprised by this, as, after all, the laws of nature physicists acknowledge also seem to be timelessly true independently of whether anyone takes them to be true. Where do they come from? Since there are somewhat mundane interpretations of what laws of nature are – including the possibility that they are accidental generalities valid in this particular universe and/or within a certain time-span – the case posed by mathematical constructs seems to be even more clear and powerful. Math, like diamonds, truly seems to be forever. If one ‘goes Platonic’ with math, one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder."

"One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them."

"How is it that mathematical ideas can be communicated in this way? I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts. ... When one 'sees' a mathematical truth, one's consciousness breaks through into this world of ideas, and makes direct contact with it ('accessible via the intellect'). I have described this 'seeing' in relation to Gödel's theorem, but it is the essence of mathematical understanding. When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'. (Indeed, often this act of perception is accompanied by words like 'Oh, I see'!) Since each can make contact with Plato's world directly, they can more readily communicate with each other than one might have expected. The mental images that each one has, when making this Platonic contact, might be rather different in each case, but communication is possible because each is directly in contact with the same externally existing Platonic world!"

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."

"The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!"

"Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an , which allows a certain amount of self-reference. And by this time everyone was pretty bored."

"In the summer of 1914 I attended Frege's course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. ...Unfortunately, his admonitions go unheeded even today."

"On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically: as is particularly noticed by Bhaskara himself, towards the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities."

"Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy."

"Proof is the idol before whom the pure mathematician tortures himself."

"Another roof, another proof."

"Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from the Book!". This viewpoint expresses... that mathematics, as the... foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God..."

"It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word."

"The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned."

"are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof's overall strategy, delightfully, is not. ...They belong to a branch of mathematics known as formal logic or mathematical logic, a field which was viewed, prior to Gödel's achievement, as mathematically suspect."

"The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth: synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths."

"The specific aim of the was to find an elementary proof of a special case of the density (DHJ)... The theorem was known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights. There were two reasons to want a new proof... First... [it] had just one—a long and complicated proof that relied on heavy mathematical machinery. An elementary proof—one that starts from first principles instead of relying on advanced techniques—would require many new ideas. Second, DHJ implies another famous theorem, called , novel proofs of which led to several breakthroughs over the past decade, so there was reason to expect that the same would happen..."

"The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of &radic;2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes."

"When we are engaged in investigating the foundations of a science, we must set up a system of s which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science... is held to be correct unless it can be derived from axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all, I wish to designate the following as the most important among numerous questions which can be asked with respect to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results."

"In a never-ending search for good mathematical problems and fresh mathematical talent, Erdős crisscrossed four continents at a frenzied pace, moving from one university or research center to the next. His modus operandi was to show up on the doorstep of a fellow mathematician, declare, "My brain is open," work with his host for a day or two, until he was bored or his host was run down, and then move on to another home. ...He did mathematics in more than 25 different countries, completing important proofs in remote places and sometimes publishing them in equally obscure journals."

"The difficulty in presenting a rigorous as well as clear statement of the theory of limits is inherent in the subject. ...If the reader has found some difficulty in grasping it he may be less discouraged when he is told that it eluded even Newton and Leibniz. ... Many contemporaries of Newton, among them ... taught that the calculus was a collection of ingenious fallacies. ... decided that he could found calculus properly... The book was undoubtedly profound but also unintelligible. One hundred years after the time of Newton and Leibniz, Joseph Louis Lagrange... still believed that the calculus was unsound and gave correct results only because errors were offsetting each other. He, too, formulated his own foundation... but it was incorrect. ...D'Alembert had to advise students of the calculus... faith would eventually come to them. This is not bad advice... but it is no substitute for rigor and proof. ... About a century and a half after the creation of calculus... Augustin Louis Cauchy... finally gave a definitive formulation of the limit concept that removed doubts as to the soundness of the subject."

"Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks, mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians."

"It was not until the nineteenth century, chiefly through... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...[I]t called attention... to the fact that a proof can be given of the impossibility of proving certain propositions within a given system. ...Gödel's paper is a proof of the impossibilty of formally demonstrating certain important propositions in number theory. ...[T]he resolution of that parallel axiom question forced the realization that Euclid was not the last word on the subject of geometry, since new systems of geometry can be constructed... incompatible with those adopted by Euclid. ...[I]t gradually became clear that the proper business of pure mathematicians is to derive theorems from postulated assumptions, and that it is not their concern whether the axioms are actually true."

"In 1920 logic was mostly a philosopher's garden. There were also a few mathematicians there, cultivating the logical roots of the mathematical tree. Today, Theory, Set Theory, and Proof Theory, logic's major subdisciplines, have become full-fledged branches of mathematics."

"We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name."

"Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors."

"Proofs must vary. . . and be differentiated according to kinds of being concerned, since mathematics is a texture of all these strands and adapts its discourse to the whole range of things."

"Proofs are for the mathematician what experimental procedures are for the experimental scientist: in studying them one learns of new ideas, new concepts, new strategies—devices which can be assimilated for one's own research and be further developed."

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ... arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof."

"The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle."

"More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence.""

"At the age of forty he was, for the first time, introduced to the works of Euclid, and at once 'fell in love with geometry,' being attracted, he says, more by the rigorous manner of proof employed than by the matter of the science. (Mathematics, we must remember, were then only beginning to be seriously studied in England. Hobbes tells us that in his undergraduate days geometry was still looked upon generally as a form of the 'Black Art,' and it was not until 1619 that the will of Sir Henry Savile, Warden of Merton College, established the first Professorships of Geometry and Astronomy at Oxford.)"

"One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time of and , the concept of 'geometric algebra'. What it amounts to is that Greek mathematics, especially after the discovery of the 'irrational'... is algebra dressed up, primarily for the sake of rigor, in geometrical garb. The reasoning... the line of attack... the solutions... etc. all are essentially algebraic... attired in geometrical accouterments. We... look for the algebraic 'subtext'... of any geometrical proof... always to transcribe... any proposition in[to] the symbolic language of modern algebra... [making] the logical structure of the proof clear and convincing, without thereby losing anything, not only in generality but also in any possible sui generis features of the ancient way of doing things. ...[i.e., that] there is nothing unique and (ontologically) idiosyncratic concerning the way... ancient Greek mathematicians went about their proofs, which might be lost... I cannot find any historically gratifying basis for this generally accepted view... those who have been writing the history of mathematics... have typically been mathematicians... largely unable to relinquish and discard their laboriously acquired mathematical competence when dealing with periods in history during which such competence is historically irrelevant and... anachronistic. Such... stems from the unstated assumption that mathematics is a scientia universalis, an algebra of thought containing universal ways of inference, everlasting structures, and timeless, ideal patterns of investigation which can be identified throughout the history of civilized man and which are completely independent of the form in which they happen to appear at a particular junction of time."

"Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic."

"An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it."

"M. Poincaré finds the answer to these questions in the so-called 'mathematical induction' which proceeds from the particular to the more general, but at the same time does so by steps of the highest degree of certitude. In this process he sees the creative force of mathematics, which leads to real proofs and not mere sterile verifications. ...No arithmetic could be built up, rejecting the axiom of mathematical induction, as the non-Euclidean geometries have been built up, rejecting the postulate of Euclid."

"Under the same assumptions made in the Chord papers, the [SIGCOMM] version of the protocol is not correct, and not one of the properties claimed invariant in [PODC] is actually invariantly true of it."

"The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it.""

"Comparatively few of the propositions and proofs in the Elements are [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions."

"The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why?"

"Proofs have gaps and are... inherently incomplete and sometimes wrong. ...There is another reason ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification."

"Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored..."

"Social pressure often hides mistakes in proofs. In a seminar lecture... interrupt the speaker... to ask for more explanation... [often] the response will be that it is "obvious" or "clear" or "follows easily..." Occasionally... a look... conveys the message that the questioner is an idiot. That's why most mathematicians sit quietly... understanding very little... and applauding politely... One of the joys of Gel'fand's seminar... he would constantly interrupt... to ask questions and give elementary examples... [T]he audience would actually learn some mathematics."

"There are... masterpieces of... exposition... Two examples... are Weil's Number Theory for Beginners... and Artin's '. Mathematics can be done scrupulously."

"Perhaps we should discard the myth that mathematics is a rigorously deductive enterprise... hand-waving is intrinsic. We try to minimize it and we can sometimes escape it, but not always, if we want to discover new theorems."

"According to the dominant view, the reflection on mathematics is the task of a specialized discipline, the philosophy of mathematics, starting with Frege, characterized by its own problems and methods, and in a sense “the easiest part of philosophy”. In this view, the philosophy of mathematics “is a specialized area of philosophy... Many of the questions that arise within it... occur within the philosophy of mathematics in an especially pure, or especially simplified, form”. ...[However,] like applied mathematics, pure mathematics draws its concepts from experience, observation, scientific theories and even economics. The questions considered by the reflection on mathematics have, therefore, all the impurity and complexity of which philosophical problems are capable."

"Philosophy of mathematics has been slow to draw the analogy from the Kuhnian sea-change in philosophy of science, but during the last decade, a growing number of younger philosophers of mathematics have turned their attention to the history of mathematics and tried to make use of it in their investigations. The most exciting of these concern how mathematical discovery takes place, how new discoveries are structured and integrated into existing knowledge, and what light these processes shed on the existence and applicability of mathematical objects."

"We are still in the aftermath of the great foundationist controversies of the early twentieth century. Formalism, and , each left its trace in the form of a certain mathematical research program that ultimately made its own contribution to the corpus of mathematics... As philosophical programs, as attempts to establish a secure foundation for mathematical knowledge, all have run their course and petered out or dried up. Yet there remains, as a residue, an unstated consensus that the philosophy of mathematics is research on the foundations of mathematics. If I find [that] uninteresting or irrelevant, I conclude that I'm simply not interested in philosophy (thereby depriving myself of any chance of confronting my own uncertainties about the meaning, nature, purpose or significance of mathematical research)."

"The doctrine that mathematical knowledge is a priori—mathematical apriorism...—has been articulated in many different ways... To name only the most prominent defenders... since the seventeenth century, Descartes, Locke, Berkeley, Kant, Frege, Hilbert, Brouwer, and Carnap... Most of the disputes... conducted in our century represent internal differences... among apriorists. ...I shall offer a picture of mathematical knowledge which rejects mathematical apriorism."

"[M]athematical apriorism... has not gone completely unquestioned. J. S. Mill attempted to argue that mathematics is an empirical science, thereby making himself the subject of Frege's biting criticism. More recently, W. V. Quine, , and Imre Lakatos have... challenged the... thesis. However, none of these have offered a systematic account of our mathematical knowledge. ...[T]he alternative...—mathematical empiricism—has never been given a detailed articulation. I shall try... I have gained much from insights of Quine and Putnam... [and] learned from Mill... My quarrel with earlier empiricists is, for the most part, that they have been incomplete rather than mistaken."

"Philosophy of mathematics appears to become a microcosm for the most general and central issues in philosophy—issues in epistemology, metaphysics, and philosophy of language—and the study of those parts of mathematics to which philosophers... most often attend (logic, set theory, aritmetic) seems designed to test the merits of large philosophical views about the existence of abstract entities of the tenability of a certain picture of human knowledge. ...[A]re [there] not other tasks ...that arise either from the current practice of mathematics or the history of the subject... the kinds of issues that occupy those who study the other branches of human knowledge... as: How does mathematical knowledge grow? What is mathematical progress? What makes some mathematical ideas (or theories) better than others? What is mathematical explanation?"

"The dialogue takes place... The class gets interested in a Problem: Is there a relation between the number of vertices V, the number of edges E and the number of faces F of a polyhedra—particularly of regular polyhedra—analogous to the trivial relation between the number of vertices and edges of polygons, namely, that there are as many edges as vertices: V = E? ...After much tiral and error they notice that for all regular polyhedra V - E + F = 2. Somebody guesses that this may apply for any polyhedron whatever. Others try to falsify [test] this conjecture... it holds good. The results corroborate the conjecture, and suggest that it could be proved. It is at this point—after the stages problem and conjecture—that we... offer a proof."

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ... arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof."

"By "philosophy of mathematics" I mean the specific set of concepts, categories, and theories employed, implicitly or explicitly, by philosophers and mathematicians in their discourse about mathematics. Understood in this way, philosophy of mathematics would include, among other things, some rather ethereal discussions on the nature of numbers by several hermetic philosophers, and the status of various notions, including number, space, infinity, according to the philosophers and mathematicians operating in the seventeenth century, as well as several other areas of investigation. Therefore I must introduce a qualification contained in the concept of "mathematical practice." ...I use this term as it is used today in mathematical logic and philosophy of mathematics, simply to indicate mathematics as it is done, not as it should be done according to some preconceived philosophical viewpoint. ...[F]ar from eliminating the philosophical questions, an interest in mathematical practice has actually extended their range. Addressing the issue of mathematical practice requires a detailed knowledge of the mathematical literature of the period."

"Philosophers and logicians have been so busy trying to provide mathematics with a "foundation" in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don't think mathematics is unclear; I don't think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs "foundations." The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intelIectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won't, but one can always hope) that the various systems of mathematicaI philosophy, without exception, need not be taken seriously."

"If one excludes the philosophy of science from the ambit of a study of its history, then one is obliged to do history with the default philosophy of science. In our case this means that one must then accept the present-day Western philosophy of mathematics, not only as a privileged philosophy, but as the only possible philosophy of mathematics."

"There is an interesting analogy... with the philosophy of the natural sciences, which has flourished under the combined influence of both general methodology and classical metaphysical questions (realism vs. antirealism, space, time, causation, etc.) interacting with detailed case studies in... (physics, biology, chemistry, etc.)... [C]ase studies both historical (studies of Einstein's relativity, Maxwell's electromagnetic theory, , etc.). By contrast, with few exceptions, philosophy of mathematics has developed without the corresponding detailed case studies."

"Already in the 1960s, first with Lakatos and later through a group of 'maverick' philosophers of mathematics (Kitcher, Tymoczko, and others), a strong reaction set in against the philosophy of mathematics conceived as foundation of mathematics. ...What these philosophers called for was an analysis of mathematics that was more faithful to its historical development."

"A characterization... of the main features of the maverick tradition could be..; a. antifoudationalism, i.e. there is no certain foundation... mathematics is... fallible... b. anti-logicism, i.e. mathematical logic cannot provide the tools for an adequate analysis of mathematics and its development; c. attention to mathematical practice: only detailed analysis and reconstruction of large and significant parts of mathematical practice can provide a philosophy of mathematics..."

"[T]he so-called ... finds applications in finding the sum of infinite series, the asymptotic value of an integral involving a large parameter, signal analysis, and imaging technique. ...In , [it] is an important tool in studying the distributions of products of two s. In particular, the Mellin transform of the product of two independent random variables equals the product of the Mellin transforms of the two variables. The Mellin transform is closely related to the two-sided Laplace transform. The so-called Mellin transform has been considered by Laplace and used by Riemann in his study of the zeta function. It was, however, Mellin who provided a systematic formation of the transform and its application to solve ODEs and to estimate the value of integrals. ... ...was a student of Mittang-Leffler and Weierstrass. The kernel for the Mellin transform is K(s,t) = t^{s-1}The Mellin transform and its inversion are defined as:F(s) = M[f(x)] = \int\limits_{0}^{\infty}x^{s-1} f(x)\, dx, f(x) = M^{-1}[F(s)] = \frac{1}{2\pi i} \int\limits_{c-i \infty}^{c+i \infty}x^{-s} F(s)\, dswhere c is a constant that lies on the right of all singularities of the kernel function. With the proper change of variables, the Mellin transform can be converted to a two-sided Laplace transform. In particular, a two-sided Laplace transform can be written asL[g(t)] = \int\limits_{-\infty}^{+\infty} g(t)e^{-st}\, dt"

"Doetsch and Bernstein, beginning from the early 1920s, worked together on the subject of Laplace transformation, integral equations and s. They published several papers together, in which the connection between the Laplace transformation and convolution, i.e., Faltung, is discussed often. ...[T]he Laplace transformation of a function f(t), denoted by \mathcal{L}(f), where f is defined for all real numbers t > 0, is the following complex function of F:F(t) = \mathcal{L}(f) = \int\limits_{0}^{\infty}e^{-tu} f(u)\, du.The relation between the Laplace transformation and convolution is...:\mathcal{L}(f*g) = \mathcal{L}(f) \cdot \mathcal{L}(g). ...In 1922, they remark, regarding the Laplace transformation: "[w]e distinguish the functions of a subfield and a field [Oberkörper], which are connected by a certain process. The operations in the subfield are actual, proper [eigentliche] ones, which are only symbolic in the field, but which in certain cases are capable of an actual analytical representation.""

"Because the links between a convolution integral and a Laplace or are so important... we briefly present Borel's (1899) work on a "Laplace like" transform. Note Mellin's work (1896)... was unknown to Borel... Borel defined two functions f(z) and g(z) by their following Laplace integrals...:f(z) = \int\limits_{0}^{+\infty}F(u)e^{-u/z}\, \frac{du}{z}; \quad g(z) = \int\limits_{0}^{+\infty}G(v)e^{-v/z}\, \frac{dv}{z}and then showed that the convolution integral is H(x) = \int\limits_{0}^{x}F(t)G(x-t)\,dt. The Laplace transform of the convolution integral H(x) reduced to a simple product of the two separate transforms f(z) and g(z). Borel failed to see all the possibilities of his theorem. Volterra... also did not see the possible uses... But... in 1920, Doetsch produced a doctoral thesis... on Borel's summability theory of diverging series. Doetsch knew Borel's proof and was able to introduce modern, proper mathematical ideas on convolution integrals and Laplace transforms. The word Faltung was first introduced by Doetsch and Bernstein in 1920. The Laplace transform... and the Fourier transform... are both adequate tools for evaluating a convolution integral. ...Doetsch would introduce the convolution integral by analogy with a between two ..."

"The Indian system of counting is probably the most successful intellectual innovation ever devised by human beings. It has been universally adopted. ...It is the nearest thing we have to a universal language."

"Medieval Indian mathematicians, such as Brahmagupta (7th century), Mahavira (9th century) and Bhüskara (19th century), made several discoveries which in Europe were not known until the Renaissance or later, They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations."

"Through the necessity of accurately laying out the open-air site of a sacrifice Indians very early evolved a simple system of geometry, but in the sphere of practical knowledge the-world owes most to India in the realm of mathematics, which were developed in Gupta times to a stage more advanced than that reached by any other nation of antiquity. ‘The success of Indian mathematics was mainly due to the fact that the Indians had a clear conception of abstract number, as distinct from numerical quantity of objects or spatial extension. While Greek mathematical science was largely based on mensuration and geometry, India transcended these conceptions quite early, and, with the aid of a simple numeral notation, devised a rudimentary algebra which allowed more complicated calculations than were possible to the Greeks, and led to the study of number for its own sake."

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately"

"It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit."

"In the Surya Siddhanta is contained a system of trigonometry which not only goes beyond anything known to the Greeks, but involves theorem which were not discovered in Europe till two centuries ago."

"We owe a lot to the Indians who taught us how to count, without which no worthwhile scientific discovery could have been made."

"In order to instill a proper and well-founded pride in Hindus, it is (once more) most important to restore the truth about Hindu history, especially about Hindu society's glorious achievements. In technology, it cannot match China, which was the world leader until a mere three, four centuries ago. But in abstract sciences like linguistics, logic, mathematics, Hindu culture has been the chief pioneer."

"In the Vedic Age, India was very religious, but it was also ahead of the rest in mathematics and astronomy. Thus, the geometry of the Shulba Sutras, geometrical appendices to the manuals of ritual (Shrauta Sutras), include the oldest known formulation of the theorem named after Pythagoras, developed in the context of Vedic altar-building. Modern Hindus are fond of recalling this scientific element in their tradition, e.g. by quoting Carl Sagan: “Hindu cosmology gives a time-scale for the earth and the universe which is consonant with that of modern scientific cosmology”, as opposed to the limited Biblical-Quranic cosmology, which was protected against more far-sighted alternatives by a vigilant religious orthodoxy."

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"In the whole history of Mathematics, there has been no more revolutionary step than the one which the Indian made when they invented the sign ‘0’ for the empty column of the counting frame."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"It is clear how much we owe to this brilliant civilization, and not only in the field of arithmetic; by opening the way to the generalization of the concept of the number, the Indian scholars enabled the rapid development of mathematics and exact sciences. The discoveries of these men doubtless required much time and imagination and, above all, a great ability for abstract thinking. These major discoveries took place within an environment which was at once mystical, philosophical, religious, cosmological, mythological and metaphysical."

"Ancient Indian culture has regarded the science of numbers as the noblest of its arts … A thousand years ahead of Europeans, Indian savants knew that zero and infinity were mutually inverse notions. In short, Indian science was born out of a mystical and religious culture and the etymology of the Sanskrit word used to describe numbers and the science of numbers bears witn The early passion which Indian civilization had for high numbers was a significant factor contributing to the discovery of the place-value system, and not only offered the Indians the incentive to go beyond the calculable physical world, but also led to an understanding much earlier than in our civilization of the notion of mathematical infinity itself."

"Because the Greeks were ‘ingenious and inventive’, they must have developed mathematics, and that is what we must assume even in the absence of evidence. And because the Hindus were backward and primitive, they could not have created mathematics, and that is what we must assume even in the presence of evidence."

"C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore."

"My confidence in our shared future is grounded in my respect for India’s treasured past—a civilization that has been shaping the world for thousands of years. Indians unlocked the intricacies of the human body and the vastness of our universe. And it is no exaggeration to say that our information age is rooted in Indian innovations—including the number zero."

"The science of numbers is used in all transactions whether earthly or Vedic, meditation or business, in the science of love, in the art of cooking, in economics and statecraft, in music and medicine, in theatre, architecture and all such things. (9-10) In prosody, in poetics and poetry, in logic, grammar and such like, in the presentation of the qualities of various arts, mathematics is held in esteem. For the movements of the sun and stars, for eclipses and planetary motion, for the three significant questions and the course of the moon, for all these connections it is used (11-12). The number, diameter and perimeter of islands, of oceans and mountains, for the dimensions of rows of buildings of the earth’s inhabitants, of the space between the worlds, of the world of light, for the worlds of devas, for the dimensions of the dwellers in hell; for measurements of all sort and voluminous evidence, all these are made out by means of computation (13- 14). The organization of living beings, the length of their lives, their eight attributes and other similar things, their travel and cohabitation and such other things, all depend on mathematics. What more can be said? All there is in the three worlds, moving or unmoving, all that cannot be described without mathematics (15-16)."

"To the Hindus is due the invention of algebra and geometry, and their application to astronomy."

"I once heard, and I think it is true, that only one man in the world—some Indian mathematician—understood the mathematics of string theory in eleven-dimensional space, and he dreamed it."

"We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards (third century).... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks."

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"Many of the key results of mathematics, some of which are at the very “core” of modern day mathematics, are of Indian origin. ...‘I wish to conclude initially by simply saying that the work of Indian mathematicians has been severely neglected by western historians.’...‘it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Indian, contributions shakes up views that have been held for hundredsof years, and challenges the very foundations of the Eurocentric ideology."

"The Hindus no less than the Greeks have shared in the work of constructing scientific concepts and methods in the investigation of physical phenomena , as well as of building up a body of positive knowledge which has been applied to industrial technique; and Hindu scientific ideas and methodology (e.g. the inductive method or methods of algebraic analysis) have deeply influenced the course of natural philosophy in Asia—in the East as well as the West—in China and Japan, as well as in the Saracen empire."

"However, Seidenberg was told by the indologists that these Sutras, or any Vedic text for that matter, were definitely written later than 1700 BC. But mathematical data cannot be manipulated just like that, and Seidenberg remained convinced of his case: “Whatever the difficulty there may be [concerning chronology], it is small in comparison with the difficulty of deriving the Vedic ritual application of the theorem from Babylonia. (The reverse derivation is easy)… the application involves geometric algebra, and there is no evidence of geometric algebra from Babylonia. And the geometry of Babylonia is already secondary whereas in India it is primary.” [To satisfy the indologists, he said that the Shulba Sutra had conserved an older tradition, and that it is from this one that the Babylonians had learned their mathematics:] “Hence we do not hesitate to place the Vedic (…) rituals, or more exactly, rituals exactly like them, far back of 1700 BC. (…) elements of geometry found in Egypt and Babylonia stem from a ritual system of the kind described in the Sulvasutras.”"

"There is not, and cannot be, number as such. There are several number worlds as there are several Cultures. We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number — each type fundamentally peculiar and unique, an expression of a specific world feeling, a symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one. ... and so it is understandable that even negative numbers, which to us offer no conceptual difficulty, were impossible in the Classical mathematic, let alone zero as a number, that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as base for a positional numeration, was nothing more nor less than the key to the meaning of existence. Negative magnitudes have no existence.... But when we are told that probably (it is at best a doubtful venture to meditate upon so alien an expression of Being) the Indians conceived numbers which according to our ideas possessed neither value nor magnitude nor relativity, and which only became positive and negative, great or small units in virtue of position, we have to admit that it is impossible for us exactly to re-experience what spiritually underlies this kind of number. For us, 3 is always something, be it positive or negative; for the Greeks it was unconditionally a positive magnitude, +3; but for the Indian it indicates a possibility without existence, to which the word “something” is not yet applicable, outside both existence and non-existence which are properties to be introduced into it. +3, -3, 1/3, are thus emanating actualities of subordinate rank which reside in the mysterious substance (3) in some way that is entirely hidden from us. It takes a Brahmanic soul to perceive these numbers as self-evident, as ideal emblems of a self-complete world form; to us they are as unintelligible as is the Brahman Nirvana, for which, as lying beyond life and death, sleep and waking, passion, compassion and dispassion and yet somehow actual, words entirely fail us. Only this spirituality could originate the grand conception of nothingness as a true number, zero, and even then this zero is the Indian zero for which existent and non-existent are equally external designations."

"I shall not now speak of the knowledge of the Hindus … of their suitable discoveries in the science of astronomy—discoveries even more ingenious than those of the Greeks and Babylonians, of their rational system of mathematics, or of their method of calculation which no words can praise strongly enough; I mean the system using nine symbols."

"I will omit all discussion of the science of the Hindus, a people not the same as the Syrians, their subtle discoveries in the science of astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe because they speak Greek, that they have reaced the limits of science should know these things, they would be convinced that there are also others who know something."

"Like the crests on the heads of peacocks, like the gems on the hoods of the cobras, mathematics is at the top of the Vedanga sastras."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"We now know that equations of degree 3 are in some kind of border area between equations of lower degree (easy) and equations of higher degree (very hard)."

"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."

"The equations of conic sections involve at most the square of x and y, never the third or higher powers. In contrast, the equation of an elliptic curve has an x^3-term; a typical example is y^2 = x^3 - x."

"For about 1500 years, from the time of Diophantus to Newton, elliptic curves were known only as curves defined by certain cubic equations. This put them just a step beyond the conic sections, and some of their geometric and arithmetic properties can in fact be viewed as generalisations of properties of conics."

"Richard Borcherds, (quote at 21:37 of 1:36:06 in video)"

"A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists (and the greatest possibility of using different consistent axiom systems)."

"The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around."

"The requisites for the axioms are various. They should be simple, in the sense that each axiom should enumerate one and only one statement. The total number of axioms should be few. A set of axioms must be consistent, that is to say, it must not be possible to deduce the contradictory of any axiom from the other axioms. According to the logical 'Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency."

"Counterexample philosophy is a distinctive pattern of argumentation philosophers since Plato have employed when attempting to hone their conceptual tools."

"Occasionally, one individual may come up with a "proof," and another with a "counterexample." Since a valid proof and counterexample cannot peacefully coexist, either the proof has some logical or mathematical flaw, or the counterexample does not faithfully represent the conditions involved, or perhaps both. This is another reason why it is so important to have good command of the underlying logic."

"Whenever the bigger theorems are stated and proven, Landau usually shows that all the hypotheses are needed by dropping each one and giving a counterexample. In some cases the counterexamples are very elaborate, such as van der Waerden’s continuous, nowhere differentiable function, and a continuous function whose Fourier series diverges."

"What is the role of counterexamples in mathematics? (Are there any in Euclid?)"

"Wavelets are everywhere nowadays. Whether in signal or image processing, in astronomy, in fluid dynamics (turbulence), or in condensed matter physics, wavelets have found applications in almost every corner of physics. Furthermore, wavelet methods have become standard fare in applied mathematics, numerical analysis, and approximation theory."

"On the one hand, the concept of wavelets can be viewed as a synthesis of ideas which originated during the last twenty or thirty years in engineering (subbing coding), physics (coherent states, renormalization group), and pure mathematics (study of Calderón-Zygmund operators). As a conseuqence of these interdiscplinary origins, wavelets appeal to scientists and engineers of many different backgrounds. On the other hand, wavelets are a fairly simple mathematical tool with a great variety of possible applications."

"Wavelets were developed independently by mathematicians, quantum physicists, electrical engineers and geologists, but collaborations among these fields during the last decade have led to new and varied applications. What are wavelets, and why might they be useful to you? The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers feel that using wavelets means adopting a whole new mind-set or perspective in processing data. Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions."

"Wavelet theory is nowadays a very active field of approximation theory with a wide impact on signal analysis, high-performance imaging applications, and adaptive transversal filter theory. It is concerned with the modeling of univariate and multivariate signals with a set of specific signals. The specific signals are just the wavelets. Families of wavelets are used to approximate a given signal (with respect to the L2 norm, say), and each element in the wavelet set is constructed from the same original window, the mother wavelet."

"Wavelets were introduced at the beginning of the 'eighties by J. Morlet, a French geophysicist at Elf-Aquitaine, as a tool for signal analysis in view of applications for the analysis of seismic data. The numerical success of Morlet prompted A. Grossmann to make a more detailed study of the wavelet transform, which resulted in a paper giving the mathematical foundations (see Grossmann & Morlet ..., where the title of the paper still shows the name wavelets of constant shape. In 1985, the harmonic analyst Y. Meyer became aware of this theory and he recognised many classical results inside it. Meyer pointed out to Grossmann and Morlet that there was a connection between their signal analysis methods and existing, powerful techniques in the mathematical study of singular integral operators. Then Ingrid Daubechies became involved, and all this resulted in the first construction of a special type of frames (see Daubechies, Grossmann & Meyer ..), (the concept frame generalizes the concept basis in a Hilbert space). It was also the start of a cross-fertilization between the signal analysis applications and the purely mathematical aspects of techniques based on dilations and translations."

"A Haar wavelet is the simplest type of wavelet. In discrete form, Haar wavelets are related to a mathematical operation called the Haar transform. The Haar transform serves as a prototype for all other wavelet transforms."

"A groundbreaking advance on the way to efficient structures in steel construction was the principle of using tetrahedra as a basic module instead of rectangular geometries. The inventor is considered to be the all-round genius Alexander Graham Bell, who became famous for the invention of the telephone and who built kites that were big enough to lift people into the air. Another engineering genius was Vladimir Grgorevic Suchov, whose genius can definitely be compared to Gustav Eiffel. In 1919 he designed towers up to 350 m high on the principle of hyperbolic paraboloids. The lightness of his constructions is seldom achieved even today, in Moscow there is a television tower with a height of 160 m. His tent constructions with suspended steel grids can be seen as the forerunners of the Olympic roof in Munich or the new Center Pompidou Metz. In , Suchov realized the first double-curved lattice shells on the floor plan of rectangular halls as early as the 19th century. Most of the structures still in existence are massively endangered by corrosion and destruction; current rescue operations are trying to preserve this legacy."

"Shukhov's water tower['s]... double curved surface... was generated by a mesh of straight members overlapping in contrary directions... supported by horizontal rings. While... constructed from steel... Shukhov's 1896 patent application... initially mentions straight wooden beams as a material option. ...[T]he ...application describes ...being able to resist extreme forces while using very little material. As a result, Shukhov's... design was used extensively throughout Russia in the first half of the twentieth century."

"The theory of minimal surfaces and surfaces of constant mean curvature is a vigorously developing branch of mathematics which has a broad range of applications in physics, chemistry and biology where it investigates, for example, soap films and bubbles, bimaterial interfaces or capillaries. Over the past decades, it was also boosted by the anticipated nanotechnology applications. Applications of minimal surfaces have been extensively explored in the sculpture (see e.g. the works by ) and in architecture (s, the Shukhov radio tower ... works by etc.) The first mention of minimal surfaces goes back to Lagrange (1798), who considered the... variational problem..."

"Construction history teaches us that excellent, huge structures like this one are invariably the result of a specific and singular solution, in which all aspects were reflected upon before or after early experiences in the building sequence. The author wants to compare the achievement of Segovia Aqueduct with more recent singular structures: the Oka electricity pylons (1929) by... Vladimir Gregoryevich Shukhov... and the roof of the 1972 Olympic Games in Munich, by a team directed by Gunter Behnish, , /, and Jürgen Linkwitz..."

"The structures of the great Russian engineer Vladimir Grigor'evič Šuchov (Shukhov) are among the world's most sophisticated and distinctive in the history of steel construction. These extremely stable structures, such as cable-stabilised arches, doubly curved grid-shells and above all lattice towers hold a great fascination... They result from the desire to achieve an engineering objective using as little material as possible. ...[T]hey are a testament to the extraordinary creativity and inventiveness of an extensively educated engineer ..."

"Many engineering structures of today were anticipated in Shukhov's works. Some of his others have no modern equivalent or have remained unmatched in their visual impact and... technical efficiency. Among these are... the lattice towers... Countless towers with this new type of construction and geometries defined by just a few parameters were built [by him]... The fine-lined tower structures served as water towers, lighthouses, power transmission masts and fire brigade watchtowers—with some of them still in use today"

"Matthias Beckh analyzes these hyperbolic lattice towers... [and] demonstrates how... Shukhov was already parameterizing his structures as part of the design... Modern methods... provide... Beckh with the tools to place Shukhov's achievements in a historical context and validate their considerable contribution to the history of . He also...demonstrate[s] their relevance to modern structures. ...Beckh was part of the first research project into Shukhov's structures ...The research project ...included in-depth studies of building history ...and detailed investigations into the way the structures were built. Later investigations were intended to provide ...knowledge about the load assumptions... also... relevant to modern structures."

"The object of this book is gain deeper knowledge on the architectural history of hyperbolic structures. The focus of the investigation is the first hyperbolic lattice towers ever built, the work of... Shukhov."

"This form of construction, which had no predecessors... is notable for its strength and economy of materials. Added to this is the high visual impact..."

"Even today, Shukhov's load-bearing system can be found in one form or another in architecture..."

"This book presents the results of the first ever extensive analysis of the way these structures work."

"The of a one-sheeted hyperboloid is resolved into three different mesh variants to create open lattices and their structural behaviour investigated."

"Particular attention is paid to evaluation and analysis of Shukhov's tower calculations and the assumptions made for the structural model. His historical calculations are compared to the results of modern calculations."

"Shukhov's design process is reconstructed and the development of the water towers... illustrated."

"Until now, the form and geometry of the hyperbolic lattice structures have only been investigated to a limited extent."

"A paper by Peter de Vries discussed the stiffness of simple hyperbolic lattice structures and highlighted a single connection between geometry and structural behaviour. The focus... is... on a simple form of hyperbolic lattice structure which always has the intermediate rings positioned at intersection points of the lattice members."

"[T]he form and structure of the Sukhov-built towers were developed not on geometrical or constructional criteria alone, their designs specifically took into account considerations."

"The chapter "Geometry and form of hyperbolic lattice structures"... deals with the form and geometry... A precise description of the mathematical principles of a one-sheeted hyperboloid precedes an explanation of the parametrisation..."

"[T]he chapter "Structural analysis and calculation methods"... considers the principle means of transfer of vertical and horizontal loads and describes the interactions between geometry and structural behaviour. Then... an explanation of the theoretical principles of determining a lattice tower's ultimate load capacity."

"The chapter "Design and analysis of Sukhov's towers"... is devoted to consideration and analysis of Sukhov's structural calculations... Sukhov's design process is reconstructed based on... calculations of five different water towers. From... tables stored in the Moscow city archives, a summary... is produced to chart the development of the towers over more than three decades..."

"An analysis of the design and construction of the NiGRES tower on the Oka... The initial ensemble of four electricity transmission masts represents the consummation of Shukhov's tower construction method. Only one... remains... today."

"The final chapter "Towers in comparison"... contains an extensive table and drawings of 18 towers."

"It is hoped that the examination of Sukhov's form of construction made public in this book will give an impetus to new applications in architecture."

"[A] new reconfigurable doubly-curved structure has been developed for a canopy roof."

"[A] series of kinetic structures which are capable of geometric transformations have been developed. ...[T]he most impressive ones are deployable bar structures with [a] single degree-of-freedom ...These structures ...may become stable and carry loads ...Therefore ...may offer viable solutions for architectural applications, especially for temporary buildings, emergency shelters, exhibition halls, outdoor recreation facilities or sporting fields ...[M]ost of them are composed of scissor like elements (SLEs) ...which is a complex structural system ...[P]resent solutions are insufficient to constitute real form flexibility, because they are limited to... forms such as singly-curved vaults and doubly-curved synclastic s."

"Due to... complexity... doubly-curved anticlastic surfaces such as hyperboloid and hyperbolic paraboloid (HP) have been rarely used for deployable structures. In fact, anticlastic structures can be easily constructed using simple straight bars rather than SLEs since their geometric forms can be generated by s..."

"[A]nticlastic structures are capable of resisting... various design loads through their curvatures and twists. Thus, their structural efficiency is more than ...others. ...[W]e have decided to use HP ...as a [deployable] canopy roof structure."

"In 1829... Nikolai Ivanovich Lobachevsky published a disproof of Euclid's fifth or using the case of a doubly curved surface, thereby establishing non-Euclidean geometry."

"According to Elizabeth C. English, there was a direct connection between the mathematical discoveries of Lobachevsky and the structural explorations of"

"The "idea of the " and the first... diagrid structure have been credited to... Shukhov. The design evolved as an efficient and easily constructed tower for carrying a large gravity load... a water tower. The "Shukhov Tower"... 1896, relies on the use of a diagonal lattice of steel angles, constrained laterally... by steel rings. ...The tower is hollow, requiring little resistance to wind loads ..."

"Shukhov is credited with the construction of some 200 towers using this method ..."

"Although the slender steel sections, when formed into the hyperbolic paraboloid shape, did undergo some bending, a diagrid shape emerges, using a combination of straight segments that are joined at their nodal points of intersection."

"The Shukhov towers tended to use much longer steel sections and overlap them at their crosspoints, rather than using the crosspoints as "nodes" in the fashion of later or spaceframe structures."

"[T]he diagrid form could support both the gravity loads and the lateral loads without requiring additional means. In comparison, the new skyscraper types that were under development at this time used a steel frame to support the gravity loads... while the central core provided the stiffness required to resist wind loads. Where additional resistance was required, it was usually the core that carried the bracing."

"The School of Engineering and Construction which came into being in Russia after the reforms of the 1860s, and in connection with large-scale railway construction towards the end of the nineteenth century, soon acquired a worldwide reputation. It produced distinguished theorists and practitioners who included Belelyubsky, Loleit, Proskuryakov, Shukhov and Yasinsky. The engineering work incorporated in the most diverse kinds of buildings in Russia during the last quarter-century before the Revolution show that this country not only kept pace with the more developed parts of Europe and the USA, but outstripped them in certain fields, in terms of engineering design, as well as the use of modern construction techniques and new materials. Thus Shukhov produced many original designs unparalleled in work done abroad. At the Nizhny-Novgorod Exhibition in 1896... several original designs by Shukhov appeared, the most important of which were latticed: suspended latticed roofs for exhibition halls with a circular, elliptical and rectangular ground plan, latticed roof vaults and hyperbolic latticed towers. This category of structure also included the dual curvature roof designed by Shukhov in 1897-98 to cover the workshop at the Vykhsusnk factory."

"Forms of spanning which have come into use in recent years were foreshadowed in various forms of latticed structures designed by Shukhov before the end of the last century—guyed structures, the spanning of dual curves by structures of standardized rods, the provision of curved 'hyperbolic' profiles by means of a straight component, using straight rods in Shukhov's case."

"Russian engineers made many new and stimulating contributions to the design and construction of multiple-span lattice metal bridges, a great number of which were needed in railway work because of the multitude of large rivers in Russia. Ferro-concrete was first used in buildings there at the beginning of the twentieth century. Russian engineers... made their own contribution to the development of structural techniques in this area."

"The creative principles of innovative trends such as Rationalism and Constructivism were well understood by engineers and constructors bent on applying the latest technological achievements. Great engineers such as Shukhov, [Artur] Loleit and [Hermann] Krasin, were happy to work with innovative architects. They jointly explored new ways of developing architecture, and engineers became active members of Asnova and Osna. ...In the twenties... Soviet architecture set tasks for the building industry which prompted the employment of new materials and structural elements, and raised building standards. Engineering technology in the building industry achieved great successes during this period, such as the metal structures by Shukhov and Krasin, Loleit's work in the field of ferro-concrete, the elaboration of modern timber work by Karlson and the production of new hyperbolic paraboloid roofing by Markarova. The very latest structures and building materials, as well as modern production methods, were applied in the construction of engineering and industrial buildings such as Shukhov's radio tower in Moscow in 1922 and Krasin's viaduct at the Shatura power station in 1925."

"[H]yperbolic s are associated with nuclear and thermal power plants... they are also used... in some large chemical and other industrial plants. [T]hey are high rise structures in the form of doubly curved thin walled shells of complex geometry..."

"The in-plane membrane actions primarily resist the applied forces and plays the secondary role in these special structures."

"The first hyperboloid shaped cooling tower was introduced by the Dutch engineers and Gerard Kuypers and built in 1918 near having 35 meter height."

"The hyperbolic geometry has advantage of a negative which makes it superior in stability against external pressures..."

"The widened bottom of the tower accommodates large installation of fill to facilitate the evaporative cooling of the thin film of circulated water. Narrowing effect of the tower accelerates the of evaporation and diverging top promotes turbulent mixing which increases the contact between hot inside air and cooler outside air."

"The... structure is made of high-strength [cement,] Reinforced... Concrete... in the form of hyperbolic thin shell standing on diagonal, meridional, or vertical supporting columns and radial supports. The shell is sufficiently stiffened by upper and lower edge members."

"[T]o achieve sufficient resistance against instability, large cooling tower shells may be stiffened by additional internal or external rings which may also be used as repair or rehabilitation..."

"The hyperbolic form of thin-walled towers provides optimum conditions for good aerodynamics, strength, and stability."

"[T]he first cooling tower shell [to be] analyzed by means of a shell bending theory [was in 1967]."

"[T]he most preferred method of the modeling and analysis of NDCT [natural draft cooling towers] is [the] (FEM)."

"Determination [utilizing nonlinear static analysis] of the ultimate strength of the cooling towers... subjected to the severe quasistatic wind loads is one of the [research] objectives... for... structural engineers. Various nonlinear factors, such as the material nonlinearities in the concrete and reinforcing steel, tensile cracking, the bond effects between concrete and steel in the cracked concrete which is known as the tension stiffening, the large displacement effects, and so on; need to be taken into account..."

"Mahmoud and Gupta... concluded that the failure of the Port Gibson Tower [in] Mississippi was caused by the circumferential in the vicinity of the throat rather than the yielding of the reinforcement, which contradicts... previous researchers."

"Deformation response and ultimate strength of RC shell structures are governed predominantly by material response of concrete and reinforcing steel, tensile cracking of concrete, [and] bond between concrete and steel.... Softening response of concrete due to quasi-brittle cracking in tension also... influences the nonlinear response by inducing loss of strength and stiffness... Due to all [of] these, analysis requires attention for realistic modeling of the layer of shell concrete confined between the reinforcement layers. ...[O]ne of the most challenging areas... is the modeling techniques using the layered elements."

"The NDCT [natural draft cooling tower] shell structures are submitted to environmental loads such as wind, earthquake and thermal gradients that are in nature. The dead loads, settlement, and construction load are... common... and various accidental loads, e.g., explosion [are]... experience[d]... in their lifetime."

"[I]n 1967, closed-form expressions [were] derived by Gould and Lee... for determining... stresses in the shell and corresponding deformations under static seismic design load."

"Abu-Sitta and Davenport... investigated the effects of dynamic earthquake loading. ...[I]nduced dynamic stresses were related to equivalent membrane stresses from static loads, resulting in... a simplified earthquake analysis procedure."

"Asadzadeh et al... studied the structural response... under static wind and pseudo static seismic forces. ...[T]wo types of... column supports at the base of the towers had been considered... The structural response... under wind and earthquake was... completely different for the towers supported on different columns."

"Asadzadeh et al... studied the effects of the inclination angle of the supporting columns on the dynamic response of the cooling towers. ...[S]tiffness of the structure increases with increase in inclination angle of the supporting columns resulting in decrease of the period therefore altering the resistance... against the earthquake loading. ...[T]he hyperbolic structure... can be optimized by finding the optimum inclination angle of the supporting columns."

"Wind is the prime lateral load and its combination with self weight of the tower shell can cause the instability leading to catastrophic failure."

"After the sudden collapse of three immense cooling towers at Ferry-Bridge Power Station in England in 1965, experimental and theoretical investigations had been done in... the stability of hyperbolic shells to study the parameters increasing the wind resistance and buckling safety..."

"The wind-induced response... is the key factor to improve safety and to reduce tower crack[ing]..."

"Form... found that the construction of the stiffening rings significantly influenced the safety factor (BSF) and structural buckling stability... He showed that adding 2, 3, or 4 stiffening rings to the cooling tower increase the BSF of the R.C. shell by a factor of 1.65, 2.32, and 2.80, respectively."

"Sabouri-Ghomi et al... investigated... buckling stability of R.C. cooling towers supported on X-shaped columns by considering... parameters such as the number, dimension, and location of stiffening rings... The stiffening rings behaved flexibly or rigidly, depending on their dimensions. The number of flexible stiffening rings required to maximize the buckling safety factor was found to be higher than the number of rigid stiffening rings..."

"Ref: S. Sabouri-Ghomi, M.H.K. Kharrazi, and P. Javidan, "Effect of stiffening rings on buckling stability of R.C. hyperbolic cooling towers," Thin Walled Structures, vol. 44, no. 2, pp. 152-158, 2006."

"Zhang et al... studied the effect of the stiffening ring... to improve the dynamic properties and the wind resistance... They... concluded that for the latitude and meridian directions, the stiffness in the latitude direction contributes more... and... the location where the maximum modal displacement of the unstiffened HCT appears will not always be the most effective place for an additional stiffening ring unless the mode shapes of the unstiffened and stiffened HCTs are similar."

"Consideration of the soil–structure interaction... is extremely important when the soil or the foundation medium is not very firm. ...[U]nder ...dynamic loadings, the structure interacts with the surrounding soil imposing soil deformations. These... change the response of the structure."

"Noorzaei et al... analyzed the cooling tower–foundation–soil system under vertical and lateral load generated due to self-weight and wind loads. In this study, the unsymmetrical wind pressure distribution in terms of [were] given..."

"The collapse of three natural draft cooling towers at the Ferry bridge power station in 1965... [was] due to the inadequate design for the wind forces..."

"In recent years... numerical simulation... has been applied to describe the collapse of structures, e.g., the collapse of cooling towers under blasting demolition... and the collapse of the World Trade Center..."

"Due to the complexity of the building procedure, uncertainties in the material properties as well as differences between the theoretical and the real geometry... reliability analysis... seems... indispensible..."

"This review is a complete collection of the studies done for cooling towers and... [gives] updated and sufficient materials for the researches in this field."

"We present a new mechanical model of interatomic bonds, which can be used to describe the elastic properties of the carbon allotropes, such as , diamond, , and s. The interatomic bond is modeled by a hyperboloid–shape structure."

"[T]here are... approaches that are closer to the field of the classical mechanics... so–called structural or discrete-continuous methods... The most straightforward example... is a modeled by the solid deformable rod... [T]he interatomic bonds are modeled as a deformable body or a construction. ...[T]hese approaches... can be implemented in standard computing packages based on the finite–element, boundary–element, or s. These methods can be considered as the bridges between the s of atomistic and continual models of the material."

"In this paper, the carbon bond model is built on the symmetry properties of the hyperboloid. These properties allow it to achieve a high ratio of the lateral and longitudinal , therefore the hyperboloid shapes are widely used in the engineering to create lightweight constructions consisting of straight beams that are known for being able to carry a large load while achieving a low use of raw materials."

"[T]he first hyperboloid tower was built by Russian engineer V.G. Shukhov in (1896)... Being widely demanded in architecture and engineering, such models still haven’t found a wide use in micro- and . It appears that the analogy drawn from the macro level will allow to describe correctly the properties of carbon materials at the micro level."

"Alongside Shukhov’s drawings and sketchpads... a typescript produced by Shukhov’s former employee Grigory Kovelman... put together an extensive overview of Shukhov’s inventions and projects, both as a biographer and as a specialist who had worked with Shukhov.... Elena Shukhova... grand-daughter... presents extensive biographical details in Vladimir Grigorevich Shukhov. The First Engineer in Russia... Art of Construction... is a valuable collection of articles about Shukhov and his various inventions, written by... specialists. It includes... examples of his calculations. Shukhov’s own book Rafters... discusses the mathematical investigations which led him to conceptualise the spatial lattice structure, describing it as 'an optimisation process'."

"In 2010, the journal Detail published an analysis of Shukhov’s constructions, calling his approach to design 'an early example of ’..."

"Vaulted gridshell constructions... were formed with thin metal arches turned away from the frontal position at a particular angle. They thus worked as one continuous resilient truss. ...Each arch was made with rigid metal strips of equal length... during the assembling process, each piece was bent equally. ...It was the first time in the world’s building practice that double-curved spatial vaults were created with single type rod elements ..."

"Shukhov’s lattice-suspended and vaulted structures represented a carrying surface, which could be shaped in any form. ...The density of the grid made it possible to attach it to the shell without additional structures. ...[T]he grids were two to three times lighter than roofs with conventional frames..."

"The final and most unusual of the gridshell structures presented at the Exhibition was the 32-metre-tall lattice hyperboloid water tower. Everything was amazing in that first Shukhov tower—everything in it was some structural and geometric puzzle: straight rods and the external silhouette double curvature, the openwork lightness below and the solid heaviness above."

"The water tower was a unique structure of its time... According to Cooper, the idea... came directly from an imaginary hyperboloid geometry, invented by... Lobachevski in 1829..."

"Grigory Kovelman writes that Shukhov told him he had been thinking about the properties of hyperboloid structures for a long time, that he had studied hyperboloid forms at the Technical School, and that apparently the moment of enlightenment came about when he saw an up-ended wicker wastepaper basket with a focus on top of his desk. According to Shukhov, this was when he understood clearly how a hyperboloid structure with its curved surface [was] generated by straight rods..."

"[T]he structure of the lattice tower was a spatial system, where the load was equally spread along the surface. ...Aiming to optimise the design process, soon after building the tower Shukhov presented the standardised elements of the tower structure in a table format... with the aid of which it became possible to design a new water tower according to a client’s requirements in twenty-five minutes..."

"The Radio Tower in Moscow is a gridshell which aims for structural efficiency. Its minimal surface and open lattice structure help in reducing the wind load, one of the main challenges in high-rise building design. ...Shukhov's design logic focuses on structural . The rings between the different segments offer additional reinforcement to create an equilibrium between minimal material consumption, structural efficiency and geometry."

"As hyperboloid structures are double curved, that is simultaneously curved in opposite directions, they are very resistant to . This means that you can get away with far less material than you would otherwise need..."

"Single curved surfaces, for example cylinders, have strengths but also weaknesses. Double curved surfaces... are curved in two directions and thus avoid... weak directions."

"[T]his is the magical part... despite the surface being curved in two directions, it is made entirely of straight lines. Apart from the cost savings of avoiding curved beams or shuttering, they are far more resistant to buckling because the individual elements are straight..."

"This is an interesting paradox: you get the best local buckling resistance because the beams are straight and the best overall buckling resistance because the surface is double curved."

"The hyperboloid is the design standard for all nuclear cooling towers and some coal-fired power plants."

"When designing... cooling towers, engineers are faced with two problems: I. The structure must be able to withstand high winds and II. They should be built with as little material as possible... The hyperbolic form solves both..."

"For a given diameter and height of a tower and a given strength, this shape requires less material than any other form. ...Hyperboloidal towers can be built from or as a steel lattice, and is the most economical such structure for a given diameter and height."

"The presented orthogonal fitting algorithm can be applied easily for ellipsoid, and sphere also other surface[s] such as paraboloid."

"In 1899... V. Shukhov... patented the principle of constructing hyperboloid gridshell structures based on a hyperboloid of revolution. ...A one-sheet hyperboloid is a connected surface having negative at each point. Through any point of this surface, two intersecting [straight] lines can be drawn, completely belonging to it. Thus the... surface can be formed by the set of straight lines. It the steel beams would be placed along these lines the shape... could be retained under... external loading... [I]n Shukhov's towers horizontal rims [rings] were used... located at different levels... The stability... was ensured by numerous riveted connections... During the installation... the straight angles of the metal profile was somewhat deformed... [at] the... intersecting elements in order to ensure maximum contact... by rivets."

"The main hazard for high-rise structures is the loads caused by wind gusts. The grid shell design... minimize[d] their influence. Open-work design... ensured... sufficient strength, high stability and low metal consumption. ...[C]onsumption of metal per unit height... was three times less than... the ..."

"... Ještěd in Czechia, Guangzhou in China... and Khan Shatir Shopping-Entertaining Center in Kazakhstan, Aspire Tower in Qatar... are vivid examples of using... the hyperboloid principle... [in] modern buildings."

"191 of the Shukhov's Towers (from known near 200...) have been irretrievably lost during... the 20th century... [Some] towers were destroyed because their continued use for water supply has become impractical. To use them for another purpose... large investments were necessary."

"Here as he walked by, on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i^2 = j^2 = k^2 = ijk = -1 & cut it on a stone of this bridge."

"Quantum theory may be formulated using s over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. ...[P]roblems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. ...There are precisely four 'normed division algebras': the real numbers \mathbb{R}, the complex numbers \mathbb{C}, the quaternions \mathbb{H} and the octonions \mathbb{O}. Roughly speaking, these are the number systems extending the reals that have an ‘absolute value’ obeying the equation |xy| = |x| |y|. Since the octonions are nonassociative [their use] proves difficult... except in a few special cases. ...[I]nstead of being distinct alternatives, real, complex and quaternionic quantum mechanics are three aspects of a single unified structure."

"The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative."

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"[Q]uaternions form the appropriate algebraic basis for a description of nature whenever we have to deal either with pseudoreal group representations or with co-representations of Wigner's Type II. The context in which quaternions arose historically, in a study of the three-dimensional rotation group, can now be seen to be an extremely special case of this general principle. Every group which admits pseudoreal representations equally admits a natural description in terms of real quaternions."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of by Monge in 1795. The quaternions of Sir William Rowan Hamilton, the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period. Whether they are destined to remain merely monuments of the ingenuity and acuteness of their authors, or are to become mighty instruments in the investigation of old and the discovery of new truths, it is perhaps impossible to predict."

"Frobenius' Theorem. Over the real number field there exist precisely three associative s, namely the real numbers, the complex numbers, and the real quaternions."

"More than a third part of a century ago, in the library of an ancient town, a youth might have been seen tasting the sweets of knowledge to see how he liked them. He was of somewhat unprepossessing appearance, carrying on his brow the heavy scowl that the "mostly-fools" consider to mark a scoundrel. In his father's house were not many books, so it was like a journey into strange lands to go book-tasting. Some books were poison; theology and metaphysics in particular they were shut up with a bang. But scientific works were better; there was some sense in seeking the laws of God by observation and experiment, and by reasoning founded thereon. Some very big books bearing stupendous names, such as Newton, Laplace, and so on, attracted his attention. On examination, he concluded that he could understand them if he tried, though the limited capacity of his head made their study undesirable. But what was Quaternions? An extraordinary name! Three books; two very big volumes called Elements, and a smaller fat one called Lectures. What could quaternions be? He took those books home and tried to find out. He succeeded after some trouble, but found some of the properties of vectors professedly proved were wholly incomprehensible. How could the square of a vector be negative? And Hamilton was so positive about it. After the deepest research, the youth gave it up, and returned the books. He then died, and was never seen again. He had begun the study of Quaternions too soon."

"My own introduction to quaternionics took place in quite a different manner. Maxwell exhibited his main results in quaternionic form in his treatise. I went to Prof Tait's treatise to get information, and to learn how to work them. I had the same difficulties as the deceased youth, but by skipping them, was able to see that quaternionics could be employed consistently in vectorial work. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalar and vectors, using a very simple vectorial algebra in my papers from 1883 onward. The paper at the beginning of vol. 2 of my Electrical Papers may be taken as a developed specimen; the earlier work is principally concerned with the vector differentiator ∇ and its applications, and physical interpretations of the various operations. Up to 1888 I imagined that I was the only one doing vectorial work on positive physical principles; but then I received a copy of Prof. Gibbs's Vector Analysis (unpublished, 1881-4)."

"Mr. McAulay asks: "What is the first duty of the physical vector analyst quâ physical vector analyst?" The answer is... to present the subject in such a form as to be most easily acquired, and most useful when acquired. ...What then is the cause of the fact ...all of us deplore? ...We need only a glance at the volumes in which Hamilton set forth his method. No wonder that physicists and others failed to perceive the possibilities of simplicity, perspicuity, and brevity... in a system presented... in ponderous volumes of 800 pages. ...[I]f we turn to his earlier papers on Quaternions in the Philosophical Magazine... we find... "On Quaternions; or on a New System of Imaginaries in Algebra," and in them we find a great deal about imaginaries and very little of a vector analysis. To show how slowly the system of vector analysis developed itself in the quaternionic nidus, we need only say that the symbols S, V, and &nabla; do not appear until two or three years after the discovery of quaternions. In short it seems to have been only a secondary object with Hamilton to express the geometrical relations of vectors... it was never allowed to give shape to his work. ...[I]s it not discouraging to be told that in order to use the quaternionic method one must give up the progress which he has already made in the pursuit of his favourite science and go back to the beginning and start anew on a parallel course? ...Whatever is special, accidental, and individual, will die, as it should; but that which is universal and essential should remain as an organic part of the whole intellectual acquisition. If that which is essential dies with the accidental, it must be because the accidental has been given the prominence which belongs to the essential. ...In Italy they say all roads lead to Rome. In mechanics, , astronomy, physics, all study leads to the consideration of certain relations and operations. These are the capital notions; these should have the leading parts in any analysis suited to the subject."

"If I wished to attract the student of any of these sciences to an algebra for vectors, I should tell him that the fundamental notions of this algebra were exactly those with which he was daily conversant. ...I should call his attention to the fact that Lagrange and Gauss used the notation (αβγ) to denote precisely the same as Hamilton by his S(αβγ) except that Lagrange limited the expression to s, and Gauss to vectors of which the length is the secant of the latitude, and I should show him that we have only to give up these limitations, and the expression (in connection with the notion of geometrical addition) is endowed with an immense wealth of transformations. I should call his attention to the fact that the notation [r_1r_2], universal in the theory of orbits, is identical with Hamilton's V(\rho_1\rho_2) except that Hamilton takes the area as a vector... I confess that one of my objects was to show that a system of vector analysis does not require any support from the notion of the quaternion, or... of the imaginary in algebra."

"Prof. Tait has spoken of the calculus of quaternions as throwing off in the course of years its early Cartesian trammels. I wonder that he does not see how well the progress in which he has led may be described as throwing off the yoke of the quaternion. A characteristic example is seen in the use of the symbol &nabla;. Hamilton applies this to a vector to form a quaternion, Tait to form a linear vector function. ...Now I appreciate and admire the generous loyalty toward one whom he regards as his master which has always led Prof. Tait to minimise the originality of his own work in regard to quaternions and write as if everything was contained in the ideas which flashed into the mind of Hamilton at the classic . But... we owe duties to our scholars as well as to our teachers, and the world is too large, and the current of modern thought is too broad, to be confined by the ipse dixit [he says] even of a Hamilton"

"I certainly admit that vectors may be used in connection with and to represent rotations. I have no objection to calling them in such cases versorial. In that sense Lagrange and Poinsot... used versorial vectors. But what has this to do with quaternions? Certainly Lagrange and Poinsot were not quaternionists. ...Does it follow that I have used a quaternion? Not at all. A quaternionic expression may represent a number. Does everyone who uses any expression for that number use quaternions? A quaternionic expression may represent a vector. Does everyone who uses any expression for that vector use quaternions? A quaternionic expression may represent a linear vector operator. If I use expression for that linear vector operator do I therefore use quaternions? My critic is so anxious to prove that I use quaternions that he uses arguments which would prove that quaternions were in common use before Hamilton was born."

"I have not been exclusively occupied by my Quaternions, but confess that they have been growing in interest upon me, and that I more and more believe they will one day justify a hope which I ventured to express... that they will constitute nothing less than "a new algebraical geometry.""

"I had been wishing for an occasion of corresponding a little with you on Quaternions: and such now presents itself, by your mentioning in your note... that you "have been reflecting on several points connected with them"... "particularly on the Multiplication of Vectors. ...No more important, or ...fundamental question, in the whole theory of Quaternions, can be proposed than that which thus inquires What is such Multiplication? What are its Rules, its Objects, its Results? What Analogies exist between it and other Operations, which have received the same general Name? And finally, what is (if any) its Utility? ...[R]eferring to an ante-quaternionic time, when you were a mere child, but had caught from me the conception of a Vector, as represented by a Triplet... I happen to be able to put the finger of memory upon the year and month—October, 1843—when... the desire to discover the laws of the multiplication referred to regained with me a certain strength and earnestness, which had for years been dormant, but was then on the point of being gratified, and was occasionally talked of with you. Every morning in the early part of the... month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me, "Well, Papa, can you multiply triplets"? Whereto I was always obliged to reply, with a sad shake of the head: "No, I can only add and subtract them." But on the 16th day of the same month… I was walking… and your mother was walking with me, along the … and although she talked with me now and then, yet an under-current of thought was going on in my mind, which gave at last a result, whereof... I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work, by myself if spared, and at all events on the part of others, if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse—unphilosophical as it may have been—to cut with a knife on a stone of , as we passed it, the fundamental formula with the symbols, i, j, k; namely,i^2 = j^2 = k^2 = ijk = -1,"

"all the numbers that have been derived from the genus are four; but number is the indefinite genus, from which was constituted, according to them, the perfect number, viz. the decade. For one, two, three, four, become ten, if its proper denomination be preserved essentially for each of the numbers. Pythagoras affirmed this to be a sacred quaternion, source of everlasting nature, having, as it were, roots in itself; and that from this number all the numbers receive their originating principle."

"The familiar proposition that all A is B, and all B is C, and therefore all A is C, is contracted in its domain by the substitution of significant words for the symbolic letters. The A, B, and C, are subject to no limitation for the purposes and validity of the proposition; they may represent not merely the actual, but also the ideal, the impossible as well as the possible. In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accordance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power."

"[O]f possible quadruple algebras the one... by far the most beautiful and remarkable was practically identical with quaternions, and... it [is] most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels."

"[T]he common solution, using three Euler's angles interpolated independently, is not ideal. The more recent (1843) notation of quaternions is proposed instead, along with interpolation on the quaternion unit sphere. Although quaternions are less familiar, conversion to quaternions and generation of in-between frames can be completely automatic, no matter how key frames were originally specified, so users don't need to know—or care—about inner details. The same cannot be said for Euler's angles, which are more difficult to use."

"While translations are well animated by using vectors, rotation animation can be improved by using the progenitor of vectors, quaternions. ...By an odd quirk of mathematics, only systems of two, four, or eight components will multiply as Hamilton desired; triples had been his stumbling block."

"Closely akin to his third and fourth propositions is Riemann's fifth proposition, that continuous quantities are coördinate with discrete quantities, both being in their nature multiples or aggregates, and therefore species of the same genus. This pernicious fallacy is one of the traditional errors current among mathematicians, and has been prolific of innumerable delusions. It is this error which has stood in the way of the formation of a rational, intelligible, and consistent theory of irrational and imaginary quantities, so called, and has shrouded the true principles of the doctrine of "complex numbers" and of the calculus of quaternions in an impenetrable haze."

"The next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions."

"I do think... that you would find it would lose nothing by omitting the word "vector" throughout. It adds nothing to the clearness or simplicity of the geometry, whether of two dimensions or three dimensions. Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell."

"Symmetrical equations are good in their place, but "vector" is a useless survival, or offshoot, from quaternions, and has never been of the slightest use to any creature. Hertz wisely shunted it, but unwisely he adopted temporarily Heaviside’s nihilism. He even tended to nihilism in dynamics, as I warned you soon after his death. He would have grown out of all this, I believe, if he had lived. He certainly was the opposite pole of nature to a nihilist in his experimental work, and in his Doctorate Thesis on the impact of elastic bodies."

"I see no possible objection to your now publishing the deferred "scrap" if you yourself approve of what is said in it in favour of quaternions. I think you are right in your use of the word "Volapuk," but I don't think you should confine it to the vector part of quaternions. The whole affair has in respect to mathematics a value not inferior to that of "Volapuk" in respect to language."

"I first became personally acquainted with Tait a short time before he was elected Professor in Edinburgh… It must have been either before his election or very soon after it that we entered on the project of a joint treatise on Natural Philosophy. He was then strongly impressed with the fundamental importance of Joule’s work… We incessantly talked over the mode of dealing with energy which we adopted in the book, and we went most cordially together in the whole affair. … We have had a thirty-eight years’ war over quaternions. He had been captivated by the originality and extraordinary beauty of Hamilton’s genius in this respect; and had accepted, I believe, definitely from Hamilton to take charge of quaternions after his death, which he has most loyally executed. Times without number I offered to let quaternions into Thomson and Tait if he could only show that in any case our work would be helped by their use. You will see that from beginning to end they were never introduced."

"Yet, though few, if any—Clerk-Maxwell perhaps only excepted—ever possessed the same almost magical quality of physical insight, none could be more strict than Lord Kelvin in requiring demonstration freed from untenable assumptions or undemonstrable hypotheses. Daring as he was, at least in his earlier days, in the application of analytical methods to the phenomena of nature, he was in several ways very conservative. For example, he never would countenance the use in physics of the method of quaternions. At the British Association Meeting at Cambridge in 1845, he had met Hamilton, who there read his first paper on Quaternions. One might have thought that the young enthusiast would have readily welcomed a new and ingenious method of symbolic analysis: but it was not so. He would not use quaternion notation or quaternion methods himself, nor did he admit the into his work."

"'(1). The word "Quaternion" requires no explanation, since... it occurs in the Scriptures and in Milton. Peter was delivered to "four quaternions of soldiers" to keep him; Adam, in his morning hymn, invokes air and the elements, "which in quaternion run." The word (like, the Latin "quaternio," from which it is derived) means simply a set of four, whether those "four" be persons or things."

"'(2). But the question arises, what special connexion has the number Four with mathematics generally, or with that branch of mathematical science in particular, to which the "Lectures on Quaternions" relate?"

"'(3). One general form of answer... is... that in the mathematical quaternion is involved a peculiar synthesis, or combination, of the conceptions of space and time; and that while TIME is usually pictured or represented by metaphysicians under the figure of a line—a single stream with its ONE current—an unique axis of progression, SPACE is, on the contrary, imagined or conceived in connexion with THREE distinct axes, three lines at right angles to each other... height, length, and breadth. In time, we have only the forward and the backward, looking before and after. In space, there is not merely the contrast between the directions of upward and downward, but also between those of southward and northward, and again between westward and eastward. Time is said to have only one dimension, and space to have three dimesions. The former is an unidimensional, the latter a tridimensional progression. The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space," or "space plus time": and in this sense it has, or at least it involves a reference to, four dimensions. In an unpublished sonnet to Sir John Herschel, entitled "The "(...Greek ...equivalent to the Latin Quaternio), the author of the Lectures introduced the two following lines... an expression of the view... in the foregoing remarks..:"And how the One of Time, of Space the Three, Might in the Chain of Symbol girdled be.""

"'(4). Those who are entirely unacquainted with mathematical science may yet derive, from what has been above remarked, a sufficient preliminary insight into the nature of the speculations and inquiries to which the "Lectures on Quaternions" relate. A philosophical, if not a technically scientific, knowledge of the author's general aim, and of the idea which has guided him, may in this way be easily attained. But a very moderate acquaintance with the conceptions of geometry will suffice to render intelligible, from another point of view, the importance which the author attaches to the number Four in mathematics."

"(5). As early as the first book of Euclid's Elements, an attentive student is (or may be) led to consider the relative length, and also the relative direction, of one straight line as compared with another. Thus when Euclid shows, in his very first proposition, how to construct on a given base AB an equilateral triangle ABC, he virtually teaches how, when one line AB is proposed or given, to draw a new line BC (or AC), which shall in length be equal to the given one, and in direction shall make with it an angle of sixty degrees, namely, the angle ABC (or BAC), which is the third part of 180 degrees, or of two right angles."

"'(6). In this elementary example, if the length of the given base AB be taken as the standard of length, and be on that account called unity, or one, then the length of the side BC (or AC) of the triangle must also be denoted by the same number, ONE; and these TWO NUMBERS, one, and sixty, serve in this view to define, or to describe, the length and direction of the new or constructed line BC; at least if the latter number (sixty) be combined with the consideration of a certain hand, or direction of rotation, towards which the old line BA may be conceived to turn, in the plane of the triangle (or of the paper)..."

"'(7). The foregoing view, although not precisely the same with that adopted by Euclid himself, in his exposition of the elements of geometry, is at least consistent therewith; and has been made the basis of an important and modern method of calculation, respecting directed lines in one plane, which seems to have been first introduced about the commencement of the present century, by Argand in France, and for which Professor De Morgan... has lately proposed the name of Double Algebra: because it recognises and employs two numerical elements (such as the numbers 1 and 60 in the foregoing example), as required for the joint determination of the length and direction of a straight line. And it is now to be shown what is the nature of the passage that has been made, by the author of the Lectures on Quaternions, from such a double system of algebraic geometry, to what may be called, by analogy and contrast, a quadruple system of calculations respecting directed lines, or a system of QUADRUPLE ALGEBRA."

"'(8). This passage from the one system to the other may be said to consist mainly in the consideration of the variable plane of an angle. If, after tracing the equilateral triangle ABC on a card, which at first rests on a horizontal table, we then lift up that card, with the figure traced thereon, and lay it on a sloping desk, the triangle in its new position takes also a new aspect; it faces a different region of space, and may be conceived to look at, or be looked at by, a new point of the heavens, which is not now the vertical point (or ), as before. This new aspect of the figure, or of the plane (or desk) on which it is now situated, is the new circumstance introduced, in the transition from Double to Quadruple Algebra. And in fact it is easy to see that this new circumstance, of the varied position of the figure, namely, of the triangle, or simply (if we choose) of the ANGLE ABC, requires the consideration of two new numerical elements. For we have now two new questions to answer, or two new things to determine: namely, 1st, the slope of the desk (or inclination of the plane), suppose forty-five degrees, conducting to a first new number, 45 ; and 2nd, the direction of the edge (or, technically speaking, the line of the nodes), where that slope meets the table, and which may deviate from the line of north and south by any other number of degrees, suppose seventy, giving thus a second new number, in this case 70.'"

"In the Green-function treatment of particle motion, a unit impulse is represented by a force F(t) = δ(t–t′) and is the analogue of the unit point source in spatial problems. The initial conditions play the role of boundary conditions."

"Inflation has attracted cosmologists because of its potential to free the standard big bang model from its worst flaw, the need for special initial conditions and, in particular, the requirement of initial acausal homogeneity. Naturally one must check whether inflation itself depends critically on initial conditions. Several “no hair” theorems and perturbation calculations have indicated that inflation is stable, and that it will take place when the initial conditions are perturbed. This has led to the belief that inflation will start in any generic universe."

"The surprising discovery of Newton’s age is just the clear separation of laws of nature on the one hand and initial conditions on the other. The former are precise beyond anything reasonable; we know virtually nothing about the latter. ,,, … how can we ascertain that we know all the laws of nature relevant to a set of phenomena? If we do not, we would determine unnecessarily many initial conditions in order to specify the behavior of the object."

"...attributes may be maintained because of deformations in fields. Such conservation laws are called topological. Thus, it may happen that a knot in a set of field lines, called a soliton, cannot be smoothed out. As a result, the soliton is prevented from dissipating and behaves much like a particle. A classic example is a magnetic monopole, which has not been found in nature but shows up as twisted configurations in some field theories. In the traditional view, then, particles such as electrons and quarks (which carry Noether charges) are seen as fundamental, whereas particles such as magnetic monopoles (which carry topological charge) are derivative. In 1977, however, Claus Montonen, now at the Helsinki Institute of Physics in Finland, and David I. Olive, now at the University of Wales at Swansea, made a bold conjecture. Might there exist an alternative formulation of physics in which the roles of Noether charges (like electrical charge) and topological charges (like magnetic charge) are reversed? In such a “dual” picture, the magnetic monopoles would be the elementary objects, whereas the familiar particles—quarks, electrons and so on—would arise as solitons."

"A method is proposed to calculate quantum numbers on solitons in quantum field theory. The method is checked on previously known examples and, in a special model, by other methods. It is found, for example, that the fermion number on kinks in one dimension or on magnetic monopoles in three dimensions is, in general, a transcendental function of the coupling constant of the theories."

"While J. Scott Russell first observed solitons in water waves in Augst 1834, a full-fledged theory of solitons has only come of age in the last decade. This advance is due primarily in the discovery of a generalization of the , the . While this method can be used to solve exactly only a certain number of nonlinear equations, many of these are relevant to broad areas in physics."

"The idea that in some sense the ordinary proton and neutron might be solitons in a non-linear sigma model has a long history. The first suggestion was made by Skyrme more than twenty years ago ... David Finkelstein and Rubinstein showed that such objects could in principle be fermions ... in a paper that probably represented the first use of what would now be θ vacua in quantum field theory. A gauge invariant version was attempted by Faddeev ... Some relevant miracles are known to occur in two space-time dimensions ... ; there also exists a different mechanism by which solitons can be fermions ..."

"The general problem of Celestial Mechanics consists in the determination of the relative motions of p bodies attracting one another according to the Newtonian law. This problem is not able to be solved directly: in order to deal with it, certain limitations must be made. ... Again, owing to the conditions under which the bodies of our solar system move, we are further able to divide the problem of p bodies into several others, each of which may be treated as a case of the problem of three particles, or, as it is generally called, the Problem of Three Bodies. The greater part of the Lunar Theory is a particular case of the Problem of Three Bodies; it involves the determination of the motion of the Moon relative to the Earth, when the mutual attraction of the Earth, Moon and Sun, considered as particles, are the only forces under consideration. When this has been found, the effects produced by the actions of the planets, the non-spherical forms of the bodies etc., can be be exhibited as small corrections to the coordinates."

"In Book I, Prop. LXVI of the first edition of Philosophiae naturalis principia mathematica, Newton (1687) discussed the dynamical problem of three bodies in a general way, and then in Book III he asserted that the vagaries of the Moon's motion could be accounted for by the gravitational attraction of the Sun. He recognized that he needed to develop the theory further, and summarized his later results in The theory of the Moon's motion of 1702 (Cohen 1975). He continued to refine his treatment up to the publication of the second edition of Principia (Newton 1712), some sections of which differ greatly from the first edition. He made almost no further changes of his own in the third edition, but added a scholium by Machin (1726) on the motion of the nodes. The published account of the rotation of the apse line, much the same in all versions, was seriously wrong, but even before 1690 Newton had developed a somewhat more satisfactory treatment, with which, however, he remained dissatisfied and never published (Whiteside 1976). (Since this article was prepared, the new English translation of the Principia by Cohen and Whitman (1999) has appeared. It is a translation of the third edition of 1726, which differs significantly in a few places from the first and second editions, as will be indicated.)"

"There are three essentially different types of lunar theory — that of de Pontécoulant, that of Delaunay, and that first developed by Hill, to which may perhaps be added that of Hansen as containing many features of more or less importance different from the others. That of de Pontécoulant and most of his predecessors consists in developing certain coordinates in periodic series of assumed form with the time or true anomaly as argument and determining the coefficients step by step as powers of the small parameters involved ; that of Delaunay consists in applying the method of the variation of parameters in the canonical form over and over in such a way as to remove the most important parts of the perturbative function ; that of Hill consists in finding very accurate particular solutions of the differential equations after the parts depending on the parallax of the sun, the eccentricity of the earth's orbit, and the latitude of the moon have been neglected, and then finding the deviations from this orbit due to general initial conditions and the neglected part of the perturbative function."

"In the , one wholeheartedly accepts traditional mathematics at face value. All questions such as the Continuum Hypothesis are either true or false in the real world despite their independence from the various axiom systems. The Realist position is probably the one that most mathematicians would prefer to take."

"... there is Brouwer's , which is utterly destructive in its results. The whole theory of the \,\aleph\,'s greater than \,\aleph_1\, is rejected as meaningless (Brouwer 1907, 569). Cantor's conjecture itself receives several different meanings, all of which, though very interesting in themselves, are quite different from the original problem. They lead partly to affirmative, partly to negative answers (Brouwer, 1907, I: 9; III: 2). Not everything in this field, however, has been sufficiently clarified. The &ldquo;semi-intuitionistic&rdquo; standpoint along the lines of H. Poincaré and H. Weyl ... would hardly preserve substantially more of set theory."

"Those who argue that the concept of set is not sufficiently clear to fix the truth-value of CH have a position which is at present difficult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertion that the meaning of CH is clear will sound more and more empty."

"One, two, buckle my shoe; Three, four, knock at the door; Five, six, pick up sticks; Seven, eight, lay them straight; Nine, ten, a big fat hen; Eleven, twelve, dig and delve; Thirteen, fourteen, maids a-courting; Fifteen, sixteen, maids in the kitchen; Seventeen, eighteen, maids in waiting; Nineteen, twenty, my plate's empty."

"The king was in his counting-house, Counting out his money;"

"Ford carried on counting quietly. This is about the most aggressive thing you can do to a computer, the equivalent of going up to a human being and saying “Blood...blood...blood...blood...”"

"If thou dost the number know Of the leaves on every bough, If thou can’st the reckoning keep Of the sands within the deep; Thee of all men will I take, And my Love’s accomptant make."

"Sooner may you count the starres, And number hayle downe pouring, Tell the Osiers of the Temmes, Or Goodwins Sands devouring, Then the thicke-showr’d kisses here Which now thy tyred lips must beare."

"Can there be any day but this, Though many sunnes to shine endeavour? We count three hundred, but we misse: There is but one, and that one ever."

"Whoe’er the number would define Of sports and joys that shall be thine, He first must count the grains of sand That spread the Erythræan strand, And every star and twinkling light That stud the glistening arch of night."

"In Riemann, Hilbert or in Banach space Let superscripts and subscripts go their ways. Our asymptotes no longer out of phase, We shall encounter, counting, face to face."

"Count the bees that on Hybla are playing, Count the flow’rs that enamel its fields, Count the flocks that on Tempe are straying, Or the grain that rich Sicily yields; Go number the stars in the heaven, Count how many sands on the shore, When so many kisses you’ve given I still shall be craving for more."

"Let’s number out the hours by blisses, And count the minutes by our kisses;"

"When I do count the clock that tells the time,"

"The axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

": {{citation"

"The Axiom of Choice is necessary to select a set from an infinite number of pairs of socks, but not an infinite number of pairs of shoes [...] Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair. [...] There is no difficulty in doing this with the boots. The pairs are given as forming an ℵ0, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible."

"The axiom gets its name not because mathematicians prefer it to other axioms."

": A. K. Dewdney from the famous April Fools' Day article in the computer recreations column of the Scientific American, April 1989."

"Isaac Newton... went to school at Grantham and in 1661 came up as a subsizar to Trinity. ...He had not read any mathematics before coming into residence but was acquainted with Sanderson's Logic, which was then frequently read as preliminary to mathematics. At the beginning of his first October term he... picked up a book on astrology, but could not understand it on account of the geometry and trigonometry. He therefore bought a Euclid, and was surprised to find how obvious the propositions seemed. He thereupon read Oughtred's Clavis and Descartes's Geometry, the latter of which he managed to master by himself though with some difficulty. The interest he felt in the subject led him to take up mathematics rather than chemistry as a serious study. His subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures. At a later time on reading Euclid more carefully he formed a very high opinion of it as an instrument of education, and he often expressed his regret that he had not applied himself to geometry before proceeding to algebraic analysis. ...He was elected to a scholarship in 1663."

"Similarly to the spread of the Indian place-value system, Indian trigonometry came to Europe via the Arab world, for example through the work on astronomy and trigonometry of Abu Abdallah Mohammad ibn Jabir al-Battani (ca. 850–929), also known as Albategnius, whose Kitab al-Zij was translated into Latin."

"[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and ... from Italy; several... as .., Thomas Harriot.., and ... were English; two... Simon Stevin... and ... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany."

"At its higher levels the golden age of Muslim civilization was both an immense scientific success and a exceptional revival of ancient philosophy. These were not its only triumphs... but they eclipse the rest. ...[T]he Saracens ...made the most original contributions [to science]. These, in brief, were nothing less than trigonometry and algebra... In trigonometry the Muslims invented the sine and the tangent. The Greeks had measured an angle only from the chord of the arc it subtended: the sine was half the chord. The Chosranian (...Mohammed Ibn-Musa) published in 820 an algebraic treatise which went as far as quadratic equations: translated into Latin in the sixteenth century, it became a primer for the West. Later, Muslim mathematicians resolved biquadratic equations. Equally distinguished were Islam's mathematical geographers, its astronomical observatories and instruments (in particular the ) and its excellent if still imperfect measurements of and , correcting the flagrant errors of Ptolemy."

"As in the rest of mathematical sciences, so in trigonometry, were the Arabs pupils of the Hindus…"

"Euler wrote... Introductio in Analysin infinitorum, 1748, which was intended to serve as an introduction to pure analytical mathematics. ...He ...showed that the trigonometrical and exponential functions are connected by the relation \cos \theta + i \sin \theta = e^{i\theta}. Here too we meet the symbol e used to denote the base of the Naperian logarithms, namely the incommensurable number 2.7182818... The use of the single symbol to denote the incommensurable number 2.7182818... seems to be due to Cotes, who denoted it by M. Newton was probably the first to employ the literal exponential notation, and Euler using the form a'z, had taken a as the base of any system of logarithms. It is probable that the choice of e for a particular base was determined by its being a vowel consecutive to a, or, still more probable because e is the initial of the word exponent."

"The development of Indian trigonometry, based on sine as against chord of the Greeks was another of 's achievements which was necessary for astronomical calculations. Because of his own concise notation, he could express the full sine table in just one couplet, which students could easily remember. For preparing the table of sines, he gave two methods, one of which was based on the property that the second order sine differences were proportional to sines themselves."

"The second part of the book... contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity."

"The idea of the logarithm probably had its source in the use of... trigonometric formulas that transformed multiplication into addition and subtraction. ...[I]f one needed to solve a triangle using the , a multiplication and division were required. ...[C]alculations were long and errors... made. Astronomers realized... multiplication and division could be replaced by additions and subtractions. To accomplish this... sixteenth century astronomers used formulas... as 2 \sin \alpha \sin \beta = cos(\alpha - \beta) - \cos (\alpha + \beta). ...A second source of the... logarithm was probably found in... algebraists as Stifel and Chuquet, who both displayed tables relating the powers of 2 to the exponents and showed that multiplication in one table corresponded to addition in the other. But because these tables had large gaps, they could not be used for necessary calculations. ...[T]wo men... independently, the Scot John Napier... and the Swiss Jobst Bürgi... came up with the idea of producing an extensive table... to multiply any... numbers... (not just powers of 2)... Napier published... first."

"We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks."

"Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations."

"Now, it is worth remarking, that this property of the table of sines, which has been so long known in the East, was not observed by the mathematicians of Europe till about two hundred years ago […] If we were not already acquainted withthe high antiquity of the astronomy of Hindostan, nothing could appear more singular than to find a system of trigonometry, so perfect in its principles, in a book so ancient as the Surya Siddhanta […]’ ‘In the progress of science […] the invention of trigonometry is to be considered as a step of great importance, and of considerable difficulty. It is an application of arithmetic to geometry […] (and) a little reflection will convince us, that he, who first formed the idea of exhibiting, in arithmetical tables, the ratios of the sides and angles of all possible triangles, and contrived the means of constructing such tables, must have been a man of profound thought, and of extensive knowledge. However, ancient, therefore, any book may be, in which we meet with a system of trigonometry, we may be assured, that it was not written in the infancy of science.’ ‘As we cannot, therefore, suppose the art of trigonometrical calculation to have been introduced till after a long preparation of other acquisitions, both geometrical and astronomical, we must reckon far back from the date of the Surya Siddhanta, before we come to the origin of the mathematical sciences in India […] Even among the Greeks […] an interval, of at least 1000 years, elapsed from the first observations in astronomy, to the invention of trigonometry; and we have surely no reason to suppose, that the progress of knowledge has been more rapid in other countries."

"Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"The sum and difference formulas are vital to building trigonometric tables finer than the traditional 24 entries per 90&deg;. ...they can also be used to generate many other identities. In particular, formulas for Sin 2&theta;, Cos 2&theta;, Sin 3&theta;, Cos 3&theta;, and higher multiples may be generated simply by writing n&theta; = &theta; + &theta; +... + &theta; and applying the sum formulas repeatedly. This was done by... Kamalākara in his Siddhānta-Tattva-Viveka (1658) up to the sine and cosine of 5&theta;; he quotes (who clearly knew this could be done) for the addition and subtraction laws. Kamalākara's sine triple-angle formula...was \mathrm{Sin} 3 \theta = \mathrm{Sin} \theta (3 - \frac{( \mathrm{Sin} \theta)^2}{(\mathrm{Sin}\,30^{\circ })^2}),equivalent to the modern formula \sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta; ...The identity ...has special significance, since it may be used to get an accurate estimate of sin 1&deg; from sin 3&deg;—provided one is able to solve cubic equations."

"After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley."

"The British mathematicians have been the greatest, nay... the only improvers of Trigonometry within these two centuries. ...[N]ot to mention the extraordinary inventions of Lord Neper... nor the analogies, wherein sines and tangents of half-arches are used, nor the applications of Trigonometry to of the sphere, for all which we are indebted... the very words of Trigonometry, cosine, cotangent, &c. have been first used by the writers of that nation."

"The most conspicuous authors of both Trigonometries amongst the British, who have been consulted by us, are Caswell, in Wallis's Works, Keil, Simpson, Robertson, Mr. , Emerson, and [Benjamin] Martin; and of those who have treated of them occasionally, Oughtred, in his Clavis Mathematica, and Circles of Proportion, Wallis, Jones, Wingate, [Henry] Sherwin, and Gardiner, in their Tables of Logarithms; Sir Isaac Newton, in his Univ. Arithm. Geometrical Problem II. Harris and Chambers in their dictionaries. Plane Trigonometry alone has been treated by [Philip] Ronayne, Mr. Thomas Simpson, Maseres, and Muller. However, the merits of some foreigners also cannot, without injustice, be suppressed. Such are Copernicus, in his Astronomia Instaurata; Balanus, a modern Greek; Simon Stevin, commented upon by '; Clavius; M. de la Caille; M. de la Lande, in his Astronomie, tom. III. de Chales; Ozanam; Segnerus; and the labours of Schottus, Tacquet, and others, are commendable. We need not mention the parents of these sciences, Theodosius, Ptolemy, Menelaus, and ."

"[N]otwithstanding the labours and exertions of so many eminent men... in this branch of mathematics many things are deficient, many superfluous; some are too general, others too particular; some are too much dwelt upon, others want a great deal of explanation; in many there is hardly any order, or connexion, or demonstration, in some too much unnecessary precision."

"[W]hat else... the reason of doubts arising in solutions... of plane and spherical triangles, but the want of accurate determinations and explanations?"

"From what have proceeded disputes in Spherical Trigonometry, not solved either by Cunn, or Ham, but from the inaccurate notion of a supplemental triangle?"

"[T]o what can this be owing, but to the want of sufficient principles, the neglect of enumerating and distinguishing: cases of a proposition, and the inattention to rendering the subject as complete as possible?"

"[T]here is as yet no classical book of Trigonometry in any language... fit to give to learners a solid foundation in them... as are Euclids Elements, Archimedes de Sphaera et Cylindro, and Dr. Hamiltons Conic Sections."

"[T]he reason, why in Algebra and Fluxions, expressions for trigonometrical lines always run out into infinite series... is because the number of arches, to which any one of such lines belongs is always infinite."

"Prop. 14. and its corollaries deserve... examination. It is hard to say, by whom they were invented, though... probably by the English; and perhaps corr. 3 & 2. of prop. 29. in spherics, have given rise to them all, as they are to be found in most books of the last age. They are all to be seen in Caswell... Wallace, Newton Univ. Arithm. Geom. Probl. 11. Thos. Simpsons Algebra, Geom. Probl. 15. Dr. Robertsons Navigation, Emerson, and [Benjamin] Martin. The analogy of the prop. in particular, is to be met with in Trigonometria Britannica, [Henry] Sherwins Tables, de la Caille, Dr. Simpson, and Ward."

"With regard to demonstration, the old and perplexed one (used formerly for the area of a triangle, and accommodated here by Dr. Simpson) is laid aside, and another... easier demonstration is substituted... after the manner of Dr. Robertson's, but greatly improved by the change of a side of the triangle; the same indeed nearly, which has been communicated in Russia some years ago by... Professor Robison of Glasgow."

"But in Book II. i.e. in Spherical Trigonometry, the greatest pains were to be taken, and the greatest difficulties to be overcome. For though... Spherical Trigonometry is not so easy as the Plane, as it wants those previous helps and that determination, which the Plane... has; yet, when out of three parts of a proposition one only or two are laid down; when a proposition is demonstrated only in one case out of several; when, on account of bad definitions, several things, wanting demonstration, are passed by, or dignified with the name of axioms; when an argument turns in circulo... when whole steps are omitted; when, instead of a direct way, we go round about; when things are scattered about without order; when a whole set of triangles is neglected, &c. &c. surely, all this is not the fault of Spherical Trigonometry."

"The whole doctrine of axes and poles is to this day both incomplete and inaccurate. The authors endeavoured to their utmost, to remedy such extraordinary defects in so important a subject. ...In truth, the subject of poles of circles, as it is laid down here, seems exhausted: several of their properties are exhibited, over and above those in Theodosius's Spherics, and Dr. Barrow's additions thereto; and this is done in a lesser number of lines, than they have pages."

"In prop. 5. and cor. the confused and inaccurate ideas of arches being measures of angles, of arches being equal to angles, and of arches being the supplements and complements of angles, and v. v. so much prevailing even among the best geometricians, are attempted to be rectified: for it is manifest enough, that nothing can be a measure of another thing, or equal to it, or a supplement and complement of it, unless it be homogeneous with it. For want of such a plain consideration, and afterwards most probably from habit, people have debased many propositions, both in their enunciations and demonstrations; and often it is not without some trouble that they are corrected."

"Prop. 7. claims particular notice on account of its use, of its application, and even on account of the disputes it has occasioned. Its application is so... extensive, that the doctrine of spherics, by means only of it, may be reduced almost to half the number of its propositions. The invention of it may be ascribed perhaps to Philip Langsberg. Vid. Simon Stevin Liv. 3. de la Cosmographie, prop. 31. & Alb. Girard in loc. ...[W]e have been obliged to form for it an enunciation entirely new; and were happy to find afterwards, that Mr. Cotes in his Æstim. Error. iem. 4. gives the first part of it exactly the same."

"It is from the vagueness of the proposition... and from misunderstanding the terms supplement and complement, that disputes have arisen in spherics: these may be seen at the end of [Samuel] Cunns Euclid, in his remarks and the appendix. Whatever be the mistakes of Mr. Heynes... the respectable names of Dr. Keil, Mr. Caswell, and Dr. Harris, whom Mr. Cunn joins in company with Heynes, are treated by him... injuriously; especially as he himself had not examined his subject with sufficient attention. His own rule... is indeed true... but it is more troublesome to the memory. Mr. Ham... awards his own rule, which, notwithstanding, is much more unmanageable... it using subtraction of natural versed sines, to whose difference therefore (and every one knows the thing is not easy) logarithms are to be accommodated. But it were time, long ago, to bury these worthless disputes in oblivion, that learners of spherics should not be discouraged by seeing them printed and reprinted so often."

"Of prop. 10. no one gives an accurate demonstration, except Menelaus : Dr. Keil's need not be mentioned, and Dr. Simpson's leaves out a case, and is at the same time very prolix. That which is offered here... seems remarkably short and easy, and is derived from Dr. Simpsons Elements of Euclid, Book XI. prop. A. ...[H]e is... to be praised for his merits in his... Euclid; but... there are still many inaccuracies..."

"Prop. 18 & 19. though... the... foundation of... Spherical Trigonometry... have not been demonstrated... except in one single case... it does not seem... long ago, that they... appear anew in their present form. V. Keil."

"Such weak foundations Spherical Trigonometry has had to this day!"

"The first considerable extension of Trigonometry, beyond its original object, was made about twenty years after the death of Newton. It was then, on the ground-work laid down by that great man, that... Clairaut, Dalembert and Euler, and ... began to establish a system of Physical Astronomy more perfect... [T]hey laid aside the Geometrical method which Newton had used... and adopted the Analytical. ...[T]hey perceived the formulæ of Trigonometry to be of continual use and recurrence, and the language, by which the process of demonstration was conducted... in a great degree, of symbols and phrases borrowed from that science. ...[T]he advancement of Trigonometry, the pure and subsidiary science, was contemporaneous with that of Astronomy, the mixed and principal one."

"Clairaut and Dalembert in their Lunar Theories... introduce... several, now commonly known, Trigonometrical formulæ. In... Thomas Simpson... the Author evidently intended the one... at p. 76, as preparatory to the... Theory of the Moon; and Euler... states as a reason for cultivating the algorithm of sines, its great utility in the mixed Mathematics."

"Spherical [Trigonometry]... has not, like... [plane Trigonometry], been extended beyond its original purpose. It has no collateral and indirect uses; it has not enriched the general language of analysis... But... its propositions are more easily established by the Analytical method than the Geometrical. ...[T]his would be the case, even if there existed no similarity and artificial connexion, between the processes by which the series of formulæ in the two branches of Trigonometry were... established. [T]he corresponding propositions can be deduced by methods so analogous, that to know the one is almost to know the other."

"We shall... find similar Algebraical derivations of formulæ from two fundamental expressions for the cosine of an angle. The principle of the derivation... is not new; it originated with Euler, who inserted in the Acta Acad. Petrop. for 1779, a Memoir entitled Trigonometria Spherica Universa, ex primis principiis breviter et dilucide derivata. Gua next, in the Memoirs of the Academy of Sciences for 1783, p. 291, deduced... Spherical Trigonometry "from the Algebraical solution of the simplest of its Problems." In 1786, Cagnoli... derived fundamental expressions for the sine and cosine of the sum of two arcs. And lastly, Lagrange and Legendre, the one in the Journal de L Ecole Polytechnique, the other in his Elemens de Geometrie, have followed and simplified Euler’s method, and instead of three fundamental expressions, have shewn one to be sufficient."

"Kepler's indefatigable inquiries, for nine years... amounts to this, that Byrgius had made some observations upon the adaptation of an arithmetical to a geometrical progression, very naturally occurring to him in trigonometrical calculations."

"Longomontanus and Byrgius, are all whom Dr Hutton can find to represent his learned calculators of the sixteenth and seventeenth centuries, who anticipated or coincided with Napier in the discovery. ...But he is contradicted by the history of science... and by every philosopher of greatest name... ...[O]ur philosopher's invention... removed a pressure, long and severely felt, and which might have crushed the temple of science, had that not possessed such a pillar as Kepler. To use the expressions of a distinguished writer, " What all mathematicians were now wishing for, the genius of Neper enabled him to discover; and the invention of Logarithms introduced into the calculations of trigonometry a degree of simplicity and ease, which no man had been so sanguine as to expect." Kepler, Ursine, Speidell, Gunter, Briggs, Vlacq, [Petrus] Cugerus, Cavalieri, Wolff, Wallis, Halley, Keill, and a host of others, all bear witness against Dr Hutton, in the honourable and enthusiastic manner they acknowledge Napier as the only author of that revolution in science."

"In continuation of... the most brilliant period of ancient geometry, the century of Euclid, Archimedes and Apollonius, recourse must again be had to the Collectio of the much later writer Pappus, for information about the lost three books of s of Euclid. ...In the Sphœrica of Menelaus, a geometer and astronomer of the first century A.D., is found the theorem (lib. III. lemma 1 p. 83, Oxon. 1758): If the sides ag, gd, da of a plane triangle be met by any transversal in the points erb respectively, thenge : ea=gr.db : rd.ba,or the product of three non-adjacent segments of the sides of the triangle by any transversal is equal to the product of the remaining three. This was... extended to spherical triangles... as a basis for the spherical trigonometry of the ancients."

"[T]he property of the six segments in plano... [led to] great results... long after, especially in the hands of Desargues."

"Claudius Ptolemseus was "le plus ceélèbre, sans contredit, mais non le plus véritablement grand astronome de toute l'antiquité." [the most famous, without a doubt, but not the truly greatest astronomer of all antiquity] Thus writes Delambre in the Biographie Universelle (vol. 36... 1823)."

"In a work on the three dimensions of bodies, Ptolemy introduced the idea of determining the position of a point in space by referring it to three rectangular axes of coordinates... His chief work, which he called a mathematical Σύνταξις [Syntax], was further described by his admirers as ή μεγύλη, and by the Arabs as ' ή μεγίστη [maximum]... In it... he reproduces the theorem of the six segments... and founds upon it a system of trigonometry, plane and spherical."

"Nicomachus turns to the discussion of proportion... which, he says, is very necessary for "natural science, music, spherical trigonometry and planimetry and particularly for the study of the ancient mathematicians.""

"Of the great Greek mathematicians, Archimedes alone (in his Circuli Dimensio) ventures to introduce actual numbers into a geometrical discussion, and to divide a line by another line. He finds the value of π and some other similar ratios but does not himself pursue such investigations further and is not followed by any other writer. Trigonometry was used only for astronomical purposes and did not form part of geometry at all."

"[T]he word 'geometry'... means 'land-measurement,' that the Egyptians gave this science to the world and that among the Egyptians... it... was confined almost entirely to the practical requirements of the surveyor. The work ["Directions for obtaining the knowledge of all dark things" in the Rhind collection] of ..., contains, beside sums in arithmetic, a great many geometrical examples... Ahmes proceeds to calculate the contents of... receptacles... The rectilineal figures of which Ahmes calculates the areas are the square, oblong, isosceles triangle and isosceles parallel-trapezium (...part of an isosceles triangle cut by a line parallel to the base). As to the last two, the areas... are incorrect. ...The errors in these cases are small... The area of a circle is found (in no. 50) by deducting from the diameter 1/9th of its length and squaring the remainder. Here π is taken = ( \frac{16}{9})^2 = 3.1604..., a... fair approximation. ...Lastly, the papyrus contains (nos. 56 to 60) some examples which seem to imply a rudimentary trigonometry. In these... the problem is to find the uchatebt, piremus or seqt of a pyramid or obelisk."

"[C]oncerning ... The Chaldees... almost contemporaneous with Ahmes... had made advances, similar to the Egyptian, in arithmetic and geometry, and were especially busy with astronomical observations. ...[T]hey had divided the circle into 360 degrees, and... obtained a fairly correct... ratio of the circumference of a circle to its diameter. They used... a notation, which the Greeks afterwards adopted for astronomical purposes. Herodotus expressly states that the polos and ' (...sundials) and the twelve parts of the day were made known to the Greeks from Babylon. Much of the trigonometry and of the later Greeks may... have been... derived from Babylonian sources."

"In 1120, Adelhard of Bath obtained in Spain a copy of Euclid's Elements and translated... into Latin. Translations from the Arabic of other Greek works, especially... Aristotle, soon followed. About 1186 Gherardo of Cremona made another translation of the Elements and... in 1260, Giovanni Campano reproduced Adelhard's translation under his own name and obtained... wide celebrity. The fruit of these translations soon followed. In 1220, Leonardo of Pisa... published... Practica Geometriae which though it deals with the calculation of areas and numerical ratios of spaces, is founded on Euclid and Archimedes and Ptolemy, and contains some trigonometry and conics."

"The century which produced Euclid, Archimedes and Apollonius was... the time at which Greek mathematical genius attained its highest development. For many centuries... geometry remained a favourite study, but no substantive work... compared with the Sphere and Cylinder or the Conics... One great invention, trigonometry, remains to be completed, but trigonometry with the Greeks remained always the instrument of astronomy and was not used in any other branch of mathematics, pure or applied. The geometers who succeed to Apollonius are professors who signalised themselves by this or that pretty little discovery or by some commentary on the classical treatises."

"'... astronomical work, Aναφορικός, does not use the trigonometry which was certainly introduced by Hipparchus, and would have been absurdly antiquated if written after Hipparchus' time..."

"The 14th Book of the Elements, or the book of on 'the Regular Solids', consists of seven propositions... The... treatise on 'Risings'... contains only six propositions, of which the first three, deal... with s... The only interesting proposition is the 4th... Divide the into 360 local degrees and the time of its revolution into 360 chronic degrees. Then, given the ratio, for any place on the earth, of the longest day to the shortest, we can deduce the number of chronic degrees for each number of local degrees. Here, for the first time in any Greek work, we find a circle divided in the Babylonian manner into 360 degrees."

"introduces the division as if it were a novelty. He does not, however, take the next step, to trigonometry. ...This was ...taken by Hipparchus ...upon whose work the whole system of Greek astronomy was founded. ...[T]hough the of Ptolemy is clearly derived almost entirely from writings of Hipparchus, none of the works of the earlier astronomer have survived, save a commentary in three books on the Phenomena of ... In the Second Book... he claims to have invented a method of solving spherical triangles for the purpose of finding the exact eastern point of the ecliptic. The treatise... is lost. Theon, in his commentary on the ... states that Hipparchus calculated a "table of chords" (i.e. practically of sines) in twelve books. ...[T]herefore ...Hipparchus was the founder of trigonometry, though we are obliged to look elsewhere for ... the progress of the Greeks in this department ..."

"Hipparchus... the following little summary, taken from Delambre, will shew what manner of man he was. ...[H]e ...determined (...not with absolute accuracy) the precession of the equinoxes, the inequality of the sun, and the place of its apogee, as well as its mean motion: the mean motion of the moon, its nodes and its apogee: the equation of the centre of the moon and the inclination of its orbit. He had discovered a second inequality of the moon (the ), of which he could not, for want of proper observations, find the period and the law. He had commenced a more regular course of observations for the purpose of supplying his successors with the means of finding the theory of the planets. He had both a spherical and a plane trigonometry. He had traced a by : he knew how to calculate eclipses of the moon and to use them for the improvement of the tables: he had an approximate knowledge of es, more correct than Ptolemy's. He invented the method of describing the positions of places by reference to and . What he wanted was only better instruments. Yet in his determination of the equations of the centres of the sun and moon and of the inclination of the moon, he erred only by a few minutes. For 300 years after his time astronomy was stationary. Ptolemy followed him with little originality. Some 800 years later the Arabs added a few more discoveries and more accurate determinations and then the science is stationary again till Copernicus, Tycho and Kepler."

"[T]hough Heron's ability is sufficiently indicated by... [his] proofs, as a general rule he confines himself merely to giving directions and formulae. ...[H]e availed himself of the highest mathematics of his time. Thus in the ', two chapters treat of the mode of drawing a plan of an irregular field and of restoring, from a plan, the boundaries of a field in which only a few landmarks remain. ... The method is closely similar to the use of latitude and longitude introduced by Hipparchus. So...Heron gives, for finding the area of a regular polygon from the square of its side, formulae which imply a knowledge of trigonometry. Suppose F_n to be the area of a regular polygon of which {a_n} is a side, and let c_n be the coefficient by which {a_n}^2 is to be multiplied in order to produce the equation F_n = c_n {a_n}^2 then it is easy to see that c_n = \frac{n}{4} \cot \frac{180^\circ}{n}. ...[H]is approximations are generally near enough. We need not be surprised... Hipparchus made a table of chords... [i.e.] the coefficients k_n were known, with the aid of which a_n = k_n r, where r is the radius. Then c_n = \frac{n}{4} \sqrt{\frac{4}{{k_n}^2}-1}, and Heron was competent to extract such square roots. But Heron does not use the sexagesimal fractions, and... sexagesimal fractions were always, as... afterwards called, astronomical fractions... [S]ave by Heron, trigonometry was generally conceived to be a chapter of astronomy and was not used for the calculation of terrestrial triangles."

"Heron was by no means a geometer of the Euclidean School. He is a practical man who will use any means to attain his end and is... untrammelled by... classical restrictions. He is... a mechanician who, unlike Archimedes, is... proud of his... ingenuity. He adds... almost nothing, to the geometry of his time but he is learned in the... bookwork. On the other hand... he is the first Greek writer who uses a geometrical nomenclature and symbolism, without the geometrical limitations, for algebraical purposes, who adds lines to areas and multiplies squares by squares and finds numerical roots for quadratic equations. Hence, for a similar reason to... de Morgan... it is now commonly believed that Heron was an Egyptian. ...[T]he ...style of his work recalls ... ... [A]lgebra was an Egyptian art and ...the symbolism of Diophantus was of Egyptian origin. ...[I]f Heron was not a Greek, he relied almost entirely on Greek learning and did not resort to the ...priestly tradition ...He is a man who writes in Greek upon Greek subjects, but who thinks in Egyptian. [Following is in the footnote.] Let it be remembered that the seqt-calcalation of Ahmes leads to trigonometry: his hau-calculation to algebra. Almost the first sign of both appears in Heron... An algebraic symbolism first appears in Diophantus, but the symbols are probably not Greek and probably are Egyptian. Both Heron and Diophantus were Alexandrians. This is all the evidence that trigonometry and algebra were of Egyptian origin, but does it not raise a shrewd suspicion? Proclus... speaks... as if Heron founded a school."

"Practically all that we know of the trigonometry of the Greeks, is derived from two chapters of the famous Μεγαλή Σύνταξις [Great Compilation] of Claudius Ptolemæus. This work contains many astronomical observations by Ptolemy... The common name μεγαλή Σύνταξις [Great Syntax] was altered by... fervent admirers into μεγίστη [maximum] and this word was adopted by the Arabs... The Arabic article was... added and the name corrupted into Almidschisti, whence is derived its common mediaeval title '."

"Ptolemy's method of calculating chords seems to be his own. The measures of the sides of regular polygons, as chords of certain arcs, were known in terms of the diameter. He next proves the proposition, now appended to Euclid VI. (D), that "the rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to both the rectangles contained by its opposite sides", and then... how from the chords of two arcs that of their sum and difference [may be found] and how from the chord of any arc that of its half may be found."

"Chapter X., which follows, is on the obliquity of the ecliptic... The next, XI., XII. contain spherical geometry and trigonometry "enough for the determination of the connexion between the sun's , and longitude and for the formation of a table of s to each degree of longitude.""

"Chap. XI. contains προλαμβανόμενα [preventable], "preliminaries to the spherical demonstrations". These begin with the lemma of Menelaus, the regula sex quantitatum, borrowed without any acknowledgement. After proving this, he gives four propositions. ...This ...contains ...the whole of Greek trigonometry. The further progress... is due mainly to the Indians and after them to the Arabians."

"The Indians never used "the chord of twice of the arc", as the Greeks... but half that chord. This they called jyârdha or ardhajyâ, but the name of the whole chord jyâ or jivâ... The Arabs... transliterated it to dschîba... later... altered for...Arabic... dschaib, which... means 'bosom' and was therefore translated 'sinus' by Plato of Tivoli in his Latin version ('De Motu Stellarum') of the astronomy of Albategnius. In this way, sine came to be a technical term of modern trigonometry."

"The applications of trigonometry in Book II. of the Almagest and the geometry of eccentric circles and epicycles in Book III. belong... by language and purpose, to the ."

"To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress."

"[N]o Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life."

"[D]uring this time practical astronomy had been making rapid strides in the hands of Eudoxus, Aristarchus, Eratosthenes and others down to Hipparchus. Now the needs of the practical astronomer are in many respects similar to those of the surveyor, the engineer and the architect. Each of these is chiefly concerned, not to find the general rules which govern all similar cases, but to find under what general rules a particular case, presented to them, falls. But the question whether an angle is acute, or a triangle isosceles, can be determined only by measurement, and hence about 130 B.C., in the time of Heron and Hipparchus, we find the results of geometry applied to measured figures, for the purpose of finding some other measurement as yet unknown. Trigonometry and an elementary algebraical method are thus introduced."

"The learning of the Greeks passed over in the 9th century to the Arabs and with them came round into the West of Europe. But no material advance was made by the Arabs in geometry and it was their arithmetic, trigonometry and algebra which chiefly interested the mediaeval Universities. In the 16th century Greek geometry again became known in the original and was studied with intense zeal for about 100 years, until Descartes and Leibnitz and Newton, the best of its scholars, superseded it."

"Ahmes then goes on to find the area of a circular field … and gives the result as (d - 1/9d)2, where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416."

"Ahmes gives some problems on pyramids. ...Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes's geometrical results were intended only as approximations correct enough for practical purposes."

"Arab missionaries who had come to China in the course of the thirteenth century, and while there introduced a knowledge of spherical trigonometry."

"The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. ...they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival|the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry."

"Archimedes... work... The following is a fair specimen of the questions considered. A solid in the shape of a paraboloid of revolution of height h and latus rectum 4a floats in water, with its vertex immersed and its base wholly above the surface. If equilibrium be possible when the axis is not vertical, then the density of the body must be less than (h - 3a)^2/h^2 (book II. prop. 4). When it is recollected that Archimedes was unacquainted with trigonometry or analytical geometry, the fact that he could discover and prove a proposition such as that... will serve as an illustration of his powers of analysis."

"The third century before Christ, which opens with... Euclid and closes with the death of Apollonius, is the most brilliant era in the history of Greek mathematics. But the great mathematicians of that century were geometricians... It was not till after... nearly 1800 years that the genius of Descartes opened the way to any further progress in geometry... [R]oughly... during the next thousand years Pappus was the sole geometrician of great ability; and... almost the only other pure mathematicians of exceptional genius were Hipparchus and Ptolemy who laid the foundations of trigonometry, and Diophantus who laid those of algebra."

"Ptolemy’s great treatise, the '... was founded on the observations and writings of Hipparchus, and from the notes there given we infer that the chief discoveries of Hipparchus, and from the notes there... we infer that the chief discoveries of Hipparchus... [H]is observations and calculations... placed the subject for the first time on a scientific basis. ...[His] theory accounted for all the facts which could be determined with the instruments then in use, and... enabled him to calculate... eclipses with considerable accuracy. ...No further advance in the theory of astronomy was made until the time of Copernicus, though the principles laid down by Hipparchus were extended and worked out in detail by Ptolemy. Investigations such as these naturally led to trigonometry, and Hipparchus must be credited with the invention of that subject. ...[I]n plane trigonometry he constructed a table of chords of arcs... practically the same as... natural sines; and... in spherical trigonometry he had some method of solving triangles: but his works are lost, and we can give no details."

"It is believed... that the elegant theorem, printed as Euc. VI. D... known as Ptolemy’s Theorem, is due to Hipparchus and was copied... by Ptolemy. It contains implicitly the addition formulæ for sin (A \pm B) and cos(A \pm B); and Carnot shewed how the whole of elementary plane trigonometry could be deduced from it."

"'... placed engineering and land surveying on a scientific basis. ...He was ...acquainted with the trigonometry of Hipparchus, but ...nowhere quotes a formula or expressly uses the value of the sine, and it is probable that like the later Greeks he regarded trigonometry as forming an introduction to, and being an integral part of, astronomy."

"[T]hroughout the first century after Christ... the only original works of any ability were... by Serenus and... Menelaus. ...Those by Serenus... were on the plane sections of the cone and cylinder... edited by E. Halley... 1710. That by Menelaus... was on spherical trigonometry, investigated in the Euclidean method... translated by E. Halley... 1758. The fundamental theorem... is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them (book III. prop. 1). Menelaus also wrote on the calculation of chords... plane trigonometry; this is lost."

"Ptolemy... produced his great work on astronomy, which will preserve his name as long as the history of science endures. This... is... the '...founded on the writings of Hipparchus, and, though it did not... advance the theory... it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise."

"Ptolemy made observations at Alexandria from the years 125 to 150... .but an indifferent practical astronomer, and the observations of Hipparchus are... more accurate..."

"The work is divided into thirteen books. ...[T]he first... treats of trigonometry, plane and spherical; gives a table of chords, i.e. of natural sines (... substantially correct and... probably taken from... Hipparchus); and explains the obliquity of the ecliptic... It became... the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be... the centre of the solar system."

"The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed... But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly... as a convenient way of expressing known facts, it is not only legitimate but convenient. It was as good a theory as with their instruments and knowledge it was possible to frame, and... it exactly corresponds to the expression of a given function as a sum of sines or cosines, a method... of frequent use in... analysis."

"Ptolemy had shewn... geometry could be applied to astronomy, but... indicated how new methods of analysis like trigonometry might be... developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence... Pappus..."

"Pappus wrote several books, but... only one which has come down to us is his Συναγωγή [Synagoge], a collection of mathematical papers... in eight books of which... part... have been lost... published by F. Hultsch... 1876—8. This collection was intended to be a synopsis of Greek mathematics... with comments and additional propositions... we rely largely on it for... knowledge of... works now lost. ...[T]he sixth [book deals] with astronomy including, as subsidiary subjects, optics and trigonometry ...His work... and... comments shew... he was a geometrician of great power; but it was his misfortune to live at a time when but little interest was taken in geometry, and... the subject, as then treated, had been practically exhausted."

"Isaac Argyrus... wrote three astronomical tracts... one on ... one on geometry... and one on trigonometry, the manuscript of which is in the Bodleian at Oxford."

"The Mathematics of the Middle Ages and the Renaissance... begins about the sixth century, and may be said to end with the invention of analytical geometry and infinitesimal calculus. The characteristic feature of this period is the creation of modern arithmetic, algebra, and trigonometry."

"Arya-Bhata... is frequently quoted by , and... many commentators [write that] he created algebraic analysis though it has been suggested that he may have seen Diophantus’s Arithmetic. ...[H]is Aryabhathiya... consists of the enunciations of... rules and propositions... in verse. There are no proofs, and the language is... obscure and concise... [I]t long defied all efforts to translate it. The book is divided into four parts: of these three are devoted to astronomy and the elements of spherical trigonometry; the remaining part... enunciations of thirty-three rules in arithmetic, algebra, and plane trigonometry. It is probable that Arya-Bhata, like and Bhaskara... regarded himself as an astronomer, and studied mathematics only so far as... was useful... in his astronomy. ...In trigonometry he gives a table of natural sines of the angles in the first quadrant, proceeding by multiples of 3 3/4° defining a sine as the semichord of double the angle. ...A large proportion of the geometrical propositions which he gives are wrong."

"... wrote a work in verse... Brahma-Sphuta-Siddhanta... system of Brahma in astronomy. ...Chaps. XII. and XVIII ...are devoted to arithmetic, algebra, and geometry... It is impossible to say whether the whole of Brahmagupta’s results... are original. He knew of Arya-Bhata’s work, for he reproduces the table of sines... and it is likely that some progress in mathematics had been made by Arya-Bhata’s... successors, and that Brahmagupta was acquainted with their works; but there seems no reason to doubt that the bulk of Brahmagupta’s algebra and arithmetic is original, although perhaps influenced by Diophantus... the origin of the geometry is more doubtful, probably some... is derived from Hero..."

"Bhaskara... is said to have been... lineal successor of Brahmagupta as head of an astronomical observatory at Ujein... sometimes written Ujjayini. He wrote an astronomy... Lilavati is on arithmetic... Bija Ganita is on algebra; the third and fourth... on astronomy and the sphere... [I]t is... probable that Bhaskara was acquainted with... Arab works... written in the tenth and eleventh centuries, and with... Greek mathematics... transmitted through Arabian sources. ...[F]rom the ...table of contents ...Arithmetical progressions, and sums of squares and cubes. Geometrical progressions. Problems on triangles and quadrilaterals. Approximate value of π. Some trigonometrical formulae. ..[T]he book ends with a few questions on combinations. This is the earliest known work which contains a systematic exposition of the decimal system of numeration. ...Chapters on algebra, trigonometry, and geometrical applications exist, and fragments of them have been translated by Colebrooke. Amongst the trigonometrical formulae is one... equivalent to... d (\sin \theta) = \cos \theta d \theta."

"Like the Greeks, the Arabs never used trigonometry except... with astronomy; but they introduced the trigonometrical expressions... now current, and worked out the plane trigonometry of a single angle. They were also acquainted with the elements of spherical trigonometry."

"The trigonometrical ratios seem to have been the invention of Albategni... who was among the earliest of the many distinguished Arabian astronomers. He wrote the Science of the Stars (published by Regiomontanus... 1537)... [where] he determined his angles by "the semi-chord of twice the angle," i.e. by the sine of the angle (taking the radius vector as unity). Hipparchus and Ptolemy... had [also] used the chord."

"Albuzani... also known as Abul-Wafa... introduced all the trigonometrical functions, and constructed tables of tangents and cotangents. He was celebrated not only as an astronomer but as one of the most distinguished geometricians of his time."

"The Arabs were at first content to take the works of Euclid and Apollonius for their text-books in geometry without attempting to comment on them, but Alhazen issued in 1036 a collection of problems something like the Data of Euclid, this was translated by Sédillot... in 1836. Besides commentaries on the definitions of Euclid and on the Almagest Alhazen also wrote a work on optics which shews that he was a geometrician of considerable power: this was published at Bale in 1572, and served as the foundation for Kepler’s treatise."

"Bhaskara, ... there is every reason to believe ...was familiar, with the works of the Arab school... and... that his writings were... known in Arabia."

"The Arab schools continued to flourish until the fifteenth century... [T]he work of the Arabs in arithmetic, algebra, and trigonometry was of a high order of excellence. They appreciated geometry and the applications of geometry to astronomy, but they did not extend the bounds of the science."

"The earliest Moorish writer of distinction... is Geber ibn Aphla... His works... chiefly... astronomy and trigonometry, were translated into Latin by Gerard... 1533. He seems to have discovered the theorem that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides."

"Leonardo ... known as Leonardo of Pisa... in 1202 published... Algebra et almuchabala (the title being taken from Alkarismi’s work) but... known as the Liber Abaci. He there explains the Arabic system of numeration, and remarks on its great advantages over the Roman system. He then gives an account of algebra, and points out the convenience of using geometry to get rigid demonstrations of algebraical formulae. He shews how to solve simple equations... All the algebra is rhetorical. ...Roger Bacon ...recommends the (...the arithmetic founded on the Arab notation) ...[B]y the year 1300, or at the latest 1350, these numerals were familiar both to mathematicians and to Italian merchants. ...He ...wrote a geometry termed Practica Geometriae ...1220. This is a good compilation and some trigonometry is introduced; among other propositions and examples he finds the area of a triangle in terms of its sides."

"The Mathematics of the Renaissance... Mathematicians had barely assimilated the knowledge obtained from the Arabs, including their translations of Greek writers, when the refugees who escaped from Constantinople after the fall of the eastern empire brought the original works and the traditions of Greek science into Italy. Thus by the middle of the fifteenth century the chief results of Greek and Arabian mathematics were accessible to European students. The invention of printing about that time rendered the dissemination of discoveries comparatively easy. ...[W]hen a mediaeval writer "published" ... the results were known to only a few of his contemporaries. This had not been the case in classical times for... until the fourth century of our era Alexandria was the... centre for the reception and dissemination of new works and discoveries. In mediaeval Europe... there was no common centre through which men of science could communicate with one another, and to this cause the slow and fitful development of mediaeval mathematics may be partly ascribed. The last two centuries of this period... described as the renaissance, were distinguished by great mental activity in all branches of learning. The creation of a fresh group of universities... testify to the... desire for knowledge. The discovery of America in 1492 and the discussions that preceded the Reformation flooded Europe with new ideas... ut the advance in mathematics was at least as well marked as that in literature and... politics. During the first part of this time the attention of mathematicians was to a large extent concentrated on syncopated algebra and trigonometry."

"was among the first to take advantage of the recovery of the original texts of the Greek mathematical works... the earliest notice in modern Europe of the algebra of Diophantus is [his] remark... that he had seen a copy... at the Vatican. He was also well read in the works of the Arab mathematicians. The fruit of this study... his De Triangulis... 1464... the earliest modern systematic exposition of trigonometry, plane and spherical, though the only trigonometrical functions introduced are... the sine and cosine. It is divided into five books. The first four... plane trigonometry... in particular... determining triangles from three given conditions. The fifth book is... spherical trigonometry. The work was printed in five volumes... 1533, nearly a century after the death of Regiomontanus."

"[A]lgebra and trigonometry were still only in the rhetorical stage of development, and when every step of the argument is expressed in words at full length it is by no means easy to realise all that is contained in a formula."

"Regiomontanus did not hesitate to apply algebra to the solution of geometrical problems. An... illustration of this is to be found in his discussion of a question... in ’s Siddhanta... to construct a quadrilateral, having its sides of given lengths, which should be inscribable in a circle. The solution given by Regiomontanus was effected by means of algebra and trigonometry: this was published by C. G. von Murr... 1786."

"Georg Purbach, first the tutor and then the friend of ... wrote a work on planetary motions... 1460; an arithmetic... 1511; a table of eclipses... 1514; and a table of natural sines... 1541."

"There are but few special symbols in trigonometry, I... however add here... all that I have been able to learn... The current division of angles is derived from the Babylonians through the Greeks. The Babylonian unit angle was the angle of an ; following their usual practice... this was divided into sixty equal parts or degrees, a degree was subdivided into sixty equal parts or minutes, and so on. The word sine was used by and was derived from the Arabs: the terms secant and tangent were introduced by Thomas Finck... in his Geometriae Rotundi, Bâle, 1583: the word cosecant was (I believe) first used by Rheticus in his Opus Palatinum, 1596: the terms cosine and cotangent were first employed by E. Gunter in his Canon Triangulorum, London, 1620. The abbreviations sin, tan, sec were used in 1626 by , and those of cos and cot by Oughtred in 1657; but these contractions did not come into general use till Euler re-introduced them in 1748. The idea of trigonometrical functions originated with John Bernoulli, and this view of the subject was elaborated in 1748 by Euler in his Introductio in Analysin Infinitorum."

"Euler wrote... in 1748 his Introductio in Analysin Infinitorum... intended... as an introduction to pure analytical mathematics. The first part... contains... the matter... found in modern text-books on algebra, theory of equations, and trigonometry. In the algebra he paid particular attention to the expansion of various functions in series, and to the summation of given series; and pointed out explicitly that an infinite series cannot be safely employed unless it is convergent. In the trigonometry, much of which is founded on F. C. Mayer’s Arithmetic of Sines... published... 1727, Euler developed the idea of John Bernoulli that the subject was a branch of analysis and not a mere appendage of astronomy or geometry: he also introduced (contemporaneously with Simpson) the current abbreviations for the trigonometrical functions, and shewed that the trigonometrical and exponential functions were connected by the relation \cos\theta + i\sin\theta = e^{i\theta}."

"Claudius Ptolemaeus, a celebrated astronomer, was a native of Egypt. ...The chief of his works are the Syntaxis Mathematica (or the ', as the Arabs call it) and the Geographica, both of which are extant. ...Ptolemy did considerable for mathematics. He created, for astronomical use, a trigonometry remarkably perfect in form. ...The fact that trigonometry was cultivated not for its own sake, but to aid astronomical inquiry, explains the rather startling fact that came to exist in a developed state earlier than plane trigonometry. ...Ptolemy has written other works which have little or no bearing on mathematics, except one on geometry. Extracts from this book made by Proclus indicate that Ptolemy did not regard the parallel-axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it."

"John Bernoulli (1667-1748)... treated trigonometry by the analytical method, studied caustic curves and trajectories."

"De Moivre enjoyed the friendship of Newton and Halley. His power as a mathematician lay in analytic rather than geometric investigation. He revolutionised higher trigonometry by the discovery of the theorem known by his name [(\cos x + i \sin x)^n = \cos nx + i \sin nx,] and by extending the theorems on the multiplication and division of sectors from the circle to the ."

"[Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \textstyle \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi} \textstyle \int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"The tables of the numerical values of the trigonometric functions had now attained a high degree of accuracy, but their real significance and usefulness were first shown by the introduction of logarithms."

"Napier is usually regarded as the inventor of logarithms, although Cantor's review of the evidence leaves no room for doubt that Bürgi was an independent discoverer. His Progress Tabulen, computed between 1603 and 1611 but not published until 1620 is really a table of antilogarithms. Bürgi's more general point of view should also be mentioned. He desired to simplify all calculations by means of logarithms while Napier used only the logarithms of the trigonometric functions."

"The greatest of Hindu astronomers and mathematicians, Aryabhata, discussed in verse such poetic subjects as quadratic equations, sines, and the value of π; he explained eclipses, solstices and equinoxes, announced the sphericity of the earth and its diurnal revolution on its axis, and wrote, in daring anticipation of Renaissance science: "The sphere of the stars is stationary, and the earth, by its revolution, produces the daily rising and setting of planets and stars.""

"The Hindus were not so successful in geometry. In the measurement and construction of altars the priests formulated the ... several hundred years before the birth of Christ. Aryabhata, probably influenced by the Greeks, found the area of a triangle, a trapezium and a circle, and calculated the value of π... at 3.1416—a figure not equaled in accuracy until the days of Purbach... Bhaskara crudely anticipated the differential calculus, Aryabhata drew up a table of sines, and the ' provided a system of trigonometry more advanced than anything known to the Greeks."

"[T]he invention of the infinitesimal calculus... foreshadowed the speedy end of trigonometry as an independent and growing branch of mathematics... By the end of the eighteenth century Leonhard Euler and others had exhibited all the theorems of trigonometry as corollaries of complex function theory."

"Three categories of people confronted or confront situations that are in many ways closely analogous. ...[T]he original inventor of a theorem or techinique... applies to the solution of his problems the mathematical tools he has inherited but slowly or quickly intuits more powerful ways... [T]he historian... seeks, hampered by his... hindsight, to retrace via texts and artifacts the thought processes of the inventor. ...[T]he student, for whom a given problem is as new as ever ...People in all three categories share... an inability to grasp the full implications of their own accomplishments. Both the historian and the student can gain understanding from the example of the inventor."

"[A]t least three thousand years ago man was employing implicity the notion of a function. For each scheme... there is a unique shadow length... Between these primitive landmarks and the road... to trigonometry there lies a great gap... The basic tool... the chord function, still tabulated in engineering handbooks, the precurser of the sine rather than the tangent."

"For serious computation a place-value system for representing numbers was requisite, but since the second millenium B.C. this had been available in the system developed in ..."

"At the end of Book I, Chapter 11, of the ' of... Ptolemy... there is a table of the function \mathrm {crd} \theta ... calculated to three significant sexigesimal places for the domain \theta = 1/2^{\circ}, 1^{\circ}, 1\ 1/2^{\circ}, \ldots , 180^{\circ } and having a column of tabular differences for . In Book I, 9, Ptolemy explains how this table was computed. ...[I]n developing the theory frequent recourse is had to the geometry and geometrical algebra... in the Elements of Euclid. ...Ptolemy is giveing a systematic exposition of... a doctrine well known in his time. This would have included... the work of Hipparchus..."

"[E]xcept for tantelizing hints... from two old cuneiform tablets, there is no way of determining how the trigonometry of chords came into being."

"The ' is a compendium of astronomy made up of cryptic rules in Sanskrit verse, with little explanation and no proofs... [N]umbers were written... in strings of words which conform to the poetic scheme... [A]lmost all...primitive sine functions [are] defined in terms of a circle whose radius, R, is not unity. We distinguish all these with from the modern sine function by the use of an intitial capital... explicitly displaying the parameter R when useful, as a subscript. In general, then, \mathrm{Sin}_R \theta \equiv R \sin \theta... and analogously for the other trigonometric functions..."

"Beginning with the ninth century, the number of people working in trigonometry increased markedly. Astronomers all, they lived in and traveled widely... from India to Spain. ...Of extraordinarily varied ethnic background—Persian, Arabic, Turkish... [etc.]—almost all shared a common faith, Islam, and a common language of science, Arabic."

"The "rule of four quanitites" marked a stage in the transition from a calculus dealing with arcs of a spherical quadrilateral to spherical trigonometry proper, involving the sides and angles of a spherical triangle. This theorem states that in a pair of right spherical triangles having an acute angle (A and A') in common or equal...\sin a/a' = \sin c/\sin c'...[A]lthough it utilizes triangles, angles are not dealt with. ...[A] proof by means of ... [is] straightforward. ...The Menelaus equation... can be stated in terms of sines by use of the identity \sin \theta \equiv \frac{1}{2} \mathrm{Crd} 2 \theta."

"By the ninth century instead of primitive schemes... tables... were common... [and] gave as a function of the sun's altitude the shadow lengths cast by it... Lengths were measured in units of a standard vertical ... These are tables of \mathrm{Cot} \theta \equiv R \cot \thetausually calculated for \theta = 1^{\circ}, 2^{\circ}, 3^{\circ}, \ldots , 90^{\circ} ."

"A few... recognized and corrected the inconvenience of the parameter R \neq 1. Since computations in the sexigesimal system were customary, this inconvenience could be minimized by putting R = 60 = 1,0 , in imitation of the Ptolemaic chord function. The astronomer Habash... tabulated... the function 1,0 \cdot \tan \theta ...[M]ultiplication or division by R becomes a... matter (as we would put it) of shifting the sexigesimal point..."

"... [i]n his unprinted manual of trigonometry... expounded the prosthaphæretic method aimed at simplifying... trigonometric computations. This method first appeared in print in 1588, in Nicolas Reimerus (Ursus)' '. It was of great value... and was going to be a direct competitor to the method of logarithms."

"... had been devised by ... and was likely brought to by... ... in 1580... This method was based on...\sin A \sin B = \frac{1}{2}[ \cos (A-B)- \cos (A+B) ], \cos A \cos B = \frac{1}{2}[ \cos (A-B)+ \cos (A+B) ].With... a table of sines, these... could... replace multiplications by additions and subtractions, something...Wittich found out, but apparently Werner didn’t realize."

"Thanks to Brahe’s manual of trigonometry, the fame of the method of prosthaphæresis spread abroad and it was brought by Wittich to Kassel in 1584... This is probably how Jost Bürgi... then an instrument maker working for the Landgrave of Kassel, learned of it."

"Bürgi not only used this method, but... improved it. He found the second formula, for Brahe and Wittich only knew the first. In addition, he improved the computation of the ...\cos c = \cos a \cos b + \sin a \sin b \cos CUsing the method of prosthaphæresis... cos a cos b and sin a sin b... could be computed but two new multiplications were... left... Bürgi realized that... prosthaphæresis could be used a second time, and... all multiplications could be replaced by additions or subtractions."

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table... Bürgi... seems to have been reluctant at publishing it and in 1592, Brahe wrote that he did not understand why he was keeping the table hidden, after... a look at it."

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table..."

"All [previous] procedures for calculating chords and sines... [were] based in principle on the method which Ptolemy had presented in his '. Totally different... is a procedure which Jost Bürgi...invented... [H]e was able to compute the sine of each angle with any desired accuracy in a... short time... Bürgi explained his procedure in ...'."

"In 1605 Bürgi went to Prague and lived there as a at the Emperor’s Chamber until 1631... [and] was... involved in astronomical observations and interpretations. ...... worked at the court. Among... temporary visitors... ... and Ursus... who had both worked with . Wittich... brought... knowledge of ... by which multiplications and divisions can be replaced by additions and subtractions of trigonometrical values... based on the identity\sin \alpha \cdot \sin \beta = \frac{1}{2}[\sin (90^\circ - \alpha + \beta) - \sin(90^\circ - \alpha -\beta)],this... achieves the same as logarithms, which were invented... decades later."

"[A] manuscript has survived in which Bürgi... explains his method. From the preface... we know that Bürgi presented it to... Rudolf II... The manuscript consists of 95 folios and contains a[n]... extensive work on by Bürgi, entitled '."

"Bürgi... found an arithmetic procedure for computing sine values with arbitrary accuracy. By dividing the right angle into 90 parts, Bürgi is able to calculate the sines of all degrees from sin 1° to sin 90°."

"When HaShem commanded him &lbrack;Moshe&rbrack; to go down to Egypt and liberate the Jewish people from bondage, he asked HaShem, "When I come to the children of Israel and I say to them, 'The God of your forefathers sent me to you,' they will ask me, 'What is His name,' What &lbrack;&rbrack; should I say to them?" ... The final letters are the name HaShem-, while the beginning letters equal "The Secret of the Name" (Sod Shem–) [30- + 40- + 300- + 40- = 60- + 6- + 4- + 300- + 30- = 410]. Therefore, the beginning and final letters make up the phrase "The Secret of the Name HaShem–.""

"Abulafia himself wrote an autobiographical account of this conversion attempt [of Pope Nicholas III] in his Sefer HaEdot (Book of Testimonies). In this account, he calls himself Raziel, an anonym that he frequently used, this being the name of the angel who taught the mysteries to Adam. Raziel () has a numerical value of 248, this being the same as that of Abulafia's first name, Abraham (). As Abulafia himself indicates, this relationship is more than a simple gematria, but it is a mishkal (balance), since both the numerical value and the number of letters in both names are equal."

"Lenny Meyer: The ancient Jews used Hebrew as their numerical system. May I? Each letter's a number. Like, the Hebrew A, Aleph &lbrack;&rbrack;, is 1. B, Bet &lbrack;&rbrack;, is 2. You understand? But look at this. The numbers are interrelated. Like, take the Hebrew word for "father", av &lbrack;&rbrack;. Aleph, Bet. 1, 2. Equals 3. All right? Hebrew word for "mother", em &lbrack;&rbrack;. Aleph, Mem &lbrack;&rbrack;. 1, 40. Equals 41. The sum of 3 and 41? 44. All right? Now, Hebrew word for "child", all right? Mother, father, child. Yeled &lbrack;&rbrack;. That's 10, 30 and 4. It's 44. Torah is just a long string of numbers. Some say that it's a code, sent to us from God. Max Cohen: That's kind of interesting."

"Sol Robeson: This is insanity, Max. Max Cohen: Or maybe it's genius! I have to get that number. Sol Robeson: Hold on! You have to slow down. You're losing it. You have to take a breath. Listen to yourself. You're connecting a computer bug I had with a computer bug you might have had and some religious hogwash. If you want to find the number 216 in the world, you will be able to find it everywhere: 216 steps from your street corner to your front door, 216 seconds you spend riding in the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere: 320, 450, 22, whatever. You've chosen 216, and you will find it everywhere in nature. But Max, as soon as you discard scientific rigor, you are no longer a mathematician—you're a numerologist."

"Let's take some extra time to talk about one: Only the number one can create all numbers with this simple equation, 111111111 × 111111111 = 12345678987654321. One, expressed nine times, multiplied by itself, produces all subsequent numbers progressively and then inversely."

"The last few decades have provided abundant evidence for physics beyond the two standard models of particle physics and . As is now known, the by far largest part of our universe's matter/energy content lies in the `dark' and consists of and . Despite intensive efforts on the experimental as well as the theoretical side, the origins of both are still completely unknown. Screened scalar fields have been hypothesized as potential candidates for dark energy or dark matter. Among these, some of the most prominent models are the , , and environment-dependent ."

"2. Are there any new elementary scalars not yet discovered with masses below the mass of the -like Higgs boson? For example, do -like particles exist? ... 8. If additional scalars are discovered, how will these discoveries impact the question of the stability of the ? 9. Do neutral (inert) scalars comprise a significant fraction of the dark matter?"

"The dependence of so many results on Riemann's challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word 'hypothesis' has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. 'Conjecture', in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann's riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated."

"Moreover, the complete details of the complication of the structure of Mandelbrot's set cannot really be fully comprehended by anyone of us, nor can it be fully revealed by any computer. It would seem that this structure is not just part of our minds, but it has a reality of its own. ... The computer is being used in essentially the same way that the experimental physicist uses a piece of experimental apparatus to explore the structure of the physical world. The Mandelbrot set is not an invention of the human mind: it was a discovery. Like Mount Everest, the Mandelbrot set is just there!"

"This present art, in which we use those twice five Indian figures, is called algorismus."

"This boke is called the boke of algorym or Augrym after lewder use. And this boke tretys the Craft of Nombryng, the quych crafte is called also Algorym. Ther was a kyng of Inde the quich heyth Algor, and he made this craft. And aft his name he called hit algory."

"‘…the transition [to the Hindu number system], far from being immediate, extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals [more precisely, Hindu numerals] were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code.’"

"‘The change did not come about without obstruction from the representatives of custom thought. An edict of A.D. 1259 forbade the bankers of Florence to use the infidel symbols, and the ecclesiastical authorities of the University of Padua in A.D. 1348 ordered that the price list of books should be prepared not in “ciphers”, but in plain letters.’"

"All the algorithms for fractions now used were invented by the Hindus. The Greek treatment of fractions never advanced beyond the level of the Egyptian Rhind papyrus. […] This inability to treat a fraction as a number on its own merits is the explanation of a practice [which] was as useless as it was ambiguous. […] When we remember that the Greeks and Alexandrians continued this extraordinary performance, there is nothing remarkable about the small progress which they achieved in their arithmetic. What is remarkable is that a few of them like Archimedes should have discovered anything at all about series of numbers involving fractional quantities."

"The cord stretched in the diagonal of an oblong produces both [areas] which the cords forming the longer and shorter sides of an oblong produce separately."

"The diagonal cord of a rectangle makes both (the squares) that the vertical side and the horizontal side make separately."

"Some of the things, which belong particularly to the history of mathematics, we will not refrain from attributing to Pythagoras himself. Among them is the Pythagorean theorem, which we want to preserve under all circumstances. (Cantor 1880: 129)"

"In the Shulba Sutra appended to Baudhayana’s Shrauta Sutra, mathematical instructions are given for the construction of Vedic altars. One of its remarkable contributions is the theorem usually ascribed to Pythagoras, first for the special case of a square (the form in which it was discovered), then for the general case of the rectangle: “The diagonal of the rectangle produces the combined surface which the length and the breadth produce separately.”"

"The first writer to attribute this proposition to Pythagoras is Vitruvius, hardly a reliable witness. From then on, this account became more widespread, but always in connection with the famous hecatomb that Pythagoras is said to have offered in celebration of discovering the proposition—an anecdote that severely undermines the credibility of the entire story. This sacrifice is incompatible with the strict prohibition of all bloody sacrifices, which writers of the same period, indeed often the very same ones who elsewhere recount the hecatomb, have handed down to us from the Pythagorean ritual laws. Cicero himself took offense at this anecdote, and in the latest Neopythagorean tradition, the bloody sacrifice is replaced by that of an ox formed from flour. For this reason, the hecatomb is not only incongruous with the Pythagoreans, but also with the Pythagoreans themselves. Proclus, an insightful writer, expresses himself remarkably vaguely: ‘When we listen to those who want to tell old stories, we find that they trace this theorem back to Pythagoras.’ As this shows, he too was unaware of any reliable source."

"Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him."

"‘These operations are all founded on a very distinct conception of what happens in the case of an eclipse, and on the knowledge of this theorem, that, in a right-angled triangle, the square on the hypotenuse is equal to the squares of the other two sides. It is curious to find the theorem of PYTHAGORAS in India, where, for aught we know, it may have been discovered, and from whence that philosopher may have derived some of the solid, as well as the visionary speculations, with which he delighted to instruct or amuse his disciples.’"

"If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast."

"It is more likely that Pythagoras was influenced by India than by Egypt. Almost all the theories, religions, philosophical and mathematical taught by the Pythagoreans, were known in India in the sixth century B.C., and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat and regarded certain vegetables such as beans as taboo" "It seems that the so-called Pythagorean theorem of the quadrature of the hypotenuse was already known to the Indians in the older Vedic times, and thus before Pythagoras."

"If we consider the results obtained together, we will not be able to doubt the conclusion to be drawn from them. The ancient priestly geometry of the Indians not only knew the Pythagorean theorem, but it even played the main role in their calculations; with its help, they constructed elements that the Greeks found in a completely different way; with its help, they also found the irrational quantities. And it was precisely these two things that Pythagoras introduced into the Greek-Italian world; these two things, according to the Greeks, he invented. Indeed, even more! The way in which Pythagoras proved his theorem was also, in all likelihood, the same as that which we find in the Vedic Shulba Sutras. After examining the Shulba Sutras, we could have said: If Pythagoras really was in India, as we previously suggested, and initiated himself into the priestly wisdom of the Brahmins, then he could have brought precisely these theorems of geometric science to Greece; — and history has been telling us for several millennia now that this was indeed the case!"

"However, considering the increasingly clear connections between India and the Babylonian cultural sphere, particularly in recent years, the Indian evidence for the study of Babylonian mathematics certainly comes into play. The difficulties which arise from assuming a direct borrowing by the Greeks from India fall away on the assumption of a common origin in Babylonia.As for Greece itself, the origin of the Pythagorean theorem is, as is well known, shrouded in mystery. However, there is no doubt that Pythagoras and his school were strongly influenced by the Orient. The possibility is therefore obvious that Pythagoras (or his school) had learned the theorem later named after him from oriental sources, which would be very compatible with seeing him as the discoverer of the 'Pythagorean method' for constructing integers that satisfy the relation a² + b² = c². This would only fit with the entire number-speculative tendency of this circle."

"Having fixed a pole, on a level piece of ground, and having described a circle by a cord attached to the pole, one should mark the points with pegs, where the shadow of the top of the pole touches the circle. The line joining these pegs is the West to East line. Having increased the length of the cord (which is equal to the distance of the two pegs fixed in the East West line) by itself, one should provide two slings at each end of the cord. Fix the slings on the two pegs (already fixed) stretch the cord with its mid point towards the south and fix a peg at the place, where the mid point of the cord touches the ground. He should similarly proceed towards the North. The line joining these pegs will be the South to North line."

"Multiply the arc by the square of the arc, and take the result of repeating that [any number of times]. Divide [each of the above numerators] by the squares of successive even numbers increased by that number [lit. the root] and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jı̄vā, as collected together in the verse beginning with “vidvān” etc."

"This is as I have said in my Arithmetic:—The area of a circle is equal to the product of the circumference by one-fourth of the diameter [πr²]. That result multiplied by 4 gives the surface of the sphere [4πr²], which is like the net surrounding a hand ball; the same (surface of a sphere) when multiplied by the diameter and divided by six [4/3πr³] becomes invariably the volume of the sphere."

"A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed.The axis of a sphere is the fixed straight line about which the semicircle revolves.The centre of a sphere is the same with that of the semicircle.The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere."

"The process of reasoning called "Mathematical Induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchman B. Pascal and P. Fermat, and the Italian F. Maurolycus. [...] By reading a little between the lines one can find traces of mathematical induction still earlier, in the writings of the Hindus and the Greeks, as, for instance, in the "cyclic method" of Bhaskara, and in Euclid's proof that the number of primes is infinite."

"The method represents a best approximation algorithm of minimal length that, owing to several minimization properties, with minimal effort and avoiding large numbers automatically produces the best solutions to the equation. The chakravala method anticipated the European methods by more than a thousand years. But no European performances in the whole field of algebra at a time much later than Bhaskara's, nay nearly equal up to our times, equalled the marvellous complexity and ingenuity of chakravala."

"This method is supreme above all praise; it is certainly the finest thing accomplished in number theory before Lagrange."