Logic

""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."

"Let no one say that in this case it is possible for “truth” in its turn by the help of the press to get the better of lies and errors. You who speak thus, do you venture to maintain that men regarded as a crowd are just as quick to seize upon truth which is not always palatable as upon falsehood which always is prepared delicately to give delight? … Or do you venture even to maintain that “truth” can just as quickly be understood as falsehood, which requires no preliminary knowledge, no schooling, no discipline, no abstinence, no self-denial, no honest concern about oneself, no patient labor?"

"And do not mix the truth with falsehood or conceal the truth while you know (it)."

"Say, "Indeed, those who invent falsehood about Allah will not succeed.""

"That is because Allah is the Truth, and that which they call upon other than Him is falsehood, and because Allah is the Most High, the Grand."

"And (they are) those who do not testify to falsehood, and when they pass near ill speech, they pass by with dignity."

"Where we desire to be informed 'tis good to contest with men above ourselves; but to confirm and establish our opinions, 'tis best to argue with judgments below our own, that the frequent spoils and victories over their reasons may settle in ourselves an esteem and confirmed opinion of our own."

""Or he might say: 'Whereas some recluses and brahmins, while living on the food offered by the faithful, engage in wrangling argumentation, (saying to one another): "You don't understand this doctrine and discipline. I am the one who understands this doctrine and discipline." — "How can you understand this doctrine and discipline?" — "You're practising the wrong way. I'm practising the right way." — "I'm being consistent. You're inconsistent." — "What should have been said first you said last, what should have been said last you said first." — "What you took so long to think out has been confuted." — "Your doctrine has been refuted. You're defeated. Go, try to save your doctrine, or disentangle yourself now if you can" — the recluse Gotama abstains from such wrangling argumentation.'"

"And there began a lang digression About the lords o' the creation."

"He'd undertake to prove, by force Of argument, a man's no horse. He'd prove a buzzard is no fowl, And that a Lord may be an owl, A calf an Alderman, a goose a Justice, And rooks, Committee-men or Trustees."

"Whatever Sceptic could inquire for, For every why he had a wherefore."

"I've heard old cunning stagers Say, fools for arguments use wagers."

"'Twas blow for blow, disputing inch by inch, For one would not retreat, nor t'other flinch."

"When Bishop Berkeley said, "there was no matter," And proved it—'twas no matter what he said."

"The only way to get the best of an argument is to avoid it."

"He whose will and desire in conversation is to establish his own opinion, even though what he says is true, should recognize that he is sick with the devil's disease."

"While the Hindu elaborates his argument, the Moslem sharpens his sword."

"A knock-down argument; 'tis but a word and a blow."

"Reproachful speech from either side The want of argument supplied; They rail, reviled; as often ends The contests of disputing friends."

"His conduct still right with his argument wrong."

"In arguing, too, the parson own'd his skill, For even though vanquished he could argue still."

"I find you want me to furnish you with argument and intellects too. No, sir, these, I protest you, are too hard for me."

"Be calm in arguing; for fierceness makes Error a fault, and truth discourtesy."

"Argument is unnecessary for an enlightened disciple. ... Argument implies a desire to win, strengthens egotism, and ties us to the belief in the idea of a self."

"I have found you an argument; but I am not obliged to find you an understanding."

"The brilliant chief, irregularly great, Frank, haughty, rash—the Rupert of debate."

"In argument with men a woman ever Goes by the worse, whatever be her cause."

"It's hard to engage in good faith with a bad faith argument without just haemorrhaging energy."

"Abba Paul the Barber and his brother Timothy lived in Scetis. They often used to argue. So Abba Paul said, 'How long shall we go on like this?' Abba Timothy said to him, 'I suggest you take my side of the argument and in my turn I will take your side when you oppose me.' They spent the rest of their days in this practice."

"The very nature of deliberation and argumentation is opposed to necessity and self-evidence, since no one deliberates where the solution is necessary or argues argues against what is self-evident."

"Like doctors thus, when much dispute has past, We find our tenets just the same at last."

"Arguments don't break chains."

"The first the Retort Courteous; the second the Quip Modest; the third the Reply Churlish; the fourth the Reproof Valiant; the fifth the Countercheck Quarrelsome; the sixth the Lie with Circumstance; the seventh the Lie Direct."

"And sheath'd their swords for lack of argument."

"There is occasions and causes why and wherefore in all things."

"For they are yet but ear-kissing arguments."

"She hath prosperous art When she will play with reason and discourse, And well she can persuade."

"Ignorantia non est argumentum."

"Ah, don't say that you agree with me. When people agree with me I always feel that I must be wrong."

"Much might be said on both sides."

"I am bound to furnish my antagonists with arguments, but not with comprehension."

"The noble Lord (Stanley) was the Prince Rupert to the Parliamentary army—his valour did not always serve his own cause."

"How agree the kettle and the earthen pot together?"

"The daughter of debate That still discord doth sow."

"I always admired Mrs. Grote's saying that politics and theology were the only two really great subjects."

"Nay, if he take you in hand, sir, with an argument, He'll bray you in a mortar."

"Seria risu risum, seriis discutere."

"There is no good in arguing with the inevitable. The only argument available with an east wind is to put on your overcoat."

"Myself when young did eagerly frequent Doctor and Saint, and heard great argument About it and about: but evermore Came out by the same door wherein I went."

"Discors concordia."

"Demosthenes, when taunted by Pytheas that all his arguments "smelled of the lamp," replied, "Yes, but your lamp and mine, my friend, do not witness the same labours.""

"In some places he draws the thread of his verbosity finer than the staple of his argument."

"In argument Similes are like songs in love: They must describe; they nothing prove."

"One single positive weighs more, You know, than negatives a score."

"Soon their crude notions with each other fought; The adverse sect denied what this had taught; And he at length the amplest triumph gain'd, Who contradicted what the last maintain'd."

"Agreed to differ."

"You can't win an argument by being right, either"

"Jolie hypothèse elle explique tant de choses."

"Sing the melody line you hear in your own head. Remember, you don't owe anybody any explanations, you don't owe your parents any explanations, you don't owe your professors any explanations."

"The overwhelming majority of theories are rejected because they contain bad explanations, not because they fail experimental tests."

"A prediction, or any assertion, that cannot be defended might still be true, but an explanation that cannot be defended is not an explanation."

"Dionysus: He who believes needs no explanation. Pentheus: What's the worth in believing worthless things? Dionysus: Much worth, but not worth telling you, it seems."

"... Dreams of a Final Theory: The Scientist’s Search for the Ultimate Laws of Nature (1992) ... belongs alongside G. H. Hardy’s ' (1940) and James Watson’s ' (1968) as one of the classic books in which a brilliant scientist has given his motivations full vent. But while Hardy and Watson portrayed science as a for personal glory, Weinberg portrayed it as a quest toward some final explanation, one that would undergird all others."

"But as our explanation will be more brief than one broken in upon by words of wonder, regret, and affection, we will proceed to it ; holding that explanation, like advice, should be of all convenient shortness."

""But surely you would explain your idea to one who asked you?" I say again, if I cannot draw a horse, I will not write THIS IS A HORSE under what I foolishly meant for one. Any key to a work of imagination would be nearly, if not quite, as absurd. The tale is there, not to hide, but to show: if it show nothing at your window, do not open your door to it; leave it out in the cold. To ask me to explain, is to say, "Roses! Boil them, or we won't have them!" My tales may not be roses, but I will not boil them. So long as I think my dog can bark, I will not sit up to bark for him."

"Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong."

"There probably is a God. Many things are easier to explain if there is than if there isn't."

"Tao mystics never talk about God, reincarnation, heaven, hell. No, they don't talk about these things. These are all creations of human mind: explanations for something which can never be explained, explanations for the mystery. In fact, all explanations are against God because explanation de-mystifies existence. Existence is a mystery, and one should accept it as a mystery and not pretend to have any explanation. No, explanation is not needed – only exclamation, a wondering heart, awakened, surprised, feeling the mystery of life each moment. Then, and only then, you know what truth is. And truth liberates."

"Denn wenn sich Jemand versteckt erklärt, so ist Nichts unhöflicher als eine neue Frage."

"Explanations are clear but since no one to whom a thing is explained can connect the explanations with what is really clear, therefore clear explanations are not clear."

"Very simple was my explanation, and plausible enough—as most wrong theories are!"

"Paradoxes explain everything. Since they do, they cannot be explained."

"Paradox is the sharpest scalpel in the satchel of science. Nothing concentrates the mind as effectively, regardless of whether it pits two competing theories against each other, or theory against observation, or a compelling mathematical deduction against ordinary common sense."

"I myself find the division of the world into an objective and a subjective side much too arbitrary. The fact that religions through the ages have spoken in images, parables, and paradoxes means simply that there are no other ways of grasping the reality to which they refer. But that does not mean that it is not a genuine reality. And splitting this reality into an objective and a subjective side won't get us very far."

"It seems a little paradoxical to construct a configuration space with the coordinates of points which do not exist."

"In the matter of reforming things, as distinct from deforming them, there is one plain and simple principle; a principle which will probably be called a paradox. There exists in such a case a certain institution or law; let us say, for the sake of simplicity, a fence or gate erected across a road. The more modern type of reformer goes gaily up to it and says, "I don't see the use of this; let us clear it away." To which the more intelligent type of reformer will do well to answer: "If you don't see the use of it, I certainly won't let you clear it away. Go away and think. Then, when you can come back and tell me that you do see the use of it, I may allow you to destroy it." This paradox rests on the most elementary common sense. The gate or fence did not grow there. It was not set up by somnambulists who built it in their sleep. It is highly improbable that it was put there by escaped lunatics who were for some reason loose in the street. Some person had some reason for thinking it would be a good thing for somebody. And until we know what the reason was, we really cannot judge whether the reason was reasonable."

"Paradoxes often arise because theory routinely refuses to be subordinate to reality."

"Since the beginning of time tricksters (the mythological origin of all clowns) have embraced life's paradoxes, creating coherence through confusion — adding disorder to the world in order to expose its lies and speak the truth."

"The more I know, the more sure I am I know so little. The eternal paradox."

"A paradox is a situation which gives one answer when analyzed one way, and a different answer when analyzed another way, so that we are left in somewhat of a quandary as to actually what should happen. Of course, in physics there are never any real paradoxes because there is only one correct answer; at least we believe that nature will act in only one way (and that is the right way, naturally). So in physics a paradox is only a confusion in our own understanding."

"The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function."

"The Greeks observed a paradox about the dyad: While it appears separate from unity, its opposite poles remember their source and attract each other in an attempt to merge and return to the state of unity. The dyad simultaneously divides and unites, repels and attracts, separates and returns."

"I learned to make my mind large, as the universe is large, so that there is room for paradoxes."

"A great many individuals ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. ...I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: ...something which is apart from general opinion, either in subject-matter, method, or conclusion. ...Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place... Phillips says paradox is "a thing which seemeth strange"—here is the old meaning...—"and absurd, and is contrary to common opinion," which is an addition due to his own time."

"Paradox, a thing that seems strange, absurd and contrary to common Opinion: In Rhetorick, Paradoxon is something cast in by the by, contrary to the Opinion or Expectation of the Auditors, and otherwise call'd Hypomone. Paradoxol or Paradoxical, belonging to a Paradox, surprizing."

"The best paradoxes raise questions about what kinds of contradictions can occur — what species of impossibilities are possible."

"Paradox is thus a much deeper and universal concept than the ancients would have dreamed. Rather than an oddity, it is a mainstay of the philosophy of science."

"The assumption that anything true is knowable is the grandfather of paradoxes."

"A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science."

"These are old fond paradoxes to make fools laugh i' the alehouse."

"You undergo too strict a paradox, Striving to make an ugly deed look fair."

"More than any other Hellenic thinker, Julian insisted on the virtue of paradox and on the importance of the search for religious truth."

"But now we come to the real paradox: that something as explosive as sexual excitement can nevertheless become a matter of habit, But then that applies to all our pleasures. We discover some new product in the supermarket, and become addicted to it. Then our tastebuds become accustomed to its flavour, and our interest fades. In the same way a honeymoon couple may find an excuse to hurry off to the bedroom half a dozen times a day; but after a month or so sex has taken its place among the many routines of their lives. They still enjoy it, but it no longer has quite the same power to excite the imagination. Sex, like every other pleasure, can become mechanical."

"PARADOX: A statement that reduces the matter at hand to complete obscurity while clarifying it. … Paradoxes are sensitive and can be routed by sneering."

"Paradoxes explain everything. Since they do, they cannot be explained."

"A Paradox is truth spelt with seven letters instead of five"

"For thence, — a paradox Which comforts while it mocks, — Shall life succeed in that it seems to fail: What I aspired to be, And was not, comforts me: A brute I might have been, but would not sink i' the scale."

"Then there is that glorious Epicurean paradox, uttered by my friend, the Historian, in one of his flashing moments: "Give us the luxuries of life, and we will dispense with its necessaries.""

"The mind begins to boggle at unnatural substances as things paradoxical and incomprehensible."

"The peril of the heavy tower, of the restless vault, of the vagrant buttress; the uncertainty of logic, the inequalities of the syllogism, the irregularities of the mental mirror,— all these haunting nightmares of the Church are expressed as strongly by the Gothic cathedral although it had been the cry of human suffering, and as no emotion had ever been expressed before or likely to find expression again."

"St. Thomas Aquinas narrates the life of St. Francis d'Assisi. O MORTAL cares insensate, what small worth, In sooth, doth all those syllogisms fill, Which make you stoop your pinions to the earth!"

"The race of prophets is dead. Europe is becoming set in its ways, slowly embalming itself beneath the wrappings of its borders, its factories, its law courts and its universities. The frozen Mind cracks between the mineral staves which close upon it. The fault lies with your mouldy systems, your logic of 2 + 2 = 4. The fault lies with you, Chancellors, caught in the net of syllogisms. You manufacture engineers, magistrates, doctors, who know nothing of the true mysteries of the body or the cosmic laws of existence."

"[In the introduction to his Middle Commentary on Aristotle's Topics, Averroes said] This art has three parts. The first part sets forth the speeches from which dialectical conversation is composed — i.e., its parts, and the parts of its parts on to its simplest components. This part is found in the first treatise on Aristotle's book. The second part sets forth the topics from which syllogisms are drawn — syllogisms for affirming something or denying it with respect to every kind of problem occurring in this art. This is the next six treatises of Aristotle's book. The third part set forth how the third part sets forth how the questioner ought to question and the answerer answer. It also sets forth how many kinds of questions and answers there are. This is in the eighth treatise of Aristotle's book."

"SYLLOGISM, n. A logical formula consisting of a major and a minor assumption and an inconsequent."

"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: Major Premise: Sixty men can do a piece of work sixty times as quickly as one man. Minor Premise: One man can dig a posthole in sixty seconds; therefore Conclusion: Sixty men can dig a posthole in one second. This may be called the syllogism arithmetical, in which, by combining logic and mathematics, we obtain a double certainty and are twice blessed."

"It is significant that in the greatest religious poem existent, the Book of Job, the argument which convinces the infidel is not (as has been represented by the merely rational religionism of the eighteenth century) a picture of the ordered beneficence of the Creation; but, on the contrary, a picture of the huge and undecipherable unreason of it. ‘Hast Thou sent the rain upon the desert where no man is?’ This simple sense of wonder at the shapes of things, and at their exuberant independence of our intellectual standards and our trivial definitions, is the basis of spirituality as it is the basis of nonsense. Nonsense and faith (strange as the conjunction may seem) are the two supreme symbolic assertions of the truth that to draw out the soul of things with a syllogism is as impossible as to draw out Leviathan with a hook."

"All important things bear the sign of death: Haven't people learned yet that the time of superficial intellectual games is over, that agony is infinitely more important than syllogism, that a cry of despair is more revealing than the most subtle thought, and that tears always have deeper roots than smiles?"

"...mathematicians say—the construction propounded will be possible too, and once more the demonstration will correspond, in the reverse order to the analysis; but if we come upon something that is admitted is impossible the problem will also be impossible. It is quite possible to accept that Plato 'discovered' the method of Analysis, in the same sense as Aristotle discovered the syllogism; that is to say, he was the first to reflect upon the process of thought involved and to describe it in contrast with the process of Synthesis."

"They may attack me with an army of six hundred syllogisms; and if I do not recant, they will proclaim me a heretic."

"In building his theory of rhetoric around the syllogism despite the problems involved in deductive inference Aristotle stresses the fact that rhetorical discourse is discourse directed toward knowing, toward truth not trickery... If rhetoric is so clearly related to dialectic, a discipline whereby we are enabled to examine inferentially generally accepted opinions on any problem whatsoever, then it is the rhetorical syllogism [i.e., the enthymeme which moves the rhetorical process into the domain of reasoned activity, or the kind of rhetoric Plato accepted later in the Phaedrus."

"The “elements” of the Great Alexandrian remain for all time the first, and one may venture to assert, the only perfect model of logical exactness of principles, and of rigorous development of theorems. If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid."

"It appears that reason is not, as sense and memory, born with us; nor gotten by experience only, as prudence is; but attained by industry: first in apt imposing of names; and secondly by getting a good and orderly method in proceeding from the elements, which are names, to assertions made by connexion of one of them to another; and so to syllogisms, which are the connexions of one assertion to another, till we come to a knowledge of all the consequences of names appertaining to the subject in hand; and that is it, men call science."

"A syllogism is valid (or logical) when its conclusion follows from its premises. A syllogism is true when it makes accurate claims--that is, when the information it contains is consistent with the facts. To be sound, a syllogism must be both valid and true. However, a syllogism may be valid without being true or true without being valid."

"Deductive reasoning moves from a generalization believed to be true or self-evident to a more specific conclusion. The process of deduction has traditionally been illustrated with a syllogism, a three-part set of statements or propositions that includes a major premise, a minor premise, and a conclusion. Major premise: All books from that store are new. Minor premise: These books are from that store. The major premise of a syllogism makes a general statement that the writer believes to be true. The minor premise presents a specific example of the belief that is stated in the major premise. If the reasoning is sound, the conclusion should follow from the two premises."

"Aristotle writes that persuasion is based on three things: the ethos, or personal character of the speaker; the pathos, or getting the audience into the right kind of emotional receptivity; and the logos, or the argument itself, carried out by abbreviated syllogisms, or something like deductive syllogisms, and by the use of example."

"Knowledge is ours only if, at the moment of need, it offers itself to the mind without syllogisms or demonstrations for which there is no time."

"Lying is not only excusable; it is not only innocent; it is, above all, necessary and unavoidable. Without the ameliorations that it offers, life would become a mere syllogism and hence too metallic to be borne."

"One horse-laugh is worth ten thousand syllogisms. It is not only more effective; it is also vastly more intelligent."

"All the sophisticated syllogisms of the ponderous volumes published by Marx, Engels, and hundreds of Marxian authors cannot conceal the fact that the only and ultimate source of Marx’s prophecy is an alleged inspiration by virtue of which Marx claims to have guessed the plans of the mysterious powers determining the course of history. Like Hegel, Marx was a prophet communicating to the people the revelation that an inner voice had imparted to him."

"...that the god in the sanctuary was finite in his power, and hence a fraud. One horse-laugh is worth ten thousand syllogisms. It is not only more effective; it is also vastly more intelligent."

"In logic, a form of deductive reasoning consisting of a major premise, a minor premise, and a conclusion. Adjective:syllogistic. Here is an example of a valid categorical syllogism: Major premise: All mammals are warm-blooded. Minor premise: All black dogs are mammals. Conclusion: Therefore, all black dogs are warm-blooded."

"I have made every effort to obtain exact information, comparing doctrines, replying to objections, continually constructing equations and reductions from arguments, and weighing thousands of syllogisms in the scales of the most rigorous logic. In this laborious work, I have collected many interesting facts which I shall share with my friends and the public as soon as I have leisure. But I must say that I recognized at once that we had never understood the meaning of these words, so common and yet so sacred: Justice,equity, liberty; that concerning each of these principles our ideas have been utterly obscure; and, in fact, that this ignorance was the sole cause, both of the poverty that devours us, and of all the calamities that have ever afflicted the human race."

"Francis Bacon not only despised the syllogism, but undervalued mathematics, presumably as insufficiently experimental. He was virulently hostile to Aristotle, but though very highly of Democrates."

"We now know that w:Limelightlimelight and a brass band do more to persuade than can be done by the most elegant train of syllogisms. It may be hoped that in time anybody will be able to persuade anybody of anything if he can catch the patient young and is provided by the State with money and equipment."

"Tim Russert told George W. Bush, on his military service that several news reporters found that there was no evidence that he had reported to military duty during the time he had claimed. Bush replied, 'Yeah, they're just wrong. There may be no evidence, but I did report. Otherwise, I wouldn't have been honorably discharged.' That's the Bush syllogism: The evidence says one thing; the conclusion says another; therefore, the evidence is false."

"For a thorough understanding of the syllogism we need to understand it not only with respect to its definition but also with respect to its divisions. Some of the divisions that must be presented apply to the syllogism in general –e.g., a syllogism is either perfect or imperfect, either affirmative or negative, and so on. However, other divisions that must be presented apply to the syllogism in such a way as to separate it into distinct types. The following division is of this kind: as syllogism is either demonstrative, or dialectical, or sophistical. A demonstrative syllogism is the one that produces scientific knowledge on the basis of necessary [premises] and the most certain reasons for the conclusion. A dialectical syllogism, however, is the one that produces opinion on the basis of probable [premises]. Finally sophistical syllogism is the one that either syllogizes on the basis of seemingly probable [premises] or seemingly syllogizes on the basis of probable [premises]; in either case it is strictly aimed at glory or victory."

"In the depth of his heart he knew he was dying, but not only was he not accustomed to the thought, he simply did not and could not grasp it. The syllogism he had learnt from Kiesewetter's Logic: “Caius is a man, men are mortal, therefore Caius is mortal” had always seemed to him correct, but he was Cauis, not an abstract man, but a creature quite, quite separate from all others."

"She certainly had a little syllogism in her head as to the Duke ruling the borough, the Duke's wife ruling the Duke, and therefore the Duke's wife ruling the Duke, and therefore the Duke’s wife ruling the borough; but she did not think it prudent to utter this on the present occasion."

"The motion picture does not outlaw the still photograph but combines a series of them according to the laws of motion. Dialectics does not deny the syllogism, but teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality."

"I believe that there never was a creator of a philosophical system who did not confess at the end of his life that he had wasted his time. It must be admitted that the inventors of the mechanical arts have been much more useful to men that the inventors of syllogisms. He who imagined a ship towers considerably above him who imagined innate ideas."

"From the viewpoint of my general purpose, I had come to believe that one way to achieve the education which leads to understanding and compassion is to take some period of the past and to immerse oneself in it so thoroughly that one could think its thoughts and speak its language. The object would be to take this chapter of vanished experience and learn to know it in three if not four dimensions. That would mean coming to understand why certain actions which in the light of retrospect appear madly irrational appeared at that time the indisputable mandate of reason; why things which had been created with pain and care were cast quickly on the gaming table of war; why men who had sat in the senate chamber and debated with syllogism and enthymeme stepped out of it to buckle on the sword against one another. Almost any book of history will give you the form of such a time, but what will give you the pressure of it? That is what I particularly wished to discover."

"In any remembering, thinking or imagining, although the object of my intending is some state of affairs or other, I am also potentially aware as I intend that what I am doing is an act of remembering, thinking, or imagining. My asserting that S is P is not an assertion of mine unless I am implicitly aware as I assert that I am asserting, not entertaining the possibility that, S is P."

"No sincere assertion of fact is essentially unaccompanied by feelings of intellectual satisfaction or of a persuasive desire and a sense of personal responsibility."

"A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended its area of applicability."

"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."

""I exist" does not follow from "there is a thought now." The fact that a thought occurs at a given moment does not entail that any other thought has occurred at any other moment, still less that there has occurred a series of thoughts sufficient to constitute a single self. As Hume conclusively showed, no one event intrinsically points to any other. We infer the existence of events which we are not actually observing, with the help of general principle. But these principles must be obtained inductively. By mere deduction from what is immediately given we cannot advance a single step beyond. And, consequently, any attempt to base a deductive system on propositions which describe what is immediately given is bound to be a failure."

"Between the workable empiricism of the early land measurers... of ancient Egypt and the geometry of the Greeks in the sixth century before Christ, there is a great chasm. ...and the chasm is bridged by deductive reasoning applied consciously and deliberately to the practical inductions of daily life. Without the strictest deductive proof from admitted assumptions, explicitly stated as such, mathematics does not exist. This does not deny that intuition, experiment, induction, and plain guessing are important elements in mathematical invention. It merely states the criterion by which the final product of all guessing, by whatever name it be dignified, is judged to be or not to be mathematics. It is not known where or when the distinction between inductive inference—the summation of raw experience—and deductive proof from a set of postulates was first made, but it was sharply recognized by the Greek mathematicians as early as 550 B.C."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...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."

"With the completion of Euclid's Elements... For the first time in history masses of isolated discoveries were unified and correlated by a single guided principle, that of rigid deduction from explicitly stated assumptions. ...Not until 1839, in the work of ...D. Hilbert, was the full impact of Euclid's methodology felt in all mathematics. Concurrently with the pragmatic demonstration of the postulational method in arithmetic, geometry, algebra, topology, the theory of point sets, and analysis which distinguished the first four decades of the twentieth century, the method became almost popular in theoretical physics in the 1930's through the work of P. A. M. Dirac. Earlier scientific essays in the method, notably by E. Mach in mechanics and A. Einstein in relativity, had shown that the postulational approach is not only clarifying but creative. Mathematicians and scientists of the conservative persuasion may feel that a science constrained by an explicitly formulated set of assumptions has lost some of its freedom... Experience shows that the only loss is denial of the privilege of making avoidable mistakes in reasoning. ...Objection to the method is neither more nor less than objection to mathematics. ...If the Pythagorean dream of a mathematized science is to be realized, all of the sciences must eventually submit to the discipline that geometry accepted from Euclid."

"Robert [Grosseteste] became much interested in science and scientific method … He was conscious of the dual approach by means of induction and deduction (resolution and composition); i.e., from the empirical knowledge one proceeds to probable general principles, and from these as premises one them derives conclusions which constitute verifications or falsifications of the principles. This approach to science was not that far removed from Aristotle ..."

"I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical."

"Experience has convinced me that the proper way of teaching is to bring together that which is simple from all quarters, and, if I may use such a phrase, to draw upon the surface of the subject a proper mean between the line of closest connexion and the line of easiest deduction. This was the method followed by Euclid, who, fortunately for us, never dreamed of a geometry of triangles, as distinguished from a geometry of circles, or a separate application of the arithmetics of addition and subtraction; but made one help out the other as he best could."

"The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another."

"The two operations of our understanding, intuition and deduction, on which alone we have said we must rely in the acquisition of knowledge."

"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. ...Weyle 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."

"Descartes was an eminent mathematician, and it would seem that the bent of his mind led him to overestimate the value of deductive reasoning from general principles, as much as Bacon had underestimated it."

"I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses."

"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."

"The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems which are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers. ...The eighteenth century doctrine of natural rights is a search for Euclidean axioms in politics. The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source."

"The Greeks... discovered mathematics and the art of deductive reasoning. Geometry, in particular, is a Greek invention, without which modern science would have been impossible."

"But in connection with mathematics the one-sidedness of the Greek genius appears: it reasoned deductively from what appeared self-evident, not inductively from what had been observed. Its amazing successes in the employment of this method misled not only the ancient world, but the greater part of the modern world also."

"It has only been very slowly that scientific method, which seeks to reach principles inductively from observation of particular facts, has replaced the Hellenic belief in deduction from luminous axioms derived from the mind of the philosopher."

"Although we acquire the skill of understanding words by experience, so that we know the correlations between them and things, between words and other words, and between words and feelings and actions, we do not do it by inductive reasoning. Nor must we think that we do it by deductive reasoning... In the main, words are cues rather than clues."

"It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers — not by logical deduction, but by something like a sense of smell."

"There is no greater evil one can suffer than to hate reasonable discourse. Misology and misanthropy arise in the same way. Misanthropy comes when a man without knowledge or skill has placed great trust in someone and believes him to be altogether truthful, sound and trustworthy; then, a short time afterwards he finds him to be wicked and unreliable, and then this happens in another case; when one has frequently had that experience, especially with those whom one believed to be one's closest friends, then, in the end, after many blows, one comes to hate all men and to believe that no one is sound in any way at all. ... This is a shameful state of affairs ... and obviously due to an attempt to have human relations without any skill in human affairs."

"I shall give you a response to what you have just recited like a magic spell, and a rebuttal to your charming ditty delivered in a bellow. Do not make me out to be an ignoramus -- I will answer you once and for all!"

"Reagan finally won the nomination by promoting "Reaganomics", an economic program based on the theory that the government could lower taxes while increasing spending and at the same time actually reduce the federal budget by sacrificing a live chicken by the light of a full moon. Bush charged that this amounted to "voo-doo economics," which got him into hot water until he explained that what he meant to say was "doo-doo economics." Satisfied, Reagan made Bush his vice-presidential nominee. The turning point in the election campaign came during the October 8 debate between Reagan and Carter, when Reagan's handlers came up with a shrewd strategy: No matter what Carter said, Reagan would respond by shaking his head in a sorrowful manner and saying: "There you go again." This was brilliant, because (a) it required the candidate to remember only four words, and (b) he delivered them so believably that everything Carter said seemed like a lie. If Carter had stated that the Earth was round, Reagan would have shaken his head, saying, "There you go again," and millions of voters would have said: "Yeah! What does Carter think we are? Stupid?"

"Sometimes people let the same problem make them miserable for years when they could just say, "So what." That's one of my favorite things to say. "So what." "My mother didn't love me." So what. "My husband won't ball me." So what. "I'm a success but I'm still alone." So what. I don't know how I made it through all the years before I learned how to do that trick. It took a long time for me to learn it, but once you do, you never forget."

"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?)"

"The sun is one, but its beams are numberless; and the effects produced are beneficent or maleficent, according to the nature and constitution of the objects they shine upon."

"It is easy to argue persuasively the truism that the lessons of history are best derived from what actually happened, rather than from what nearly happened. It should be added, however, that what happened becomes more fully comprehensible in the light of the contending forces that existed at moments of decision. Understanding of the total historical setting is bound to contribute to a clearer view of the actual course of affairs."

"I learned this bit of wisdom from a principle of William Blake's which I discovered early and followed far too assiduously the first half of my aesthetic life, and from which I have happily released myself and this axiom was: "Put off intellect and put on imagination; the imagination is the man." From this doctrinal assertion evolved the theoretical axiom that you don't see a thing until you look away from it which was an excellent truism as long as the principles of the imaginative life were believed in and followed. I no longer believe in the imagination."

"Where this age differs from those immediately preceding it is that a liberal intelligentsia is lacking. Bully-worship, under various disguises, has become a universal religion, and such truisms as that a machine-gun is still a machine-gun even when a "good" man is squeezing the trigger — and that in effect is what Mr Russell is saying — have turned into heresies which it is actually becoming dangerous to utter."

"It is a truism that success in science comes to the individuals who ask the right questions."

"Decide on some imperfect Somebody and you will win, because the truest truism in politics is: You can’t beat Somebody with Nobody."

"Quot homines tot sententiæ (Translation: There are as many opinions as there are people who hold them)."