Mathematics

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

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

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

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

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

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

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

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

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

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

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

"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."

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"Buzz Lightyear: To infinity and beyond!"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"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."

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

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