Mathematics
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Mathematics is the body of knowledge centered on concepts such as quantity, structure, space, and change, and the academic discipline which studies them.

CONTENTS
Quotes by mathematicians and philosophers: A, B, C, D, E, F, G, H, IJ, K, L, M, N, OP, Q, R, S, T, UW, YZ
Quotes that mention mathematics: A'C', D'L', M'Z', Misattributed, See also
Quotes by mathematicians and philosophers[edit]
A[edit]
 Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means.
 Maurice Allais, La formation scientifique, Une communication du Prix Nobel d’économie, Maurice Allais, address to the Académie des Sciences Morales et Politiques (1997).
 Mathematics is not a careful march down a wellcleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
 W. S. Anglin, in Mathematics and History, elucidating the symmetry between the creative and logical aspects of mathematics.
 It was mathematics, the nonempirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances.
 Hannah Arendt, The Life of the Mind (1971), p. 7.
B[edit]
 If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
 Roger Bacon Opus Majus, Book 1, Chapter 4.
 If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.
 John D. Barrow, The Artful Universe (1995)
 Where there is life there is a pattern, and where there is a pattern there is mathematics. Once that germ of rationality and order exists to turn a chaos into a cosmos, then so does mathematics. There could not be a nonmathematical Universe containing living observers.
 John D. Barrow, The Artful Universe (1995)
 We say that the string is 'random' if there is no other representation of the string which is shorter than itself. But we will say that it is 'nonrandom' if there does exist such an abbreviated representation. ...In general, the shorter the possible representation... the less random... On this view we recognize science to be the search for algorithmic compressions. ...It is simplest to think of mathematics as the catalogue of all possible patterns. ...When viewed in this way, it is inevitable that the world is described by mathematics. ...In many ways the search for a Theory of Everything is a manifestation of a faith that this compression goes all the way down to the bedrock of reality...
 John D. Barrow, New Theories of Everything (2007)
 Mathematics became an experimental subject. Individuals could follow previously intractable problems by simply watching what happened when they were programmed into a personal computer. ...The PC revolution has made science more visual and more immediate ...by creating films of imaginary experiences of mathematical worlds. ...Words are no longer enough.
 John D. Barrow, Cosmic Imagery: Key Images in the History of Science (2008)
 A narrative of the decisive epochs in the development of mathematics was wanted. ...Numerous professionals... know from hard experience what mathematical invention means. ...Whoever has himself attempted to advance mathematics is inclined to be more skeptical than the average spectator toward any alleged anticipation of notable progress. ...often what looks like an anticipation ...was not even aimed in the right direction. ...when at length progress started; it proceeded along lines totally different from those which, in retrospect, it 'should' have followed.
 Eric Temple Bell, The Development of Mathematics (1940)
 Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed.
 Eric Temple Bell, The Development of Mathematics (1940)
 The analytical geometry of Descartes and the calculus of Newton and Leibniz have expanded into the marvelous mathematical method—more daring than anything that the history of philosophy records—of Lobachevsky and Riemann, Gauss and Sylvester. Indeed, mathematics, the indispensable tool of the sciences, defying the senses to follow its splendid flights, is demonstrating today, as it never has been demonstrated before, the supremacy of the pure reason.
C[edit]
 I would myself say that the purely imaginary objects are the only realities, the ὂντως ὂντα, in regard to which the corresponding physical objects are as the shadows in the cave; and it is only by means of them that we are able to deny the existence of a corresponding physical object; and if there is no conception of straightness, then it is meaningless to deny the conception of a perfectly straight line.
 Arthur Cayley, Presidential address British Association for the Advancement of Science, Southport (1883) as quoted by Nicholas Murray Butler, The Meaning of Education: And Other Essays and Addresses (1898)
 This statistical regularity in moral affairs fully establishes their being under the presidency of law. Man is now seen to be an enigma only as an individual; in the mass he is a mathematical problem.
 Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic.
 Robert Chambers, Chambers's Information for the People (1875) Vol. 2
 There is probably no other science which presents such different appearances to one who cultivates and one who does not, as mathematics. To [the nonmathematician] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple of bloom of vigorous youth, everywhere stretching out after the "attainable but unattained," and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable bypaths that end in a wilderness.
 C. H. Chapman, Review of Sophus Lie's Theorie der Transformationsgruppen (1892) Bulletin of the New York Mathematical Society 2, p. 61
 The study of mathematics is the indispensable basis for all intellectual and spiritual progress.
 F.M. Cornford (18741943), summarizing a central tenet of Pythagoreanism.[1]
D[edit]
 The progress of mathematics has been most erratic, and... intuition has played a predominant rôle in it. ...It was the function of intuition to create new forms; it was the acknowledged right of logic to accept or reject these new forms, in whose birth in had no part. ...the children had to live, so while waiting for logic to sanctify their existence, they throve and multiplied.
 Between the philosopher's attitude towards the issue of reality and that of the mathematician there is this essential difference: for the philosopher the issue is paramount; the mathematician's love for reality is purely platonic.
 Tobias Dantzig, Number: The Language of Science (1930).
 The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. ...The conic sections, invented in an attempt to solve the problem of doubling the alter of an oracle, ended by becoming the orbits followed by the planets... The imaginary magnitudes invented by Cardan and Bombelli describe... the characteristic features of alternating currents. The absolute differential calculus, which originated as a fantasy of Reimann, became the mathematical model for the theory of Relativity. And the matrices which were a complete abstraction in the days of Cayley and Sylvester appear admirably adapted to the... quantum of the atom.
 Tobias Dantzig, Number: The Language of Science (1930).
 Underpinning everything... are the laws of physics. These remarkably ingenious laws are able to permit matter to selforganize to the point where consciousness emerges in the cosmos—mind from matter—and the most striking product of the human mind is mathematics. This is the baffling thing. Mathematics is... produced by the human mind. Yet if we ask where mathematics works best, it is in areas like particle physics and astrophysics, areas of fundamental science that are very, very far removed from everyday affairs. ...at the opposite end of spectrum of complexity from the human brain. ...a product of the most complex system we know in nature, the human brain, finds a consonance with the underlying, simplest and most fundamental level, the basic building blocks that make up the world.
 Paul Davies, Are We Alone?: Philosophical Implications of the Discovery of Extraterrestrial Life (1995)
 It suggests to me that consciousness and our ability to do mathematics are no mere accident, no trivial detail, no insignificant byproduct of evolution that is piggybacking on some other mundane property. It points to what I like to call the cosmic connection, the existence of a really deep relationship between minds that can do mathematics and the underlying laws of nature that produce them. We have a closed system of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can encode... the very laws of physics that gave rise to it.
 Paul Davies, Are We Alone?: Philosophical Implications of the Discovery of Extraterrestrial Life (1995)
 Physicists have been drawn to elegant mathematical relationships that bind the subject together with economy and style, melding disparate qualities in subtle and harmonious ways. But this is to import a new factor into the argument—questions of aesthetics and taste. We are then on shaky ground indeed. It may be that M theory looks beautiful to its creators, but ugly to N theorists, who think that their theory is the most elegant. But then the O theorists disagree with both groups...
 Paul Davies, Cosmic Jackpot: Why Our Universe Is Just Right for Life (2007)
 The nature of mathematical demonstration is totally different from all other, and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us.
 Augustus De Morgan, On the Study and Difficulties of Mathematics (1831) p. 2
 With a view to summon myself to the search for a science of mathematics in general, I asked myself... what precisely was the meaning of this word mathematics, and why arithmetic and geometry only, and not also astronomy, music, optics, mechanics, and so many other sciences, should be considered as forming a part of it; for it is not enough here to know the etymology of the word. In reality the word mathematics meaning nothing but science, those which I have just named have as much right as geometry to be called mathematics; and nevertheless there is no one, however little instructed, who cannot distinguish at once what belongs to mathematics... from what belongs to the other sciences. But... all the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics...
 René Descartes, Rule IV: "Necessity of Method in the Search for Truth," "Rules for the Direction of the Mind," Part I of Discourse Upon Method in The Philosophy of Descartes: In Extracts from His Writing pp. 7172, Tr. Henry A. P. Torrey
 It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.
 Paul Dirac, "The Evolution of the Physicist's Picture of Nature," Scientific American (May, 1963)
E[edit]
 I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission.
 Carl Eckart, Our Modern Idol: Mathematical Science (1984), p. 95.
 Proof is the idol before whom the pure mathematician tortures himself.
 Arthur Eddington, The Nature of the Physical World (1928)
 There is no royal road to geometry. (μή εἶναι βασιλικήν ἀτραπόν ἐπί γεωμετρίαν, Non est regia [inquit Euclides] ad Geometriam via)
 Euclid, alleged reply when Ptolemy I Soter asked him if there was a shorter road to learning geometry than through Euclid's Elements.
F[edit]
 If you are interested in the ultimate character of the physical world, or the complete world, and at the present time our only way to understand that is through the mathematical type of reasoning... the great depth of character of the universality of the laws, the relationships of things... I don't know any other way to do it, we don't know any other way to describe it accurately... or to see the interrelationships without it... don't misunderstand me, there are many, many aspects of the world that mathematics is unnecessary for... but we were talking about physics... to not know mathematics is a severe limitation in understanding the world.
 Richard Feynman, "The Rules of the Game," The Pleasure of Finding Things Out (1999)
 The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things.
Considered from this point of view, mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.
Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them. Joseph Fourier, The Analytical Theory of Heat (1878) Preliminary Discourse, p.7
 Numbers exist only in our minds. There is no physical entity that is number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe.
 John B. Fraleigh, Raymond A. Beauregard, Linear Algebra (1995)
 I united the majority of wellinformed persons into a club, which we called by the name of the Junto, and the object of which was to improve our understandings. ... The first members of our club were...
Thomas Godfrey, a selftaught mathematician, and afterwards inventor of what is now called Hadley's dial; but he had little knowledge out of his own line, and was insupportable in company, always requiring, like the majority of mathematicians that have fallen in my way, an unusual precision in everything that is said, continually contradicting, or making trifling distinctions—a sure way of defeating all the ends of conversation. He very soon left us.
G[edit]
 What has philosophy got to do with measuring anything? It's the mathematicians you have to trust, and they measure the skies like we measure a field.
 Galileo Galilei, Concerning the New Star (1606).
 Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.
 Galileo Galilei, The Assayer (1623) as quoted by Edwin Arthur Burtt in The Metaphysical Foundations of Modern Physical Science (1924), p. 64.
 Humane wisdom understandeth some propositions so perfectly, and is as absolutely certain thereof, as Nature herself; and such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few comprehended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the necessity thereof, than which there can be no greater certainty.
 Galileo Galilei, Dialogo sopra i due Massi Sistemi del Mondo (1632) as quoted in the Salusbury translation, The Systeme of the World: in Four Dialogues (1661).
 Even God cannot make two times two not make four.
 Hugo Grotius, as quoted in Delbert D. Thiessen (ed.), A Sociobiology Compendium: Aphorisms, Sayings, Asides, p. 18
H[edit]
 Mathematical methods present... two advantages. Their terminology is precise and concentrated, in a fashion which ordinary language cannot afford to adopt. Further, the symbols which result from their employment have implications which, when brought to light, yield new knowledge. This is deductively reached, but it is none the less new knowledge. With greater precision than is usual, ordinary language may be made to do some, if not a great deal, of this work for which mathematical methods are alone quite appropriate. If ordinary language can do part of it an advantage may be gained. The difficulty that attends mathematical symbolism is the accompanying tendency to take the symbol as exhaustively descriptive of reality. Now it is not so descriptive. It always embodies an abstraction. It accordingly leads to the use of metaphors which are inadequate and generally untrue. It is only qualification by descriptive language of a wider range that can keep this tendency in check.
 The idea that theorems follow from the postulates does not correspond to simple observation. If the Pythagorean theorem were found to not follow from the postulates, we would again search for a way to alter the postulates until it was true. Euclid's postulates came from the Pythagorean theorem, not the other way around.
 Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly 87 (2), February 1980, pp. 8190.
 I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
 G. H. Hardy, . Quotations by Hardy. Gap.dcs.stand.ac.uk. Retrieved on 27 November 2013..
 The number 1729 is known as the Hardy–Ramanujan number after a famous anecdote of Hardy regarding a visit to the hospital to see S. Ramanujan.
 Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. "Immortality" may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
 G. H. Hardy, A Mathematician's Apology (London 1941).. Quotations by Hardy. Gap.dcs.stand.ac.uk. Retrieved on 27 November 2013..
 A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
 G. H. Hardy, A Mathematician's Apology (London 1941).. Quotations by Hardy. Gap.dcs.stand.ac.uk. Retrieved on 27 November 2013.
 Mathematicians have constructed a very large number of different systems of geometry, Euclidean or nonEuclidean, of one, two, three, or any number of dimensions. All these systems are of complete and equal validity. They embody the results of mathematicians' observations of their reality, a reality far more intense and far more rigid than the dubious and elusive reality of physics. The oldfashioned geometry of Euclid, the entertaining sevenpoint geometry of Veblen, the spacetimes of Minkowski and Einstein, are all absolutely and equally real. ...There may be three dimensions in this room and five next door. As a professional mathematician, I have no idea; I can only ask some competent physicist to instruct me in the facts.
The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subjectmatter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. ...Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers. G. H. Hardy, "The Theory of Numbers," Nature (Sep 16, 1922) Vol. 110 p. 381
 Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.
 Robert Heinlein, Time Enough For Love (1973), and again in The Notebooks of Lazarus Long (1978).
 An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.
 David Hilbert, Mathematical Problems (1900) address, International Congress of Mathematicians at Paris, Tr. Maby Winton Newson, Bulletin of the American Mathematical Society 8 (1902)
 The art of doing mathematics consists in finding that special case which contains all the germs of generality.
 David Hilbert, as quoted by Constance Reid, "Hilbert" (1970)
 The doctrine of Right and Wrong, is perpetually disputed, both by Pen and the Sword: Whereas the doctrine of Lines, and Figures, is not so; because men care not, in that subject what be truth, as a thing that crosses no mans ambition, profit, or lust. For I doubt not, but if it had been a thing contrary to any mans right of dominion, or to the interest of men that have dominion, That the three Angles of a Triangle, should be equall to two Angles of a Square; that doctrine should have been, if not disputed, yet by the burning of all books of Geometry, suppressed, as far as he whom it concerned was able.
 Thomas Hobbes, Leviathan (1651)
 Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination.
 E. W. Hobson, Presidential Address British Association for the Advancement of Science (1910) Nature Vol. 84 p. 290 as quoted by Robert Édouard Moritz, Memorabilia Mathematica; Or, The Philomath's Quotationbook (1914) p. 184.
 Mathematics is the language of size, shape and order and that it is an essential part of the equipment of an intelligent citizen to understand this language. If the rules of mathematics are the rules of grammar, there is no stupidity involved when we fail to see that a mathematical truth is obvious. The rules of ordinary grammar are not obvious. They have to be learned. They are not eternal truths. They are conveniences without whose aid truths about the sorts of things in the world cannot be communicated from one person to another.
 Lancelot Hogben, Mathematics for the Million (1936).
 As soon as a thought or word becomes a tool, one can dispense with actually ‘thinking’ it, that is, with going through the logical acts involved in verbal formulation of it. As has been pointed out, often and correctly, the advantage of mathematics—the model of all neopositivistic thinking—lies in just this ‘intellectual economy.’ Complicated logical operations are carried out without actual performance of the intellectual acts upon which the mathematical and logical symbols are based. … Reason … becomes a fetish, a magic entity that is accepted rather than intellectually experienced.
 Max Horkheimer, Eclipse of Reason (1947), p. 23.
IJ[edit]
 From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.
 Sir James Jeans, The Mysterious Universe (1930) p. 165.
 The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit.
 Sir James Jeans, The Mysterious Universe (1930)
 Clearly, if electric action is to be explained in mechanical terms, the mechanism must be supposed to be attached to the electric charges, and to move through space with them. It must extend through the whole of space, because the attraction and repulsion of an electron extend through the whole of space, and it must be the same for all directions in space. Further, as the pattern of events is unaltered by motion, the mechanism must be the same when the electron is in motion as when it is at rest. But experiment shows that an electron in motion exerts additional forces which are not the same for all directions in space; if we picture this electron as moving headforemost through space, these forces surround it like a belt around its waist.
Thus direct experimental evidence shows that the forces exerted by an electron (or... any charged body) can neither be attributed to any mechanism attached to the body, nor through action transmitted through an ether or any medium surrounding the body. We have a perfect specification of the pattern of events written... in the language of mathematics, but this does not admit of interpretation in mechanical terms, or indeed in any terms other than those of mathematics. Sir James Jeans, Physics and Philosophy (1942) pp. 121122
K[edit]
 I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for... science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. ...the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics.
 Immanuel Kant, Preface, The Metaphysical Foundations of Natural Science (1786) Tr. Ernest Belfort Bax (1883)
 The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience.
 Immanuel Kant, Critique of Pure Reason (1781) Tr. Max Müller (1881) p. 610.
 Those who thought they could distinguish philosophy from mathematics by saying that the former was concerned with quality only, the latter with quantity only, mistook effect for cause. It is owing to the form of mathematical knowledge that it can refer to quanta only, because it is only the concept of quantities that admits of construction, that is, of a priori representation in intuition, while qualities cannot be represented in any but empirical intuition.
 Immanuel Kant, Critique of Pure Reason (1781) Tr. Max Müller (1881) p. 612.
 There is a familiar formula—perhaps the most compact and famous of all formulas—developed by Euler from a discovery of De Moivre: e^{iπ} + 1 = 0. ...It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
 Edward Kasner, James R. Newman, Mathematics and the Imagination (1940)
 It is impossible, and it has always been impossible, to grasp the meaning of what we nowadays call physics independently of its mathematical form.
 Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968)
 There is no doubt... that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgement has caused mathematicians to equate elegance and conciseness at the cost of intelligibility.
 Morris Kline, Mathematics and the Physical World (1959)
 Electromagnetic theory is entirely a mathematical theory illustrated by a few crude physical pictures. These pictures are no more than the clothes that dress up the body of mathematics and make it appear presentable in the society of sciences. ...Though he [James Clerk Maxwell] had tried desperately to build a physical account of electromagnetic phenomena, in his classic Treatise on Electricity and Magnetism he omitted most of this material and emphasized the highly polished, complex mathematical theory. ...Radio waves and light waves operate in a physical darkness illuminated only for those who would carry the torch of mathematics.
 Morris Kline, Mathematics and the Physical World (1959) pp. 360361
 One of the curious things about mathematics that clearly emerges... is that mathematics which is concerned with reasoning nevertheless creates processes which can be applied almost mechanically, that is, without reasoning. The thinking is, so to speak, mechanized and this mechanization enables us to solve complicated problems in no time. We think up processes so that we don't have to think.
 Morris Kline, Mathematics for the Nonmathematician (1967) p. 118.
L[edit]
 It is a wellknown experience that the only truly enjoyable and profitable way of studying mathematics is the method of "filling in details" by one's own efforts.
 Cornelius Lanczos, Applied Analysis, PrenticeHall, 1956.
 There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.
 Nikolai Ivanovich Lobachevsky, as quoted in George Edward Martin, The Foundations of Geometry and the NonEuclidean Plane, Springer (1998 [1975]), p. 225; also in Stanley Gudder, A Mathematical Journey, McGrawHill (1976), p. 36.
M[edit]
 Think of it: of the infinity of real numbers, those that are most important to mathematics—0, 1, √2, e and π—are located within less than four units on the number line. A remarkable coincidence? A mere detail in the Creator's grand design? I let the reader decide.
 Eli Maor, e: The Story of a Number (1994)
 One should not be deceived by philosophical works that pretend to be mathematical, but are merely dubious and murky metaphysics. Just because a philosopher can recite the words lemma, theorem and corollary doesn't mean that his work has the certainty of mathematics. That certainty does not derive from big words, or even from the method used by geometers, but rather from the utter simplicity of the objects considered by mathematics.
 Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or "mathematics envy."
 Marvin Minsky, "Music, Mind, and Meaning" (1981).
 The thorough analysis of even simple problems in arithmetic may require the application of advanced mathematics.
 Lloyd Motz & Jefferson Hane Weaver, Conquering Mathematics: From Arithmetic to Calculus (1991)
 Mathematics is good for the soul, getting things right enlivens a sense of truth, efforts to understand automatically purify desires.
 Iris Murdoch, Metaphysics As A Guide To Morals, p. 178.
 Mathematics is the source of a wicked intellect that, while making man the lord of the earth, also makes him the slave of the machine.
 Robert Musil, The Man Without Qualities.
 Mathematics is the bold luxury of pure reason, one of the few that remain today.
 Robert Musil, “Mathematical man” (1913).
 In their field they [mathematicians] do what we ought to be doing in ours. Therein lies the significant lesson … of their existence. They are an analogy for the intellectual of the future.
 Robert Musil, “Mathematical man” (1913).
N[edit]
 Wer ein mathematisches Buch nicht mit Andacht ergreift, und es wie GottesWort liest, der versteht es nicht.
 Whoever seizes a mathematical book without devotion, whoever does not read it as if it were God's word, doesn't understand it.
 Novalis, Fragmente vermischten Inhalts.
 Whoever seizes a mathematical book without devotion, whoever does not read it as if it were God's word, doesn't understand it.
OP[edit]
 [Mathematics is] the science that draws necessary conclusions.
 Benjamin Peirce, "Linear Associative Algebra", American Journal of Mathematics, Vol. 4, No. 1/4 (1881).
 Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention, they are interested in form alone.
 Henri Poincaré, Science and Hypothesis (1901)
 What we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all; this common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony then which is the sole objective reality, the only truth we can attain; and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better.
 Henri Poincaré, The Value of Science (1905) Tr. George Bruce Halsted, p. 14
 Mathematics have a triple aim. They must furnish an instrument for the study of nature. But that is not all: they have a philosophic aim and, I dare maintain, an esthetic aim. They must aid the philosopher to fathom the notions of number, of space, of time. And above all, their adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms; they marvel when a new discovery opens to them an unexpected perspective; and has not the joy they thus feel the esthetic character, even though the senses take no part therein? Only a privileged few are called to enjoy it fully, it is true, but is not this the case for all the noblest arts?
This is why I do not hesitate to say that mathematics deserve to be cultivated for their own sake, and the theories inapplicable to physics as well as the others. Even if the physical aim and the esthetic aim were not united, we ought not to sacrifice either. Henri Poincaré, The Value of Science (1905) Tr. George Bruce Halsted pp. 7576
 You're not supposed to have equations in a public lecture, so think of this as a piece of art.
 Joseph Polchinski, introducing the Dirac equation as improvement upon the Schrödinger equation in "Spacetime versus the Quantum" (November 25, 2014) 59th Annual Faculty Research Lecture, University of California, Santa Barbara
 Number is the ruler of forms and ideas, and the cause of gods and daemons.
 Pythagoras, as quoted in Life of Pythagoras (c. 300) by Iamblichus of Chalcis, as translated by Thomas Taylor (1818).
 Variants:
 Number rules the universe.
 As quoted in Desmond MacHale, Comic Sections (1993).
 There is geometry in the humming of the strings, there is music in the spacing of the spheres.
 Pythagoras, as quoted in The Mystery of Matter (1965) edited by Louise B. Young
Q[edit]
 The more advanced the sciences have become, the more they have tended to enter the domain of mathematics, which is a sort of center towards which they converge. We can judge of the perfection to which a science has come by the facility, more or less great, with which it may be approached by calculation.
 Adolphe Quetelet (ca. 18251826) as quoted by Frank H. Hankins, "Adolphe Quetelet as Statistician" in Studies in History Economics and Public Law (1908) Vol. 31 p. 443
R[edit]
 An equation means nothing to me unless it expresses a thought of God.
 S. Ramanujan in **. Quotations by 60 Greatest Indians. Dhirubhai Ambani Institute of Information and Communication Technology. Retrieved on 27 November 2013.
 Mathematics has the dubious honor of being the least popular subject in the curriculum … Future teachers pass through the elementary schools learning to detest mathematics. They drop it in high school as early as possible. They avoid it in teachers' colleges because it is not required. They return to the elementary school to teach a new generation to detest it.
 Report of the Educational Testing Service, Princeton, N. J., as quoted in TIME magazine (18 June 1956), cited by George Pólya, How to Solve It, Page ix in Expanded Princeton Science Library Edition (2004), ISBN 069111966X
 Extension and abstraction without apparent direction or purpose is fundamental to the discipline. Applicability is not the reason we work, and plenty that is not applicable contributes to the beauty and magnificence of our subject.
 Peter Rowlett, "The unplanned impact of mathematics", Nature 475, 2011, pp. 166169.
 Trying to solve realworld problems, researchers often discover that the tools they need were developed years, decades or even centuries earlier by mathematicians with no prospect of, or care for, applicability.
 Peter Rowlett, "The unplanned impact of mathematics", Nature 475, 2011, pp. 166169.
 There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.
 Peter Rowlett, "The unplanned impact of mathematics", Nature 475, 2011, pp. 166169.
 Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true … If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.
 Bertrand Russell, Recent Work on the Principles of Mathematics, published in International Monthly, vol. 4 (1901).
 It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true.
 Bertrand Russell, in a letter to Gilbert Murray, April 3, 1902.
 Mathematics, rightly viewed, possesses not only truth, but supreme beauty —a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
 Bertrand Russell, Study of Mathematics (1902)
 Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the world, but every possible world, must conform.
 Bertrand Russell, Study of Mathematics (1902).
 Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
 Bertrand Russell, Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 3.
 The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
 Bertrand Russell, Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 5.
 I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe – because, like Spinoza's God, it won't love us in return.
 Bertrand Russell, in a letter to Lady Ottoline Morrell, March, 1912, as quoted in Gaither's Dictionary of Scientific Quotations (2012), p. 1318.
 Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say.
 Bertrand Russell, The Scientific Outlook (1931).
 In universities, mathematics is taught mainly to men who are going to teach mathematics to men who are going to teach mathematics to.... Sometimes, it is true, there is an escape from this treadmill. Archimedes used mathematics to kill Romans, Galileo to improve the Grand Duke of Tuscany's artillery, modern physicists (grown more ambitious) to exterminate the human race. It is usually on this account that the study of mathematics is commended to the general public as worthy of State support.
 Bertrand Russell, Human Society in Ethics and Politics (1954), p. 54.
S[edit]
 is a long way from infinity.
 Daniel Shanks, Solved and Unsolved Problems in Number Theory, 3rd edition, chapter IV, page 217.
 Computer calculation even up to a big number can't really say much about asymptotic behaviour.
 I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was deBourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes.
 Alexander Stepanov, . Bjarne Stroustrup: Evolving a language in and for the real world: C++ 19912006. ACM HOPLIII. June 2007.. Archived from the original on 20071120. Retrieved on 20080425..
 It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the notreal; its meaning does not reside in formal abstractions, but neither is it tangible. ...it is the great strength of mathematics—what I have elsewhere called its "unreal reality." Mathematics links the abstract world of mental concepts to the real world of physical things without being located in completely in either.
 Ian Stewart, Preface to second edition of What is Mathematics? by Richard Courant and Herbert Robbins, revised by Ian Stewart (1996)
 So, nat'ralists observe, a flea
Hath smaller fleas that on him prey,
And these have smaller still to bite 'em
And so proceed ad infinitum. Jonathan Swift, On Poetry: A Rhapsody
T[edit]
 The subject matter of mathematics has increased so rapidly and extensively that there is some element of truth in maintaining that mathematics is not so much a subject as a way of studying any subject, not so much a science as a way of life.
 George Frederick James Temple, 100 Years of Mathematics: a Personal Viewpoint (1981)
 Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.
 Nikola Tesla, "Radio Power Will Revolutionize the World", in Modern Mechanics and Inventions (July 1934)
 I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.
 William Thomson (Lord Kelvin), Lecture on "Electrical Units of Measurement" (3 May 1883), published in Popular Lectures Vol. I, p. 73 (1889)
 Mathematics is the only true metaphysics.
 William Thomson, as quoted by Silvanus Phillips Thompson, The Life of William Thomson, Baron Kelvin of Largs (1910) Vol. 2, p. 1124
 Mathematical reasoning may be regarded rather schematically as the exercise of a combination of... intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. ...We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions.
 Alan Turing, "Systems of Logic Based on Ordinals" (1938) in Proceedings of the London Mathematical Society (1939) Series 2, Vol. 45
UW[edit]
 The Koch Curve  Triangles outside triangles outside triangles ad infinitum the Koch curve goes, it's infinitely infinitesimal, this selfsimilarity shows. A length too long to measure, an area too small to see, what else can this contradiction be, behold fractal geometry.
 Bernt Wahl, The Adventures of the Fractal Explorer, about the beauty of Fractal Geometry.
 By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race.
 Alfred North Whitehead, Introduction to Mathematics (1911), Ch. 5. Notably quoted in Florian Cajori, A History of Mathematical Notations (1929), Vol. 2, p. 332.
 Nothing is more impressive than the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned back to earth with a corresponding growth of importance for the analysis of concrete fact. ...The paradox is now fully established that the utmost abstractions are the true weapons with which to control our thought of concrete fact.
 Alfred North Whitehead, Science and the Modern World (1925)
YZ[edit]
 Math is perfect (in principle), but mathematicians are not (because they are humans), hence the mathematics that (human) mathematicians do is influenced by the weltanschauung of the people around them.
 Doron Zeilberger "Computerized Deconstruction". Appeared in Adv. Appl. Math. v. 31 (2003), 532543.
Quotes that mention mathematics[edit]
A'C'[edit]
 The land of easy mathematics where he who works adds up and he who retires subtracts.
 Núria Añó, 2066. Beginning the age of correction.
 I can’t help it, gas escapes from my fundament on the least pretext, it’s hard not to mention it now and then, however great my distaste. One day I counted them. Three hundred and fifteen farts in nineteen hours, or an average of over sixteen farts an hour. After all it’s not excessive. Four farts every fifteen minutes. It’s nothing. Not even one fart every four minutes. It’s unbelievable. Damn it, I hardly fart at all, I should never have mentioned it. Extraordinary how mathematics help you to know yourself.
 Samuel Beckett, Molloy.
 They who study mathematiks only to fix their minds, and render them the steadyer to apply to all other things, as there are many who profess to do, are as wise as those who think by rowing boats, to learn to swim.
 Samuel Butler, Prose Observations (Oxford: 1979), p. 4.
 Yeah, Silver and his math are jokes, because math has a liberal bias. After all, math is the reason Mitt Romney's tax plan doesn't add up.
 Stephen Colbert, The Colbert Report (20121105)[2]
 I was an atheist, finding no reason to postulate the existence of any truths outside of mathematics, physics and chemistry. But then I went to medical school, and encountered life and death issues at the bedsides of my patients. Challenged by one of those patients, who asked "What do you believe, doctor?", I began searching for answers.
 Francis Collins, a geneticist who led the U.S. government’s effort to decipher the human genome (DNA). cnn.com
D'L'[edit]
 A man with all the algebra in the world is often only an ass when he knows nothing else. Perhaps in ten years society may derive advantage from the curves which these visionary algebraists will have laboriously squared. I congratulate posterity beforehand. But to tell you the truth I see nothing but a scientific extravagance in all these calculations. That which is neither useful nor agreeable is worthless. And as for useful things, they have all been discovered; and to those which are agreeable, I hope that good taste will not admit algebra among them.
 Frederick the Great, Letters of Voltaire and Frederick the Great (New York: Brentano's, 1927), transl. Richard Aldington, letter 93 from Frederick to Voltaire, 16 May 1749.
 As to your Newton, I confess I do not understand his void and his gravity; I admit he has demonstrated the movement of the heavenly bodies with more exactitude than his forerunners; but you will admit it is an absurdity to maintain the existence of Nothing.
 Frederick the Great, Letters of Voltaire and Frederick the Great (New York: Brentano's, 1927), transl. Richard Aldington, letter 221 from Frederick to Voltaire, 25 November 1777.
 Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!
 Frederick the Great, Letters of Voltaire and Frederick the Great (New York: Brentano's, 1927), transl. Richard Aldington, letter 221 from Frederick to Voltaire, 25 November 1777.
 The Devil: Okay, boys...tonight's homework. Algebra. Xn + Yn = Zn. You're never gonna use that, are you? Imperialism and the First World War. What was done is done. No point thinking about it now. German, French, Spanish. Ja, ja, oui, oui, s, s. It's nonsense. Everyone speaks English anyway. And if they don't, they ought to. So, no homework tonight. But I want you to watch a lot of TV, don't neglect your video games...and I'll see you in the morning. Shall we say 10:00, 10:30? No point in getting up too early.
 Larry Gelbart, Harold Ramis and Peter Tolan, Bedazzled (2000 film)
 Mathematics is really a liberal art if you look at it from a slightly different point of view.
 Steve Jobs, Steve Jobs Lost Interview  A must watch for any entrepeneur (1990)
 The mathematical genius can only carry on from the point which mathematical knowledge within his culture has already reached. Thus if Einstein had been born into a primitive tribe which was unable to count beyond three, lifelong application to mathematics probably would not have carried him beyond the development of a decimal system based on fingers and toes.
 Ralph Linton, The Study of Man (1936)
M'Z'[edit]
 Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite.
 Ezra Pound, The Spirit of Romance (1910), p. 5
 Halakhic man, well furnished with rules, judgments, and fundamental principles, draws near the world with an a priori relation. His approach begins with an ideal creation and concludes with a real one. To whom may he be compared? To a mathematician who fashions an ideal world and then uses it for the purpose of establishing a relationship between it and the real world. ... The essence of the Halakhah, which was received from God, consists in creating an ideal world and cognizing the relationship between that ideal world and our concrete environment.
 Joseph B. Soloveitchik, Halakhic Man (1983), p. 19
 I had been to school most all the time and could spell and read and write just a little, and could say the multiplication table up to six times seven is thirtyfive, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics anyway.
 School children and students who love God should never say: “For my part I like mathematics”; “I like French”; “I like Greek.” They should learn to like all these subjects, because all of them develop that faculty of attention which, directed toward God, is the very substance of prayer.
 Simone Weil, Waiting for God (1951), p. 105.
 The number 2 thought of by one man cannot be added to the number 2 thought of by another man so us to make up the number 4.
 Simone Weil, Oppression and Liberty (1958), p. 82
Misattributed[edit]
 The good Christian should beware of mathematicians, and all those who make empty prophesies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
 Misattributed to Augustine of Hippo. This is a very bad mistranslation of De genesi ad litteram libri XII, book 2, 17.37. 'Mathematici' in Latin means astrologers, not mathematicians, and the book makes repeated attacks on astrology. The text really reads: For which reason both astrologers and those impiously making divinings, as the truth says emphatically, must be avoided by the good Christian, lest after making a pact of agreement they entangle their soul in a hidden partnership with demons.
See also[edit]
 Foundations of mathematics
 History of logarithms
 History of mathematics
 Mathematics education
 Mathematical induction
 Mathematicians
 Statistics