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.
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Quotes by mathematicians and philosophers[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
 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.
 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
 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]
 Mathematics is not something that you find lying around in your back yard. It’s 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.
 Paul Davies, How Unique You Are!; Is There a Creator Who Cares About You?, published by Jehovah's Witnesses.
 It suggests to me that consciousness and our ability to do mathematics are no mere accident, no trivial detail, no insignificant byproduct of evolution.
 Paul Davies, Are We Alone?
 Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum,
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and greater still, and so on. Augustus De Morgan, A Budget of Paradoxes.
 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.
 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.
 There is no royal road to geometry. (μή εἶναι βασιλικήν ἀτραπόν ἐπί γεωμετρίαν, Non est regia [inquit Euclides] ad Geometriam via)
 Euclid's alleged reply when the ruler Ptolemy I Soter asked him if there was a shorter road to learning geometry than through Euclid's Elements.
 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.
 Linear Algebra by Fraleigh/Beauregard.
 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 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.
 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).
 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.
 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).
 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)
 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.
 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, pg. 165.
 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 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.
 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)
 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).
 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).
 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.
 [Mathematics is] the science that draws necessary conclusions.
 Benjamin Peirce, "Linear Associative Algebra", American Journal of Mathematics, Vol. 4, No. 1/4 (1881).
 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
 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.
 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
 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)
 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
 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.
Quotes that mention mathematics[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]
 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)
 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.