Mathematics is the body of knowledge centered on concepts such as quantity, structure, space, and change, and the academic discipline which studies them. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

Sourced

• 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 well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
  • W.S. Anglin, in Mathematics and History, elucidating the symmetry between the creative and logical aspects of mathematics.

• If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.
  • Roger Bacon Opus Majus, bk.1, ch4.

• 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

• 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

• 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 0-691-11966-X

• 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.

• 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. 81-90

• ... 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.

• [Mathematics is] the science that draws necessary conclusions.
  • Benjamin Peirce, "Linear Associative Algebra", American Journal of Mathematics, Vol. 4, No. 1/4 (1881).

• 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, in Study of Mathematics, about the beauty of Mathematics

• 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 de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise. I still believe in abstraction, but now I know that one ends with abstraction, not starts with it. I learned that one has to adapt abstractions to reality and not the other way around. Mathematics stopped being a science of theories but reappeared to me as a science of numbers and shapes.
  • Alexander Stepanov,

• 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

Unsourced

• Mathematics is music for the mind; Music is mathematics for the soul.
  • Stanley Gudder

• Mathematics is man's attempt on understanding nature.
• A mathematician is a blind man in a dark room looking for a black cat which isn't there.
• Mathematics seems to endow one with something like a new sense.
  • Charles Darwin

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

• As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
• Do not worry about your problems with mathematics, I assure you mine are far greater.
• God does not care about our mathematical difficulties. He integrates empirically.
• I don't believe in mathematics.
• Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
  • Albert Einstein

• As long as algebra is taught in school, there will be prayer in school.
  • Cokie Roberts

• God does arithmetic.
  • Karl Friedrich Gauss

• God made the natural numbers, all the rest is the work of man.
  • Leopold Kronecker

• He who can properly define and divide is to be considered a god.
• The knowledge of which geometry aims is the knowledge of the eternal.
• Mathematics is like checkers in being suitable for the young, not too difficult, amusing, and without peril to the state.
  • Plato

• I admit that mathematical science is a good thing. But excessive devotion to it is a bad thing.
  • Aldous Huxley

• I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate.
  • David Mumford

• I have no faith in political arithmetic.
  • Adam Smith

• If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.
  • Alfréd Rényi

• In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
  • G. H. Hardy

• In mathematics you don't understand things. You just get used to them.
  • John Von Neumann

• It is easy to lie with statistics. It is hard to tell the truth without it.
  • Andrejs Dunkels

• Life is good for only two things, discovering mathematics and teaching mathematics.
  • Siméon Poisson

• Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning.
  • John Locke

• Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.
  • Bernard Le Bouyer de Fontenelle

• [Mathematics] is an independent world created out of pure intelligence.
  • William Wordsworth

• Mathematics is not yet capable of coping with the naïveté of the mathematician himself.
  • Abraham Kaplan

• Mathematics is the art of giving the same name to different things.
  • Jules Henri Poincaré

• Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
  • Max Dehn

• Mathematics is the science of what is clear by itself.
  • Carl Jacobi

• 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. (referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequences thereof are then logically derived)
• Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.
• How dare we speak of the laws of chance? Is not chance the antithesis of all law?
  • Bertrand Russell

• Measure what is measurable, and make measurable what is not so.
  • Galileo Galilei

• No human investigation can be called real science if it cannot be demonstrated mathematically.
  • Leonardo da Vinci

• Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important...The knowledge doesn't make life any sweeter or happier, does it?
  • Helen Keller

• One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.
  • Heinrich Rudolf Hertz

• Perfect numbers like perfect men are very rare.
  • René Descartes

• Pure mathematics, may it never be of any use to anyone.
  • Henry John Stephen Smith

• Sex is the mathematics urge sublimated.
  • M. C. Reed

• The art of doing mathematics consists in finding that special case which contains all the germs of generality.
• The infinite! No other question has ever moved so profoundly the spirit of man.
• Mathematics is a game played according to certain simple rules with meaningless marks on paper.
  • David Hilbert

• The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.
  • Gottfried Wilhelm Leibniz

• The mathematician has reached the highest rung on the ladder of human thought.
  • Havelock Ellis

• 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

• The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificed his life.
  • Ernest Renan

• There are three kinds of lies: lies, damned lies, and statistics.
  • Leonard H. Courtney. Often attributed (including by Mark Twain) to Benjamin Disraeli.

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

• In real life there is no such thing as algebra.
  • Fran Lebowitz

• There is something I don't understand about algebra: It has been around for thousands of years, yet no one has ever found out what the value of "x" or "y" really is.
  • Richard van der Merwe

Misattributed

• 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 St. Augustine. This is a very bad mistranslation of De genesim 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.