Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?
Standard community wiki rules apply: one quote per post.
Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wiki rules apply: one quote per post. 

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I heard this one while taking a differential geometry class in Mexico City. I love it. "Groups, as men, will be known by their actions". Guillermo Moreno. 


"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."  John von Neumann. (From a 1947 ACM keynote, recalled by Alt in this 1972 CACM article.) 


“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.” Lucien Szpiro during Algebra 1 lecture. 


"Mathematics is the art of giving the same name to different things." Henri Poincaré. (This was in response to "Poetry is the art of giving different names to the same thing.") 


We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?"  GianCarlo Rota 


"The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps..." Alexander Grothendieck, writing to Ronald Brown 


Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he's scribbled on, scrunches them all up, and throws them in the trash can. J. von Neumann's housekeeper, describing her employer. 


Beginning of A. Douady's thesis. Quoted by Michèle AUdin in her Conseils aux auteurs de textes mathématiques. In a less barbarous language: The purpose of this thesis is to obtain the degree of Doctor for its author. 


"The Axiom of Choice is obviously true, the wellordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona 


It's hard to beat John Stembridge's page of quotes. My single favorite one on this page: "If I have not seen as far as others, it is because there were giants standing on my shoulders."  Hal Abelson. 


D. Hilbert, talking about an exstudent. I'd love to remember where I got this from! 


"The art of doing mathematics is finding that special case that contains all the germs of generality."  David Hilbert 


Algebra is the offer made by the devil to the mathematician...All you need to do, is give me your soul: give up geometry Michael Atiyah 


"Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate."  David Mumford 


Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass: "I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance." Grothendieck, of course, always pioneered this approach, and considered for example that JeanPierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way. 


"God exists since mathematics is consistent, and the Devil exists since we cannot prove it." André Weil 


In mathematics you don't understand things. You just get used to them. John von Neumann, reply to a physicist at Los Alamos who had said "I don't understand the method of characteristics."  footnote on page 226 of Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics, Rider, London, 1990. (taken from Warren Dicks' Home Page) 


"The shortest path between two truths in the real domain passes through the complex domain."  Jacques Hadamard 


"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein (from Reed & Simon: Methods of modern mathematical physics) 


"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics."  Weyl 


GianCarlo Rota, in an interview with David Sharp. 


"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." Pierre de Fermat 


Not famous yet, maybe from now on! At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate. Terence Tao 


"The question you raise, "how can such a formulation lead to computations?" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand  and it always turned out that understanding was all that mattered."  Grothendieck 


"Wir müssen wissen, wir werden wissen."  Hilbert. Translation: We must know, we will know. 


" Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?" " Serge Lang 


"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies." Stefan Banach "Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies." Stanislaw M. Ulam quoting Stefan Banach 


Hermann Weyl, Philosophy of Mathematics and Natural Science. 


"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out." Emil Artin, Geometric Algebra 


Dieudonné in "Foundations of Modern Analysis, Vol. 1":


