Question

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.

Answer 1

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 Jean-Pierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way.

Answer 2

"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

Answer 3

"There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms." - Eichler

Answer 4

(Caveat for all of mine: I've not hunted down primary sources to check that they're properly attributed)

"Manifolds are a bit like pornography: hard to define, but you know one when you see one." -S. Weinberger

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

The purpose of computing is insight, not numbers.

— Richard Hamming (1962)

The attitude adopted in this book is that while we expect to get numbers out of the machine, we also expect to take action based on them, and, therefore we need to understand thoroughly what numbers may, or may not, mean. To cite the author's favorite motto,

“The purpose of computing is insight, not numbers,” although some people claim,

“The purpose of computing numbers is not yet in sight.”

There is an innate risk in computing because “to compute is to sample, and one then enters the domain of statistics with all its uncertainties.”

– Richard W. Hamming, Introduction to applied numerical analysis, McGraw-Hill 1971, p.31.

Answer 10

My favorite math quote will probably always be Paul Gordan's response to Hilbert's proof of his Basis Theorem: "This is not Mathematics. This is Theology."

Along with his redaction after he came to accept the method: "I have convinced myself that even theology has its merits."

Answer 11

«Allez en avant, et la foi vous viendra.»

Free translation: keep going, faith will come later.

Jean-le-Rond D'Alembert, to his students (quoted by Florian Cajori in A history of mathematics)

Answer 12

A mathematician is a device for turning coffee into theorems. —Alfréd Rényi, but often attributed also to Paul Erdős

Answer 13

La vie est étrange. En fait, en géometrie, on ne se représente pas de la même manière une droite complexe affine (par exemple pour le teorème de Ceva dans un triangle) et le corps des complexes x+iy. Quand j'y songe, les points imaginaires de la géometrie sont gris, les points réels noirs, et l'intersection de deux droites imaginaires conjuguées est un point réel noir. La belle conique ombilicale est argentée, les droites et cônes isotropes sont plutôt roses

Laurent Schwartz, Un mathematicien aux prises avec le siècle.

Free translation: «Life is strange. In fact, in geometry, we do not think in the same way of a complex affine line (for example in the theorem of Ceva in a triangle) and of the field of complex numbers x+iy. When I think about this, imaginary points in geomtry are gray, the real points are black, and the intersection of two conjugate imaginary lines is a black real point. The beautiful umbilical conic is silver, the lines and isotropic cones are mostly pink.»

Answer 14

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

Answer 15

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.

Answer 16

"Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."- Paul Erdős

Answer 17

“The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.”- Godfrey Harold Hardy

Answer 18

Dirichlet allein, nicht ich, nicht Cauchy, nicht Gauß, weiß, was ein vollkommen strenger Beweis ist, sondern wir lernen es erst von ihm. Wenn Gauß sagt, er habe etwas bewiesen, so ist es mir sehr wahrscheinlich, wenn Cauchy es sagt, ist ebensoviel pro als contra su wetten, wenn Dirichlet es sagt, ist es gewiß; ich lasse mich auf diese Delikatessen lieber gar nicht ein.

C. G. J. Jacobi, writing to von Humboldt, in 1846.

Without pretty ßs: Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters.

Answer 19

You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine.

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

Answer 20

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." - Henry Poincaré

Answer 21

"You don't have to believe in God, but you should believe in The Book." --- Paul Erdős. describing the Book held by the God that contains the most beautiful proofs to all the theorems

Answer 22

"My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful."- Herman Weyl

Answer 23

"A mathematician who is not also something of a poet will never be a perfect mathematician"- Karl Weierstraß

Answer 24

"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)

Answer 25

Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.

Gian-Carlo Rota, in an interview with David Sharp.

Answer 26

Someone once told me that Grothendieck said "a sheaf of groups is a group of sheaves," although I have been unable to find a real reference. Can anyone substantiate this?

Answer 27

" 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

Answer 28

"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

Answer 29

"A mathematical truth is neither simple nor complicated in itself, it is." - Émile Lemoine

Answer 30

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)