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

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

Answer 2

"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

Answer 3

"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.")

Answer 4

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

Answer 5

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 6

“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.

Answer 7

Le but de cette thèse est de munir son auteur du titre de Docteur.

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.

Answer 8

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

Answer 9

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 10

"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.)

Answer 11

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 12

"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 13

"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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

"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 18

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 19

" 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 20

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)

Answer 21

"As every mathematician knows, nothing is more fruitful than these obscure analogies, these indistinct reflections of one theory into another, these furtive caresses, these inexplicable disagreements; also nothing gives the researcher greater pleasure... The day dawns when the illusion vanishes; intuition turns to certitude; the twin theories reveal their common source before disappearing; as the Gita teaches us, knowledge and indifference are attained at the same moment. Metaphysics has become mathematics, ready to form the material for a treatise whose icy beauty no longer has the power to move us." - Andre Weil

Answer 22

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.

Answer 23

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 24

The introduction of numbers as coordinates is an act of violence.

Hermann Weyl, Philosophy of Mathematics and Natural Science.

Answer 25

"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 26

"Wir müssen wissen, wir werden wissen." - Hilbert.

Translation: We must know, we will know.

Answer 27

"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 28

"Later mathematicians will regard set theory as a disease from which one has recovered." Henri Poincaré.

Answer 29

"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

Answer 30

"The price of metaphor is eternal vigilance." Norbert Wiener.