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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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Why did a question with so much positive feedback get closed? –  Romeo Nov 28 '10 at 23:21
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Diminishing marginal utility. –  Qiaochu Yuan Jan 31 '11 at 2:46
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Closing this solved what problem? –  Matt Brin Jan 18 '12 at 18:35
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@Matt: standards for what kind of questions people want on MO have changed over time, and keeping this question opens gives a false impression to new users of what kind of questions we want on MO. It's less confusing to close it. This happens on other SE sites as well; many of the most popular questions on StackOverflow, for example, are also closed. There's also the more practical issue that if it's open people keep adding answers and, again, the marginal utility of each additional answer is decreasing. –  Qiaochu Yuan Nov 12 '13 at 3:03

97 Answers 97

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 théorème 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 mathématicien 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.»

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Reminds me of the synaesthetic experiences of Feynman: "When I see equations, I see the letters in colors – I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students." –  Todd Trimble Jun 9 '13 at 14:42

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 zu 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.

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Oh I never get tired of cribbing about German along with Mark Twain .. baetzler.de/humor/the_awful_german_language.var .. –  Anweshi Feb 6 '10 at 16:20

Here you have one of my all-time favorites:

" The ultimate goal of Mathematics is to eliminate any need for intelligent thought."

  • R. L. Graham (?)

Can any of you guys tell me where that quote first appeared? Same thing for the quote of Atiyah entered by Petrunin.

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I hope that's not a true quote. Graham, Knuth & Patashnik's Concrete Mathematics quotes him as saying in "Technical Education and its Relation to Science and Literature" among other things "Civilization advances by extending the number of important operations which we can perform without thinking about them." Which I like much more. –  Mio Mar 26 '10 at 3:06
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I remember trying to track this down; this is what I found. Although WZ quote it as being from Concrete Mathematics by Graham, Knuth and Patashnik and attribute it to the authors, in the book it is just a margin comment left by one of the students of the Stanford class. (The book is full of those.) It follows Whitehead's quote ("It is a profoundly erroneous truism [...] Civilization advances by extending the number of important operations which we can perform without thinking about them": www-history.mcs.st-and.ac.uk/Quotations/Whitehead.html) and some student must have "extended" it. –  shreevatsa Oct 22 '12 at 4:41
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Well, one could reasonably say that the sentiment is overblown (as are most aphorisms, almost by definition), but another take on it might be that mathematical understanding is full and ripe when every step, every argument, feels natural and inevitable -- eliminating traces of cleverness which appear as if out of nowhere. Such cleverness being felt as jarring in a way, and indicating that there is something left which hasn't yet been truly and deeply understood. –  Todd Trimble Jun 9 '13 at 14:32
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I'd like to paraphrase Orcar Wilde and say "Every Mathematician Kills the Problem He Loves" –  Pietro Majer Nov 10 '13 at 21:42

“This remarkable conjecture relates the behaviour of a function L, at a point where it is not at present known to be defined, to the order of a group \Sha, which is not known to be finite.”

-John Tate on the Birch-Swinnerton-Dyer Conjecture

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Thank you! I've been trying to track down the wording of this quote for awhile now. –  Qiaochu Yuan Jan 16 '10 at 14:23

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.

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"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it's the exact opposite!" -- Paul Dirac (some people attribute it to Franz Kafka!?)

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Another quote from Dieudonné's "Foundations of Modern Analysis, Vol. 1":

The reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the "Riemann integral". It may well be suspected that, had it not been for its prestiguous name, this would have been dropped long ago, for (with due respect to Riemann's genius) it is certainly quite clear for any working mathematician that nowadays such a "theory" has at best the importance of a mildly interesting exercise [...]. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.

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

There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

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I wish I could up-vote this a few more times (I know, I'm really slow reading this one)! –  Jeremy West Feb 1 '11 at 2:47

‘Life is complex: it has both real and imaginary components.” (I don't know who said this...)

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Was it mentioned? Where? –  Jonas Meyer Apr 28 '10 at 1:06

Like many people, I am fascinated by the quote from Weyl (already listed here), that

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

But I can see why people are puzzled by the quote, so I'd like to add some more information (too much to put in a comment) as another answer.

First, what is the context? The quote occurs in Weyl's paper Invariants in Duke Math. J. 5 (1939), pp. 489--502, the first page of which can be seen here. This page includes most of what Weyl has to say on algebra v. geometry, though the quote itself does not occur until p.500. Then on p.501 Weyl explains his discomfort with algebra as follows

In my youth I was almost exclusively active in the field of analysis; the differential equations and expansions of mathematical physics were the mathematical things with which I was on the most intimate footing. I have never succeeded in completely assimilating the abstract algebraic way of reasoning, and constantly feel the necessity of translating each step into a more concrete analytic form.

Second, why the image of angel and devil? According to V.I Arnold, writing here, Weyl had a particular image in mind, namely, the Uccello painting "Miracle of the Profaned Host, Episode 6", which can be viewed here.

Arnold describes this painting as "representing an event that happened in Paris in 1290." "Legend" is probably a better word than "event," but in any case it is a very strange origin for a famous mathematical quote.

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"Life is good for only two things, discovering mathematics and teaching mathematics." -- Simeon Poisson

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Mathema est ars et scientia, discenda

Aquinas?

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Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the computation because sometimes you find out that more is there than you realized."

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Mathematicians are born, not made. -- Henri Poincare

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I disagree,Henri.Most strongly. –  Andrew L Apr 28 '10 at 0:34

At the risk of overloading an already bloated thread, I found a rather large collection here. Example:

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

Richard W. Hamming, in N. Rose's Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

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This reminds me of Körner's wonderful discussion, "Why go further", discussing reasons for using Lebesgue's theory while countering Dieudonné's extreme opposition to Riemann integrals. Available at books.google.com/…. –  Jonas Meyer Feb 28 '10 at 23:41
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@Jonas: nice link! I especially enjoyed Korner's remark later on: "It is frequently claimed that Lebesgue integration is as easy to teach as Riemann integration. This is probably true, but I have yet to be convinced that it is as easy to learn." –  Thierry Zell Nov 28 '10 at 6:27

"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." V.I.Arnold

http://pauli.uni-muenster.de/~munsteg/arnold.html

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"The noblest ambition is that of leaving behind something of permanent value."

-G.H. Hardy, A Mathematicians Apology

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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

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"Maybe at times I like to give the impression, to myself and hence to others, that I am the easy learner of things of life, wholly relaxed, "cool" and all that - just keen for learning, for eating the meal and welcome smilingly whatever comes with it's message, frustration and sorrow and destructiveness and the softer dishes alike. This of course is just humbug, an images d'Epinal which at whiles I'll kid myself into believing I am like. Truth is that I am a hard learner, maybe as hard and reluctant as anyone."

Grothendieck in Pursuing stacks (letters to Quillen).

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

-Emil Artin, Geometric Algebra

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I once read, in an autobiographical piece, what the author said to his high-school teacher upon graduation; my recollection is:

"Poincaré has written that geometry is the art of making a correct argument from incorrectly drawn figures. For you, sir, it is the opposite."

I would love to know the correct quote, and an accurate source. I've seen a version attributed to Poincaré, but couldn't verify that.

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Welcome to MO, Danny. –  Ryan Budney Feb 6 '10 at 5:40

Who can does; who cannot do, teaches; who cannot teach, teaches teachers.

Paul Erdos.

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Terrible. If you enjoy mathematics, why wouldn't you want to share that joy with others? If you are going to have children some day, why not make sure their teachers are going to be educated about what mathematics really is? –  Steven Gubkin Jan 19 '10 at 18:45
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I think the quote is about how things are, not how things are supposed to be. –  darij grinberg Mar 21 '10 at 18:22
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There is a variant of the quote: "Those who can't do teach, those who can't teach teach gym."-Red Dwarf –  Sean Tilson Jan 31 '11 at 6:32

And each man hears as the twilight nears, to the beat of his dying heart, the devil tap on the darkening pane, "You did it, but is it art?"

Epigraph to Hille-Phillips, "Functional analysis and semigroups"

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"In mathematics you don't understand things. You just get used to them."

John von Neumann

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This was already posted. –  Qiaochu Yuan Jan 16 '10 at 14:21
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"Ambiguous nonsense. Look, I can do it too: 'The sun darkens, but lo! Here comes the dawn!'" - Jowan, Dragon Age: Origins. –  darij grinberg Mar 22 '10 at 1:45

"There are, therefore, no longer some problems solved and others unsolved, there are only problems more or less solved, according as this is accomplished by a series of more or less rapid convergence or regulated by a more or less harmonious law. Nevertheless an imperfect solution may happen to lead us towards a better one."

Henri Poincare

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Boy, was he wrong or what? –  Harry Gindi Jan 15 '10 at 22:46
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The quote could benefit by being put into a better context. Here it is: books.google.ca/… but Poincare was making a pretty important point that IMO you've missed. –  Ryan Budney Feb 6 '10 at 5:50

Do not ask whether a statement is true until you know what it means. -- Errett Bishop

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"Mathematics consists of proving the most obvious thing in the least obvious way." - George Polya

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No. –  darij grinberg Mar 21 '10 at 18:24
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Darij, this is a sentiment with which one can agree or disagree (I agree about the content of many undergraduate mathematics courses, but disagree about much mathematics beyond that), but surely it's courteous to offer a bit more than “No”? –  L Spice Mar 21 '10 at 23:54

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater. —- Albert Einstein

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As many people say it is a example of false modesty, it is a fact that Einstein was poor mathematician. And physics of his times do not require very abstract knowledge. But it was very deep thinker, and very consequent one. –  kakaz Feb 28 '10 at 19:30

For general continuous curves, it's not that a simple proof [of the Jordan curve theorem] is not possible, it's that it's not desirable. The true content of the result is homology theory, which proves the separation result in n dimensions. There are special proofs in 2D that are simpler, but every such proof that I have seen feels like a one-night stand.

Greg Kuperberg, in a comment to a MO question

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"Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel

A declaration of war by a Platonist.

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