## Famous mathematical quotes [closed]

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 2010 at 23:21
Diminishing marginal utility. – Qiaochu Yuan Jan 31 2011 at 2:46
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## 97 Answers

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.

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"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

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Does one have to be French to write lines like that? It's as if Weil were channeling Proust or something. – Todd Trimble Jan 31 2011 at 1:53

“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 2010 at 14:23

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

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"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

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+1 I cannot tell you how frequently I use this quote in explaining to others (or to avoid explaining to others) how wonderful mathematics is. – Jesse Madnick Jul 11 2010 at 3:32

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 2011 at 2:47

(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

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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.

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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 2010 at 16:20
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This one has to do with the quote by Rota that appears in the first post of C. Siegel:

" The essence of Mathematics is proving theorems and so, that is what mathematicians do: they prove theorems. But to tell the truth, what they really want to prove once in their lifetime, is a lemma, like the one by Fatou in Analysis, the lemma of Gauss in Number Theory, or the Burnside-Frobenius lemma in Combinatorics.

Now what makes a mathematical statement a true lemma? First, it should be applicable to a wide variety of instances, even seemingly unrelated problems. Secondly, the statement should, once you have seen it, be completely obvious. The reaction of the reader might well be one of faint envy: Why haven't I noticed this before? And thirdly, on an esthetic level, the lemma including its proof should be beautiful!"

• Aigner & Ziegler in Proofs from the Book.
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A mathematician is a device for turning coffee into theorems. —Alfréd Rényi, but often attributed also to Paul Erdős

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Ken Ribet once told me the story of how he was sent a freebie book "for possible use in your undergraduate classes" that he looked at and decided he didn't want, so took it to the second hand bookstore in his lunch break, sold it, and bought lunch with the proceeds. On the way back to the math department he realised he'd turned theorems into coffee. – Kevin Buzzard Nov 29 2009 at 22:42
A comathematician is a device for turning cotheorems into ffee. A cotheorem is of course what one deduces from a rollary. – Saul Glasman Nov 30 2009 at 19:02
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If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.

P. Turan, "The Work of Alfred Renyi", Matematikai Lapok 21, 1970, pp 199-210

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

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Taken out of context, this would seem to be accurate (the concept of evil with respect to higher categories), but one must remember that Poincar\'e was not against axiomatic set theory per se, but axiomatic theories in general. – Harry Gindi Nov 30 2009 at 6:33
Actually this is not a quote. It is attributed to Poincare in various sources, but it is quite likely that this is a misinterpretation of something that he actually wrote (something to the effect that the diseases of set theory (such as Russell's paradox, for example) will one day be overcome). See J. Gray, Did Poincaré say Set theory is a disease''?, Math. Intell. 13 (1991), 19--22 – Franz Lemmermeyer Feb 15 2010 at 16:21

"[Mathematics consists of] true facts about imaginary objects." Philip Davis and Reuben Hersh.

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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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“Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions.”

– Th. Bröcker & K. Jänich, Introduction to differential topology (p.25)

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I see it just the other way around! – Jose Brox Dec 8 2009 at 15:50
A good definition can occassionally be worth a thousand proofs, I think, but frankly, elementary differential geometry seems to me to fit the “desert of definitions” pretty well. Not that there is anything wrong in that. Some times you have to suffer loads of boring definitions in order to see the depth and beauty of the subject. (And besides, deserts can be beautiful too.) – Harald Hanche-Olsen Dec 8 2009 at 18:19

"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

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You wanted probably this: "You don't have to believe in the S.F., but you should believe in The Book." – Péter Komjáth Nov 8 2010 at 13:41

Farkas Bolyai to his son Janos, speaking about attempts to study Euclid's Vth postulate on parallel lines:

"You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of the parallels alone... I thought I would sacrifice myself for the sake of truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction.... I turned back when I saw that no man can reach the bottom of the night. I turned back unconsoled, pitying myself and all mankind.

I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness - aut Caesar aut nihil."

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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 2010 at 14:21
"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 2010 at 1:45
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"A mathematician who is not also something of a poet will never be a perfect mathematician"- Karl Weierstraß

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"So far as the theories of mathematics are about reality, they are not certain; so far as they are certain, they are not about reality." - Albert Einstein

(Personally, I'd take certainty over being about reality any day)

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In the biographical piece on Grothendieck a couple of years ago in the Notices <<http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf>> the author says "One thing Grothendieck said was that one should never try to prove anything that is not almost obvious". It's not a quote, but it is a nice succinct way of putting his 'nut' analogy given above.

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«Jusqu'à quand les pauvres jeunes gens seront-ils obligés d'écouter ou de répéter toute la journée? Quand leur laissera-t-on du temps pour méditer sur cet amas de connaissances, pour coordonner cette foule de propositions sans suite, de calculs sans liaison? … Mais non, on enseigne minutieusement des théories tronquées et chargées de réflexions inutiles, tandis qu'on omet les propositions les plus brillantes de l'algèbre…». Evariste Galois

(My poor translation: For how long will young people be forced to listening or memorizing during whole days? When will they be allowed time to ponder on this mass of knowledge, to coordinate the multitude of unconnected propositions, of unrelated calculations? … Instead, they are carefully taught truncated theories, loaded with unnecessary reflections, while omitting the most brilliant propositions of algebra…)

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

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Described as apocryphal in at least one source --- see other question. – Greg Kuperberg Nov 29 2009 at 20:44

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."

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To find out how misunderstood is the relationship between Gordan and Hilbert read McLarty's 'Theology and its discontents'. cwru.edu/artsci/phil/… – David Corfield Mar 31 2010 at 7:47

“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

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«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)

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I have put this quote in the front of my thesis:

There are two kinds of mathematical contributions: work that's important to the history of mathematics, and work that's simply a triumph of the human spirit - Paul Cohen, Stanford, 1996

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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?

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This is the best way to think of a sheaf of groups though (IMO)- as a group object in the category of sheaves of sets. – Steven Gubkin Nov 29 2009 at 21:38

A mathematician is a blind man in a dark room looking for a black cat which isn't there.

Attributed to Charles Darwin

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To Anna Varvak: the usual way to continue is that a theologian is a blind man in a dark room looking for a black cat which isn't there, and finding it. – Zsbán Ambrus Apr 16 2010 at 18:39
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