# 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.

• There are 10 kinds of people in the world, those that understand binary and the other 9. Nov 30 '09 at 5:29
• Why did a question with so much positive feedback get closed? Nov 28 '10 at 23:21
• Diminishing marginal utility. Jan 31 '11 at 2:46
• Closing this solved what problem? Jan 18 '12 at 18:35
• @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. Nov 12 '13 at 3:03

"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

• So, category theorists? Nov 29 '09 at 22:55
• @Qiaochu: Higher category theorists! Nov 29 '09 at 22:59
• Who can see analogies between analogies between analogies? Nov 28 '10 at 4:28
• @timur The analogy operator is idempotent ;P Oct 11 '17 at 11:15
• @timur 2-category theorists Mar 27 '18 at 17:20

“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

• Thank you! I've been trying to track down the wording of this quote for awhile now. Jan 16 '10 at 14:23
• Last year our distinguished Cantrell Lecture series was given by Dick Gross. I had been booked as a speaker in the graduate student seminar, which took place immediately before Gross's first lecture. I decided to give an introduction to elliptic curves, including BSD and Gross-Zagier. I made sure to include this quote of Tate. So did Gross in his first lecture. Mar 26 '16 at 20:17

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

• I didn't know that you were a constructivist :p
– Amr
Mar 26 '16 at 20:02
• @Amr: Well, I'm certainly not. On the one hand, a quotation is not an endorsement. On the other hand, although you perceive -- I believe correctly -- constructivist principles in the provenance of the quote, it is a good enough quote that it can mean other things as well. As a piece of general mathematical advice, I find it quite sound. Mar 26 '16 at 20:12

"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

• Does one have to be French to write lines like that? It's as if Weil were channeling Proust or something. Jan 31 '11 at 1:53

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.

• "The purpose of computing is numbers — specifically, correct numbers." Greengard 2000. Aug 6 '11 at 15:26

"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

• +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. Jul 11 '10 at 3:32

(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

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

“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)

• I see it just the other way around! Dec 8 '09 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.) Dec 8 '09 at 18:19

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.

• Shouldn't that be "er hat etwas bewiessen"? (I'd edit it had I the power.) Nov 29 '09 at 21:15
• @ Cory Knapp: no, it should be "Beweis", a noun as attested by the "er" ending of the preceding adjective "strenger". The present perfect you suggest is impossible here, and moreover its correct spelling would be "er hat bewiesen" (only one "s"). Freundliche Grüsse. Nov 29 '09 at 22:26
• Very interesting quote, Mariano. For Jacobi to write that when Gauss claims to have a proof, it is just "sehr wahrscheinlich" sounds like the height of "chutzpah" ! Nov 29 '09 at 22:37
• The conjugation "er habe" is correct. It is a subjunctive mood indicating that it only a personal (Gauß's) opinion is expressed. It might also be possible that in former times (when spelling was different) they might have written it "er have". Nov 30 '09 at 6:13
• Oh I never get tired of cribbing about German along with Mark Twain .. baetzler.de/humor/the_awful_german_language.var .. Feb 6 '10 at 16:20

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.

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

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

• 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. Nov 30 '09 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 Feb 15 '10 at 16:21

"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!?)

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

• 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. Nov 29 '09 at 22:42
• A comathematician is a device for turning cotheorems into ffee. A cotheorem is of course what one deduces from a rollary. Nov 30 '09 at 19:02
• I first heard this on Numb3rs. Fantastic quote! Dec 2 '09 at 3:49
• I think "A mathematician is a device for turning 'coffee', wink wink, into theorems" is more likely to be authentically Erdos, no? Dec 22 '13 at 19:26

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

"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.

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

«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)

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.

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

• It is hard to believe that there aren't third (and possibly further) categories of mathematical contributions that are neither important nor triumphal. May 10 '11 at 17:12
• I agree in general, though I suppose it could be argued that if one takes all mathematical contributions that aren't "important to the history of mathematics", then collectively they are a triumph of the human spirit (but I don't claim that this is what Cohen actually meant). When I first answered this question, I had just finished proving most of the main theorems for my PhD thesis, so the romantic notion of "triumph of the human spirit" kind of captured my mood at the time. Also, Cohen said this when interviewed for A Beautiful Mind, so he probably simplified things a bit for the public. May 11 '11 at 5:58

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

• Described as apocryphal in at least one source --- see other question. Nov 29 '09 at 20:44

«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…)

"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

• You wanted probably this: "You don't have to believe in the S.F., but you should believe in The Book." Nov 8 '10 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."

"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)

• If this quote were from a mathematician, it would probably not contain the same information twice... Aug 14 '18 at 20:40

"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).

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."

"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

“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