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.
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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“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 BirchSwinnertonDyer Conjecture 


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


— Richard Hamming (1962)
– Richard W. Hamming, Introduction to applied numerical analysis, McGrawHill 1971, p.31. 


"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 


Greg Kuperberg, in a comment to a MO question 


(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 





"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 


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 fiftyfifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters. 


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


P. Turan, "The Work of Alfred Renyi", Matematikai Lapok 21, 1970, pp 199210 


"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!?) 


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 BurnsideFrobenius 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!"



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


“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) 


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


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


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." 


In the biographical piece on Grothendieck a couple of years ago in the Notices <<http://www.ams.org/notices/200410/feagrothendieckpart2.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. 


"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) 


"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 


«Jusqu'à quand les pauvres jeunes gens serontils obligés d'écouter ou de répéter toute la journée? Quand leur laisseraton 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…) 


"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. 


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


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


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'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 


Free translation: keep going, faith will come later. JeanleRond D'Alembert, to his students (quoted by Florian Cajori in A history of mathematics) 


Attributed to Charles Darwin 


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


