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.
"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."
"Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies."
--Stanislaw M. Ulam quoting Stefan Banach
"The price of metaphor is eternal vigilance." Norbert Wiener.
Do not ask whether a statement is true until you know what it means. -- Errett Bishop
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.
"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
"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
(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
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.
“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)
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
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!"
"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!?)
"Later mathematicians will regard set theory as a disease from which one has recovered." Henri Poincaré.
A mathematician is a device for turning coffee into theorems. —Alfréd Rényi, but often attributed also to Paul Erdős
"[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
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.
"There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms." - Eichler
«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)
«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
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
"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)
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."
“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
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
A mathematician is a blind man in a dark room looking for a black cat which isn't there.
Attributed to Charles Darwin