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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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.» 


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. 


Here you have one of my alltime favorites: " The ultimate goal of Mathematics is to eliminate any need for intelligent thought."
Can any of you guys tell me where that quote first appeared? Same thing for the quote of Atiyah entered by Petrunin. 


“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 


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. 


"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!?) 


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



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



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


Like many people, I am fascinated by the quote from Weyl (already listed here), that
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. 489502, 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
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. 


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


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." 


Mathematicians are born, not made.  Henri Poincare 


At the risk of overloading an already bloated thread, I found a rather large collection here. Example:
Richard W. Hamming, in N. Rose's Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988. 


"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 


"The noblest ambition is that of leaving behind something of permanent value." G.H. Hardy, A Mathematicians Apology 


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 


Grothendieck in Pursuing stacks (letters to Quillen). 


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


I once read, in an autobiographical piece, what the author said to his highschool 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. 


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


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 HillePhillips, "Functional analysis and semigroups" 


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


"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 





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


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


Greg Kuperberg, in a comment to a MO question 


"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. 


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." 

