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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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? 


Grothendieck in Pursuing stacks (letters to Quillen). 


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. 


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." 


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. 


"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 


"My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful." Herman Weyl 


"Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether." David Hilbert 


Apart from the most elementary mathematics, like arithmetic or high school algebra, the symbols, formulas and words of mathematics have no meaning at all. The entire structure of pure mathematics is a monstrous swindle, simply a game, a reckless prank. You may well ask: "Are there no renegades to reveal the truth?" Yes, of course. But the facts are so incredible that no one takes them seriously. So the secret is in no danger.  T. Kaczynski 


"We can only see a short distance ahead, but we can see plenty there that needs to be done." Alan Turing 


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



`The human is just a creature for doing slower (and unreliably) (a small part of) what we already know (or soon will know) to do faster. All pretensions of human superiority should be withdrawn if humans want to survive in the future. Shalosh B. Ekhad (i.e., Doron Zeilberger) 


"Why is this a good idea?"



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" 


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. 


"We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things."  Anonymous quote from Bernt Øksendal's "Stochastic Differential Equations". 


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. 


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


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.» 


"The case for my life, then, or for that of anyone else who has been a mathematician in the same sense which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them."  G.H. Hardy 


"the zeros of the zeta function are like the Fourier transform of the primes" As related in Karl Sabbagh's book on the Riemann Hypothesis. (Amazon reference) From the relevant page in the Google book, it might be Samuel Patterson. 


"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 


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


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


"A mathematical truth is neither simple nor complicated in itself, it is."  Émile Lemoine 


"'Imaginary' universes are so much more beautiful than this stupidly constructed 'real' one; and most of the finest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts."  G.H. Hardy 


Dunno if it's appropriate, but: "Now, I've often thought of writing a mathematics textbook someday, because I have a title that I know will sell a million copies. I'm going to call it: Tropic of Calculus"  Tom Lehrer, New Math 


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


Explanation given by Newton to Leibniz in response to the latter's request for details about Newton's newly developed method of fluxions and fluents, in the form of an anagram for «Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire, et vice versa». Who's not shared the feeling that Leibniz must have felt at getting this response when reading obscure explanations in the literature? :P 


C. G. J. Jacobi writing (in French) to Legendre Translation as given in Additive number theory: inverse problems and the geometry of sumsets, vol. 2, by M. B. Nathanson: «It is true that Fourier believed that the principal goal of mathematics is the public welfare and the understanding of nature, but as a philosopher he should have understood that the only goal of science is the honor of the human spirit, and, in this regard, a problem in number theory is as important as a problem in physics.» The translation sadly loses much of the tone... 

