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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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closed as no longer relevant by Scott Morrison Apr 28 '10 at 17:51

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Why did a question with so much positive feedback get closed? –  Romeo Nov 28 '10 at 23:21
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Diminishing marginal utility. –  Qiaochu Yuan Jan 31 '11 at 2:46
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Closing this solved what problem? –  Matt Brin Jan 18 '12 at 18:35
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@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. –  Qiaochu Yuan Nov 12 '13 at 3:03

97 Answers 97

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?

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This is the best way to think of a sheaf of groups though (IMO)- as a group object in the category of sheaves of sets. –  Steven Gubkin Nov 29 '09 at 21:38

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."

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"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

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"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

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Happily, the science of mathematics has moved on from what Hilbert thought... –  Todd Trimble Jun 9 '13 at 20:16

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

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

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At the risk of overloading an already bloated thread, I found a rather large collection here. Example:

Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

Richard W. Hamming, in N. Rose's Mathematical Maxims and Minims, Raleigh NC: Rome Press Inc., 1988.

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This reminds me of Körner's wonderful discussion, "Why go further", discussing reasons for using Lebesgue's theory while countering Dieudonné's extreme opposition to Riemann integrals. Available at books.google.com/…. –  Jonas Meyer Feb 28 '10 at 23:41
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@Jonas: nice link! I especially enjoyed Korner's remark later on: "It is frequently claimed that Lebesgue integration is as easy to teach as Riemann integration. This is probably true, but I have yet to be convinced that it is as easy to learn." –  Thierry Zell Nov 28 '10 at 6:27

"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

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`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)

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This is definitely misattributed. Ekhad might be good at proving theorems in combinatorics but I don't think he's quite sentient enough to come up with something like this. [he's a computer] –  Kevin Buzzard Nov 30 '09 at 6:52

"Why is this a good idea?"

  • Bill Ralph, on the most important question to ask yourself when doing (or studying) mathematics.
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"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".

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"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).

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"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

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"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.

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"The noblest ambition is that of leaving behind something of permanent value."

-G.H. Hardy, A Mathematicians Apology

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

In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

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. 489--502, 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

In my youth I was almost exclusively active in the field of analysis; the differential equations and expansions of mathematical physics were the mathematical things with which I was on the most intimate footing. I have never succeeded in completely assimilating the abstract algebraic way of reasoning, and constantly feel the necessity of translating each step into a more concrete analytic form.

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.

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Here you have one of my all-time favorites:

" The ultimate goal of Mathematics is to eliminate any need for intelligent thought."

  • R. L. Graham (?)

Can any of you guys tell me where that quote first appeared? Same thing for the quote of Atiyah entered by Petrunin.

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I hope that's not a true quote. Graham, Knuth & Patashnik's Concrete Mathematics quotes him as saying in "Technical Education and its Relation to Science and Literature" among other things "Civilization advances by extending the number of important operations which we can perform without thinking about them." Which I like much more. –  Mio Mar 26 '10 at 3:06
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I remember trying to track this down; this is what I found. Although WZ quote it as being from Concrete Mathematics by Graham, Knuth and Patashnik and attribute it to the authors, in the book it is just a margin comment left by one of the students of the Stanford class. (The book is full of those.) It follows Whitehead's quote ("It is a profoundly erroneous truism [...] Civilization advances by extending the number of important operations which we can perform without thinking about them": www-history.mcs.st-and.ac.uk/Quotations/Whitehead.html) and some student must have "extended" it. –  shreevatsa Oct 22 '12 at 4:41
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Well, one could reasonably say that the sentiment is overblown (as are most aphorisms, almost by definition), but another take on it might be that mathematical understanding is full and ripe when every step, every argument, feels natural and inevitable -- eliminating traces of cleverness which appear as if out of nowhere. Such cleverness being felt as jarring in a way, and indicating that there is something left which hasn't yet been truly and deeply understood. –  Todd Trimble Jun 9 '13 at 14:32

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 Hille-Phillips, "Functional analysis and semigroups"

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I once read, in an autobiographical piece, what the author said to his high-school 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.

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Welcome to MO, Danny. –  Ryan Budney Feb 6 '10 at 5:40

"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

http://pauli.uni-muenster.de/~munsteg/arnold.html

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

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Was it mentioned? Where? –  Jonas Meyer Apr 28 '10 at 1:06

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

The reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the "Riemann integral". It may well be suspected that, had it not been for its prestiguous name, this would have been dropped long ago, for (with due respect to Riemann's genius) it is certainly quite clear for any working mathematician that nowadays such a "theory" has at best the importance of a mildly interesting exercise [...]. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.

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La vie est étrange. En fait, en géometrie, on ne se représente pas de la même manière une droite complexe affine (par exemple pour le teorème de Ceva dans un triangle) et le corps des complexes x+iy. Quand j'y songe, les points imaginaires de la géometrie sont gris, les points réels noirs, et l'intersection de deux droites imaginaires conjuguées est un point réel noir. La belle conique ombilicale est argentée, les droites et cônes isotropes sont plutôt roses

Laurent Schwartz, Un mathematicien 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.»

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"'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

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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

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

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As many people say it is a example of false modesty, it is a fact that Einstein was poor mathematician. And physics of his times do not require very abstract knowledge. But it was very deep thinker, and very consequent one. –  kakaz Feb 28 '10 at 19:30

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

Paul Erdos.

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Terrible. If you enjoy mathematics, why wouldn't you want to share that joy with others? If you are going to have children some day, why not make sure their teachers are going to be educated about what mathematics really is? –  Steven Gubkin Jan 19 '10 at 18:45
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I think the quote is about how things are, not how things are supposed to be. –  darij grinberg Mar 21 '10 at 18:22
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Makes me sick because the quote sums up the attitude of most high end universities in America. –  Andrew L Apr 28 '10 at 0:56
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There is a variant of the quote: "Those who can't do teach, those who can't teach teach gym."-Red Dwarf –  Sean Tilson Jan 31 '11 at 6:32

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

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

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Il est vrai que M. Fourier avait l'opinion que le but principal des mathématiques était l'utilité publique et l'explication des phénomènes naturels; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c'est l'honneur de l'esprit humain, et que sous ce titre, une question de nombres vaut autant qu'une question du système du monde.

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...

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