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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2$\begingroup$ There are 10 kinds of people in the world, those that understand binary and the other 9. $\endgroup$– Jonathan KarivCommented Nov 30, 2009 at 5:29
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46$\begingroup$ Why did a question with so much positive feedback get closed? $\endgroup$– RomeoCommented Nov 28, 2010 at 23:21
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22$\begingroup$ Diminishing marginal utility. $\endgroup$– Qiaochu YuanCommented Jan 31, 2011 at 2:46
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14$\begingroup$ Closing this solved what problem? $\endgroup$– Matt BrinCommented Jan 18, 2012 at 18:35
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11$\begingroup$ @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. $\endgroup$– Qiaochu YuanCommented Nov 12, 2013 at 3:03
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94 Answers
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A mathematician is a blind man in a dark room looking for a black cat which isn't there.
Attributed to Charles Darwin
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$\begingroup$ I believe the reference is to a theologian. $\endgroup$ Commented Dec 10, 2009 at 12:33
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26$\begingroup$ To Anna Varvak: the usual way to continue is that a theologian is a blind man in a dark room looking for a black cat which isn't there, and finding it. $\endgroup$ Commented Apr 16, 2010 at 18:39
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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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$\begingroup$ There is also a Wikipedia article on the painting: The Miracle of the Desecrated Host. $\endgroup$ Commented Apr 20, 2022 at 7:30
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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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1$\begingroup$ Worth noting that this is from a Rudyard Kipling poem, The Conundrum of the Workshops. https://poets.org/poem/conundrum-workshops $\endgroup$– e4494sCommented Jul 23 at 15:39
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"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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2$\begingroup$ tihs is somehow reminiscent of Dostoevsky's statement "if someone were to prove to me that Christ was outside the truth, and it was really the case that the truth lay outside Christ, then I should choose to stay with Christ rather than with the truth" $\endgroup$ Commented May 29, 2010 at 20:13
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$\begingroup$ I've often thought about this quote of Weyl's, and I remain uncertain as ever what to make of it. Inevitably, for me, it becomes not just a question of the Beautiful and the True, but of the Good: what are the ethical implications of this cryptic utterance? $\endgroup$ Commented Jun 9, 2013 at 14:21
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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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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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2$\begingroup$ 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/…. $\endgroup$ Commented Feb 28, 2010 at 23:41
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13$\begingroup$ @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." $\endgroup$ Commented Nov 28, 2010 at 6:27
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1$\begingroup$ I've seen that quotation of Hamming, and I think: well, so what? Did somebody say there had to be a physical significance? Wouldn't Hamming agree that the difference has mathematical significance? And that counts for something, right? $\endgroup$ Commented Dec 5, 2014 at 3:42
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1$\begingroup$ In my undergrad, the joke was that physics students took measure theory expecting to learn to integrate more functions. They come out being able to integrate fewer. $\endgroup$ Commented Aug 14, 2018 at 20:45
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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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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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6$\begingroup$ 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. $\endgroup$ Commented Nov 29, 2009 at 21:38
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$\begingroup$ @StevenGubkin, I would be glad if further explanation is provided. I'm quite familiar with the categorical terminology so feel free to use them. $\endgroup$– FNHCommented Apr 21, 2017 at 22:30
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$\begingroup$ @StevenGubkin, I would be glad if further explanation is provided. I'm quite familiar with the categorical terminology so feel free to use them. $\endgroup$– FNHCommented Apr 21, 2017 at 22:30
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1$\begingroup$ Read en.wikipedia.org/wiki/Group_object, and apply the definition to a group object in the category of sheaves of sets. $\endgroup$ Commented Apr 29, 2017 at 1:51
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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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"In mathematics you don't understand things. You just get used to them."
John von Neumann
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$\begingroup$ Oh, I'm so sorry ... I didn't acknowledge ... $\endgroup$– AxiomCommented Jan 16, 2010 at 14:39
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8$\begingroup$ "Ambiguous nonsense. Look, I can do it too: 'The sun darkens, but lo! Here comes the dawn!'" - Jowan, Dragon Age: Origins. $\endgroup$ Commented Mar 22, 2010 at 1:45
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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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$\begingroup$ I somehow think it was Ulam, in a riposte to one of his teachers who had given him a bad grade. But I'm having trouble tracking this down myself. $\endgroup$ Commented Jun 9, 2013 at 22:27
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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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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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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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4$\begingroup$ 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. $\endgroup$– MioCommented Mar 26, 2010 at 3:06
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2$\begingroup$ 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. $\endgroup$ Commented Oct 22, 2012 at 4:41
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7$\begingroup$ 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. $\endgroup$ Commented Jun 9, 2013 at 14:32
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10$\begingroup$ I'd like to paraphrase Orcar Wilde and say "Every Mathematician Kills the Problem He Loves" $\endgroup$ Commented Nov 10, 2013 at 21:42
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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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4$\begingroup$ Happily, the science of mathematics has moved on from what Hilbert thought... $\endgroup$ Commented Jun 9, 2013 at 20:16
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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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8$\begingroup$ 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] $\endgroup$ Commented Nov 30, 2009 at 6:52
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$\begingroup$ I added Doron Zeilberger in parentheses. $\endgroup$ Commented Nov 30, 2009 at 20:31
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$\begingroup$ I absolutely disagree with that. $\endgroup$ Commented Jan 20, 2014 at 2:08
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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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$\begingroup$ Just in case the reader doesn't know: T. Kaczynski is better (and unfortunately) known as Unabomber. $\endgroup$ Commented Nov 18, 2019 at 15:42
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"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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"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
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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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1$\begingroup$ 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. $\endgroup$– kakazCommented Feb 28, 2010 at 19:30
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2$\begingroup$ There are lots of ways in which one could read this, aside from false modesty. Some being aided by the fact that Einstein wasn't a poor mathematician -- rather, he wasn't a mathematician at all! $\endgroup$ Commented Jun 9, 2013 at 14:46
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$\begingroup$ Riemann did all the heavy lifting $\endgroup$ Commented May 6, 2017 at 4:51
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$\begingroup$ He is just saying that his math is not so good but still managed to turn everything in physics on its head. $\endgroup$– timurCommented Jan 1, 2018 at 5:03
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"Life is good for only two things, discovering mathematics and teaching mathematics." -- Simeon Poisson
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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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$\begingroup$ Almost certainly this was already mentioned. It might be a little unreasonable to ask people to read every answer to this question before they submit their own, but I think that's just an indication that this question already has enough answers... $\endgroup$ Commented Mar 26, 2010 at 0:38
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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 théorème 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 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.»
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3$\begingroup$ Reminds me of the synaesthetic experiences of Feynman: "When I see equations, I see the letters in colors – I don't know why. As I'm talking, I see vague pictures of Bessel functions from Jahnke and Emde's book, with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around. And I wonder what the hell it must look like to the students." $\endgroup$ Commented Jun 9, 2013 at 14:42
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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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"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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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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$\begingroup$ What an excellent observation (lossy translation notwithstanding)! $\endgroup$ Commented Feb 1, 2011 at 5:18
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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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$\begingroup$ Well OK there's no reason, with the web at hand, to be that lazy. Was it Samuel Patterson who actually said this? books.google.com/… $\endgroup$– patflaCommented Dec 8, 2009 at 4:22
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$\begingroup$ I can't find anything else, but I edited that in. Also, nice. $\endgroup$ Commented Dec 8, 2009 at 8:01
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$\begingroup$ Thanx Elizabeth. What would be even nicer is if it were, in some sense, true. $\endgroup$– patflaCommented Dec 8, 2009 at 17:17
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Who can does; who cannot do, teaches; who cannot teach, teaches teachers.
Paul Erdos.
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32$\begingroup$ 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? $\endgroup$ Commented Jan 19, 2010 at 18:45
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18$\begingroup$ I think the quote is about how things are, not how things are supposed to be. $\endgroup$ Commented Mar 21, 2010 at 18:22
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2$\begingroup$ There is a variant of the quote: "Those who can't do teach, those who can't teach teach gym."-Red Dwarf $\endgroup$ Commented Jan 31, 2011 at 6:32
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"A mathematical truth is neither simple nor complicated in itself, it is." - Émile Lemoine
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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