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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.

Michael Lugo
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Sam Derbyshire
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  • Closed. Let's put this one to rest. – Scott Morrison Apr 28 '10 at 17:51
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    In order to motivate examples in the first class in congruence theory, my teacher remarked that the beginning chapters of the Holy Bible mathematically said entail the following:

    "Let the days of the week be congruent modulo seven."

     – Unknown Jun 12 '10 at 17:07
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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 02:46
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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 03:03
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    There are 10 kinds of people in the world, those that understand binary and the other 9. – Jonathan Kariv Nov 30 '09 at 05:29
  • In Wikiquote there is one that I like very much due to Paul Dirac. – user142929 Sep 18 '19 at 22:27

94 Answers94

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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.

Thierry Zell
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Csar Lozano Huerta
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  • I recall reading about this quote on a book, but I couldn't find any reference about its origin.. can you provide one? – Dinisaur Jun 12 '23 at 14:48
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"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." --- John von Neumann. (From a 1947 ACM keynote, recalled by Alt in this 1972 CACM article.)

mic
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    I love this quote very much. I myself have compared our life with mathematics I am doing and reached the same conclusion. – Sunni Mar 26 '10 at 03:31
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“The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.” Lucien Szpiro during Algebra 1 lecture.

user2204
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    This reminds me of Surely You're Joking, Mr. Feynman!, in which the physics grad students at Princeton propose the theorem that mathematicians only prove trivial theorems. – Pete L. Clark Jan 08 '10 at 06:27
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    I once heard Henry McKean say of mathematical models in physics that "First, they’re pretty disgraceful. Second, they work extremely well...One of the faults of mathematicians is: when physicists give them an equation, they take it absolutely seriously." (I wrote this down on the spot.) – Steve Huntsman Apr 23 '10 at 15:48
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    Notable counterexample: The existence of the monster group. I think Conway has said that no one quite understands "why" it exists. – Akiva Weinberger Mar 07 '17 at 20:55
  • @user2204 - I used to tell my students that one of the hardest things about teaching mathematics was that once you understand something it is obvious, but, for the student, before you understand it is as clear as mud. – Chris Leary Mar 10 '21 at 15:48
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"Mathematics is the art of giving the same name to different things." Henri Poincaré.

(This was in response to "Poetry is the art of giving different names to the same thing.")

José Figueroa-O'Farrill
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SixWingedSeraph
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We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?" - Gian-Carlo Rota

Charles Siegel
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Oh, he seems like an okay person, except for being a little strange in some ways. All day he sits at his desk and scribbles, scribbles, scribbles. Then, at the end of the day, he takes the sheets of paper he's scribbled on, scrunches them all up, and throws them in the trash can. --J. von Neumann's housekeeper, describing her employer.

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

There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices.

Konrad Waldorf
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"The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps..."

-Alexander Grothendieck, writing to Ronald Brown

Akhil Mathew
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    Grothendieck was French, and I suppose the correct translation would be "digit" not "cipher". – Tom Ellis Feb 01 '11 at 09:39
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    As far as I know, Grothendieck is still French. – José Figueroa-O'Farrill Nov 07 '11 at 15:23
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    As far as I know, Grothendieck does not hold any nationality. But I may well be wrong. – Jonathan Chiche Oct 19 '12 at 08:52
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    I think that Grothendieck was (alas!) French citizen since the 1980s. See Cartier: http://xahlee.info/math/i/Alexander_Grothendieck_cartier.pdf Footnote 12. – Lennart Meier Apr 20 '15 at 17:41
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    The question is, what "childish steps" do today's mathematicians need to take? – Akiva Weinberger Sep 02 '15 at 02:20
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    @TomEllis Cipher is also an old-fashioned word for zero in English, and is the most similar to the Arabic sifr. Since Grothendieck was talking about the introduction of zero, and how it would have seemed at the time, he may have used the old name for it on purpose for rhetorical effect (this is how I interpreted it). Incidentally, his Muttersprache was German. – Robert Furber Jul 02 '18 at 14:43
  • Thank you Akhil. This kind of dubious quote should be accompanied by a reference. Alexander Grothendieck probably said something like that, but i guess that the rigorous minds dwelling in MO would appreciate exact quotes, with context -not translations, especially inappropriate ones. It would also be interesting to be able to discuss the assertions in this quote, which look to me highly questionable, but this is too much to ask. – plm Jun 20 '23 at 08:52
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"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona

Malik Younsi
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    Obviously there is something wrong with ZF for proving their equivalence. I blame infinity. – Andrew Critch Nov 30 '09 at 01:36
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    Zorn's Lemma seems the most intuitive out of the three, but well-ordering isn't so counter-intuitive, since all it comes down to is being able to well-order a set with cardinality strictly larger than the natural numbers. Thinking about well-ordering the reals gives a false impression of the difficulty, since the well-ordering only has to do with the underlying set. – Harry Gindi Nov 30 '09 at 06:25
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    @Harry: I can't really agree with either statement. Naively, if we could truly "see" a well-ordering of a set of continuum cardinality, then intuitively we should be able to compare it to $\aleph_1$ and "see" whether it is larger. – Todd Trimble Jul 03 '11 at 12:07
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It's hard to beat John Stembridge's page of quotes. My single favorite one on this page: "If I have not seen as far as others, it is because there were giants standing on my shoulders." - Hal Abelson.

Greg Kuperberg
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    Along those lines, R. W. Hamming said "Mathematicians stand on each other’s shoulders while computer scientists stand on each other’s toes." – John D. Cook Nov 30 '09 at 03:02
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    My version of this quote, after working my way backwards through a series of papers, each relying on the previous ones: If I can't see a darn thing it's because I stand on the shoulders of giants... – Ehud Friedgut Dec 02 '09 at 05:49
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    What was Gell-Mann's version? Something like, "If I have seen farther than others, it's because I'm surrounded by pygmies?" – Todd Trimble Jan 31 '11 at 01:15
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    The way I heard it: "If I have not seen as far as others, it is because I have stood in the footprints of giants." – bof Dec 05 '14 at 01:35
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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

Jose Brox
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You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine.

D. Hilbert, talking about an ex-student. I'd love to remember where I got this from!

Mariano Suárez-Álvarez
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  • This sounds like it's from the Apology. – Qiaochu Yuan Nov 29 '09 at 21:08
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    You might have gotten it from the Monthly in 1993, or a mailing list or blog that derives from that source. Google Books traces it back to a book about Ernst Cassirer published in 1949. Since it is quoted there as Cassirer's verbal story, it's less clear what Hilbert himself said. http://www.archive.org/stream/philosophyoferns033109mbp/philosophyoferns033109mbp_djvu.txt – Greg Kuperberg Nov 29 '09 at 22:14
  • I seem to recall it being in one of the 10^150 popular books on the Riemann Hypothesis or Fermat's Last Theorem I read when I was younger, but I don't know that I can narrow it down any further than that. – Harrison Brown Nov 29 '09 at 22:58
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    This story is mentioned in Constance Reid's lovely biography "Hilbert". – Lea M May 16 '11 at 04:27
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Le but de cette thèse est de munir son auteur du titre de Docteur.

Beginning of A. Douady's thesis. Quoted by Michèle AUdin in her Conseils aux auteurs de textes mathématiques.

In a less barbarous language: The purpose of this thesis is to obtain the degree of Doctor for its author.

Mariano Suárez-Álvarez
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"The art of doing mathematics is finding that special case that contains all the germs of generality." -- David Hilbert

John D. Cook
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Algebra is the offer made by the devil to the mathematician...All you need to do, is give me your soul: give up geometry --Michael Atiyah

Anton Petrunin
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    A quote that obviously inspired the soul theorem. http://en.wikipedia.org/wiki/Soul_theorem – Greg Kuperberg Dec 18 '09 at 07:24
  • This comment was in an article on 20th century mathematics which did not contain the words "category", but did talk about the unification of mathematics, which has been one of the major contributions of category theory. There is not really a duality between algebra and geometry, as Grothendieck has shown, but there is a search for underlying processes. – Ronnie Brown Aug 09 '12 at 15:37
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    @Ronnie, maybe there is no such duality in mathematics, but there is duality in our perception. – Anton Petrunin Aug 09 '12 at 23:35
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    @GregKuperberg: do you have any clue that why Cheeger and Gromoll called it soul? – C.F.G Sep 09 '20 at 12:36
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    I think that "soul" amounts an alternative word for "core" in this context. "Soul" or "core" just means a canonical, usually smaller middle part, like an apple core. Elsewhere in geometry and topology, people talk about compact cores of spaces, in particular compact cores of 3-manifolds, which is a similar albeit not identical construction. Another construction which is again similar but not identical is the maximal compact subgroup of a Lie group. "Soul" happens to be a bit pompous compared to other names for this type of thing. – Greg Kuperberg Sep 10 '20 at 17:04
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"God exists since mathematics is consistent, and the Devil exists since we cannot prove it."- André Weil

Spinorbundle
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    So, mathematics is consistent? I think we can more or less rely on dividing by three, for the rest we shouldn't be that sure... – Jose Brox Dec 08 '09 at 15:42
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    @Jose: I don't know if you were referring to this paper indirectly, but you should check out http://www.math.dartmouth.edu/~doyle/docs/three/three.pdf, where it is proven that division by three is possible. – Steven Gubkin Apr 23 '10 at 12:42
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Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.

Gian-Carlo Rota, in an interview with David Sharp.

Qiaochu Yuan
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  • Qiaochu, do you have a source for this interview? I'd be interested in reading it. – Michael Lugo Nov 30 '09 at 00:34
  • "Mathematics, Philosophy and Artificial Intelligence," Los Alamos Science, 1985. I can't seem to find an online copy. – Qiaochu Yuan Nov 30 '09 at 00:58
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    Here is an online copy: http://www.fas.org/sgp/othergov/doe/lanl/pubs/00326965.pdf – Jose Capco Nov 30 '09 at 11:11
  • like physics versus mathematics – Yoo Jan 07 '10 at 18:55
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    like blue collar versus cubicle – Yoo Jan 07 '10 at 18:56
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    Combinatorics is discrete functional analysis in my world view, while functional analysis is applied combinatorics. – Bill Johnson Mar 21 '10 at 19:28
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    Bill, would you mind elaborating? As someone not particularly familiar with either field, I can imagine that by combinatorics being "discrete functional analysis" you mean e.g. generating function methods, or perhaps the general ambition of associating some sort of linear operator to combinatorial objects (e.g. adjacency matrix). But what do you mean by functional analysis being applied combinatorics. – Erik Davis Apr 15 '10 at 01:01
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    It takes balls to do combinatorics. – Todd Trimble Jan 31 '11 at 01:45
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"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain."

Pierre de Fermat

AFK
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    This one should definitely have more votes! – Jose Brox Dec 08 '09 at 15:48
  • The question is that, if someone provides evidence that all tools available at Fermat's time were not able to prove his conjecture, then Fermat's 'marvellous proof' was wrong. – Sunni Mar 26 '10 at 04:01
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"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."

-Emil Artin, Geometric Algebra

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Grothendieck comparing two approaches, with the metaphor of opening a nut: the hammer and chisel approach, striking repeatedly until the nut opens, or just letting the nut open naturally by immersing it in some soft liquid and let time pass:

"I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance."

Grothendieck, of course, always pioneered this approach, and considered for example that Jean-Pierre Serre was a master of the "hammer and chisel" approach, but always solving problems in a very elegant way.

Sam Derbyshire
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    A friend who I won't name at the moment once told me this, paraphrased: An excellent problem-solver might not always be a great mathematician, while a bad problem-solver can still be an okay mathematician. On the other hand, a good Grothendieck is a great mathematician, while a bad Grothendieck is really terrible! – Greg Kuperberg Dec 18 '09 at 07:20
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    Greg, the symmetry in that statement would be nicer if you replaced "problem-solver" with "Erdos." – Qiaochu Yuan Dec 27 '09 at 08:11
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    That's a good suggestion, but I can't edit comments. So let me just rephrase the aphorism: "An excellent Erdos might reach certain limits as a mathematician, while a bad Erdos can still be an okay mathematician. On the other hand, a good Grothendieck can be a truly great mathematician, while a bad Grothendieck is really terrible." – Greg Kuperberg Jan 19 '10 at 19:36
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"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions", F. Klein (from Reed & Simon: Methods of modern mathematical physics)

Marko Budisic
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"Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate." - David Mumford

Charles Siegel
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"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard

John D. Cook
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  • I saw this quote first at Klein's . However, evidence I have come across is insufficient to show this (shortest path). It is a good question to convince people like me the truth of Sir Hadamard's assertion. – Sunni Mar 26 '10 at 03:43
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    Here's a simple example. What is the radius of convergence for the power series of 1/(x^2 + 1) centered at 0? Looking only at the real line there's no apparent reason for the radius to be only 1. But in the complex plane, you can see that the radius is 1 because that's the distance from the center to the singularity at i.

    Another example would be using contour integration to compute integrals over the real line.

     – John D. Cook Mar 26 '10 at 18:35
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In mathematics you don't understand things. You just get used to them.

--John von Neumann, reply to a physicist at Los Alamos who had said "I don't understand the method of characteristics."

---- footnote on page 226 of Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics, Rider, London, 1990.

(taken from Warren Dicks' Home Page)

mathreader
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    I don't know how much I agree with that. It may be true about physics, but not math. – Harry Gindi Nov 30 '09 at 06:20
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    I thought the quote was aiming at talking about abstract concepts, that we never really understand. – Ohdarkdevil Dec 01 '09 at 07:30
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    I feel that it's true of both mathematics and physics, but when talking about mathematics it's a much deeper statement. – Dan Piponi Dec 02 '09 at 19:37
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    I may have a surprise for you... "Understanding"="getting used to"! :) – M.G. Dec 04 '09 at 21:20
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    I constantly try to check myself to make sure that I am really understanding, and am not just "getting used to" the things that I am learning. It is very difficult, but I think that I am gaining understanding of the basics little by little. – Steven Gubkin Dec 07 '09 at 02:04
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    I totally disagree with this quote ... – Martin Brandenburg Feb 02 '10 at 15:45
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    Different understandings of the quote lead to different styles of math-study. People who agree are doing 'potential infinity' mathematics, while people who disagree are doing 'actual infinity' mathematics. – Sunni Mar 26 '10 at 03:56
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    I'm curious what you mean with that. what does potential infinity mean? – Martin Brandenburg Apr 26 '10 at 05:20
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    'Potential infinity' and 'actual infinity' are concepts dating back, I think, to Aristotle. A potential infinity is like a basket that always holds another apple, no matter how many apples you take, whereas an actual infinity is like the totality of points in a circle drawn on paper. One (the actual infinity) is a completed whole to be apprehended at once, whereas the other (potential infinity) can never be apprehended in total. This strikes me as the distinction between algebra and geometry. – Jon Bannon Nov 21 '12 at 01:03
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    It took some time to get used to this quote, but now I understand it… – jmc Dec 16 '14 at 14:18
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"The question you raise, "how can such a formulation lead to computations?" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered." - Grothendieck

Sam Derbyshire
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"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." - Weyl

Charles Siegel
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    I never understood this quote, I mean, there's nasty algebra and nice algebra, so isn't that statement a bit strong? – Harry Gindi Nov 30 '09 at 06:28
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    I actually think it refers to something Mephistophelian about algebra, as if such heights of abstraction are meant not for mortals, or such symbolic calculation lacks intuitive "soul". (That would be a tendentious way of putting it!) – Todd Trimble Jan 31 '11 at 01:39
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    Oh, and I swear that I wrote that before seeing Anton Petrunin's contribution! – Todd Trimble Jan 31 '11 at 01:42
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    Weyl surely loves to talk about the devil. – Niemi Nov 18 '12 at 11:45
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" Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?" "- Serge Lang

Spinorbundle
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"The price of metaphor is eternal vigilance." Norbert Wiener.

SixWingedSeraph
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“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 Birch-Swinnerton-Dyer Conjecture

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    Thank you! I've been trying to track down the wording of this quote for awhile now. – Qiaochu Yuan Jan 16 '10 at 14:23
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    Last year our distinguished Cantrell Lecture series was given by Dick Gross. I had been booked as a speaker in the graduate student seminar, which took place immediately before Gross's first lecture. I decided to give an introduction to elliptic curves, including BSD and Gross-Zagier. I made sure to include this quote of Tate. So did Gross in his first lecture. – Pete L. Clark Mar 26 '16 at 20:17
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The introduction of numbers as coordinates is an act of violence.

Hermann Weyl, Philosophy of Mathematics and Natural Science.

Mariano Suárez-Álvarez
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"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."

--Stefan Banach

"Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies."

--Stanislaw M. Ulam quoting Stefan Banach

grshutt
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"Wir müssen wissen, wir werden wissen." - Hilbert.

Translation: We must know, we will know.

Sam Derbyshire
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Charles Siegel
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Do not ask whether a statement is true until you know what it means. -- Errett Bishop

Pete L. Clark
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  • I didn't know that you were a constructivist :p – Amr Mar 26 '16 at 20:02
  • @Amr: Well, I'm certainly not. On the one hand, a quotation is not an endorsement. On the other hand, although you perceive -- I believe correctly -- constructivist principles in the provenance of the quote, it is a good enough quote that it can mean other things as well. As a piece of general mathematical advice, I find it quite sound. – Pete L. Clark Mar 26 '16 at 20:12
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"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

Steven Gubkin
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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.

Harald Hanche-Olsen
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lhf
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(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

Charles Siegel
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"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

Spinorbundle
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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

Community
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Mariano Suárez-Álvarez
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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.

Lennart Meier
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Mariano Suárez-Álvarez
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  • Shouldn't that be "er hat etwas bewiessen"? (I'd edit it had I the power.) – Cory Knapp Nov 29 '09 at 21:15
  • @ Cory Knapp: no, it should be "Beweis", a noun as attested by the "er" ending of the preceding adjective "strenger". The present perfect you suggest is impossible here, and moreover its correct spelling would be "er hat bewiesen" (only one "s"). Freundliche Grüsse. – Georges Elencwajg Nov 29 '09 at 22:26
  • Very interesting quote, Mariano. For Jacobi to write that when Gauss claims to have a proof, it is just "sehr wahrscheinlich" sounds like the height of "chutzpah" ! – Georges Elencwajg Nov 29 '09 at 22:37
  • @Georges Elencwajg: "Bewiessen" was a indeed typo, but I was correcting the word have (the original said "er have..." Perhaps it is my inadequate German, but "er habe.." still seems to be an incorrect conjugation. – Cory Knapp Nov 30 '09 at 05:42
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    The conjugation "er habe" is correct. It is a subjunctive mood indicating that it only a personal (Gauß's) opinion is expressed. It might also be possible that in former times (when spelling was different) they might have written it "er have". – Philipp Lampe Nov 30 '09 at 06:13
  • Haha, this is awesome. –  Dec 09 '09 at 05:12
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    Oh I never get tired of cribbing about German along with Mark Twain .. http://baetzler.de/humor/the_awful_german_language.var .. – Anweshi Feb 06 '10 at 16:20
  • This is structurally similar to the quote describing how one should feel if Spassky, Tal, or Petrosian sacrifices a piece against you in a game of chess. – PrimeRibeyeDeal Aug 04 '22 at 14:49
36

“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)

Harald Hanche-Olsen
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    I see it just the other way around! – Jose Brox Dec 08 '09 at 15:50
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    A good definition can occassionally be worth a thousand proofs, I think, but frankly, elementary differential geometry seems to me to fit the “desert of definitions” pretty well. Not that there is anything wrong in that. Some times you have to suffer loads of boring definitions in order to see the depth and beauty of the subject. (And besides, deserts can be beautiful too.) – Harald Hanche-Olsen Dec 08 '09 at 18:19
36

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!"

  • Aigner & Ziegler in Proofs from the Book.
José Hdz. Stgo.
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"Later mathematicians will regard set theory as a disease from which one has recovered." Henri Poincaré.

SixWingedSeraph
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    Taken out of context, this would seem to be accurate (the concept of evil with respect to higher categories), but one must remember that Poincar'e was not against axiomatic set theory per se, but axiomatic theories in general. – Harry Gindi Nov 30 '09 at 06:33
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    Actually this is not a quote. It is attributed to Poincare in various sources, but it is quite likely that this is a misinterpretation of something that he actually wrote (something to the effect that the diseases of set theory (such as Russell's paradox, for example) will one day be overcome). See J. Gray, Did Poincaré say ``Set theory is a disease''?, Math. Intell. 13 (1991), 19--22 – Franz Lemmermeyer Feb 15 '10 at 16:21
32

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

Mariano Suárez-Álvarez
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"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!?)

innerproduct
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"[Mathematics consists of] true facts about imaginary objects." Philip Davis and Reuben Hersh.

SixWingedSeraph
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A mathematician is a device for turning coffee into theorems. —Alfréd Rényi, but often attributed also to Paul Erdős

Steve Flammia
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    Ken Ribet once told me the story of how he was sent a freebie book "for possible use in your undergraduate classes" that he looked at and decided he didn't want, so took it to the second hand bookstore in his lunch break, sold it, and bought lunch with the proceeds. On the way back to the math department he realised he'd turned theorems into coffee. – Kevin Buzzard Nov 29 '09 at 22:42
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    A comathematician is a device for turning cotheorems into ffee. A cotheorem is of course what one deduces from a rollary. – Saul Glasman Nov 30 '09 at 19:02
  • I first heard this on Numb3rs. Fantastic quote! – Gabriel Benamy Dec 02 '09 at 03:49
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    I think "A mathematician is a device for turning 'coffee', wink wink, into theorems" is more likely to be authentically Erdos, no? – Benjamin Lindqvist Dec 22 '13 at 19:26
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"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.

abcdxyz
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«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)

Georges Elencwajg
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Mariano Suárez-Álvarez
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"A mathematician who is not also something of a poet will never be a perfect mathematician"- Karl Weierstraß

Spinorbundle
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  • As pointed out by Jordan Ellenberg in his review of Once Upon a Prime (Sarah Hart), "the quote cuts off the part of the sentence specifying that the problematically nonpoetic mathematicians were especially to be found among “those from the semitic tribe.”" – E G Mar 23 '24 at 22:05
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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.

Joel Dodge
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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

Philip Brooker
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    It is hard to believe that there aren't third (and possibly further) categories of mathematical contributions that are neither important nor triumphal. – LSpice May 10 '11 at 17:12
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    I agree in general, though I suppose it could be argued that if one takes all mathematical contributions that aren't "important to the history of mathematics", then collectively they are a triumph of the human spirit (but I don't claim that this is what Cohen actually meant). When I first answered this question, I had just finished proving most of the main theorems for my PhD thesis, so the romantic notion of "triumph of the human spirit" kind of captured my mood at the time. Also, Cohen said this when interviewed for A Beautiful Mind, so he probably simplified things a bit for the public. – Philip Brooker May 11 '11 at 05:58
22

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."

Ben
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"There are five elementary arithmetical operations: addition, subtraction, multiplication, division, and… modular forms." - Eichler

Sam Derbyshire
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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).

user2146
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«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…)

quim
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"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

Boris Bukh
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"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)

Zev Chonoles
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“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

Spinorbundle
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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."

Mariano Suárez-Álvarez
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Cory Knapp
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    To find out how misunderstood is the relationship between Gordan and Hilbert read McLarty's 'Theology and its discontents'. http://www.cwru.edu/artsci/phil/Theology%20and%20its%20discontents.pdf – David Corfield Mar 31 '10 at 07:47
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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

Spinorbundle
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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

Escualo
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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"

Carl Weisman
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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.

Qiaochu Yuan
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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 http://books.google.com/books?id=H3zGTvmtp74C&lpg=PP1&client=firefox-a&pg=PA195#v=onepage&q=&f=false. – 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 06:27
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    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? – Todd Trimble Dec 05 '14 at 03:42
  • 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. – R. van Dobben de Bruyn Aug 14 '18 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."

Bill Johnson
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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.

The Amplitwist
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John Stillwell
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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

Jan
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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

Spinorbundle
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    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" – Pietro Majer May 29 '10 at 20:13
  • 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? – Todd Trimble Jun 09 '13 at 14:21
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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?

Jon Yard
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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
  • @StevenGubkin, I would be glad if further explanation is provided. I'm quite familiar with the categorical terminology so feel free to use them. – FNH Apr 21 '17 at 22:30
  • @StevenGubkin, I would be glad if further explanation is provided. I'm quite familiar with the categorical terminology so feel free to use them. – FNH Apr 21 '17 at 22:30
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    Read https://en.wikipedia.org/wiki/Group_object, and apply the definition to a group object in the category of sheaves of sets. – Steven Gubkin Apr 29 '17 at 01:51
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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".

Rhys Gibson
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"In mathematics you don't understand things. You just get used to them."

John von Neumann

Axiom
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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.

Danny Ruberman
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  • 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. – Todd Trimble Jun 09 '13 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

Holzinger Raphael
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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.

Konrad Waldorf
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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.

José Hdz. Stgo.
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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 03:06
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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": http://www-history.mcs.st-and.ac.uk/Quotations/Whitehead.html) and some student must have "extended" it. – shreevatsa Oct 22 '12 at 04:41
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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 09 '13 at 14:32
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    I'd like to paraphrase Orcar Wilde and say "Every Mathematician Kills the Problem He Loves" – Pietro Majer Nov 10 '13 at 21:42
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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)

Akhil Mathew
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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

Harry Gindi
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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

santker heboln
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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.
James
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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

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

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

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

Abhishek Parab
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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
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    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! – Todd Trimble Jun 09 '13 at 14:46
  • Riemann did all the heavy lifting – Forever Mozart May 06 '17 at 04:51
  • He is just saying that his math is not so good but still managed to turn everything in physics on its head. – timur Jan 01 '18 at 05:03
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‘Life is complex: it has both real and imaginary components.” (I don't know who said this...)

Hanxiong Zhang
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  • 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... – Qiaochu Yuan Mar 26 '10 at 00:38
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    Was it mentioned? Where? – Jonas Meyer Apr 28 '10 at 01:06
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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.»

Community
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Mariano Suárez-Álvarez
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    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." – Todd Trimble Jun 09 '13 at 14:42
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Who can does; who cannot do, teaches; who cannot teach, teaches teachers.

Paul Erdos.

Pandelis Dodos
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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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    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 06:32
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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

Harry Gindi
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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

Harry Gindi
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7

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...

Mariano Suárez-Álvarez
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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.

Elizabeth S. Q. Goodman
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patfla
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  • Well OK there's no reason, with the web at hand, to be that lazy. Was it Samuel Patterson who actually said this?

    http://books.google.com/books?id=IFIc07eswukC&printsec=frontcover&dq=karl+sabbagh&ei=m9IdS7H7KaGqkATztIy2CQ&cd=1#v=snippet&q=fourier%20transform%20of%20the%20primes&f=false

     – patfla Dec 08 '09 at 04:22
  • I can't find anything else, but I edited that in. Also, nice. – Elizabeth S. Q. Goodman Dec 08 '09 at 08:01
  • Thanx Elizabeth. What would be even nicer is if it were, in some sense, true. – patfla Dec 08 '09 at 17:17
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Mathematicians are born, not made. -- Henri Poincare

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

Spinorbundle
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6

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

Gabriel Benamy
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6a cc d æ 13e ff 7i 3l 9n 4o 4q rr 4s 9t 12vx

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

Mariano Suárez-Álvarez
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  • Ah! I've always assumed the anagram was something more clever, and have wondered what that sentence can possibly anagram to! So he just gives a list of letter counts... that's so cheap! – Willie Wong Mar 21 '10 at 19:47
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"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

Victor Miller
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    Boy, was he wrong or what? – Harry Gindi Jan 15 '10 at 22:46
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    The quote could benefit by being put into a better context. Here it is: http://books.google.ca/books?id=QPXa2UgysLQC&lpg=PA38&ots=PIs4AINBvb&dq=There%20are%2C%20therefore%2C%20no%20longer%20some%20problems%20solved%20and%20others%20unsolved%2C%20there%20are%20only%20problems%20more%20or%20less%20solved%2C%20according&pg=PA38#v=onepage&q=There%20are,%20therefore,%20no%20longer%20some%20problems%20solved%20and%20others%20unsolved,%20there%20are%20only%20problems%20more%20or%20less%20solved,%20according&f=false but Poincare was making a pretty important point that IMO you've missed. – Ryan Budney Feb 06 '10 at 05:50
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"Mathematics consists of proving the most obvious thing in the least obvious way." - George Polya

kolistivra
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    No. – darij grinberg Mar 21 '10 at 18:24
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    Darij, this is a sentiment with which one can agree or disagree (I agree about the content of many undergraduate mathematics courses, but disagree about much mathematics beyond that), but surely it's courteous to offer a bit more than “No”? – LSpice Mar 21 '10 at 23:54