most recent 30 from http://mathoverflow.net 2013-05-20T02:58:08Z http://mathoverflow.net/feeds/question/7155 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7155/famous-mathematical-quotes Sam Derbyshire 2009-11-29T20:18:38Z 2011-02-01T05:06:28Z

<p>Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of?</p> <p>Standard community wiki rules apply: one quote per post.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7156#7156 Sam Derbyshire 2009-11-29T20:19:20Z 2009-11-29T20:19:20Z

<p>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:</p> <p>"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."</p> <p>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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7157#7157 Sam Derbyshire 2009-11-29T20:19:54Z 2009-11-29T20:19:54Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7158#7158 Sam Derbyshire 2009-11-29T20:20:54Z 2009-11-29T20:20:54Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7159#7159 Charles Siegel 2009-11-29T20:23:21Z 2009-11-29T20:23:21Z

<p>(Caveat for all of mine: I've not hunted down primary sources to check that they're properly attributed)</p> <p>"Manifolds are a bit like pornography: hard to define, but you know one when you see one." -S. Weinberger</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7160#7160 Charles Siegel 2009-11-29T20:23:47Z 2009-11-29T20:23:47Z

<p>"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." - Weyl</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7161#7161 Charles Siegel 2009-11-29T20:24:05Z 2009-11-29T20:24:05Z

<p>We often hear that mathematics consists mainly of "proving theorems." Is a writer's job mainly that of "writing sentences?" - Gian-Carlo Rota</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7162#7162 John D. Cook 2009-11-29T20:26:11Z 2009-11-29T20:26:11Z

<p>"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7163#7163 John D. Cook 2009-11-29T20:26:50Z 2009-11-29T20:26:50Z

<p>"The art of doing mathematics is finding that special case that contains all the germs of generality." -- David Hilbert</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7164#7164 lhf 2009-11-29T20:35:58Z 2009-11-30T03:17:25Z

<blockquote> <p>The purpose of computing is insight, not numbers.</p> </blockquote> <p>— Richard Hamming (1962)</p> <blockquote> <p>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,</p> <p>“The purpose of computing is insight, not numbers,” although some people claim,</p> <p>“The purpose of computing numbers is not yet in sight.”</p> <p>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.”</p> </blockquote> <p>– Richard W. Hamming, <em>Introduction to applied numerical analysis</em>, McGraw-Hill 1971, p.31.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7165#7165 Cory Knapp 2009-11-29T20:37:35Z 2009-11-29T20:38:29Z

<p>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."</p> <p>Along with his redaction after he came to accept the method: "I have convinced myself that even theology has its merits."</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7166#7166 Mariano Suárez-Alvarez 2009-11-29T20:37:46Z 2009-11-29T22:03:33Z

<blockquote> <p>«Allez en avant, et la foi vous viendra.»</p> </blockquote> <p>Free translation: keep going, faith will come later.</p> <p>Jean-le-Rond D'Alembert, to his students (quoted by Florian Cajori in A history of mathematics)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7167#7167 Steve Flammia 2009-11-29T20:38:35Z 2009-11-29T20:38:35Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7168#7168 Mariano Suárez-Alvarez 2009-11-29T20:46:11Z 2009-11-29T21:59:43Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7171#7171 Spinorbundle 2009-11-29T20:49:04Z 2009-11-29T20:49:04Z

<p>"God exists since mathematics is consistent, and the Devil exists since we cannot prove it."- André Weil</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7172#7172 Greg Kuperberg 2009-11-29T20:50:14Z 2009-11-29T20:50:14Z

<p>It's hard to beat <a href="http://www.math.lsa.umich.edu/~jrs/plans.html" rel="nofollow">John Stembridge's page of quotes</a>. 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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7173#7173 Spinorbundle 2009-11-29T20:50:35Z 2009-11-29T20:50:35Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7174#7174 Spinorbundle 2009-11-29T20:52:21Z 2009-11-29T21:06:27Z

<p>“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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7175#7175 Mariano Suárez-Alvarez 2009-11-29T20:56:32Z 2009-11-29T22:13:39Z

<blockquote> <p>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 su wetten, wenn Dirichlet es sagt, ist es gewiß; ich lasse mich auf diese Delikatessen lieber gar nicht ein.</p> </blockquote> <p>C. G. J. Jacobi, writing to von Humboldt, in 1846.</p> <p>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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7176#7176 Mariano Suárez-Alvarez 2009-11-29T20:58:19Z 2009-11-29T20:58:19Z

<blockquote> <p>You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine.</p> </blockquote> <p>D. Hilbert, talking about an ex-student. I'd love to remember where I got this from!</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7177#7177 Spinorbundle 2009-11-29T21:03:40Z 2009-11-29T21:03:40Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7178#7178 Boris Bukh 2009-11-29T21:04:49Z 2009-11-30T15:33:15Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7179#7179 Spinorbundle 2009-11-29T21:07:40Z 2009-11-29T21:07:40Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7180#7180 Spinorbundle 2009-11-29T21:10:54Z 2009-11-29T21:10:54Z

<p>"A mathematician who is not also something of a poet will never be a perfect mathematician"- Karl Weierstraß</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7181#7181 Marko Budisic 2009-11-29T21:15:45Z 2009-11-29T21:15:45Z

<p>"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 &amp; Simon: Methods of modern mathematical physics)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7182#7182 Qiaochu Yuan 2009-11-29T21:18:22Z 2009-11-29T21:18:22Z

<blockquote> <p>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.</p> </blockquote> <p>Gian-Carlo Rota, in an interview with David Sharp.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7183#7183 Jon Yard 2009-11-29T21:20:57Z 2009-11-29T21:20:57Z

<p>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?</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7184#7184 Spinorbundle 2009-11-29T21:21:21Z 2009-11-29T21:21:21Z

<p>" 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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7186#7186 YBL 2009-11-29T21:27:28Z 2009-11-29T21:27:28Z

<p>"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." </p> <p>Pierre de Fermat</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7187#7187 Spinorbundle 2009-11-29T21:32:58Z 2009-11-29T21:32:58Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7188#7188 mathreader 2009-11-29T21:34:00Z 2009-11-29T21:34:00Z

<p><strong>In mathematics you don't understand things. You just get used to them.</strong> </p> <p>--John von Neumann, reply to a physicist at Los Alamos who had said "I don't understand the method of characteristics."</p> <p>---- footnote on page 226 of Gary Zukav, The Dancing Wu Li Masters: An Overview of the New Physics, Rider, London, 1990.</p> <p>(taken from Warren Dicks' Home Page)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7189#7189 Steven Gubkin 2009-11-29T21:37:03Z 2009-11-29T21:37:03Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7190#7190 SixWingedSeraph 2009-11-29T21:37:31Z 2009-11-30T05:44:40Z

<p>"Mathematics is the art of giving the same name to different things." Henri Poincaré.</p> <p>(This was in response to "Poetry is the art of giving different names to the same thing.")</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7191#7191 SixWingedSeraph 2009-11-29T21:40:53Z 2009-11-29T21:40:53Z

<p>"Later mathematicians will regard set theory as a disease from which one has recovered." Henri Poincaré.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7193#7193 grshutt 2009-11-29T21:53:47Z 2009-11-29T21:53:47Z

<p>"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."</p> <p>--Stefan Banach</p> <p>"Good mathematicians see analogies between theorems or theories. The very best ones see analogies between analogies."</p> <p>--Stanislaw M. Ulam quoting Stefan Banach</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7194#7194 Malik Younsi 2009-11-29T21:53:52Z 2009-11-29T21:53:52Z

<p>"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7195#7195 SixWingedSeraph 2009-11-29T21:54:53Z 2009-11-29T21:54:53Z

<p>"[Mathematics consists of] true facts about imaginary objects." Philip Davis and Reuben Hersh.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7196#7196 SixWingedSeraph 2009-11-29T21:57:22Z 2009-11-29T21:57:22Z

<p>"The price of metaphor is eternal vigilance." Norbert Wiener.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7199#7199 Akhil Mathew 2009-11-29T22:25:15Z 2009-11-29T22:25:15Z

<p>"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..."</p> <p>-Alexander Grothendieck, writing to Ronald Brown</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7203#7203 Anton Petrunin 2009-11-29T22:45:28Z 2009-12-04T20:43:03Z

<p><em>Algebra is the offer made by the devil to the mathematician...All you need to do, is give me your soul: give up geometry</em> --Michael Atiyah</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7205#7205 Charles Siegel 2009-11-29T22:48:10Z 2009-11-29T22:48:10Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7206#7206 Charles Siegel 2009-11-29T22:49:04Z 2009-11-30T00:27:31Z

<p>"Wir müssen wissen, wir werden wissen." - Hilbert.</p> <p>Translation: We must know, we will know.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7207#7207 Chris Godsil 2009-11-29T22:54:12Z 2009-11-29T22:54:12Z

<p>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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7208#7208 Mariano Suárez-Alvarez 2009-11-29T23:07:16Z 2009-11-29T23:07:16Z

<blockquote> <p>6a cc d æ 13e ff 7i 3l 9n 4o 4q rr 4s 9t 12vx</p> </blockquote> <p><em>Explanation</em> 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».</p> <p>Who's not shared the feeling that Leibniz must have felt at getting this response when reading obscure explanations in the literature? :P</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7216#7216 Akhil Mathew 2009-11-30T01:21:27Z 2009-11-30T20:30:47Z

<p>`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.</p> <p>--Shalosh B. Ekhad (i.e., Doron Zeilberger)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7222#7222 Joel Dodge 2009-11-30T02:07:09Z 2009-11-30T02:07:09Z

<p>In the biographical piece on Grothendieck a couple of years ago in the Notices &lt;&lt;<a href="http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf" rel="nofollow">http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf</a>>> 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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7230#7230 Harald Hanche-Olsen 2009-11-30T03:31:09Z 2009-11-30T03:31:09Z

<p>“Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions.”</p> <p>– Th. Bröcker &amp; K. Jänich, <em>Introduction to differential topology</em> (p.25)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7233#7233 Zev Chonoles 2009-11-30T04:19:32Z 2009-11-30T04:19:32Z

<p>"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</p> <p>(Personally, I'd take certainty over being about reality any day)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7238#7238 Arrieta 2009-11-30T05:11:20Z 2009-11-30T05:11:20Z

<blockquote> <p>A mathematician is a blind man in a dark room looking for a black cat which isn't there.</p> </blockquote> <p>Attributed to <em>Charles Darwin</em></p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7239#7239 Mariano Suárez-Alvarez 2009-11-30T05:28:17Z 2009-11-30T05:28:17Z

<blockquote> <p>If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.</p> </blockquote> <p>P. Turan, "The Work of Alfred Renyi", Matematikai Lapok 21, 1970, pp 199-210</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7240#7240 Jonathan Kariv 2009-11-30T05:29:59Z 2009-11-30T05:29:59Z

<p>There are 10 kinds of people in the world, those that understand binary and the other 9.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7246#7246 Harry Gindi 2009-11-30T06:42:14Z 2009-11-30T06:42:14Z

<p>"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."</p> <p>-David Hilbert</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7257#7257 Harry Gindi 2009-11-30T11:08:37Z 2009-11-30T11:08:37Z

<p>"'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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7259#7259 Harry Gindi 2009-11-30T11:28:29Z 2009-11-30T11:28:29Z

<p>"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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7306#7306 mic 2009-11-30T19:09:22Z 2009-11-30T19:09:22Z

<p>"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 <a href="http://portal.acm.org/citation.cfm?id=361528" rel="nofollow">1972 CACM article</a>.)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7342#7342 unknown (yahoo) 2009-12-01T02:07:01Z 2009-12-01T02:07:01Z

<p>“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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7377#7377 Mariano Suárez-Alvarez 2009-12-01T05:17:10Z 2009-12-01T05:17:10Z

<blockquote> <p>The introduction of numbers as coordinates is an act of violence. </p> </blockquote> <p>Hermann Weyl, <em>Philosophy of Mathematics and Natural Science</em>.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7549#7549 Philip Brooker 2009-12-02T03:32:43Z 2009-12-02T03:32:43Z

<p>I have put this quote in the front of my thesis:</p> <blockquote> <p>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 - <em>Paul Cohen, Stanford, 1996</em></p> </blockquote>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7552#7552 Mariano Suárez-Alvarez 2009-12-02T04:05:15Z 2009-12-02T04:05:15Z

<blockquote> <p>Le but de cette thèse est de munir son auteur du titre de Docteur.</p> </blockquote> <p>Beginning of A. Douady's thesis. Quoted by Michèle AUdin in her <a href="http://www-irma.u-strasbg.fr/~maudin/newhowto.ps" rel="nofollow">Conseils aux auteurs de textes mathématiques</a>.</p> <p>In a less barbarous language: The purpose of this thesis is to obtain the degree of Doctor for its author.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7553#7553 Mariano Suárez-Alvarez 2009-12-02T04:18:41Z 2009-12-02T04:18:41Z

<blockquote> <p>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.</p> </blockquote> <p>C. G. J. Jacobi writing (in French) to Legendre</p> <p>Translation as given in <em>Additive number theory: inverse problems and the geometry of sumsets</em>, 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...</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7557#7557 Gabriel Benamy 2009-12-02T05:31:48Z 2009-12-02T05:31:48Z

<p>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, <em>New Math</em></p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7578#7578 quim 2009-12-02T13:26:17Z 2009-12-02T13:26:17Z

<p>«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 </p> <p>(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…)</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7598#7598 santker heboln 2009-12-02T17:57:15Z 2009-12-02T17:57:15Z

<p>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</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/7759#7759 Csar Lozano Huerta 2009-12-04T06:32:04Z 2010-11-28T02:42:47Z

<p>I heard this one while taking a differential geometry class in Mexico City. I love it.</p> <p><strong>"Groups, as men, will be known by their actions"</strong>.</p> <p>-Guillermo Moreno.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8159#8159 patfla 2009-12-08T04:08:30Z 2009-12-08T08:00:51Z

<p>"the zeros of the zeta function are like the Fourier transform of the primes"</p> <p>As related in Karl Sabbagh's book on the Riemann Hypothesis. (<a href="http://www.amazon.com/Riemann-Hypothesis-Greatest-Unsolved-Mathematics/dp/0374529353/ref=ntt_at_ep_dpi_2" rel="nofollow">Amazon reference</a>)</p> <p>From <a href="http://books.google.com/books?id=IFIc07eswukC&amp;printsec=frontcover&amp;dq=karl+sabbagh&amp;ei=m9IdS7H7KaGqkATztIy2CQ&amp;cd=1#v=snippet&amp;q=fourier%20transform%20of%20the%20primes&amp;f=false" rel="nofollow">the relevant page in the Google book</a>, it might be Samuel Patterson.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8162#8162 Jan 2009-12-08T05:08:03Z 2009-12-08T05:08:03Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8233#8233 J. H. S. 2009-12-08T19:34:27Z 2009-12-08T19:34:27Z

<p>This one has to do with the quote by Rota that appears in the first post of C. Siegel:</p> <p>" The essence of Mathematics is proving theorems and so, that is what mathematicians do: they prove theorems. <strong>But to tell the truth</strong>, 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.</p> <p>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!"</p> <ul> <li>Aigner &amp; Ziegler in <em>Proofs from the Book</em>.</li> </ul>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8238#8238 J. H. S. 2009-12-08T19:55:49Z 2011-02-01T05:06:28Z

<p>Here you have one of my all-time favorites:</p> <p>" The ultimate goal of Mathematics is to eliminate any need for intelligent thought."</p> <ul> <li>R. L. Graham (?)</li> </ul> <p>Can any of you guys tell me where that quote first appeared? Same thing for the quote of Atiyah entered by Petrunin.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8394#8394 James 2009-12-09T22:59:40Z 2009-12-09T22:59:40Z

<p><strong>"Why is this a good idea?"</strong></p> <ul> <li>Bill Ralph, on the most important question to ask yourself when doing (or studying) mathematics.</li> </ul>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8841#8841 Rhys Gibson 2009-12-14T04:28:14Z 2009-12-14T04:28:14Z

<p>"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".</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/8869#8869 unknown (google) 2009-12-14T11:34:28Z 2009-12-14T11:34:28Z

<p>Without Mathematics Earth is just a Big Zero - Chacko Mash</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/9180#9180 Ben 2009-12-17T15:55:23Z 2009-12-17T15:55:23Z

<p>Farkas Bolyai to his son Janos, speaking about attempts to study Euclid's Vth postulate on parallel lines:</p> <p>"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.</p> <p>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."</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/9409#9409 Tran Chieu Minh 2009-12-20T06:36:10Z 2009-12-20T06:36:10Z

<p>"Either mathematics is too big for the human mind or the human mind is more than a machine." - Kurt Gödel</p> <p>A declaration of war by a Platonist.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/9800#9800 Mariano Suárez-Alvarez 2009-12-26T16:38:32Z 2009-12-26T16:38:32Z

<blockquote> <p>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. </p> </blockquote> <p>Greg Kuperberg, in a comment to a <a href="http://mathoverflow.net/questions/8521/nice-proof-of-the-jordan-curve-theorem" rel="nofollow">MO question</a></p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/9867#9867 Abhishek Parab 2009-12-27T07:45:47Z 2009-12-27T07:45:47Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/11900#11900 kolistivra 2010-01-15T18:59:20Z 2010-01-15T18:59:20Z

<p>"Mathematics consists of proving the most obvious thing in the least obvious way." - George Polya</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/11917#11917 Pete L. Clark 2010-01-15T20:34:16Z 2010-01-15T20:34:16Z

<blockquote> <p>Do not ask whether a statement is true until you know what it means. -- Errett Bishop</p> </blockquote>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/11919#11919 Ben Linowitz 2010-01-15T20:58:58Z 2011-01-31T01:02:17Z

<p>“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.”</p> <p>-John Tate on the Birch-Swinnerton-Dyer Conjecture</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/11935#11935 Victor Miller 2010-01-15T22:28:51Z 2010-01-15T22:28:51Z

<p>"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."</p> <p>Henri Poincare</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/11983#11983 Soheil Malekzadeh 2010-01-16T13:50:31Z 2010-01-16T13:50:31Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/12219#12219 Carl Weisman 2010-01-18T19:03:55Z 2010-01-19T14:21:52Z

<p>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?"</p> <p>Epigraph to Hille-Phillips, "Functional analysis and semigroups"</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/12328#12328 Pandelis Dodos 2010-01-19T18:37:28Z 2010-01-19T18:37:28Z

<p><em>Who can does; who cannot do, teaches; who cannot teach, teaches teachers</em>.</p> <p>Paul Erdos.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/12600#12600 Danny Ruberman 2010-01-22T03:06:01Z 2010-01-22T03:06:01Z

<p>I once read, in an autobiographical piece, what the author said to his high-school teacher upon graduation; my recollection is:</p> <p>"Poincaré has written that geometry is the art of making a correct argument from incorrectly drawn figures. For you, sir, it is the opposite."</p> <p>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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/12989#12989 Ben Linowitz 2010-01-26T00:11:06Z 2010-01-26T00:11:06Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/13113#13113 philip314 2010-01-27T09:45:27Z 2010-01-27T09:45:27Z

<blockquote> <p>"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."</p> </blockquote> <p>Grothendieck in Pursuing stacks (letters to Quillen).</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/14347#14347 Jose Brox 2010-02-06T01:53:29Z 2010-02-06T01:53:29Z

<p>Not famous yet, maybe from now on!</p> <p><em>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</em>.</p> <p>Terence Tao</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/14396#14396 Holzinger Raphael 2010-02-06T15:56:54Z 2010-02-06T15:56:54Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/15332#15332 kakaz 2010-02-15T13:55:55Z 2010-02-15T13:55:55Z

<p>"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</p> <p><a href="http://pauli.uni-muenster.de/~munsteg/arnold.html" rel="nofollow">http://pauli.uni-muenster.de/~munsteg/arnold.html</a></p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/16725#16725 Qiaochu Yuan 2010-02-28T23:22:25Z 2010-02-28T23:22:25Z

<p>At the risk of overloading an already bloated thread, I found a rather large collection <a href="http://math.furman.edu/~mwoodard/mqs/mquot.shtml" rel="nofollow">here</a>. Example:</p> <blockquote> <p>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.</p> </blockquote> <p>Richard W. Hamming, in N. Rose's <em>Mathematical Maxims and Minims</em>, Raleigh NC: Rome Press Inc., 1988.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/18941#18941 Ady 2010-03-21T18:09:23Z 2010-03-21T18:09:23Z

<p>Mathematicians are born, not made. -- Henri Poincare</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/18949#18949 Bill Johnson 2010-03-21T19:35:44Z 2010-03-21T20:05:42Z

<p>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."</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/18963#18963 F Zaldivar 2010-03-21T23:50:12Z 2010-03-21T23:50:12Z

<p>Mathema est ars et scientia, discenda</p> <p>Aquinas?</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/18972#18972 Amy Glen 2010-03-22T02:02:26Z 2010-03-22T02:02:26Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/18973#18973 John Stillwell 2010-03-22T04:04:34Z 2010-03-22T04:04:34Z

<p>Like many people, I am fascinated by the quote from Weyl (already listed here), that</p> <blockquote> <p>In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.</p> </blockquote> <p>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. </p> <p>First, what is the context? The quote occurs in Weyl's paper <em>Invariants</em> in Duke Math. J. 5 (1939), pp. 489--502, the first page of which can be seen <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdffirstpage_1&amp;handle=euclid.dmj/1077491405" rel="nofollow">here.</a> 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</p> <blockquote> <p>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.</p> </blockquote> <p>Second, why the image of angel and devil? According to V.I Arnold, writing <a href="http://books.google.com/books?id=hLrptG7XxvwC&amp;lpg=PP1&amp;pg=PA3#v=onepage&amp;q=&amp;f=false" rel="nofollow">here,</a> Weyl had a particular image in mind, namely, the Uccello painting "Miracle of the Profaned Host, Episode 6", which can be viewed <a href="http://www.aiwaz.net/panopticon/miracle-of-the-profaned-host-episode-6/gi4610c360" rel="nofollow">here.</a> </p> <p>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.</p>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/19369#19369 Hanxiong Zhang 2010-03-26T00:27:51Z 2010-03-26T00:27:51Z

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

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/19885#19885 Konrad Waldorf 2010-03-30T22:34:35Z 2010-03-30T22:34:35Z

<p>Dieudonné in "Foundations of Modern Analysis, Vol. 1":</p> <blockquote> <p>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.</p> </blockquote>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/19887#19887 Konrad Waldorf 2010-03-30T22:45:36Z 2010-03-30T22:45:36Z

<p>Another quote from Dieudonné's "Foundations of Modern Analysis, Vol. 1":</p> <blockquote> <p>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. </p> </blockquote>

http://mathoverflow.net/questions/7155/famous-mathematical-quotes/22780#22780 innerproduct 2010-04-27T21:06:25Z 2010-04-27T21:06:25Z

<p>"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!?)</p>