If I wanted to make a somewhat bold and rather vague claim in print that it is widely acknowledged among mathematicians that knowledge of mechanics (in the sense in which physicists understand that word) sheds light on geometry and perhaps kindred areas of mathematics, and that mathematicians qua mathematicians can benefit from such knowledge, what published papers or books could I cite in support of that statement?
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15$\begingroup$ This reminds me of V. I. Arnold. $\endgroup$– Z. MCommented May 19 at 19:33
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8$\begingroup$ If you go sufficiently far back in time the disciplines are not considered separate. Even around Newton, inventing new math and doing physics were all interconnected ideas. $\endgroup$– Sidharth GhoshalCommented May 19 at 19:47
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7$\begingroup$ Google will find plenty of answers to this question. I don’t think it is appropriate for MO, and I’ve voted to close. $\endgroup$– Andy PutmanCommented May 19 at 20:19
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5$\begingroup$ @TimothyChow: Doesn’t it seem trivial that knowing the origins of a subject is useful? That’s hardly enough meat for a MO question. And as the answers we have gotten show, what this question is really going to attract are the usual “inspirational quotes” about how amazing physics is. $\endgroup$– Andy PutmanCommented May 20 at 13:12
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6$\begingroup$ @TimothyChow: Chapter 1.1 of McDuff-Salamon (the standard textbook on symplectic geometry) is literally called “Hamiltonian Mechanics”. I think it really is a cliche that people working in that field should know something about mechanics. More generally, I think that collecting quotes from famous people about the importance of a field is a form of hero-worship that is inappropriate for MO. $\endgroup$– Andy PutmanCommented May 20 at 17:25
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5 Answers
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Quoting the first two paragraphs of V. I. Arnol'd, On teaching mathematics, Uspekhi Mat. Nauk 53 (1998) 229-234, translated to English in Russian Math. Surveys 53 (1998) 229-236 (a transcription may also be found here if you want to go around the paywall):
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.
The Jacobi identity (which forces the altitudes of a triangle to meet in a point) is an experimental fact in the same way as the fact that the earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.
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1$\begingroup$ Does Arnol'd explain why the Jacobi identity forces this geometric consequence? $\endgroup$– LSpiceCommented May 20 at 13:24
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3$\begingroup$ Unfortunately he doesn't, at least not in this reference... this specific fact refers to the Jacobi identity for the cross (vector) product in $\mathbb{R}^3$. A proper outline of this argument can be found e.g. in khudian.net/Etudes/Geometry/jacidentandheights2.pdf $\endgroup$ Commented May 20 at 16:41
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2$\begingroup$ Triple cross products always do in the case of three non-collinear but linearly dependent vectors, that's the point. $\endgroup$ Commented May 20 at 18:18
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2$\begingroup$ It is interesting that the quote in the link in another answer by Jacobi indicates that he does not entirely concur. $\endgroup$– KapilCommented May 21 at 2:46
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Michael Atiyah, On the Work of Edward Witten:
In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found.
Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics.
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1$\begingroup$ Besides Witten, I would add Isaac Newton and Richard Feynman $\endgroup$ Commented May 20 at 2:57
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2$\begingroup$ @Jorge, no doubt, but I think OP wants citations, not just names. $\endgroup$ Commented May 20 at 2:59
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1$\begingroup$ This does not answer the very specific question asked by the OP (actually, what Atiyah is saying here is quite different to what was requested by the OP). $\endgroup$ Commented May 20 at 17:35
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Saunders Mac Lane:
The recent fruitful interchange of ideas (connections, fiber bundles, etc.) with physics (quantum gravity and all that) has been a decided stimulus and a source of new ideas and reapplication of old ones. (page 192)
He continues by pointing out: "It's great, but involves some of the current weaknesses of physics" and elaborates further on this point. The source is
Atiyah, Michael; Borel, Armand; Chaitin, G. J.; Friedan, Daniel; Glimm, James; Gray, Jeremy J.; Hirsch, Morris W.; Mac Lane, Saunders; Mandelbrot, Benoit B.; Ruelle, David; Schwarz, Albert; Uhlenbeck, Karen; Thom, René; Witten, Edward; Zeeman, Christopher Responses to: A. Jaffe and F. Quinn, "Theoretical mathematics: toward a cultural synthesis of mathematics and theoretical physics'' [Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 1–13]. Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 178–207. doi:10.1090/S0273-0979-1994-00503-8, arXiv:math/9404229
In the same collection, David Ruelle writes:
One point that perhaps deserves being stressed is the usefulness of cultural cross-fertilization in mathematics. Feigenbaum's cultural background in theoretical physics has allowed him to discover a new generic bifurcation of smooth dynamical systems, which would not have been encountered soon by following standard mathematical paths. Similarly, the physical ideas of equilibrium statistical mechanics have richly contributed to the mathematical theory of smooth dynamical systems (with the concepts of entropy, Gibbs states, etc., see my note in BAMS(NS) 19 (1988), 259-268.) (page 197).
Richard Courant wrote:
... the life blood of our science rises through its roots; these roots reach down in endless ramification deep into what might be called reality, whether this "reality" is mechanics, physics, biological form, economic behavior, geodesy, or, for that matter, other mathematical substance already in the realm of the familiar. (Courant as quoted in Carrier, G.; Courant, R.; Rosenbloom, P.; Yang, C.; Greenberg, H. "Applied Mathematics: What is Needed in Research and Education." SIAM Rev., 4(4) (1962), 297--320.)
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Perhaps it is correct that this opinion is "widely acknowledged" but it is not unanimous. Here I collected some statements of some prominent mathematicians related to this question.
Some of those statements are quoted here:
Profound study of nature is the most fertile source of mathematical discoveries. (J. Fourier, Analytic theory of heat, 1830)
It is true that Mr. Fourier had the opinion that the principal purpose of mathematics was the benefit of the society and the explanation of phenomena of nature; but a philosopher like he should know that the sole purpose of science is the honor of the human mind, and under this title, a question about numbers is as valuable as a question about the system of the world. (C. G. Jacobi, Letter to Le Gendre, 1830)
In practice this is of course not at all important, because it is negligible for the largest triangle on earth that can be measured; however the dignity of science requires that we understand clearly the nature of this inequality... (Gauss, in a letter to a friend on a correction of .001" in the measurement of triangles on the earth surface).
It is completely clear to me which conditions caused the gradual decadence of mathematics, from its high level some 100 years ago, down to the present hopeless nadir. Degeneration of mathematics begins with the ideas of Riemann, Dedekind and Cantor which progressively repressed the reliable genius of Euler, Lagrange and Gauss. Through the influence of textbooks like those of Hasse, Schreier and van der Waerden, the new generation was seriously harmed, and the work of Bourbaki finally dealt the fatal blow. (C. L. Siegel, Letter to A. Weil, 1959. German original is cited in the lecture of H. Grauert, in: Mathematics and Theoretical Physics, ch 1993, ed.: Minaketan Behara, Rudolf Fritsch, Rubens G. Lintz, W. de Gruyter, 1995.) What would Siegel write today?
All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA). (V. Arnold, in: Mathematics: Frontiers and Perspectives, AMS 2000)
You cannot have both. I mean a fish having a meeting one billion years ago under the water and saying: "We are fish, we have all this power, now it is time to conquer the land". But you cannot conquer the land while remaining fish. You can't go to the real world remaining mathematicians; that's absurd. Either you study real problems in the real world - it's a remarkable intellectual challenge, or you remain a mathematician. (M. Gromov, Dead Sea discussions, in: GAFA 2000. Visions in Mathematics)
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1$\begingroup$ The first of these statements, by Fourier, seems to be exactly what the OP is looking for! $\endgroup$– VincentCommented May 20 at 13:59
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9$\begingroup$ These are great quotes. Why not paste them into your MathOverflow answer, rather than putting them on a separate linked webpage? More people will read them if they don't have to leave MO. $\endgroup$ Commented May 20 at 14:06
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2$\begingroup$ @Vincent Is it? The quote from Fourier sounds pretty generic and non-specific; cf. Andy Putman's comments under the question, and also the comment by Sam Hopkins under Trunk's answer (now converted to a comment). $\endgroup$ Commented May 20 at 14:33
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2$\begingroup$ The quote by Fourier is specific imo: he is saying that the deepest mathematical insights always come from studying physical problems. $\endgroup$ Commented May 20 at 17:39
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3$\begingroup$ @HollisWilliams Hardly specific enough for the purposes of the question, which is about how mechanics in particular informs geometry; to me it's along the very general lines of Andy Putman's "And as the answers we have gotten show, what this question is really going to attract are the usual “inspirational quotes” about how amazing physics is." (Also I find your assertion of specificity here at odds with your comment below Carlo Beenakker's answer.) $\endgroup$ Commented May 20 at 18:47
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Mark Levi's book The Mathematical Mechanic: Using Physical Reasoning to Solve Problems is full of concrete examples of applying physical intuition in geometry, including even a proof of Gauss Bonnet theorem.
Eugene Wigner wrote that "... the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena."
Riemann in his habiltation lecture refers to the physical experience both at the large and small scales as a source of development for geometry: "A decision upon these questions can be found only by starting from the structure of phenomena that has been approved in experience hitherto, for which Newton laid the foundation, and by modifying this structure gradually under the compulsion of facts which it cannot explain". One could say that he was anticipating developments which we know today as quantum mechanics and general relativity.
Finally one might even cite Newton's Principia, which of course is as much about physics as it is about math, or what was called "natural philosophy" at the time. It is clear that Newton's physical intuition guided the development of Calculus. If anyone tried to ask him whether he was a mathematician or physicist, I think that he would be mystified by the question. He wrote that "The description of right lines and circles, upon which geometry is founded, belongs to mechanics", which indicates that physics and geometry have been interlinked all the way back to Euclid.
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1$\begingroup$ I think the last paragraph on Mark Levi may answer the question, provided that Mark Levi is a mathematician (I haven't checked). @SamHopkins Perhaps this paragraph should be placed first instead of last. $\endgroup$ Commented May 21 at 7:43
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1$\begingroup$ Mark Levi is at the Math Department at Penn State, which certainly qualifies him as a "mathematician". $\endgroup$ Commented May 21 at 8:22