# What are some historical examples in physics of heuristic proofs of mathematical results?

In the proceedings of the XIth International Congress of Mathematical Physics Edward Witten wrote (p. 704)

[$$\dots$$] when a mathematical result is really relevant to a physics problem it often happens that, turning things around, the result can be deduced from the behavior of the physics problem.

Witten's own work perfectly exemplifies this method. This method can be traced back to Archimedes, famous for his mechanical style.
I would be interested in learning about any piece of work in this vein that happened between Archimedes and Witten. In particular, is there any example from the work of natural philosophers in the XVIIIth or XIXth century? Is there any striking example from the first half of the XXth century with a mathematical result whose proof had been first sketched by physicists, using a heuristic proof originating in general relativity or quantum mechanics, and later made rigorous by mathematicians, possibly using very different methods?
Please provide references either to original pieces of work or to historical accounts.

Edit: as it should be clear from Witten's quote the question is not so much about physical insights leading to new fields of mathematics, but rather about physical insights leading to sketches of proofs of conjectures in well established mathematical fields or leading to new results altogether in a well established mathematical theory.

• There are too many. Bernoulli's famous solution to the brachistochrone was based on Fermat's least time principle in optics. Feynman came up with a formula for solution to parabolic equations in terms of his path integral, which was later made rigorous by Kac, see Feynman–Kac formula. Making Feynman integrals rigorous in QFT is still an open problem. – Conifold Oct 1 '20 at 8:18
• I upvoted for your Bernoulli example. – Ansonī Bōdo Oct 1 '20 at 8:30
• @Conifold Was the Feynman-Kac formula used to solve an important problem in math? Or at least used to shed a new light on such a problem? – Ansonī Bōdo Oct 2 '20 at 16:37
• Of course, it allows to solve PDE by simulating a stochastic process. Not to mention that the problem of extending non-trivial integration to infinite-dimensional spaces when no corresponding sigma-additive measure exists (which is typical) is of purely mathematical interest. One approach that emerged as a result was combining Feynman-Kac with analytic continuation, and the idea was imitated in more complex situations. – Conifold Oct 2 '20 at 17:09
• @Conifold Combining your Bernoulli and Feynman-Kac examples you may be able to write a nice answer. – Ansonī Bōdo Oct 2 '20 at 17:48

The first and most famous example was the book of Archimedes which is usually called The Method (more complete title is The Method of mechanical theorems), where he uses mechanics (statics) to compute volumes of various bodies. Unfortunately, this book was lost and found again only in the beginning of 20th century. Meanwhile what he did there was rediscovered in 17 century by people like Stevin, Fermat, Kepler and Cavalieri.

Archimedes (a pure mathematician of the highest rank) writes very clearly that the method is not rigorous. It took 2 centuries of development of Calculus/Analysis in 17-19 centuries to make it rigorous.

Another example is Maxwell's Treatise on electricity and magnetism, where he anticipates a lot of 20th century mathematics, including such things as differential forms, cohomology theories and extremal length. Unfortunately, mathematicians of 19th century did not appreciate Maxwell. (There is a nice paper about this, by Freeman Dyson, titled "Lost opportunities").

For example, Maxwell's discussion of electric resistance (Ch. VIII, art. 306-309) of conductors contains a method of estimating this resistance. Maxwell mentions Rayleigh as the author of the idea. This method was rediscovered by Ahlfors and Bers in 1950s under the name Extremal length which became one of the main working tools in the theory of conformal mappings. They do not refer on Rayleigh or Maxwell: the earliest predecessor they refer to is Courant, who wrote in 20th century.

Examples from 20th century are abundant: whole new areas of mathematics were developed to lay a rigorous foundation for the insights of Maxwell, Boltzmann and Gibbs in statistical mechanics. 20th century mathematicians are more inclined to talk to physicists and read their writings.

For example see Wikipedia articles Ergodic hypothesis and Ergodic theory. All statistical mechanics was developed by Maxwell, Boltzman and Gibbs on the "pysical level of rigor", and mathematicians are still busy with converting their "laws" into theorems. There is still a large gap between the laws of statistical mechanics and rigorously proved results. On the other hand, considerations from statistical mechanics led to discovery of new mathematical theorems not related to physics directly. See for example, D. Ruelle, Is our mathematics natural? or this paper. Such examples are really numerous.

Finally let me mention Fourier and his remarkable book Analytic theory of heat, whose main points were that a) every periodic function can be expanded into Fourier series, and b) every reasonable function on the real line can be represented by Fourier integral. He gives all kinds of ingenious arguments in favor of these statements (including experimental evidence with heated metal rings!). It took more than a century to mathematicians to rigorously state and justify his main assertions. Some of them were proved only recently, MR1769725 Ki, Haseo and Kim, Young-One, On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture, Duke Math. J. 104 (2000), no. 1, 45–73.

• Are you aware of a mathematical result first conjectured by a mathematician whose proof has been sketched by Maxwell or another physicist using his electromagnetic theory? Same question with Boltzmann and Gibbs you mentioned. I edited my question to avoid a misunderstanding. – Ansonī Bōdo Sep 30 '20 at 8:07
• @Ansoni Bodo: The simplest example is "ergodic hypothesis". But there are plenty. – Alexandre Eremenko Sep 30 '20 at 13:42
• Thanks. Could you please elaborate a bit on this example, maybe in your answer? – Ansonī Bōdo Sep 30 '20 at 20:37
• @Ansonī Bōdo: I did expand, but volumes can be written about this, and they are actually written. – Alexandre Eremenko Sep 30 '20 at 23:30
• Then please cite these volumes! I upvoted your answer for the Fourier example and for the Ruelle's reference. Thanks. – Ansonī Bōdo Oct 1 '20 at 8:42

Dyson proposed that a good understanding of quasicrystals would help solving the Riemann hypothesis.

My suggestion is the following. Let us pretend that we do not know that the Riemann Hypothesis is true. Let us tackle the problem from the other end. Let us try to obtain a complete enumeration and classification of one-dimensional quasicrystals. That is to say, we enumerate and classify all point distributions that have a discrete point spectrum...We shall then find the well-known quasi-crystals associated with PV numbers, and also a whole universe of other quasicrystals, known and unknown. Among the multitude of other quasi-crystals we search for one corresponding to the Riemann zeta-function and one corresponding to each of the other zeta-functions that resemble the Riemann zeta-function. Suppose that we find one of the quasi-crystals in our enumeration with properties that identify it with the zeros of the Riemann zeta-function. Then we have proved the Riemann Hypothesis and we can wait for the telephone call announcing the award of the Fields Medal.

In the same vein Salvatore Torquato, who is studying hyperuniformity, has outlined a proposal for the distribution of primes. His work is presented in Quanta with refs., most of them in the arxiv.

• Interesting, though these are mere proposals, not successful historical examples. – Ansonī Bōdo Sep 30 '20 at 9:55

Witten gave a spinorial proof of the positive energy theorem in GR. This was originally conjectured by Arnowitt, Deser and Misner in the early 60s. Special cases were then shown by a great many people with the general theorem finally established by Schoen and Yau.

Witten also gave a super-symmetric physics proof of the Atiyah-Singer index theorem. This had already been established by Atiyah & Singer.

Unfortunately, so far, all experiments have shown that supersymmetry is not an option taken by the real universe as opposed to the physically speculative universes dreamt up by physically orientated mathematicians. After all, Witten win a Fields prize for mathematics and not a Nobel prize for physics ...

As for string theory - well that was mostly Schwartz and Green's work who showed that various anomalies cancelled as so string theory was a viable project. Witten hadn't bothered to work on string theory until then.

Rather like Picasso, who looked over other people's work, deciding what he could 'steal' and work on himself. After all, Picasso did state:

good artists borrow, great artists steal ...

As you might gather from this description, I tend to think of Wittens work as being over-hyped as far as its significance to actual living and breathing physics is concerned.

• Dear Mozibur, could you roughly specify the physics tools/models used (just the basic ingredients really) in the supersymmetric physics proof of the Atiyah-Singer index theorem? – Ansonī Bōdo Nov 13 '20 at 15:26