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Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about "The Unreasonable Effectiveness of Physics in Mathematics".

What can be the reasons for it ?

Do physicists have some tools/ideas/techniques which allow them to make insights, which are not seen for mathematicians? Or it is just because Witten&K are very ... very smart ?

If, yes, what are these tools/ideas ? How to learn/absorb/(put into math. framework) them?

What can be the further applications of these ideas ?

Being a mathematician, but working in physicists surrounding for many years, I have thought on this questions for quite a while. The recent MO question Mathematician trying to learn string theory prompts me to ask it here.

I would think, that yes, there are such "ideas". But from some outstanding mathematicians I've heard an opposite opinion.

My vague feeling is that quantum field theory and string theory it is something like an analysis/differential geometry on infinite-dimensional manifolds. But these manifolds are not abstract, say Banach modeled manifolds, which theory is not so rich, but kind of maps from one finite-dim. manifold to another, which has certain specific structures which are not fully revealed by mathematicians. E.g. vertex operator algebras, arise from maps of circle to manifold, if we map not circle but something higher dimensional there should be something more complicated. Another issue is about Feynman integral, which allows physicist to use integration techniques in geometric problems, it is not well-defined mathematically, but it might be it cannot be defined in very general form of infinite-dimensional integrals, but again physicist have an intuition where it can be defined, where cannot, and proper mathematical theory should first clarify the setup where it exists, rather than trying to build general theory which might not exist. These words are probably very vague, so might be answers help to me clarify.

I think everybody knows the influence of physics happened from 80-ies, but for completeness let me mention just a few.

Donaldson used instanton moduli spaces in his study of 4-folds.

Faddeev, Drinfeld et. al. created quantum groups

Representation theory of infinite-dimensional algebras have been large influenced by conformal field theory developments.

Witten's contributions are numerous his Fields Medal says more than I can say.

Mirror Symmetry, quantum cohomology etc...

The works of Fields Medalist Kontsevich and Okounkov are largely influenced by physics.

So on an so forth...

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    $\begingroup$ Dijkgraaf has been giving talks with the same title as yours for several years now. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 16, 2012 at 15:53
  • $\begingroup$ @Scott thank you, I did not know this. Googling I can find 1-2pages with abstracts of his talks, but I do not see any the slides/paper of talk(s) itself. $\endgroup$
    – Alexander Chervov
    Commented Dec 16, 2012 at 16:02
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    $\begingroup$ In order to really address the question of your title, one probably needs to consider that this did not start in the 1980's. Certainly it goes back to the 19th century (Riemann, Gauss, Poincare etc.), even back to the 17th century (Newton etc.), and even back to the first half-millenium B.C. (Archimedes, Euclid etc.) $\endgroup$
    – Lee Mosher
    Commented Dec 16, 2012 at 16:11
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    $\begingroup$ @Alexander Chervov The Dyson quote is from his "Missed Opportunities" Gibbs lecture in 1972. He says "As a working physicist, I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce. Discussing this divorce, the physicist Res Jost remarked the other day, `as usual in such affairs, one of the two parties has clearly got the worst of it.'" Dyson clearly thought that physics had got the worst of the divorce. $\endgroup$
    – Jeff Harvey
    Commented Dec 17, 2012 at 0:01
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    $\begingroup$ Dyson's lecture was published in Bulletin of the AMS (which is freely available). Here's the link: ams.org/journals/bull/1972-78-05/S0002-9904-1972-12971-9/… $\endgroup$
    – Pablo Lessa
    Commented Dec 17, 2012 at 14:21

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My opinion is that physicists transferred from study of "individual objects" to that of "large systems" where the order arises from limit probability laws rather than from simple deterministic formulae and from the study of something "readily observable" to something that is, essentially, "a purely mathematical object" invisible to a direct experiment. This brought them to the realm traditionally reserved for pure mathematicians. And, of course, with their eagerness to use whatever tools they have available in any way that is short of total lunacy, they went on to make predictions, many of which could be confirmed experimentally, leaving a long trail of successes and failures in their wake for mathematicians to explain.

I do not know the situation with the string theory and low dimensional topology but I have some idea about what's going on in random matrices (thanks to Mark Rudelson and his brilliant series of lectures) and in percolation/random zeroes (thanks to Stas Smirnov and Misha Sodin). The thing that saves physicists from making crude mistakes there is various "universality laws".

Here is a typical physicist's argument (Bogomolny and Schmidt). You want to study the nodal domains of a random Gaussian wave $F$ (the Fourier transform of the white noise on the unit sphere times the surface measure). Let's say, we are in dimension 2 and want just to know the typical number of nodal lines (components of the set $\{F=0\}$) per unit area. The stationary random function $F$ has only a power decay of correlations. However, we ignore that and model it with a square lattice that has the same length per unit area as $F$ (this is a computable quantity if you use some standard integral geometry tricks). Now, at each intersection of lattice lines, we choose one of the two natural ways to separate them (think of the intersection as of a saddle point with the crossing lines being the level lines at the saddle level). Then, we get a question (still unresolved on the mathematical level, by the way) about a pure percolation type model. Thinking by analogy once more, we get a numerical prediction.

From the viewpoint of a mathematician, this all is patented gibberish. There is no way to reduce one process to another (or, at least, no one has the slightest idea how this could be done as of the moment of this writing). Still, the Nature is kind enough to make the answers the same or about the same for all such processes and Mathematics is powerful enough to provide an answer (or a part of an answer) for some models, so the physicists run a simulation, and, voila, everything is as they predicted and we are left with 20 years or so worth of work to figure out what is really going on there.

I'm not complaining here, quite the opposite: this story is really quite exciting and the work mentioned is both real and fascinating. We are essentially back to the days when Newton tried to explain the nature of gravity looking at Kepler's laws trying various options and separating what works from what doesn't. I'm only saying that the famous "physicists' intuition", which is so overrated, is actually just the benevolence of Nature. Why should the Nature be so benevolent to us remains a mystery and I know neither a physicist, nor a mathematician, who could shed any light on that. The best explanation so far is contained in Einstein's words "God is subtle, but not malicious", or, in a slightly less enigmatic form, "Nature conceals her mystery by means of her essential grandeur, not by her cunning".

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  • $\begingroup$ Especially good observations at the end! Also, this reminded me that one could have been very unhappy (and I think some were, though I do not recall specifics) at Newton's postulation of gravity's "action at a distance" (mediated by what???) to "explain" planetary orbits. Not to mention that Newtonioan "fluxions" or Leibnizian "infinitesimals" were viewed with immense skepticism by non-enthusiasts (such as Bishop Berkeley) for 150+ years, despite the amazing results. At the time, this was counted "mathematics", but disparaged for dubious proof mechanisms, since (until after 1800 I think) ... $\endgroup$
    – paul garrett
    Commented Dec 17, 2012 at 0:57
  • $\begingroup$ ... "real proof" status was only granted to things resembling Euclid. "Analysis" and "algebra" were not legit mathematics by a strict standard. I am also reminded that until only 100-200 years ago, "natural philosophy" was part of philosophy (in the human pigeon-holing of things), but (strict) "philosophy" gradually rejected (what we now call) "science" until "strict philosophy" nowadays partly has the negative definition of reference only to that which cannot be tested by experiment? :) $\endgroup$
    – paul garrett
    Commented Dec 17, 2012 at 1:00
  • $\begingroup$ You only make mention of the study of "macroscopic systems", and in that case I would say it's nature's size (seen as made up by so many tiny elements), rather than "nature's benevolence", as you call it, what lies behind the success of physics in such kind of problems (I mean, the fact that one can think about laws of large numbers and deviation principles, which hold so well). As for the other examples in the OP, i don't know how "nature's kindness" is evident. $\endgroup$
    – Camilo Sarmiento
    Commented Dec 18, 2012 at 8:59
  • $\begingroup$ (You could always say it's logic's benevolence what underlies progress in, say, algebraic geometry, but that's a matter of taste, or "metaphysical stance", if you like) $\endgroup$
    – Camilo Sarmiento
    Commented Dec 18, 2012 at 9:01
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    $\begingroup$ It is not the sheer size that matters. There are very many genuinely different large scale random processes that are not reducible to each other. Still, they somehow split into few classes for which the answer seems to be the same in the whole class but proved only for one or two simplest representative processes in the class. That's what I call "universality". There was some recent progress in its understanding, but we are very far from grasping its mechanism yet. Even the CLT arises in 5-6 completely different ways in the proofs. As to other examples, I am talking only of what I know... $\endgroup$
    – fedja
    Commented Dec 18, 2012 at 14:32
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This question is very nicely discussed in the paper of Ruelle,

Ruelle, David(F-IHES) Is our mathematics natural? The case of equilibrium statistical mechanics. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 259–268.

But he does not really have an answer, and perhaps nobody has an exact answer.

My own opinion is that a large part of mathematics (some say all of it, but I disagree) actually comes from physics. If this is so, it is not surprising that physics helps in discovering new theorems. And there is nothing unreasonable in this.

Using Ruelle's example, if we ever discover an extraterrestrial civilization, its mathematics (most of it) must be equivalent to a large part of our mathematics. I am convinced that this is so. Most of mathematics is somehow determined by the laws of nature, that is by physics.

A completely different explanation (with which I cannot completely agree) is that some physicists are just very clever people, and once they start thinking of mathematics, they discover new things, and when they (physicists) publish these things everyone says that these results "came from physics". I agree that this happens sometimes, but on my opinion this is a small part of the picture.

I don't want to give examples (I am sure, many will be given), but I just want to make the point that this phenomenon ("unreasonable" effectiveness of physics in mathematics) occurs from the very beginning of mathematics (and physics). I mean the remarkable surviving work of Archimedes, A method of mechanical theorems, where he uses physical reasoning to prove or guess purely mathematical theorems. This example can be used to defend any of the two above explanations:-)

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It seems to me the answer to: "Do physicists have some tools/ideas/techniques which allow them to make insights, which are not seen for mathematicians?" is indeed yes. Not only such a tool exists but, in my opinion, it is also unique: functional integrals. Predictions based on that tool are what mathematicians have a hard time reproducing and justifying using well-established rigorous theories. Imagine a world where we still would not know how to define an ordinary integral rigorously, e.g., via Riemann sums, but where physicists, engineers etc. use them on a daily basis and with great success. This would be quite similar to the situation today with the heuristic theory of functional integrals developed by physicists. Also, there is an area of mathematics which aims at constructing and studying these objects rigorously: constructive quantum field theory or rigorous renormalization group theory.

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    $\begingroup$ Not really. The idea that the least entropy or the least action point is what you should look at can be traced to Laplace and Leibnitz and is by no means uncommon in the enumerative combinatorics and analysis. If you ask me (as I was asked once), what is a typical probability distribution on $[0,1]$ with the expectation $E$, I can answer as confidently as anyone else that it is $ae^{-bx}$ with appropriately chosen $a$ and $b$. That we do not blatantly use the tool we cannot justify doesn't mean we are not aware of it ;). $\endgroup$
    – fedja
    Commented Dec 17, 2012 at 14:51
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    $\begingroup$ @Fedja: what does "Not really" apply to, and how does the following soliloquy support that assertion? $\endgroup$
    – Abdelmalek Abdesselam
    Commented Dec 17, 2012 at 15:20
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    $\begingroup$ It applies to the statement that mathematicians do not have that tool at their disposal to make predictions. They do but predictions without rigorous justifications aren't worth much in mathematics. The following passage is just an example of a question and an answer about the average of the space of all functions at the most basic level. Yes, we can make such predictions and statements if pushed against the wall. However we prefer not to and it is due to our free choice, not to our lack of knowledge. Also it is a decent example to work through to understand what the real gap is there. $\endgroup$
    – fedja
    Commented Dec 17, 2012 at 15:43
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    $\begingroup$ How does that apply to using the specific tool I mentioned which is functional integrals? Do you know many mathematicians who, pushed against the wall, pull out such an integral to reach a desired conclusion? In any case it is unrelated to the point I made which was about proving results and not just making predictions. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Dec 17, 2012 at 16:18
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    $\begingroup$ I'm not. I'm just answering the question as asked. Of course, what the physicists are doing with functional integrals relates to my example in the same way as, say, solving a stochastic ODE relates to finding the area under the graph of $x^2$, but I'm completely serious when I'm saying that it is a good example to work out and that it sheds light on what the problem is. If it sounds in any way condescending or impolite, I sincerely apologize, but the point of the message remains. $\endgroup$
    – fedja
    Commented Dec 18, 2012 at 14:39
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After upvoting and seconding "Hollowdead" and Alexandre E's answers (and various comments)... One might continue further on the idea that the alleged distinction between "mathematics" and "physics" is more a collection of conventions, artifacts of universities' development, artifacts of the human impulse to "classify", and so on. Another peculiar element that seems to have driven the "need" to distinguish these subjects is the modern-times mathematical fixation on a certain style of "proof" (despite the actual importance of heuristics, e.g., from physics) as the sine-qua-non of "mathematics". Thus, heuristic arguments/computations in physics often don't "qualify" as "mathematics" ... even while in the best-case scenarios Witten and other "physicists" have been led to suspect that proof-defined mathematics could proceed in a certain way which proved fruitful.

My own perception of my own "intuition" for mathematics is that it is "physical" in an informal sense, even on topics that are not part of the physics canon, nor "mechanics", nor... For me, this "physicality" includes many topics in "number theory" or "Galois theory" or "linear algebra" that are sometimes counted as "abstract mathematics" rather than "physical reality", although I disagree with the claims that this is "abstract".

Yes, of course, it is possible to employ a narrative style that specifically disassociates itself from "physical intuition", and there were historical motivations for at least setting up the possibility of reasoning that did not depend on "physicality", since some incorrect conclusions were reached by relying on "physical intuition" in situations that stretched our/human limitations a bit too far.

Another notion (which I do partly endorse) is that, since humans (and aliens) are in the physical world, the notion that mathematics (or anything else) would/could completely transcend or disconnect from this world seem unlikely, or pointless, or tautologically impossible, depending...

And there is the "false coincidence" pseudo-paradox notion: we do not notice cases that X is irrelevant to Y, nor remark upon it. :)

Thus, one way or another, I do think that the "suprising effectiveness of physics in mathematics" is due to a misrepresentation of the underlying ideas, and artificial stylization and compartmentalization of the official subjects, together with a not-so-unusual coincidence-fallacy amplifying it all.

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I think that since theoretical physics is really near the edge in its quest to understand the building blocks of our universe, it chooses the "right" mathematical structures for its development. Quantum field theory and String theory are a laboratory of great mathematical ideas, so the inappropriate mathematical structures (those which are not suitable for the description of reality) are put aside and by natural selection the appropriate and useful mathematical structures, tools and ideas are emphasized. Your question transforms itself then to the following: what is the interplay between the world of mathematics and the real world, and does mathematics really exist in some way?

It follows from the above considerations that there is no way to absorb the physical ideas into the framework of mathematics, unless mathematics itself changes its objective from seeking the absolute truth to seeking the truth about nature, i.e. unless mathematics transforms into mathematical physics.

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Perhaps less well-known, but no less significant is Dirac's:

http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html

essay on physics, mathematics and beauty.

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If you accept Max Tegmark's view that "our physical world is an abstract mathematical structure" (per https://arxiv.org/abs/0704.0646), then it would follow that the "effectiveness" of physics in mathematics is not unreasonable at all. It's exactly what you would expect.

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Physicist can build mathematical or mechanical models to describe a system. It is really a matter of choice. Mechanical models use mathematics but the processes and transformations describing the system are highly visual. Mathematicians are shaping physics. Mathematician’s influence on physics became profound during the 1920’s with the birth of modern quantum mechanics. Hilbert, Born, Heisenberg, Dirac and others produced mathematical models for quantum mechanics that are inherently abstract. This abstractness was not the norm prior to the 1920’s. Mechanical models were the preferred tools of physicist because in these models, processes and transformations are not abstract and can be manipulated to build new technologies. Mathematical models describe systems but what is actually occurring as it relates to processes and transformations is unclear and does not lend itself very well to manipulation at the deepest levels. Physicist can only bring new insights to geometry because unlike other branches of mathematics, geometry is talking about something real (space). Space is real and not abstract. Witten is smart and so are many other individuals who have failed to give any new testable insights into how real space works beyond Einstein.

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It is my experience that physics motivates and allows the consideration of objects, with some degree of rigor, whose nature would be very difficult to understand mathematically with no motivation and prior training.

Things like manifolds and line integrals are early examples of objects whose involvement in physical theories allowed me to understand their workings through context long before I had developed sufficient mathematical language and machinery to really articulate clearly what it was I understood.

I spent the last semester and the beginning of this summer constructing $\mathbb{R}$ (and larger real-closed fields up to the Surreals) directly out of cuts in a subclass of $O_n\times O_n\times O_n\times O_n~\;$ using set theoretical axioms (https://arxiv.org/abs/1706.08908*). Two of my math professors were reviewing the paper with me once a week, an analyst and an algebraist by training, and both of them repeatedly wondered what made me think of trying something like this. I was motivated by the fact that infinities seem to be inextricably working their way into the leading edge of problems in physics -- for example, it is my understanding that one of the disagreements between GR and QM is the existence some nasty divergent series describing a particles position and momentum states in a quantized setting, something that is typically handled through renormalization but which can't be here.

I figured that maybe these series would converge to some value in a larger, denser real-closed field, and I wanted to build those fields in the same way that $\mathbb{R}$ can be built out of $\mathbb{N}$ since so much of quantum mechanics relies on analysis together with indexing over some well defined discrete ordered subset. To do so I constructed larger discrete ordered rings that contained the integers out of $\delta$-numbers greater than $\omega$, then created fields of fractions for them and 'cut up' the fields to real-close them. I also algebraically closed all of these fields so we have an analogous 'non-Archimedean' version of $\mathbb{C}$ to work with. All of this was motivated by happenings in physics, or as another answer elegantly put it 'considerations of Nature'.

*currently under review for publication

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