2011-09-24 17:46:05 8 Comments

The purpose of this question is to ask about the role of mathematical rigor in physics. In order to formulate a question that can be answered, and not just discussed, I divided this large issue into five specific questions.

**Update February, 12, 2018:** Since the question was put yesterday on hold as too board, I ask future to refer only to questions one and two listed below. I will ask a separate questions on item 3 and 4. Any information on question 5 can be added as a remark.

What are the most important and the oldest insights (notions, results) from physics that are still lacking rigorous mathematical formulation/proofs.

The endeavor of rigorous mathematical explanations, formulations, and proofs for notions and results from physics is mainly taken by mathematicians. What are examples that this endeavor was beneficial to physics itself.

What are examples that insisting on rigour delayed progress in physics.

What are examples that solid mathematical understanding of certain issues from physics came from further developments in physics itself. (In particular, I am interested in cases where mathematical rigorous understanding of issues from classical mechanics required quantum mechanics, and also in cases where progress in physics was crucial to rigorous mathematical solutions of questions in mathematics not originated in physics.)

The role of rigor is intensely discussed in popular books and blogs. Please supply references (or better annotated references) to academic studies of the role of mathematical rigour in modern physics.

(Of course, I will be also thankful to answers which elaborate on a single item related to a single question out of these five questions. **See update**)

Related Math Overflow questions:

Examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics (related to question 1);

Examples-of-using-physical-intuition-to-solve-math-problems;

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## 4 comments

## @Urs Schreiber 2011-09-27 23:23:28

Rigor is clarity of concepts and precision of arguments. Therefore in the end there is no question that we want rigor.

To get there we need freedom for speculation, first, but for good speculation we need...

in the words of our review, which is all about this issue.

Sometimes physicists behave is if rigor is all about replacing an obvious but non-precise argument with a tedious and boring proof. But more often than not rigor is about identifying the precise and clear definitions such that the obvious argument becomes also undoubtly correct.

There are many historical examples.

For instance the simple notion of differential forms and exterior derivatives. It's not a big deal in the end, but when they were introduced into physics they not only provided rigor for a multitude of vague arguments about infinitesimal variation and extended quantity. Maybe more importantly, they clarified structure. Maxwell still filled two pages with the equations of electromagnetism at a time when even the concepts of linear algebra were an arcane mystery. Today we say just $d \star d A = j_{el}$ and see much further, for instance derive the charge quantization law rigorously with child's ease. The clear and precise concept is what does this for us.

And while probably engineers could (and maybe do?) work using Maxwell's original concepts, the theoreticians would have been stuck. One can't see the subtleties of self-dual higher gauge theory, for instance, without the rigorous concept of de Rham theory.

There are many more examples like this. Here is another one: rational CFT was "fully understood" and declared solved at a non-rigorous level for a long time. When the rigorous FRS-classification of full rational CFT was established, it not onyl turned out that some of the supposed rational CFT construction in the literature did not actually exist, while other existed that had been missed, more importantly was: suddenly it was very clear why and which of these examples exist. Based on the solid ground of this new rigor, it is now much easier to base new non-rigorous arguments that go much further than one could do before. For instance about the behaviour of rational CFT in holography.

Rigor is about clarity and precision, which is needed for seeing further. As Ellis Cooper just said elsewhere:

## @Abhimanyu Pallavi Sudhir 2013-08-09 15:51:23

Rigour is certainly

notclarity, etc. , but in fact the suffocation of clarity .## @Urs Schreiber 2013-08-09 15:53:55

You haven't seen clarity yet.

## @Ron Maimon 2013-08-21 03:54:06

@UrsSchreiber: Maybe Dimension10 has seen clarity in some cases. The main issue with rigor is that it is subject to a ton of human arbitrariness, where a certain development path is chosen from the infinite number of possible development paths and declared "the rigorous path", and the benefits of time saved from the standardization means that all mathematicians go down this road, and anyone going down another road is "unrigorous". This introduces human social annoyances into the evaluation of ideas, when the structure you are defining is actually much more universal. You don't want sociology.

## @timur 2016-03-21 23:07:17

@RonMaimon: There is no "the rigorous path," nobody would say your proof is "unrigorous" just because you used a different path. One theorem can be proved by two different ways, both of them completely rigorous.

## @Ron Maimon 2012-05-28 09:14:09

Rigorous arguments are very similar to computer programming--- you need to write a proof which can (in principle) ultimately be carried out in a formal system. This is not easy, and requires defining many data-structures (definitions), and writing many subroutines (lemmas), which you use again and again. Then you prove many results along the way, only some of which are of general usefulness.

This activity is extremely illuminating, but it is time consuming, and tedious, and requires a great deal of time and care. Rigorous arguments also introduce a lot of pedantic distinctions which are extremely important for the

mathematics, but not so important in the cases one deals with in physics.In physics, you never have enough time, and we must always have a only just precise enough understanding of the mathematics that can be transmitted maximally quickly to the next generation. Often this means that you forsake full rigor, and introduce notational short-cuts and imprecise terminology that makes turning the argument rigorous difficult.

Some of the arguments in physics though are pure magic. For me, the replica trick is the best example. If this ever gets a rigorous version, I will be flabbergasted.

Here are old problems which could benefit from rigorous analysis:

There are a bazillion problems here, but my imagination fails.

There are a few examples, but I think they are rare:

pure logic, where they were able to demonstrate that an appropriate tiling with complicated matching edges could do full computation. The number of tiles were reduced until Penrose gave only 2, and finally physicists discovered quasicrystals. This is spectacular, because here you start in the most esoteric non-physics part of pure mathematics, and you end up at the most hands-on of experimental systems.This has happened several times, unfortunately.

In addition to this, there are countless no-go theorems that delayed the discovery of interesting things:

There are several examples here:

I can't do this, because I don't know any. But for what it's worth, I think it's a bad idea to try to do too much rigor in physics (or even in some parts of mathematics). The basic reason is that rigorous formulations have to be

completely standardizedin order for the proofs of different authors to fit-together without seams, and this is only possible in very long hindsight, when the best definitions become apparent. In the present, we're always muddling through fog. So there is always a period where different people have slightly different definitions of what they mean, and the proofs don't quite work, and mistakes can happen. This isn't so terrible, so long as the methods are insightful.The real problem is the massive barrier to entry presented by rigorous definitions. The actual arguments are always much less daunting than the superficial impression you get from reading the proof, because most of the proof is setting up machinery to make the main idea go through. Emphasizing the rigor can put undue emphasis on the machinery rather than the idea.

In physics, you are trying to describe what a natural system is doing, and there is no time to waste in studying sociology. So you can't learn all the machinery the mathematicians standardize on at any one time, you just learn the ideas. The ideas are sufficient for getting on, but they aren't sufficient to convince mathematicians you know what you're talking about (since you have a hard time following the conventions). This is improved by the internet, since the barriers to entry have fallen down dramatically, and there might be a way to merge rigorous and nonrigorous thinking today in ways that were not possible in earlier times.

## @Yvan Velenik 2011-09-25 09:38:42

I still think that it's not the right place for this type of questions. Nevertheless, the topic itself is interesting, and I'll also have a shot at it. Since I'm neither a philosopher of science, nor an historian (and there are probably very few such people on this site, one of the reasons this question might not be suitable),

I'll focus on my own restricted field, statistical physics.1) There are many. For example, a satisfactory rigorous derivation of Boltzmann equation, the best result to this day remaining the celebrated theorem of Lanford proved in the late 1970s. In equilibrium statistical mechanics, one of the major open problems is the proof that the two-dimensional $O(N)$ models have exponentially decaying correlations at all temperatures when $N>2$ (there is supposedly a close relationship between such models and four-dimensional gauge models, and this problem might shed light on the issue of asymptotic freedom in QCD, see this paper for a critical discussion of these issues). Of course, there are many others, such as trying to understand why naive real-space renormalization (say, decimation) of lattice spin systems provides reasonably accurate results (even though such transformations are known to be generally ill-defined mathematically); but it seems to me that it's unlikely to happen, which does not mean that the philosophy of the renormalization group cannot find uses in mathematical physics (it already has led to several profound results).

2) Well, one major example was Onsager's rigorous computation of the free energy of the 2d Ising model, which showed that all approximation schemes used by physicists at that time were giving completely wrong predictions. Rigorous results can also lead to (i) new approaches to old problems (this is the case recently with SLE), (ii) new results that were not known to physicists (this is the case with, e.g., the results of Johansson and others on growth models), (iii) a much better understanding of some complicated phenomena (e.g., the equilibrium properties of fixed magnetization Ising models), (iv) settling controversies in the physics literature (a famous example was the problem of determining the lower critical dimension of the random-field Ising model, which was hotly debated in the 1980s, and was rigorously settled by Bricmont and Kupiainen).

3) None that I know of. Although, one might say that the "paradoxes" raised against Boltzmann's theory by Zermelo and Loschmidt were both of mathematical nature (and thus criticized the apparent lack of of rigour of Boltzmann's approach), and did delay the acceptance of his ideas.

4) Not sure about this point. Certainly the numerous conjectures originating from physics, in particular striking predictions, provide both motivation, and sometimes some degree of insight to the mathematicians... But I am not sure that's what you're asking for.

5) There are many papers discussing such issues, e.g.:

and references therein.

## @Yvan Velenik 2011-09-27 14:58:50

@András Bátkai : yes, I find it one of the most compelling example. But it should be pointed out that it took until the 1960s for (most) physicists to take it really seriously (remember that to most people, this was a non-realistic, two-dimensional toy model, and they were more willing to associate the disagreements with predictions from their approximation schemes to pathologies of the model rather than to failure of these approximations.

## @Ron Maimon 2013-08-21 03:55:59

The philosophy article is a bit of a problem--- the issue with rigor is the social structure around it, there are people who will tell you that you aren't rigorous when you are, for example, with a path integral, which can be defined very well, but mathematicians won't accept the definition, because they don't like constructions based on probability. One issue here is that the formal development of measure theory is screwed up completely in mathematics, and you can't expect physicists to change. but this is Davey's point. Davey is also misrepresenting Dirac's view on delta-distributions.

## @Joe Fitzsimons 2011-09-25 05:02:38

I can by no means claim to give a full answer on this question, but perhaps a partial answer is better than no answer at all.

As regards (1) perhaps the most famous example is the Navier-Stokes equation. We know it produces extremely good results for modeling fluid flow, but we can't even show that there always exists a solution. Indeed, there is a Clay prize going for proving the existence of smooth solutions on $\mathbb{R}^3$ (problem statement here).

An example of (2) is that the study of topological quantum field theory has been motivated at least in part by mathematics.

As regards (3) I don't really think this has ever happened. However, by this, I do not mean that demanding rigor would not prevent or slow the progression of physics, but rather that it seems extremely hard to find an example of a case where a relatively large community has not simply ignored any such demand. Certainly it is true that mathematically rigorous formulations often follow far behind the current state of the art in physics, but there is nothing unexpected about this.

I do not currently have any good answers as regards the remainder of your question.

There is a relatively interesting essay on this (C. Vafa - On the future of mathematics/physics interaction) in Mathematics: Frontiers and Perspectives, which also mentions the TQFT example.