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The Role of Rigor

+ 23 like - 0 dislike

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.

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

2) 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.

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

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

5) The role of rogor is intensly 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.)

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; Demonstrating-that-rigour-is-important

This post has been migrated from (A51.SE)

asked Sep 24, 2011 in Theoretical Physics by Gil Kalai (325 points) [ revision history ]
retagged Mar 24, 2014 by dimension10
I'll post the link I posted on Meta here as well, it contains a few very nice answers to Part 1 of your question: http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics

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Dear Moshe, By no mean did I mean to belittle the quality of discussions on blogs and popular books on this matter. Some such discussions are of good quality! Part 5 of my question was just asking for studies of this issue on professonal academic journals. Parts 1-4 are questions (hopefully answerable) aimed at mathematical physicists and theoretical physcisits. Also, I completely agree with not being sure if the question is within the purview of the forum.Dear Michael, Thanks! I had also another MO question in mind http://mathoverflow.net/questions/37610/demonstrating-that-rigour-is-important

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Moshe, in any case I will be very interested to see specific answers (not discussions) on what notions/results from physics still lack mathematical rigorous description; on cases where rigorous mathematical framework came from advances in physics and where rigorous mathematical framework proposed in mathematics was useful in physics.

This post has been migrated from (A51.SE)
As I said this is just my gut feeling and I am happy to be proven wrong, certainly this set of issues is very interesting.

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I think that the problem is that two notions of rigour are mixed: An insistence on laying open people's assumptions is probably always good, an insistence on sticking to universally accepted assumptions not so much. I think that Archimedes' paper on calculating volumes using "calculus" (that is, weighing infinitely small slices and comparing them) is a good example to see that the ancient Greeks could have advanced a lot if they had not insisted of staying within the accepted framework of finite numbers.

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3 Answers

+ 17 like - 0 dislike

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.

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answered Sep 25, 2011 by Yvan Velenik (1,110 points) [ no revision ]
2) is a great example. This is why this question should stay...

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@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.

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+ 13 like - 0 dislike

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...

...solid ground, which is the only ground that serves as a good jumping-off point for further speculation.

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 probaby 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:

Rigor cleans the window through which intuition shines.

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answered Sep 27, 2011 by Urs Schreiber (5,305 points) [ no revision ]
Great asnwer! ${}$

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+ 11 like - 0 dislike

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.

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answered Sep 25, 2011 by Joe Fitzsimons (3,555 points) [ no revision ]

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