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The purpose of this question is to supplement an earlier question The Role of Rigor and 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. This question asks for answers to Questions 3 and 4.

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

  2. 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.)

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    $\begingroup$ Much like your original post, this appears to be a list-based question which is generally considered off-topic as too broad. $\endgroup$ – Kyle Kanos Feb 12 '18 at 12:52
  • $\begingroup$ To be (slightly) flippant about it, why the concentration on math? Physics is empirical, you can possibly totally nail any pet theoretical idea in math formulation, but one experiment can kill it. $\endgroup$ – user184116 Feb 12 '18 at 14:13
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    $\begingroup$ The interest in math exists because it can signpost the way to useful experimental lines of inquiry and can unify multiple seemingly disparate phenomena. Even as an experimentalist I appreciate that. And these sorts of questions are interesting, but they seem to call for more back-and-forth than our engine is well suited to provide. $\endgroup$ – dmckee Feb 12 '18 at 17:43
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    $\begingroup$ @dmckee This is indeed close to the edge of the envelope in terms of the amount of back-and-forth that our engine can handle without breaking down into an unreadable mess - but the previous incarnation produced four thoughtful, high-quality contributions without crumbling into unreadable back-and-forth. We don't close questions because they might push the engine a bit, we close them because they have a high likelihood of becoming unreadable. This patently hasn't happened with the previous version. $\endgroup$ – Emilio Pisanty Feb 13 '18 at 9:56
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As to (3), this sort of thing is a loaded question. First, it's hard to actually give a good definition of rigor, but I'll just assume by "non-rigor" we mean "a state of affairs where a mathematical procedure can be shown to not work in all cases, but no attempt is made to show the cases where it's applied are valid".

Second, it's almost impossible to point out when rigor has "slowed down" physics because physicists are non-rigorous, a revolution that seems to date back to Dirac. It's basically taken as a given that not worrying about being logically consistent or putting everything on rigorous footing saves you time. In physics, we worry about getting the right answer and having a procedure that gives the right answer in at least some other cases.

That said, I can provide some anecdotes which at least show massive amounts of progress can be made without rigor, if you'll take that 'focusing on rigor' would slow down these investigations then these might be what you're looking for:

  1. Infinite Series -- This is actually an example from mathematics, but it was when mathematicians and physicists were the same job. The notion of an infinite series goes back at least as early as the Greeks. But arguments as to, "whether you can in fact add up infinite elements" were not really settled until the time of Newton. Later Fourier discovered his series expansion in sines and cosines, which prompted Gauss, Cauchy, and Abel to finally put notions of convergence on rigorous footing. Could the Greeks have done the work of Gauss, Cauchy, and Abel without the non-rigorous work of Fourier?

  2. QFT/The Dirac Equation -- To the best of my knowledge, canonical quantization does not really have a solid mathematical foundation. Perhaps someone in the world understands what the hell you are doing when you start with a classical theory, and then somehow just reinterpret an equation that describes the bulk motion of the center of mass of charged bodies, that are orders and orders of magnitude larger than any elementary particle, as an operator acting on the wave-function of the tiniest things in existence. That has always seemed pulled out of thin air and unquestioned to me.

    I have not ever found a translation of the Klein's paper which Dirac cites as containing the idea, nor have I been able to study the criticisms of the idea at the time, but it seems clear that Dirac was working by the seat of his pants here, asking himself "how to take the square root of an operator." All he cared about was pushing symbols in systematic way that had some cohesion, not the existence of the mathematical objects his calculations described. To this day, quantum field theory is not on rigorous foundations, but physicists have been using Dirac's procedure, and worse renormalization, to get the right answer. If the entirety of the physics community was worried about rigor, no physics would have been done in the past 100 years.

  3. The measurement problem -- Again in quantum foundations, we have the measurement problem. This is the issue that all quantum evolution is unitary, but representing the outcome of measurements requires non-unitary operators. In other words, quantum theory says "when you're not measuring something, use this procedure (evolution via the Schrodinger equation), when you go to measure something, use this procedure (the born rule), don't worry that you're a quantum system".

    This has long been pointed out as something that desperately needs some philosophical consistency, but as of yet there is no explanation for why we need two theories to describe the universe based on the arbitrary notion of "measurement." I am not sure whether you would call this non-rigorous, as there is a defined mathematical procedure in both cases, but this has the same flavor as non-rigor in that instead of hashing out the edge case (measurements), physicists simply brush the point under the rug and say "use this workaround".

As to (4), again, I think this is hard to answer. Physical systems are a subset of all mathematical systems. There has been plenty of cases where physics revealed that certain aspects of mathematics were very important, for instance

  1. General Relativity and Reimannian Manifolds,
  2. Newtonian Mechanics and Differential Equations.

And there are plenty of cases where physical experiments answer questions that we could not figure out how to calculate from first principles (e.g. any numerical integration done on a computer). But I am not sure if there are any cases where someone was sitting in a lab with a laser that changed color in medium, and then immediately realized that the number of holes in a 2 dimensional surface is enough to classify all 2D spaces.

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