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This question is motivated by something from the set-theorist Hugh Woodin, a prediction he has made and styled as empirical, according to which a subtheory that he uses will not be shown inconsistent for centuries to come. That's a more "purely mathematical" example of this family of reasoning, though; what I have more specifically in mind right now seemed to come up in one of the essays I was reading about whether set theory has any special value in the production of physics theories.

But so it seems like you could make a trivial prediction like, "Someone will somehow find a way to use something from set theory, to come up with a new meta-explanation/grounding for physics." That wouldn't be a matter of falsity but vacuous success, then.

Perhaps a known(?) example of a better (less trivial) prediction, or something that could have been a prediction, along these lines, concerns the role of the Dehorney order in the background understanding of anyons. I mean, I don't actually know how much that order is involved in understanding anyons, maybe I'm misinterpreting the information I've looked over.

On the personal side of things, I have been trying for a few years now to work out a mathematical formalism for Lee Smolin's changing-laws-of-physics thesis. Offhand, I think I can see a way to model a physical universe in terms of well-founded sets of fields such that the "edges" of such a universe change (with spatial expansion and temporal progression) so as to "reprogram" the contents of spacetime for that universe. But I don't know if this will really "go anywhere," i.e. I don't know how to come up with a meaningful prediction on this basis. I would like to say, "I predict that we will be able to nontrivially define laws of physics in terms of infinitary logic and elementary embeddings, such that..." and go from there, but would that really be a scientific prediction, a meta-scientific prediction, or not really a "prediction" in much of a useful sense at all? Because at other times, when I think over the thousands of pages of set-theory material I've read, and contemplate the daunting content of nLab (over 18,000 entries! I hardly know where to begin with it), it seems as if there's probably no way that I myself will be able to find purchase in something that could not be easily adapted, like an all-purpose theory-engine oil, to make up whatever random "model" I'd be pleased to, and then it seems as if I'd be better off spending my time on something besides untestable conjectures about physics.

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    When X mathematical theory is understood by x,000 people, someone is y% likely will write a novel(la) about it, e.g. Flatland? Are you relegating usefulness to physics/science? Is this too trivial? en.wikipedia.org/wiki/Fourth_dimension_in_literature
    – J Kusin
    Commented Jul 20, 2023 at 4:04
  • Hmm. Well, on a "sideways level" of analysis, it benefits pure science/mathematics when they have more and more contributors, so one might reflect on how appealing science fiction can inspire various people to do pure science and sometimes pure mathematics too, and this materially advances the production of scientific economic value. So it could be both that science fiction has an aesthetic utility (so to speak) as well as a proto-theoretical (or methodological) one. Commented Jul 20, 2023 at 4:20
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    What you seem to be getting at is not prediction but what the French call aperçu, in developed form, research program. Fruitful, not necessarily correct. Its "non-triviality" is not a function of the area, math or not, but of the aperçu itself. If it is "something or other will help somewhere" then the problem is not that math is too malleable. Researchers often "predict" by heavily investing into a specific line of inquiry with no assurances that it is not a dead end. Hamilton "predicted" that Ricci flow will crack geometrization, Perelman kept at it and it finally paid off.
    – Conifold
    Commented Jul 20, 2023 at 5:23

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The answer is almost certainly no. There have been several celebrated cases in physics where a math tool, invented by a starving mathematician and then lying unused for dozens of years, was discovered to be exactly what was needed to solve an important physics problem.

Non-Abelian group theory was discovered to furnish the correct formalism for organizing subatomic particles into families; so powerful was the result that the physicists using it were able to predict the existence of undiscovered particles in the form of unoccupied spots within the family structure. Subsequent searches turned up those particles with exactly the right characteristics that the family structure dictated. Nobel prizes resulted.

Another example was Riemann's work on the characterization of curved spaces, 50 years before Einstein made practical use of Riemannian geometry in developing general relativity.

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  • Hmm. Might we say then that math-physics connections can never be predicted but can only be discovered, or even (worse?) they involve some kind of "random epistemic function" (from our point of view, anyway)? Commented Jul 20, 2023 at 3:00
  • I believed (as i guess most do) what this answer is saying about Riemanm creating the geometry and Einstein fitting it to 'reality' a ½ century later. Thus recent answer corrects this view showing that it's actually Herbart-Philosophy → Riemann-math→Einstein-physics. We tend to ascribe 'reality' to only the last and 'creativity' to the others. But that's just the physicalism/scientism dominant world view at play. To me personally it's simply an arbitrary culture choice that Beethoven created while Newton discovered
    – Rushi
    Commented Jul 20, 2023 at 3:46
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    @KristianBerry, I do not know. the picture is complicated by mathematical physicists like Alain Connes, who strategically invents mathematical systems like noncommutative algebras, to use as tools to study physics. Commented Jul 20, 2023 at 4:19
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    Sorry I didnt indicate the above comments were directed at @KristianBerry. As for your query, it's best raised with Katz who made that other answer who quotes a book by Nowak Gregory. BTW Katz is one of the high rep users on mathexchange and mathoverflow.
    – Rushi
    Commented Jul 20, 2023 at 4:45
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    @nielsnielsen, Nowak devotes a whole book to the issue. I provided further details in my mathscinet review in case you have access to that. Commented Jul 20, 2023 at 6:52

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