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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.

Kristian Berry
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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? https://en.wikipedia.org/wiki/Fourth_dimension_in_literature – J Kusin Jul 20 '23 at 04:04
  • 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. – Kristian Berry Jul 20 '23 at 04:20
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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 Jul 20 '23 at 05:23

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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.

niels nielsen
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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)? – Kristian Berry Jul 20 '23 at 03:00
  • 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](https://philosophy.stackexchange.com/a/100810/37256) 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 Jul 20 '23 at 03:46
  • An alternate view https://philosophy.stackexchange.com/a/64745/37256 – Rushi Jul 20 '23 at 03:46
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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. – niels nielsen Jul 20 '23 at 04:19
  • @Rushi, show me exactly how Herbart's work inspired that of Riemann. – niels nielsen Jul 20 '23 at 04:22
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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 Jul 20 '23 at 04:45
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    @nielsnielsen, Nowak devotes a whole book to the issue. I provided further details in my [mathscinet review](https://mathscinet.ams.org/mathscinet/article?mr=3469866) in case you have access to that. – Mikhail Katz Jul 20 '23 at 06:52
  • @nielsnielsen, a short answer to your question is that it involved a sustained intellectual effort to break out of the cave of absolute space: Herbart was able to accomplish this, and Riemann followed suit. With hindsight this might seem easy but it wasn't at the time. The same applies to the idea that Lie theory is an integral part of mathematics, which seemed extremely counterintuitive to folks at Berlin at the time :-) – Mikhail Katz Jul 20 '23 at 07:49
  • @mikhail Katz, thank you for your comments, I will have a look at the Nowak reference. which of the many novaks and nowaks is the right one? -NN – niels nielsen Jul 20 '23 at 16:34
  • @nielsnielsen, the reference is provided in [my answer here](https://philosophy.stackexchange.com/a/100810/27522). It's an article, not a book. – Mikhail Katz Jul 24 '23 at 07:38