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I've asked the question below on mathexchange here about 2 weeks ago. while I did not satisfied with the comments and answer there specially because the lack of examples and references that I was demanded for and also IMO it did not logically well discussed there, I've decided to put it as an inquiry on Physics community.

"We know that mathematical models of physical phenomena need to becomes more and more sophisticated as our observations become more precise and comprehensive by utilizing more advanced instruments that we upgrade over time. Also extending the mathematical models by means of mathematical tools and logical inferences lead us to predict and control physical incidents until observations confirm validity of the predictions. The problem arises where we couldn't find a comprehensive consistent mathematical model which fits over observations data and justify all of them. In such a case, the necessity of evolving mathematical theories seems crucial.

Questions here come to mind considering examples related to content above, imagine the observation data of Mercury’s movement that could not be justified by Newtonian mechanics achieved sooner than the appearance of non-Euclidean geometry, then wouldn’t it be influential on the advent of a different fifth postulate of geometry? Maybe the physics(or in more general term "the mother nature") can lead us for example to an axiom in order to make a mathematical hypothesis provable by adding it to our axiomatic system? How and under what circumstances can mathematics benefit from physics’ experimental achievements in order to evolve mathematical theories in a way that is not just be helpful for modelling physics but also useful for proving a mathematical hypotheses or altering a fundamental postulate of an axiomatic system?

Question : To what extent can advances in physics measurements and observations (not just ideas) aid mathematicians with proving a mathematical hypothesis that is based on an effectively generated and consistent axiomatic system like PA or ZFC?

For example consider dimensionless physical constant $\alpha$ “fine-structure constant” characterizing the strength of the electromagnetic interaction between elementary charged particles ,which Michael Atiyah has stated in his attempt to prove the Riemann hypothesis. Any precise explanation of this and other examples or references related to the topic are welcome."

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closed as primarily opinion-based by AccidentalFourierTransform, Jon Custer, Qmechanic Nov 2 '18 at 1:47

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I believe I remember in highschool reading in Brian Greene's "The Elegant Universe" about how string theorists solved some sphere packing problem that mathematicians could not figure out. That is as much detail as I can give though... $\endgroup$ – Aaron Stevens Nov 1 '18 at 20:55
  • $\begingroup$ To be fair string theory isn't really physics. It's not based on observation or empirical evidence. It is not the same as suggesting that Mercury's orbit may have inspired a new movement in geometry. $\endgroup$ – ggcg Nov 1 '18 at 21:01
  • $\begingroup$ @ggcg That's a way to alienate any theoretical physicist trying to come up with new theories of the universe. I'm not one to really care about string theory, but it's still based on trying to explain observations and empirical evidence. Just because it's not on the same footing as your example doesn't mean it's not an example where attempted advances in physics helped mathematicians. $\endgroup$ – Aaron Stevens Nov 1 '18 at 23:20
  • $\begingroup$ @AaronStevens, sorry you feel that way. I am in fact a theorist. It's been a long time since I studied string theory, and it really goes back to Dirac and other early particle QFT theorists trying to describe hadron physics before we had color (SU(3) qcd) theory. But modern string theory doesn't seem to be motivated by empiricism whereas the early attempts were. It is not my intent to "alienate a theorist" but a theorist should be a scientist first. $\endgroup$ – ggcg Nov 1 '18 at 23:51
  • $\begingroup$ Welcome to the physics stack exchange! This is an interesting question, but many might agree that it's more a question of philosophy rather than either physics or mathematics. Physicists and engineers use mathematical frameworks to help describe, predict physical phenomena but if you look closer in some respect you'll more than likely find imperfections in the framework. George Box said "all models are wrong ... some are useful" Einstein looked closer and found a better model than Newton, but GR breaks down at the quantum level. $\endgroup$ – docscience Nov 2 '18 at 0:15
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"Questions here come to mind considering examples related to content above, imagine the observation data of Mercury’s movement that could not be justified by Newtonian mechanics achieved sooner than the appearance of non-Euclidean geometry, then wouldn’t it be influential on the advent of a different fifth postulate of geometry?"

I am pretty sure that this was the case, this observation was known before Riemann and Einstein (though I cannot cite a reference it may be discussed in 300 years of Gravitation). But why would the failure of Newtonian gravity and physics to describe Mercury's orbit lead to a questioning of Euclid's laws (if I understood your thesis correctly)? This would not mean that the postulates of Euclidean geometry are wrong, merely that the laws of motion and gravity are misunderstood.

On another note I do recall that techniques developed by physicists are sometimes useful in proving mathematical theorems or as computational tools. An example of this is super symmetric quantum mechanics and the use of operator ordering to generate solutions to new ODE's that may have previously been unsolved. Again, I cannot recall a specific citation but in graduate school I knew a group of theorists who were very good at this. They used QM principles and SUSY QM to (1) find exact solutions to previously unsolved ODE's, and (2) use QM Postulates to develop proofs of theorems in ODE theory. But I would not call this physics leading or inspiring mathematics as these results could have and eventually would have been found in math.

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  • $\begingroup$ If we consider the observations of mercury orbit were precise enough and one were persistent enough to change and reformulate heavily the Newtonian laws in Euclidean geometry then it may not be required to a different geometry of space time, but the clever one prefers to change the geometry first and then write the rules in its new language in order to solve the newtonian issue. Here the problem is not about finding the true geometry one, but to find a geometry that eases our formulation and calculation of new rules which justify the more precised observations $\endgroup$ – MasM Nov 1 '18 at 22:21
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    $\begingroup$ I really don't understand what you are trying to say. "If we consider the observations...". What I am pointing out is that they were not. They were known to be off a long time before diff geom and GR, yet they did not inspire the thought you are provoking. Perhaps I am completely missing the motive of your post. Since you have not received a satisfying response (as per your post) try to re-frame it. $\endgroup$ – ggcg Nov 1 '18 at 23:48

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