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 can require for example an axiom in order to make a 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 the “fine-structure constant” Atiyah used in his attempt to prove the Riemann hypothesis. Any precise explanation of other examples or references related to the topic are welcome.