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.

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    $\begingroup$ IMHO mathematicians can at most be inspired/stimulated by physics (or other sciences). Physicians need mathematicians but not vice versa. Characteristic is that someone who knows "nothing" about any (other) science can still be a good mathematician. I wrote parentheses here because in my view mathematics is no science itself. It needs no phenomena outside mathematics and is more an art. $\endgroup$
    – drhab
    Oct 16 '18 at 11:01
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    $\begingroup$ @drhab I disagree with this viewpoint. Mathematics as a human endeavour is as dependent on human intuition as anything else. Mathematicians need ideas to succeed. Skill at writing proofs is not sufficient, they also need to discover new mathematical truths, in order to have something interesting to prove. Physics as a source of ideas for mathematicians has been successful time and time again. Having said that, I leave this as a comment because it does not address the specific question about physics experiments as opposed to physics ideas. $\endgroup$
    – Lee Mosher
    Oct 16 '18 at 12:06
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    $\begingroup$ New observations need new theories to fit them, and this often requires new mathematics, alot of mathematics was discovered this way, but if you think some observation will immediately help mathematicians prove some theorem then thats not quite how it works. But it can go the route new observation->new theory->new mathematics->new tools to prove things, which can take many many years. $\endgroup$
    – badmf
    Oct 17 '18 at 10:09
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    $\begingroup$ An example is Wittens proof of the Morse inequalities, which is based on tools from quantum physics. Or Mirror symmetry. $\endgroup$
    – badmf
    Oct 17 '18 at 10:10
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    $\begingroup$ Experts agree that Atiyahs proof is wrong and his fine-structure computation doesnt even give the right value $\endgroup$
    – badmf
    Oct 18 '18 at 12:15

To begin, I would like to establish what to me seems like a key point. Modern physics concerns itself with creating mathematical models of the universe. However, one mustn't confuse the model for the universe in much the same way that there is a difference between a map and an actual place. Statements about the mathematical models are sometimes purely mathematical. For example, Newtonian physics models $F=\frac{GMm}{r^2}$ and $F=ma$ assuming constant $m$. Then one can solve that within this model $a=\frac{GM}{r^2}$, which is a purely algebraic truth. However, some statements about the model aren't mathematical. For example, the statement that Newton's laws of motion accurately describe the universe. The distinction is that the mathematical statements operate purely within the model. We have defined some quantity called the force and that the acceleration is the second derivative of position and that they are related in such and such a way and so on, and from those definitions, we can derive mathematical truths.

With that out of the way, it seems like you could be asking two questions. The first is whether physical measurements can lead to new mathematical statements in a broad sense. The answer to that is clearly yes. For example, look at Noether's theorem. There is some purely mathematical statement about certain types of functions and integration. However, that mathematical statement was clearly inspired by what was seen in the universe. The second form of the question is whether or not physical observations can be definitive proof of a mathematical statement, and to me, it seems the answer is no. For example, suppose we have some model of the universe such that an observable quantity is 1 if the Reimann hypothesis is true and 2 if it is false. Experiment determines that the value in the world is 1. That leads to several possible conclusions. The first is that the model is right and the Reimann hypothesis is true. The second is that the model is wrong and the Reimann hypothesis could be either true or false. In order to narrow down to only the first option, you would have to know that the model is perfectly accurate. However, a model is perfectly accurate if and only if all of its predictions are correct. To know that, we must know if the Reimann hypothesis is true as otherwise, we don't know about the accuracy in this particular situation. As such, we can't really decide ahead of time whether the model is accurate and therefore whether the Reimann hypothesis is true based on the observation. This type of problem extends to any physical observation as the correspondence between it and a mathematical value depend on the model being used. The fact that things in the universe act in a relativistic way don't undermine the basic algebraic statements one uses to derive the equations above and similarly the accuracy of statements about relativity don't prove the mathematical statements relativity is based upon. Instead, they prop up that model being used as accurate.


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