Reference Request: Mathematical Foundations of Physics I am looking for reading on examples, or preferably a comprehensive summary on how the foundations of mathematics are related to physical theory. I would like to know whether basic set-theoretic and logical assumptions that underlie the higher-level mathematical techniques such as linear algebra, various branches of analysis, etc. have any bearing on the interpretation of theories in physics. For example, would it be of any relevance at all to physics if the axiom of choice was removed from the set of assumptions with which we operate? If anyone has any knowledge of examples themselves, that would also be highly appreciated.
 A: I think you have the relationship between physics and mathematics back to front.
Physicist choose the mathematical tools that they need to support models that explain their empirical observations. If the necessary mathematical tools do not yet exist, or are not quite in the form the physicist wants, then they will develop new tools (as when Newton and Leibniz developed calculus to integrate equations of motion and handle the dynamics of extended objects).
Developments in mathematics do not drive physics. Instead, physical theories are modified as a result of (a) new empirical evidence or (b) a synthesis of ideas that replaces several unconnected models with a single unifying theory (as when Newton's law of universal gravity replaced Kepler's laws of planetary motion, or Maxwell's equations replaced and subsumed Gauss's law, Faraday's law etc.).
A: Since you asked for a specific reference: You might be interested in the work of Günther Ludwig. He did a lot of research in the logical foundation of physical theories. He also wrotes a book about it. I haven't read it myself so I can't say much about it.
Unfortunately, I am not sure if this book is also available in English. The original one is in German: "Die Grundstrukturen einer physikalischen Theorie" from Springer. However, it seems that there is also a French translation with the title "Les structures de base d'une theorie physique".
But in any case, maybe you find some articles from this author on the subject.
