I was wondering is there a branch of mathematical physics which studies the underlying logical structure of physics as a mathematical object per se?
Let me explain what I mean by that.
I'm interested in a theory which takes an axiomatic approach to physics, clearly distinguishing between what is postulated or in any other way "swept under the rug" and what is clearly a logical consequence of what is taken to begin with.
I imagine something like theory of axiomatic systems in mathematics, which studies both internal consistency of an axiomatic system and relative consistency of systems. Obviously, I imagine it would also be necessary to integrate experimental physics to it with fields like probability theory in an attempt to quantitatively express the certainty of some consequences in physical theory.
In an ideal case, that theory would produce results which could perhaps allow us to say how certain or how speculative some physical theory is, but not just qualitatively.
I believe you can see where I am going. Is there a field like that out there? Or something similar to it?