Studying the logical structure of physics as a mathematical object per se? 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?
 A: There isn't any such "field" per se, but some physicists and more often mathematicians do work with such such an approach. Just to give you a couple of examples from hep-th, consider 


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*Wightmann, who worked on axiomatic QFT

*Atiyah and also Segal who were involved in formulating topological QFT, and then many more mathematicians after that. In fact, a lot of mathematicians now work off axiomatic formulations of field theory.

*Recently, Kapustin released a preprint considering the axiomatic formulation of quantum mechanics, using the language of category theory.


There will be many examples in areas like statistical mechanics on the physics side, differential equations on the math side and dynamical systems on both sides; I can't give examples off the top of my head since it's not my primary field of study.
A: It sounds like you're interested in the philosophy of physics.
The introduction to Gordon McCabbe's Structure and Interpretation of the Standard Model reviews the "logical structure" of physics, and various interpretations of the mathematics (structuralism, etc.).
I don't have the book in front of me, but he discusses how physics works with first order or second order logic, various features to studying physics using model theory (i.e., studying physics as a model), etc.
This sounds exactly like what you're after! 
Some references McCabbe cites (again, this is from memory, so there may be more):


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*Oliver Pooley, "Points, particles, and structural realism". Eprint 2005

*John Earman, "Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity." Eprint 2002

