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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?

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I think there are 2 problems. The first is that before we have an axiomatic version of physics, we really need an axiomatic version of mathematics. Experts are still debating the axioms of set theory, so we aren't there yet.\ –  Kevin Driscoll Apr 3 '13 at 14:14
    
The second problem is that to consider evidence and certainty quantitatively we would probably have to use Bayesian probability. There is still controversy about whether to interpret this method in an objectivist way (exemplified by Cox's theorem) or in a subjectivist way and exactly what axioms we should accept. –  Kevin Driscoll Apr 3 '13 at 14:26
    
Hilbert's desire to axiomatize math certainly spilled over into physics. My understanding is that there was an attempt to create a formal field for this kind of thing in the 1900s, but it never really got the momentum it needed. (As Kevin says, probably the fact that there are still issues on the math side hindered things.) But there is much out there. To add to Siva's list, I would mention people like Mario Bunge and his "Foundations of Physics" -- although it's a bit dated now. –  Andrew Gibson Apr 3 '13 at 15:38
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Hi @Schlomo Steinbergerstein: Philosophy-like tags are not allowed, cf. this meta Phys.SE post. –  Qmechanic Apr 3 '13 at 16:49
    
I respectfully disagree and I think people who want this tag removed have no idea what philosophy of science actually is, and that also includes philosophy of physics. I advice you to look up the difference between analytic philosophy and continental philosophy and instead of removing the tag, maybe you should moderate constructively by closing the questions that aren't really a part of philosophy of science or analytic philosophy in general. It's a shame that people want that tag removed because the majority doesn't know what philosophy actually is. Sadly, the moderators aren't an exception. –  Schlomo Steinbergerstein Apr 3 '13 at 18:08
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closed as not constructive by dmckee May 2 '13 at 5:52

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2 Answers

up vote 2 down vote accepted

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):

  1. Oliver Pooley, "Points, particles, and structural realism". Eprint 2005
  2. John Earman, "Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity." Eprint 2002
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You're right, it is closest to what I am after, perhaps it is precisely what I'm after, I'll investigate it further, thank you! I'd just like to add that I cannot help notice how moderators often remove tags without even thinking about why are they doing it, this IS philosophy of science AND the question IS soft (if it's not, I don't know what is). This is not the first time I see the same moderator nonchalantly exercising his/her power without being constructive. –  Schlomo Steinbergerstein Apr 3 '13 at 16:06
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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

  1. Wightmann, who worked on axiomatic QFT
  2. 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.
  3. 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.

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Readers interested in point 3. above might want check out publications from Bob Coecke's group in Oxford; they are working on formulations of physical theories, particularly quantum mechanics, using category theory. –  Mark Mitchison Apr 3 '13 at 15:44
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