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I know this question had been asked many times before but maybe not in this form. So I really need the exact axiomatization of Physics. I have been looking for it for a long time. Precise logical axioms written in a first order (or maybe higher order) language. So not just a couple of differential equations but the pure skeleton of Physical theory itself. I need the axioms from which important theorems of chemistry and Physics and maybe Biology etc. can be derived logically. I really think that formalizing problems can lead us to much better understanding and I am hungry for that kind of knowledge. Can you link ANYTHING that answers my question? Is there anybody (Physicist, mathematician, philosopher, logician or any kind of scientist) whos professional field is similar to that?

I'm adding some concrete notes and questions(to unlock the topic):

  1. I have started to learn some physics just for fun and I found that lots of proofs use infinitesimals. Those proofs are heuristic and I think they can be made more precise and exact by using infinitesimal analysis (that is in fact part of logic). Does anyone know any books with this approach?

  2. I am just looking for the axioms really, like Newton's axioms etc. Because I find it fascinating that from a couple of axioms we can get so many things. Are there books or papers which emphasize this kind of logical structure of Physics? (Like they write the axioms and theorems they use from Geometry and then put some Physical axioms next to it.)

  3. Any books on the "meta" side of Physics, like problems of determinism or locality (I have read a few about that in wikipedia but still know next t nothing about it) and their formalization? Thank you!

Re-open please?

MetaLogicianWannabe
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The problem of deriving all of physics from axioms is one of Hilbert's famous problems (specifically, Hilbert's Sixth Problem). Currently, there is no experimentally supported unification of quantum field theory and general relativity, so this problem remains unsolved.

There have, however, been attempts at this problem. The Wightman axioms (https://en.wikipedia.org/wiki/Wightman_axioms) are the closest we have come to an axiomatic treatment of quantum field theory, and general relativity also has an approximately axiomatic treatment, though there's a debate there about what is strictly "necessary" to describe the field (see Is there an accepted axiomatic approach to general relativity?).

You might think that if we have a set of assumptions for quantum field theory and a set of assumptions for general relativity, we can just concatenate the two to get assumptions for the theory of everything. But this is unfortunately not the case, since the assumptions of QFT are often incompatible with those of GR.

probably_someone
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There is something called axiomatic field theory. Interest in this stemmed from the Wightman axioms that set operators for fields on a spatial surface as commuting operators, which recover their role in Hilbert space with quantum commutators on the light cone. This lead to a number inquiries into the fundamental structure of quantum field theory. In particular with the analytical continuation with $\tau~=~it$ to a Euclidean metric operators under the Wightman conditions are studies according to analytic functions in complex spaces.

The t conditions partition propagators of fields into two parts, those on the future part of the light cone and those on the past. The standard computation is the modulus square of a quantum field $\phi(x)$, or for any function of a quantum field $f(\phi(x))$. A path integral of this in Euclidean form is a partition function over two sets $\{\phi_+,~\phi_-\}$. A distribution of this function of fields is then $$ \int {\cal D}\phi f(\phi(x))\overline{f(\phi(x))}e^{-S[\phi]}~=~\sum_{\phi_\pm~=~\phi_0}\int {\cal D}\phi_+ f(\phi_+(x))e^{-S[\phi_+]}\int {\cal D}\phi_- \overline{f(\phi_-(x))}e^{-S[\phi_-]}. $$ The relativistic condition on the propagation of fields is a sum of fields on the positive and negative half spaces.

The over all success of this and related programs has been unimpressive. This was a very active area of theoretical research in the 1960-70s. It has largely never managed to accomplish its main goal of reducing QFT to an axiomatic system capable of reducing all quantum field theory or even quantum gravitation as something computable from these axioms.

Lawrence B. Crowell
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Mathematics underpins physics, so in my opinion, you are looking in the wrong place. Formalising physics will do the opposite of what you want it to do, it will constrain it, rather than allow it from finding the experimentally verifiable truth.

The formalism and axioms will not last. They never have. We have constantly tweaked and replaced them as new discoveries appear. Dumping the old axioms such as absolute space and time has not held us back, quite the reverse.

If mathematics cannot be put on a solid, absolutely self consistent basis, (which it can't), and physics relies on mathematics, then physics can't either.

A well known example of this AFAIK, is the abstract spaces used in the standard model of particle physics. Group theory is essential to this, and it is founded on mathematics.

Another example is the problem of describing the particles themselves, not their properties so much, but their actual "nature", which depends on mathematics to describe, as physical visualisation is impossible.

You can't get away with saying they are some form of energy, as there is no general!y accepted definition of energy. Look up Wikipedia for the amount of different attempts at pinning down the notion of energy.

In the future, to solve the fusion of GR and QFT, it does seem obvious to me that a new set of axioms and formalism will be needed again.

Sorry about the rant, I would like to ask you why formalism is that important in physics but this is not the place for it.