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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?

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  • $\begingroup$ "The next great era of awakening of human intellect may well produce a method of understanding the qualitative content of equations. Today we cannot. Today we cannot see that the water flow equations contain such things as the barber pole structure of turbulence [...]. Today we cannot see whether Schrödinger’s equation contains frogs, musical composers, or morality—or whether it does not. We cannot say whether something beyond it like God is needed, or not. And so we can all hold strong opinions either way." Thus saith Feynman... $\endgroup$
    – NickD
    Aug 7 '17 at 0:53
  • $\begingroup$ ... (who would not agree with you that formalizing Physics will lead to better understanding). $\endgroup$
    – NickD
    Aug 7 '17 at 0:53
  • $\begingroup$ Yes those who say that formalizing is not part of Physics have a point. But still I think we must formalize scientific theories to get a better understanding and synthesis of the topics. I am suprised that I haven't found more information on this. $\endgroup$ Aug 7 '17 at 0:59
  • $\begingroup$ Have you read Godel's Incompleteness Theorems, I am sure you have. (Apologies I can't get the spelling right on this machine). I think that took the wind out of the sails of many people, possibly that's why there is less discussion of the subject. Hasn't math broken into two parts, one based on the idea that the Incompleteness idea is true, and one that well, just ignores it, basically? +1 for the question though. $\endgroup$
    – user163104
    Aug 7 '17 at 1:26
  • $\begingroup$ @Countto10 The majority of mathematicians just ignore the incompleteness theorems. They just don't care really. They are hardcore Platonists in the sense that they think truth is given and truth "exists" even if we can't characterize it using the concept of proof (1st incompleteness theorem). I am not that pessimistic about the incompleteness theorems and ToE. The Physical world is not the abstract world. In fact if the Physical world is discrete and finite then completeness is possible. Incompleteness comes with natural (or real) numbers but in Physics we don't need every number. $\endgroup$ Aug 7 '17 at 1:41
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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.

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  • $\begingroup$ Thank you for the fast answer! Those axioms seem beautiful! I really want to understand the logic behind these Physical theories! $\endgroup$ Aug 7 '17 at 0:47
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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.

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  • $\begingroup$ That's all right! I know this topic is just as much about logic as about Physics but who knows maybe somebody will know something about it. I find it interesting at the least. $\endgroup$ Aug 7 '17 at 1:20
  • $\begingroup$ I would not be answering (or waffling on really), if I was not as interested as you are. :) Best of luck with it. Again, as you know probably better than I, Hilbert was not pleased to be kicked out of the Garden of pure math when the incompleteness idea came along. $\endgroup$
    – user163104
    Aug 7 '17 at 1:28
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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.

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  • $\begingroup$ Thank you very much for that answer, which I never paused to think of , or worse yet, research, during my uniformed diatribe :). I just had an inbuilt unthinking view that math used axioms, and physics had empirically derived laws, ( the next level up, so to speak). But your answer has made me reconsider that QM has axioms and Relativity has postulates,. But I will stick to my guns, in the sense that axioms are only good until the next big thing comes along. Thanks again, for an informative answer that will make me think about physics in a new light. $\endgroup$
    – user163104
    Aug 7 '17 at 22:29
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    $\begingroup$ It is my hypothesis that quantum mechanics and general relativity are categorically equivalent. This might mean postulates of one system have a map to the postulates of the other. The one thing that has to be remembered is that mathematics has axioms, but physics is not strictly a mathematics. Physics is an empirical subject. As a result physics probably can't be reduced completely to an axiomatic system. $\endgroup$ Aug 7 '17 at 22:35
  • $\begingroup$ One last question. I am extremely lucky in being almost totally ignorant regarding string theory, so I get less stressed than some on the topic. But it does seem to me to be very math based and I wonder because of this, is it potentially the area of physics most likely to be axiom based. Again, I would stress my opinion of ST validity is worthless, but I was influenced by Woit, Penrose and Smolin's books, all of whom question string theory on the basis that the math behind it is not completely proven, and still bring discovered). These 3 names can cause interesting reactions on the site :) $\endgroup$
    – user163104
    Aug 7 '17 at 23:05

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