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As a disclaimer, I come from a pure math background, and I've only done a minimal amount of mathematical philosophy and epistemology, so the philosophy of physics is certainly not my strong suit.

That being said, I came across Hilbert's Sixth Problem a while back and become curious into the philosophy behind it, so I started reading up on the works of different philosophers. That being said, there's something in my mind that I just can't shake or explain. Why do we even think we can axiomatize physics?

My reason for asking this problem is as follows; suppose you have a set of axioms that accurately describes reality. How would you verify it? You would need to know everything about reality to ensure that your axiomatic system isn't inconsistent with observed reality. This to me implies one of two things: either the set of axioms must be infinite, or there is a disconnect between humans and the axioms; there are rules that explain reality but humans could simply never verify them, and so we would never know whether or not what we've found are actually "the rules".

In this answer, it is said that "Quantum Mechanics, together with a finite cut-off QED, explains all of chemistry and nearly everything in Physics". How can that be so?* Do they intend to say "nearly everything in modern physics", as in everything we know of? What about string theory and all that stuff? How do we know that QM explains everything consistently, and if we know that it's true then why are we still even trying to learn about the universe?

From what I can tell (though I don't know the lurid details of it; again, I'm from pure math hello from the dark side) it seems that the axioms proposed by Weyl and Dirac are only approximations, and hence we don't know that they are axioms.

I'm very confused, as you can probably tell. I'm trying not to say too much for fear of being inconsistent (I've read in exactly 50% of the places I've looked that we haven't axiomatized physics, and in the other 50% that we have), but to sum up my main question is this:

Why do we think we can axiomatize physics if we could never fully verify an axiomatization? Are we simply trying to show that an axiomatization could exist, and not necessarily to write it down?

* - I'm not doubting the claim here, I simply want to know how that conclusion was arrived at.

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  • The first objective of axiomatization is "structural" : to clarify all the assumptions (and to fully state them) necessary to prove the known theorems of a theory. – Mauro ALLEGRANZA Dec 4 '16 at 8:11
  • Regarding Hilbert, you can see : Leo Corry, David Hilbert and the Axiomatization of Physics (1898–1918) (2004). – Mauro ALLEGRANZA Dec 4 '16 at 8:12
  • But the role of the axiomatic method in math is hardly the same in other sciences: in math can be a powerful tool for "discovery"; I do not think so for empirical sciences. – Mauro ALLEGRANZA Dec 4 '16 at 8:13
  • For some ref to the axiomatic method in math, you can see this post. – Mauro ALLEGRANZA Dec 4 '16 at 8:15
  • You can consider that axiomatization is "tipically" adopted when a theory showa some paradox; see Euclid's axiomatization of geometry following the discovery of the irrational nu mbers and the axiomatization of set theory after the discovery of Russel's Paradox. The same for physical theories; following the "paradoxical" aspect of quantum theory the need for a clear axiomatization emerged. – Mauro ALLEGRANZA Dec 4 '16 at 10:47
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I share your concern. Physics and mathematics deal with two different subjects. While we all agree that our mathematics is a human creation, physics covers a subject that man is not really in a position to change at will and that has existed long before man.

An axiom is a statement that we consider as a foundation of our knowledge. While in mathematics we are largely free to add new definitions or to alter an axiom in order to derive new reasonings, physics is constrained by experiment (this constrains the choice of axioms).

As a result, the research procedure is distinct in both fields. In mathematics, the general idea is to pose a problem (e.g. verify a conjecture) and try to find a solution within the confines of that mathematical world we are able to define quite accurately. In the extreme, mathematics does not care about the semantics of the objects described, aside of the mathematical properties which are explicitly laid down.

In physics, there is an additional constraint: whatever is expressed must have a (near-)equivalent in experience. Hence a mathematically expressed law must result in an a (near-) accurate prediction in a series of experiment.

The rules for validating a law in physics are also very different: it is impossible to demonstrate a physical law as true (in the logical sense), because it would mean exhausting all possibilities of the contrary. What we call "certainty" is relative and depends on many experiments that matched the theory and no credible counter-example. Hence the nearest mathematical equivalent of a physical law is a conjecture. And while in mathematics it is generally possible to turn a conjecture into a theorem, physics has to live with that degree of uncertainty.

Hence, could physics be described by axioms? To a relative degree, yes: as the motion laws of Newton were the foundation of classical mechanics (but had to be altered by Einstein), or principles of QED are a foundation of modern sub-atomic physics. Then physicists use the tools of mathematics to try to match reality as best as they can (the theory could be conceived as a copy of reality on trace paper, it is not the original). Hence limitation of whatever axioms we use in physics is the usefulness and accuracy of the results according to experiment.

For the conclusion about the enthusiastic claims on "ultimate laws of physics", here is what physicist Richard Feynman said about this (emphasis added):

People say to me, “Are you looking for the ultimate laws of physics?” No I am not. I am just looking to find out more about the world. And if it turns out there is a simple ultimate law that explains everything so be it. That would be a very nice discovery. If it turns out it’s like an onion with millions of layers and we're just sick and tired of looking at the layers then that’s the way it is! But whatever way it comes out, it’s nature, it’s there, and she’s going to come out the way she is.

And he points at a possible aporia:

And therefore when we go to investigate we shouldn’t pre-decide what it is we are trying to do, except to find out more about it. (...) If you thought that you are trying to find out more about it, because you are going to get an answer to some deep philosophical question, you may be wrong; and may be that you can’t get an answer to that particular question by finding out more about the character of the nature.

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Meant as a comment, but too long: my background is physics, not math, per se, and I think you're conflating "physics" and "reality". Physics >>is<< a mostly-mathematical axiomatic model of observable physical behavior (of observable reality). As Jammer puts it (and explains in more detail) in https://www.scribd.com/doc/127985402/Jammer-QUantum-mechanics-philosophy physical theory is "a partially interpreted formal system".

What you're questioning is whether or not "all of reality" (whatever that is) can be axiomatized (in more or less the way physics is axiomatized). I suppose you're right that would be the limiting case (limiting goal) of physical theorizing, but nobody claims to know whether or not that's achievable (that the limit exists, so to speak). Okay, I suppose some people would claim it's definitely achievable, maybe even some well-known physicists, but read Jammer and others for a more thoughtful view.

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