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.