Can axioms be used to prove anything in physics?

Question

Are there any laws of physics that use axioms as their base, and all the other laws are built on those? Or are the laws just derived through experimental evidence? Is there any statement in physics that can be accepted as completely true?

Answer 1

The practice of physics works more or less as follow. (1) Make postulates, of mathematical nature, and then (2) derive mathematical results. So far, this is what you have in mind, using a different vocabulary. But, most essentially, those results, which are called predictions, shall be testable experimentally. So, (3) perform the experiments and here comes the key fork. If the experimental results contradict the predictions, scrap the postulates we started from and go back to (1) with new postulates. If the experimental results agree with the predictions, go back to (2) to derive new predictions, and proceed as long as you can, or until the predictions are falsified by experiments.

As a result, no, there are no "completely true" statement in physics. We have only statements which have not yet been proved wrong by experiments. Sure, at any point, we have theories, which are sets of consistent mathematical postulates, whose predictions have been widely confirmed by experiments. So within these theories, barring the fact that physicists take liberty with mathematical rigour, we can say whether a statement is true or false, by checking whether we can derive it from the postulates, or whether we can derive its negation from the postulates. But the whole edifice of the theory itself has only a transient truth value, because of the Damocles sword of an experimental falsification.

Answer 2

A physical law has no absolute validity, it is always part of a model, that is, it is relative to a certain mathematical way to describe reality. The model experimentally fits the natural world more or less precisely, and at specific scales (of time, space, energy, complexity).

Physics has many models, all mostly mutually compatible, but not always. When the ranges of two badly compatible models overlaps, we get a research field where so-called "new physics" is to be expected. That is, we know a new model is around the corner that will provide the conceptual links allowing the previous models to be considered compatible. This may at times be akin to a conceptual revolution, introducing a brand new way to look at things.

There is a philosophical standpoint called reductionism which (I am vastly simplifying here) claims that eventually all models can be considered as approximations of the true one description which is fundamental and closest to the real nature of things. This would be the closest to the idea that physical laws proceed from axioms, the fundamental model providing all "axioms".

Answer 3

I agree with the answers by Stephane and Luc, this is an extension too long for a comment.

There exists the platonic theory of the world that can be extended to the world of physics

The theory of Forms or theory of Ideas is Plato's argument that non-physical (but substantial) forms (or ideas) represent the most accurate reality.

In this sense, one can postulate that "mathematics exists" and is the mold for anything physical. Then one could be discussing of a physics structure starting from axioms that would constrain and prove any function fitting physical data.

The opposite argument is that "nature exists" and mathematics is a tool used to approximate measurements and observations and , impotant, predict new ones; using postulates and laws as extra axioms to chose from the mathematical solutions the subset which applies to data.

The present mainstream view is the second one. Maybe if, in the future, theoretical research comes up with a theory of everything doubters will be convinced on the platonic ideal ;).