Art is more concerned with human experience and seems like a different thing.
I am a strong proponent of distinguishing „truth“ from „certainty“. That is to say, a theory (philosophical or not) might be true in the sense that it corresponds to the things it says something about. But since all human beings are fallible we can never be certain not to have made mistakes. So a philosophy can be true. But we can not be certain about it. EDIT: If OP writes that "there is no way to refute or evaluate philosophy" they might imply that no philosophy is true. Yet I would only agree that no philosophy is certain.
> I think of it as the search for truths that we can’t empirically prove, contrasted to science and math which are purely logical.
What (empirical) scientists usually mean when they call a theory or hypothesis „empirically proven“ is that there is evidence that meets some arbitrary criterion of strength. What mathematicians usually mean when they call a theorem proven is that it can be logically deduced from a set of axioms. So not only is the word "proof" used very differently in (empirical) science and mathematics, the way to obtain such proofs is usually very different as well. To lump these procedures together under the term "purely logical" seems to me to mix up important differences.
> But math and science have their bases in philosophy through universal laws and propositions, which are unprovable but necessary.
Neither the laws and propositions in math nor in the (empirical) sciences are necessary in the sense that practitioners are forced to assume them. Nothing prevents anyone from assuming something else whenever they wish to and important discoveries have been made that way (e.g. non-euclidean geometry, theory of relativity). Maybe you refer to some other kind of necessity though.
> Art is more concerned with human experience and seems like a different thing.
Isn't the same true for philosophy? Part of the reason why philosophy is a meaningful field of study as opposed to complete charlatanry is that it "points" towards parts of human experience that aren't quite captured by math/science/models. Philosophers trying to be natural scientists would make terrible philosophy and terrible science.
Minor nitpick:
> science and math
Science and math are very different beasts. Math is a meaningless game you play with symbols. I'm not serious, but I'm not joking either. There are zero assumptions of any kind mathematics makes (except primitive concepts and axioms, which are themselves mathematical objects), as the old joke goes: science talks about the universe, math talks about a universe. Math is kind of unique in this respect, no other discipline is quite like it.
DNA and quantum mechanics are 2 areas where the boundary between philosophy and science has shifted over time. All the modern talk about multiverses, universe simulations, and AGI seem philosophical, but might become scientific in the future. But people who believe in these concepts will take different actions than people who don’t, so you can’t say it just doesn’t matter because we can’t prove it yet.
I wasn’t trying to lump math and science together and agree they’re very different, although both based on core propositional truths that are taken on faith.
> All the modern talk about multiverses, universe simulations, and AGI
None of the things you mention are "philosophical truths". The first is one of many interpretations of QM, the second is a meme, the third is a vague, unclear teleological goal of computer science. I'm not objecting to the fact that you can speculate, I myself said that pointing to new things, as vague and undefined they may be, is generally useful and stimulating. I'm objecting to the idea that you can have "truths you can't prove".
> I wasn’t trying to lump math and science together and agree they’re very different, although both based on core propositional truths that are taken on faith.
Sorry, my fault. Tomato, tomato.
Law of causation, Doomsday argument, complex processes that can't be recreated, historical processes which leave faint empirical markers, etc.