> Math, of course, is that stuff which can’t be invalidated by observations.

This is a misunderstanding of what math is, I think. You can invent a perfectly valid mathematical theory that conflicts with observations. Math is just a sequence of "if this, then this" and if the conclusions follow from the premises, it's math. But if you demonstrate that in the universe we occupy, a premise isn't true, then the mathematical theory isn't any less valid, it's just not sound in our universe.

For example, there's a ton of completely valid mathematical work on the correspondence between anti-de Sitter spaces and Conformal Field Theory. However, much of this mathematics has no application in our universe, because our universe seems to be a de Sitter space (positive cosmological constant/expanding), not an anti-de Sitter space (negative cosmological constant/contracting). That doesn't make their math invalid, it just makes it not real.

You can also do a lot of math in Minkowski spaces, which are flat. But our universe isn't flat, it's curved. Doesn't mean it's not math, just that it's not real in our universe.

I think this is a disagreement about the word invalidate, and maybe it was a bad pick on my part. The observations don’t make the math wrong, maybe less useful.

But, I say the math can’t be invalidated by observations; and you describe some cases where the math is valid but might or might not be applicable to certain physical cases. So actually, I’m not clear as to what you are saying I’m misunderstanding.

Ok, I see what you mean now.

I took your word "invalidate" to mean "be proven false" whereas I meant "valid" as in "logically coherent, even if not true."

> I say the math can’t be invalidated by observations

I meant, isn't a counter-example or proof by contradiction an invalidation by a type of observation?

No, proof by contradiction is a logical construction, has nothing to do with the real world. Proof by counter-example could or could not be observation, depending on whether your example comes from observation or from pure logic.

I suppose even in pure math, if you postulate a set of axioms, then proceed to to prove some theorems, you're still at risk of someone providing something like a counter-example showing that your axioms are not consistent, and that not all of them can be true at the same time.

Meaning that even if the inductive logic is 100% correct, the theory can be incorrect due to using mutually exclusive (in some non-obvious way) axioms.

“Prove” could mean many things. For physics, prove what?

That a given model is the one and only model that accurately explains the known universe? Then I agree. Observation won’t get you there. Asymptotic at best.

But by “prove”, that it accurately or usefully makes predictions with respect to certain constraints (which may not be known)?

That’s a more modest use of “prove”, where observation is certainly a key factor.

Interesting linguistic aside: "the exception that proves the rule" makes use of this more old fashioned use of prove, to test. So it's the exceptional case that tests the rule.

With that in mind, "prove" is perfectly fine to use in the context of science.

Maybe proof should be used for that more modest concept. In the sense of “bullet proof.” Similar to a piece of armor with a bullet proof; the ding from where it was shot for testing, a physics theory could be described as “microscope proof,” haha. We hit it with a bunch of microscopes and it didn’t fall apart.

In Physics, you just remove all possible theories that are inconsistent with observations.

For instance, you can prove that Newton's laws are not 100% accurate.

> You can’t prove anything by observation.

Nice observation.

Epistemology is a harsh mistress.