[n4r9]

You can prove Noether's Theorem in a mathematical sense, but you cannot conclusively prove that a specific physical force is conservative or that a specific physical symmetry of action is continuous.

Likewise, we assume at an operational level that temperature and barometric pressure are continuous functions (as assumed in Borsuk-Ulam), but it's not something you can conclusively prove aobut reality.

[seanhunter]

Sure but that doesn’t matter for my examples. The parent of my comment said “science never proves a positive” and I gave a couple of examples of proving implications. Proving “If A then certainly B” is definitively proving a positive whether or not we can prove A.

[n4r9]

I guess this comes down to what you mean by "science". Some would say that science is the process of testing hypotheses about reality. Mathematical facts exist in an abstract sense apart from reality, and so mathematics is not really science to those people.

There's an argument that you are still doing science if you construct a logical proof showing that "if the world is like X, then it will behave like Y". A lot of theoretical physics is like this, and people call that science. But I think there's truth to what OP is saying in that science does not conclusively positively prove things about reality.