Of course not. It's a branch of mathematics. Always has been.
„A branch“ - so we are back to trees then?
Mathematics and computer science are both in the category of formal science.
Yes, it’s a formal science.
The other category of science is called formal science.
Is proving something internal to a system of ad-hoc rules that cannot (to the same standard of proof) be proven to have any correspondance with the natural world really proving something? Sincere question.
I think so, yes. You start with axioms, and the proof follows from those.
You can argue whether that’s interesting or not, but IMO it’s as much a proof as anything can be, and IMO it’s very interesting.