Epistemical simplicity is one of the things I love about math, but it's hard to extend it wholesale to physics. GR will almost certainly be proved wrong some day, at least in some regime (Einstein was ironically one of the first to feel it needed something more, as an alternative to the inelegant cosmological constant). If GR had been mathematically proven, this would be terrible, as it would prove our mathematics are inconsistent!

As for the existence objective reality, that's another thing that seems hard to prove conclusively. I'd suggest looking at some of the basic epistemology surrounding modern science (e.g. Popper[1]) for some thoughts on this.

As an interesting aside, while it's true that mathematical proofs (idealizing here and assuming incorrect proofs are never accepted by the mathematical community, because they on occasion are) are absolute statements of truth, they may not be stating precisely the truth you expect. Thanks to Godel[2], we know that it is not possible for any consistent mathematical theory rich enough to talk about addition and multiplication of natural numbers to prove it's own consistency. As a result, we may have a proof that 2+2 != 5, but that actually doesn't exclude the possibility that there is a proof of 2+2 = 5. In fact, his result shows we will never be able to prove such a thing does not exist (since it would imply the consistency of our mathematical system). So our absolute truths from proofs of X are actually of the form ZFC implies X, where ZFC is the background theory which is generally taken to underly modern mathematical works unless otherwise specified. So things are not so clear cut even here.

[1]: http://plato.stanford.edu/entries/popper/ [2]: https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...