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...
It seems you're more interested in dicing up and attacking what I said, instead of what I mean.
I'll re-articulate it for you once more: We don't have a way to mathematically prove a drug for depression or to explain psychological phenomena. The studies in the article are talking about those which rely on empirical observation.
Go home.
How many problems have you worked through in GR? Have you gone through the proofs that GR recreates Newtonian gravity in the low-energy limit? If not, don't go around talking about how Einstein proved GR was a true description of reality with naught but mathematical proof when he didn't think he did that.
Physics has the same requirements for empiric evidence as the life sciences, it's just that we work within regimes where we can apply fundamental, empirically validated mathematical theories directly. If you don't believe me, go ask another physicist.
My point is this: Something is true whether one accepts it or not. E.g. the earth was proven to be round before anyone actually went around it.
Another example is climate change deniers. To them there's no proof of; 1) Climate change; 2) That it's man made.
Another example is people who believe the earth is 10,000 years old (I'm not joking millions of ppl believe this). They will deny any evidence you put in front of them.
That's the beauty of maths. If you can prove it mathematically, it's true. Whether or not you believe it is irrelevant.