You bring up an interesting example in special relativity. I am a physicist, and there isn't really a "proof" in the mathematical sense of special relativity. Googling it provides this quora answer:
https://www.quora.com/What-is-the-proof-for-Special-Relativi...
There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Your example of limiting cases of special relativity is how I wish statistics texts were taught - it isn't based on the "proof" of special relativity. Of course I know when special relativity is applicable and when it isn't (when beta = v / c is close to one, special relativistic effects are important, and gamma = 1 / sqrt(1 - beta*beta) indicates how good the approximation is).
But I do agree that proofs are useful and needed, I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Of course this is just my personal preference.
> There is no proof of special relativity, we believe it because it makes experimentally verified predictions.
Right, that's because special relativity is a scientific theory. A scientific theory has two components: a mathematical theory (in which you can perform a priori physically meaningless calculations) and a physical interpretation (which turns the results of such calculations into predictions).
However, statistics isn't a scientific theory. It's just math, and like all math, it's about itself and nothing else. It doesn't make sense to “experimentally confirm a mathematical theory”, because, without a physical interpretation, math doesn't make any predictions about the real world.
> I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant.
Then you're looking for books on applications. That's fine. But a book whose subject matter is a mathematical theory (it even has “foundations” in the title!) can't relegate proofs to appendices.
There is no proof of special relativity, we believe it because it makes experimentally verified predictions. Your example of limiting cases of special relativity is how I wish statistics texts were taught - it isn't based on the "proof" of special relativity. Of course I know when special relativity is applicable and when it isn't (when beta = v / c is close to one, special relativistic effects are important, and gamma = 1 / sqrt(1 - beta*beta) indicates how good the approximation is).
But I do agree that proofs are useful and needed, I think that complicated proofs could easily be placed in appendices and looked at after understanding why the theorem is relevant. Of course this is just my personal preference.