[wastewaste]

> the Foundations of Mathematics should give a precise definition of what a mathematical statement is and what a mathematical proof is.

I don't see a difference between precise definition and mathematical statement in this context, so your question would be paradox. Is it wrong to ask though? I think higher logic is supposed to deal with these types of paradoxes. Your question, if taken rhetorical, is a statement about mathematical statements, a metapher.

The in/-completeness theorems show that higher order logics like that are either inconsistent or incomplete. So a theory like you are asking for, is ab ever growing set of proofs that can't fit in a single book. Taken with a sense of philosophy in mind, a book can only be an introduction to think further.

I was glad to read at least the intro to Univalent foundations mentioned in a previous comment. Russels paradox defies a set of all sets in first order logic. FOL is otherwise said to be complete, by the way, a fact often overlooked when talking about the related liars paradox. I was pleasently surprised to read that they just avoid the paradox by embedding theory into theory (Universes, tbh). To me that means we can't draw elemens from an arch-set, but have to build facts from the ground up.