> A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
Unlike all the other sciences, mathematics is abstract enough to allow for things being fundamentally simple and true.
As a mathematical object, a proof is a series of derivations from axioms to a conclusion. That is well defined in various systems, can be studied as a branch of mathematics.
In fact, most mathematics isn't done this way (though there's interesting work in the Lean community). Mathematical proof as it is actually done is far messier. Here's the paper that first came to mind (Probabilistic Proofs and Transferability, by Kenny Easwaran), though I'm sure there is a more canonical source: https://ucfc6eb8f695030deb8332de441a.dl.dropboxusercontent.c...
P.S. You're free to adopt a way of speaking in which you say the majority of graduate mathematics textbooks do not involve any proofs, but I think that's a revision of the term.
I think it's fair to say that the state of the art of proofs is the equational form. Recorde revolutionized mathematical writing with a couple parallel line segments. Generalizing that to proofs using the equivalence, consequence, and implication is an equally (hoho) great leap forward.
> A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
Unlike all the other sciences, mathematics is abstract enough to allow for things being fundamentally simple and true.