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

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.com/cd/0/inline2/BbTZopOAzf1VbSX0lAsDAl5GOKwAq1-960-jlCzC_AqYhQbFgU1Y3eicCW5UxEigaTU5t-1uWJKU7bRIOC5yXkTtM-lp6JRVwTWIAGc_iyCbSbR9lzKU4QjGrYD6h0dOtl3FstnRhM7UKfxxx91UX71ur3qxNVvR1bTH0scimARr-aKyRYQ_mmRkWKLsdSrtQ27hNrEcmJlrsyIkzE-HFAlOW_1CID5ErjA7_6qLLWlJ8mOyuNPcw1YlcEKQs10zek6qmCQ5d-02UvrsxYeJg4gBV-VijWlzOaznH9zKrJgmCPckciuTVChJBGLVjnwaAHvzDL3PVHfIr4iW-nXvOzaZyB2yftcmEttaHbsS1vh64aVFvemb1Jm8i6CzSoK-PyU/file 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.