As for what I know that I know: I'd limit that to mathematics of a particular formal sort, that I've proved myself, along the lines of "given these axioms, and these rules of logic, this conclusion follows". I'd like to see a project to prove all known maths in this way, using declared rules of logic and steps small enough to permit computer verification - it boils down to nothing but symbolic manipulation. (I even registered "axiomattic.org" for a while, hoping to start the project myself.)
What you describe sounds like Hilbert's Program. Unfortunately, Godel's Incompleteness Theorem states that we can never find an all encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods.
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