Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them. I want to begin with a classical formulation of purity, due to Hilbert: In modern mathematics one strives to preserve the purity of the method, i.e. to use in the proof of a theorem as far as possible only those auxiliary means… CONTINUE READING