Logical and Semantic Purity

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 that are required by the content of the theorem. A pure proof of a theorem, then, is one that draws only on what is “required by the content of the theorem”. I want to continue by distinguishing two ways of understanding “required by the content of [a] theorem”, and hence of understanding what counts as a pure proof of a theorem. I’ll then provide three examples that I think show how these two understandings of contentrequirement, and thus of purity, diverge.