Summary: 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 pu-
rity 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.1
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 "re-
quired 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 content-
requirement, and thus of purity, diverge.
1. Logical purity
The first way of understanding purity that I want to consider takes