Logical and Semantic Purity
Andrew Arana, ProtoSociology Project · ProtoSociology · 2008
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. 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 “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. 1. Logical purity The first way of understanding purity that I want to consider takes what is “required by the content of [a] theorem ” to be just what suffices for proving that theorem. The ideal is what Hilbert pursued in his Grundlagen der Geometrie: to determine which of the axioms he gave for geometry are sufficient for proving interesting geometric theorems, such that if any of those axioms were left out, the theorem would no longer follow. 2 As an first approximation, then, this ideal can be