Elements of Purity

This Element is the first comprehensive discussion of a new, critical subject in the philosophy of mathematics, purity of proof.

This Element explores the preference for pure proofs in mathematics since antiquity, focusing on geometry and number theory. It discusses different types of purity, reasons for preferring pure proofs, and the importance of local purity. It also touches on translation issues and the relationship between purity and local considerations.A proof of a theorem can be said to be pure if it draws only on what is 'close' or 'intrinsic' to that theorem. In this Element we will investigate the apparent preference for pure proofs that has persisted in mathematics since antiquity, alongside a competing preference for impurity. In Section 1, we present two examples of purity, from geometry and number theory. In Section 2, we give a brief history of purity in mathematics. In Section 3, we discuss several different types of purity, based on different measures of distance between theorem and proof. In Section 4 we discuss reasons for preferring pure proofs, for the varieties of purity constraints presented in Section 3. In Section 5 we conclude by reflecting briefly on purity as a preference for the local and how issues of translation intersect with the considerations we have raised throughout this work.