Impurity in Contemporary Mathematics

Abstract

Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the examples of algebraic invariants and of considering a mathematical object from several different perspectives to illustrate that the methods used in contemporary mathematics favor impurity. Then I consider an example from category theory which demonstrates how unification and generality are related to impurity and that impure solutions can be explanatory. In light of this discussion, we see that purity only has marginal value within contemporary mathematics which instead prioritizes the epistemic values associated with impurity.