A Definition of Mathematical Beauty and Its History

Abstract

Synopsis I define mathematical beauty as cognisability and trace the import of this notion through several episodes from the history of mathematics. 1. Beauty is cognisability. The definition of mathematical beauty that I propose to defend in this essay is the following: A beautiful proof is one which the mind can play its way through with a natural grace, as if it were created for this very purpose. We grasp a beautiful proof as a whole, yet see the role of every detail; it is vivid and transparent; we are its masters and its connoisseurs, like a conductor directing a symphony. I call this type of proof cognisable for short. An ugly proof resorts to computations, algorithms, symbolic manipula-tion, ad hoc steps, trial-and-error, enumeration of cases, and various other forms of technicalities. The mind can neither predict the course nor grasp the whole; it is forced to cope with extra-cognitive contingencies. The mind's task is menial: it can only grasp one step at a time, checking it for logical adequacy. It can become convinced of the results but it is not happy since all the work was being done outside of it. Our memory is strained, our mind distorted to accommodate some artificial logic, like a student struggling with a foreign grammar.