An application of abstract nonsense to surface area

In this paper we present a brief history of attempts to calculate surface area for those surfaces whose area cannot be calculated using standard techniques of integral calculus, and we show the role played by abstract category theory, affectionately known as “abstract nonsense”, in this very concrete subject. Wethen show that the technique used to find the length of a curve, or “arc length”does not work when generalized to find surface area, give a valid technique that is based on a “parametrizaion” of the surface, and investigate the problem of whether different parametrizaions result in different values for the surface area. It is in this that “abstract nonsense” plays an important role.