Review

Abstract

Riemann’s integral is a major achievement of civilization. It solves in a rigourous way the ancient problems of squaring the disk and straightening the circle. It provides a solid foundation for Newtonian Physics, on which modern theories of physics and engineering can grow. Nevertheless, it has shortcomings: its limit theorems are not good enough, and not sufficiently many functions are Riemann–integrable. Consequently it is an insufficient tool for quantum physics and the associated partial differential equations. These shortcomings can be overcome, though, by a careful analysis of its concepts and a surprisingly slight improvement on them.