Discovery and Purity in Archimedes

Abstract

Philosophers of mathematics wonder about the applicability of mathematics to scientific explanations of the physical world. My inquiry is in the opposite direction: why and how can the study of physical phenomena be helpful to the advancement of mathematics? The value of purity has long driven changes and progress in mathematics. According to the ideal of purity, mathematics should be purged of ideas of an extraneous source, because they do not amount to true explanations and can be misleading. Meanwhile, mathematicians, including those who underscore purity, acknowledge the fruitfulness of borrowing foreign ideas to help with mathematical discovery. This dissertation studies Archimedes’ Method, a work that highlights the fruitfulness of geometric discovery through mechanical imaginations. I argue that Archimedes brings out the heuristic potential of mechanics in two ways: one is to develop new methods that incorporate non-rigorous techniques inspired by the study of the physical world into rigorous mathematical demonstrations, the other is to envisage an art of discovery through mechanics, of which his Method provides starting points. In this dissertation I show that a dialogue between Archimedes’ vision with regard to discovery and the ideal of mathematical purity can shed light on both the thought of Archimedes and the study of the history of mathematics.