A process of generalization

Abstract

This article examines Ernst Kummer’s creation of ideal factors, which provides an interesting example of generalization within the set of complex numbers. Kummer developed a theory of ideal numbers in order to generalize arithmetical properties of natural numbers by extending these properties to certain complex numbers. His goal was to make complex numbers analogous to natural ones. This article first considers Kummer’s use of several analogies, primarily with arithmetic and chemistry, to come up with ideal factors of complex numbers. It then situates Kummer’s investigations on complex numbers with respect to Carl Friedrich Gauss’s work and compares his theory of ideal factors with Richard Dedekind’s ideals theory. It shows that Kummer’s method of generalization is premised on the distinction he articulated between ‘permanent’ and ‘accidental’ properties of complex numbers. This distinction draws from Kummer’s conception of mathematics, which was essentially different from those espoused by Gauss and Dedekind.