The Evolution of Number

Abstract

Abstract How is the historical development of mathematics to be understood sociologically? The writings of Mannheim and Wittgenstein suggest two contrasting approaches which may be evaluated through a detailed analysis of the development of the number concept. Five main number systems can be distinguished of which the most basic is the system of whole numbers and, given that importance attaches to generalizing power, there can be shown to be a tendency for the number concept to be progressively augmented. This process may be understood by reference to the basic pre-Darwinian evolutionary idea of “unfolding.” Complementing this intrinsic element, in actual historical instances a variety of extrinsic factors may be seen to be operative, facilitating or retarding development. To explain historical change fully an elaborate normative structure must be analyzed incorporating a macroinstitutional level, the “rules of the mathematical game,” notational considerations, and relations internal to mathematics itself...