Whatever Happened to Group Theory?

Abstract

It is a little-known fact that Russell, in his very earliest attempts to apply Peano’s logic, attempted (and quite successfully) to apply it to group theory. That this is little-known is not surprising since Russell quickly abandoned his treatment of group theory and left it unpublished. It formed part of the first draft of his important paper, ‘The Logic of Relations’ (LOR), which he wrote in 1900 immediately after his return from the Paris Congress at which he learnt of Peano’s work for the first time. This paper has long been rightly celebrated as a landmark in Russell’s development: it marks his first use of modern symbolic logic, the creation of a formal theory of relations, the first presentation of his definition of the cardinal numbers, a treatment of cardinal and ordinal addition, a construction of the rational numbers from operations on the natural numbers, a generalization of Cantor’s construction of the real numbers from sequences of rational numbers, and Russell’s first efforts to establish the truth of logicism. It was, in short, the first serious, constructive step towards Principia Mathematica — the first step that was not subject to almost immediate retraction — and as the above partial list of its contents makes clear, it was a large first step. But LOR, as it was eventually published in Peano’s Revue de mathématiques in 1901, was very different from the paper Russell drafted in October 1900;¹ in particular, the treatment of group theory was dropped without replacement and without comment before publication.