Logicism in Logical Empiricism

Abstract

Logicism presents one of the cornerstones of logical empiricism. At the same time, the views defended by Carnap, Hahn, and Hempel, among others, differ significantly from Frege’s original thesis. The present chapter will focus on several accounts of logicism developed in logical empiricism between 1920 and 1940. The aim here is twofold. The first aim is to survey how the classical logicist thesis was modified during the period in question. As we will show, this concerns not only a radically revised conception of the underlying logic but also a new focus on non-arithmetical mathematical theories to be reduced to logic. More specifically, philosophers such as Carnap aimed to formulate a generalized logicism valid for all branches of pure mathematics, including different theories of geometry, topology, and algebra. As will be shown, his and related accounts are best described as a form of conditional logicism based on an if-thenist reconstruction of mathematics. The second aim in this chapter is to clarify how these contributions are related to other developments in the foundations of logic and mathematics at the time. Specifically, the focus here will be on Carnap’s attempts to reconcile logicism with a structuralist account of mathematics.