Philosophical Foundations of Russell's Logicism

Abstract

Logicism is the doctrine that mathematics is reducible to logic. It is usually presented in two theses: (1) Every mathematical concept is definable in terms of logical concepts. (2) Every mathematical theorem is deducible from logical principles. In this paper, I am not concerned with the truth or falsity of (1) and (2). Rather, I am concerned with the underlying philosophical system. Logicism is connected with the names of Frege, Russell (and Whitehead). But the logicism which is familiar to most philosophers is not the original logicist system of Russell. Instead, we read either Russell's later Introduction to Mathematical Philosophy or articles by Carnap and Hempel. My purpose in this paper is to return to the original logicist system of Russell. This system, at least in essentials, lasts through the publication of the First Edition of Principia Mathematica. I believe that examination of this system will shed light on why certain difficulties arose in later logicism, including the logicist views of the Logical Empiricist movement. Further, the issues are closely connected with general doctrines on the nature of philosophical analysis.