Russell and the Neo-Logicists

Abstract

For the Neo-Fregeans, logicism is first and foremost a means to meet Benacerraf's challenge. The contention is that Hume's Principle provides us with an attractive semantic and epistemological theory, which avoids both extreme Platonism and fictionalism. This answer does not extend, however, to earlier versions of logicism - the ones defended by the historical Frege and Russell, which do not use any abstraction principle. From the neo-logicist perspective, the old versions of logicism no longer constitute credible philosophies of mathematics.

In this paper, I suggest that the central position occupied today by the Benaceraff's dilemma blinds us to the possibility of other forms of philosophical agenda, which the ancient logicists attempted to fulfill. Focusing on geometry and the theory of reals, I show that, beside the unification and reduction of all mathematics to logic, another issue was at stake in The Principles as in Principia: how to carve mathematics at its joint? Russell wanted to arbitrate between the various conceptions of mathematical architecture, and found a rational way of doing this. If both mathematics and logic have changed since Russell's time, there is reason to believe that the architectonic issue is still alive today.