Volume 34, 1998
Philosophy of Mathematics
The Neo-Kantians and the ‘Logicist’ Definition of Number
The publication of Russell's The Principles of Mathematics (1903) and Couturat's Les principes des mathematiques (1905) incited several prominent neo-Kantians to make up their mind about the logicist program. In this paper, I shall discuss the critiques presented by the following neo-Kantians: Paul Natorp, Ernst Cassirer and Jonas Cohn. They argued that Russell's attempt to deduce the number concept from the class concept is a petitio principii. Russell replied that the sense in which every object is 'one' must be distinguished from the sense in which 'one' is a number. I claim that Russell was wrong in dismissing the neo-Kantian argument as an elementary logical error. To accept Russell's distinction would be to accept at least part of Russell's logicist program. The expression 'a class with one member' would presuppose the number 'one' only if one simultaneously accepted the analysis which mathematical logic provides for it (the class u has one member when u is not null and 'x and y are us' implies 'x and y are identical'). My point is that the aforementioned analysis provided by mathematical logic was something that the neo-Kantians were not ready to accept.