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The Neo-Kantians and the 'Logicist' Definition of Number

ABSTRACT: 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.

Although Frege published the first informal exposition of his 'logicist' programme in Die Grundlagen der Arithmetik (1884), his thesis that all mathematics follows from logic was almost completely neglected in Germany for a long time. Frege remained an isolated figure whose works were either strongly criticised or completely neglected by German philosophers. Frege's ideas started to have an impact in Germany only in the first decade of the twentieth century. In particular, the publication of Bertand Russell's The Principles of Mathematics (1903) and Louis Couturat's Les principes des mathématiques (1905) incited several prominent German philosophers to state their opinion about mathematical logic and the logicist programme. In this paper I shall discuss how the neo-Kantians Paul Natorp (1854-1924), Ernst Cassirer (1874-1945) and Jonas Cohn (1869-1947) criticised Russell's and Frege's theories of number. The study of their criticism will also throw some light on the historical origins of the current situation in philosophy, that is, on the split between analytic and Continental philosophy.

1. The 'logicist' definition of number as a class of classes

According to Russell, the goal of the logicist programme is to show that

all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903: v).

ABSTRACT: 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.

Although Frege published the first informal exposition of his 'logicist' programme in Die Grundlagen der Arithmetik (1884), his thesis that all mathematics follows from logic was almost completely neglected in Germany for a long time. Frege remained an isolated figure whose works were either strongly criticised or completely neglected by German philosophers. Frege's ideas started to have an impact in Germany only in the first decade of the twentieth century. In particular, the publication of Bertand Russell's The Principles of Mathematics (1903) and Louis Couturat's Les principes des mathématiques (1905) incited several prominent German philosophers to state their opinion about mathematical logic and the logicist programme. In this paper I shall discuss how the neo-Kantians Paul Natorp (1854-1924), Ernst Cassirer (1874-1945) and Jonas Cohn (1869-1947) criticised Russell's and Frege's theories of number. The study of their criticism will also throw some light on the historical origins of the current situation in philosophy, that is, on the split between analytic and Continental philosophy.

1. The 'logicist' definition of number as a class of classes

According to Russell, the goal of the logicist programme is to show that

all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903: v).

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