The Neo-Kantians and the Logicist Definition of Number :: Mathematics Math Mathematical Papers

The Neo-Kantians and the Logicist Definition of NumberABSTRACT The takings of Russells The Principles of Mathematics (1903) and Couturats Les principes des mathematiques (1905) incited several prominent neo-Kantians to make up their mind roughly the logicist program. In this paper, I shall address the critiques presented by the following neo-Kantians Paul Natorp, Ernst Cassirer and Jonas Cohn. They argued that Russells attempt to deduce the number concept from the carve up concept is a petitio principii. Russell replied that the sense in which e rattling(prenominal) object is one must be distinguished from the sense in which one is a number. I state that Russell was wrong in dismissing the neo-Kantian argument as an elementary discursive error. To accept Russells distinction would be to accept at least part of Russells logicist program. The expression a crystalize with one member would presuppose the number one altogether if one concurrently accepted the analysis which mat hematical logic provides for it (the class u has one member when u is not null and x and y be 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 create the first informal exposition of his logicist programme in gnarl 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 potently criticised or completely neglected by German philosophers. Freges ideas started to have an impact in Germany only in the first decade of the twentieth century. In particular, the publication of Bertand Russells The Principles of Mathematics (1903) and Louis Couturats Les principes des mathmatiques (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 Russells and Freges 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 soften between analytic and Continental philosophy. 1. The logicist definition of number as a class of classes correspond to Russell, the goal of the logicist programme is to show thatall pure mathematics deals exclusively with concepts definable in terms of a very small number of entire logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles (Russell 1903 v).The Neo-Kantians and the Logicist Definition of Number Mathematics Math numeric PapersThe Neo-Kantians and the Logicist Definition of NumberABSTRACT The publication of Russells The Principles of Mathematics (1903) and Couturats Les principes de s 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 Russells 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 Russells distinction would be to accept at least part of Russells 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 mathematica l 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. Freges ideas started to have an impact in Germany only in the first decade of the twentieth century. In particular, the publication of Bertand Russells The Principles of Mathematics (1903) and Louis Couturats Les principes des mathmatiques (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 Russells and Freges 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 classesAccording to Russell, the goal of the logicist programme is to show thatall 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).