The philosophy of mathematics of Wittgenstein's 'Tractatus Logico -Philosophicus': Explication and analysis of Wittgenstein's early philosophy of mathematics.

摘要

Whether Wittgenstein intended his Tractatus Logico-Philosophicus to support logicism is a question on which disparate views have been advanced. The views advanced on the question are that either (i) Wittgenstein was a logicist in the same sense as were Frege and Russell or (ii) he was a logicist in a different sense or (iii) he was not a logicist in any sense. There are two reasons for the disparity of views. (i) What Wittgenstein intended to say in the Tractatus regarding mathematics has never been adequately explained. (ii) Wittgenstein's understanding of logicism, particularly his understanding of the purpose that he believed logicism was supposed to achieve, has never been evaluated. To explain what Wittgenstein intended to say in the Tractatus regarding mathematics, I have tried to elucidate various arguments concerning mathematics both from his earlier writings and from those later writings which explain views he held soon after his return to philosophy in 1929. The writings on which I have based my conclusions are found in Wittgenstein's early letters, dictations, notebooks and the Prototractatus, and also in transcripts made of Wittgenstein's conversations with Schlick and Waismann during 1929 and 1930, Waismann's 1930 lecture at Konigsberg on Wittgenstein's philosophy of mathematics and the Philosophical Remarks. On the basis of these writings, I have been able to give an interpretation of the philosophy of mathematics of the Tractatus which explains Wittgenstein's assertions on series of forms, the ancestral relation, the natural number series, the concept of number, arithmetical equations, classes in mathematics and the generality of mathematics. Furthermore, the resulting interpretation of the philosophy of mathematics of the Tractatus includes an explanation of what Wittgenstein understood to be the similarity between logic and arithmetic, namely, that the respective methods of using tautologies and equations are analogous. Hence, given this explanation, I have been able to decide whether Wittgenstein intended the Tractatus to support logicism. I contend that he did not. Not only did he believe that Frege's, and Whitehead and Russell's logicism misrepresent the similarity between logic and arithmetic, he regarded the very problem which logicism was supposed to solve as a pseudo-problem.