Chapter 8 - Arithmetic in the Tractatus Logico-Philosophicus

Summary

This chapter discusses Wittgenstein’s remarks on mathematics in sections 6.02–6.031 and 6.2–6.241. These remarks are limited to arithmetic, with definitions of the natural numbers (6.02) and of multiplication (6.241). In the first part, we discuss Wittgenstein’s criticism of the theory of types involved in Russell’s rival ‘logicist’ account, with specific criticisms of Russell’s axioms of infinity (5.535) and reducibility (6.1232). The second part presents Wittgenstein’s positive account of natural numbers in terms of ‘repeated applications of an operation’ (based on his remarks on operations at 5.2–5.254) and of arithmetical calculations. Limited parallels with the lambda calculus are brought to the fore, while explanations that presuppose a scheme of primitive recursion are criticized. The third section discusses related philosophical remarks about the centrality of the ‘method of substitution’ (6.24), arithmetical equations as ‘pseudo-propositions’ (6.241 & 6.2), and the claim that the identity of the two sides of an equation is merely perceived, not assertable (6.2322). Looking ahead, the fourth and final part discusses briefly Wittgenstein’s reasons for abandoning this approach in the early ‘middle period’.