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date: 25 January 2022

Abstract and Keywords

The philosophy of mathematics was one of Ludwig Wittgenstein's central concerns from the beginning of his philosophical career until close to the end: it is one of the issues addressed in Tractatus Logico-Philosophicus. When Wittgenstein returned to philosophy and to Cambridge in 1929, it was the problem of the infinite that he started to consider — first of all in concert with Frank Ramsey. As far as one can judge, Wittgenstein and Ramsey seem to have moved together towards finitism, that is, a rejection of the extensional view of generalisation in the case of infinitely many propositions. Wittgenstein deduces that the sense of an arithmetical generalisation is its inductive proof. In addition, Wittgenstein argues that there is no problem of inconsistency until we actually find an inconsistency. He attributes the existence of hidden contradictions to ambiguity in the rules. As early as 1930, Wittgenstein was explicit about the ‘unbridgeable gulf between rule and application’, which is the essence of the rule-following argument.

Keywords: Ludwig Wittgenstein, philosophy, mathematics, Tractatus Logico-Philosophicus, Frank Ramsey, finitism, generalisation, proof, inconsistency, rule-following argument

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