Quantification and Finitism

Abstract

 My aim is to clarify Wittgenstein's foundational outlook. I shall argue that he was neither a strict fmitist, nor an intuitionist, but a finitist (Skolem and Goodstein.) In chapter I, I argue that Wittgenstein was a revisionist in philosophy of mathematics. In chapter II, I set up a distinction between Kronecker's divisor-theoretical approach to algebraic number theory and the set-theoretic style of Dedekind's ideal-theoretic approach, in order to show that Wittgenstein's remarks on existential proofs and the Axiom of Choice are in the constructivist tradition. In chapter in, I give an exposition of the logicist definitions of the natural numbers by Dedekind and Frege, and of the charge of impredicativity levelled against them by Poincare, in order to show, in chapter IV, that Wittgenstein's definition of the natural number in the Tractatus Logico-Philosophicus was constructivist. I also discuss the notions of generality and quantification, and Wittgenstein's later criticisms of the notion of numerical equality. In chapter V, after discussing the current strict finitist literature, I reject the contention that Wittgenstein's remarks give support to such a programme, by showing that he adhered to a potentialist view of the infinite, and, moreover, that his grammatical approach provides him with an argument against strict finitism. In chapter VII, I also reject the identification of his remarks about surveyability with the strict finitist insistence on feasibility. I n chapter VI, I describe the Grundlagenstreit about the status of Π 0 1 -statements. Wittgenstein views on generality, induction, and the quantifiers lead to a rejection of quantification theory which sets him apart from intuitionism, and closer to finitism. I also examine Wittgenstein's argument against the Law of Excluded Middle. In the last chapter, I discuss Wittgenstein's prescriptions for the formation of real numbers, showing that they imply a constructivization of the Cauchy sequences of the type of Bishop or of the finitist recursive analysis, and the rejection of the intuitionistic notion of choice sequences.