Wittgenstein on Truth and Necessity in Mathematics

Dissertation, Columbia University (1994)

Marc A. Joseph
Mills College
This essay investigates the nature of necessary truth in logic and mathematics through a discussion of Wittgenstein's writings on logic and mathematics. My principal scholarly interest in the dissertation is Wittgenstein' s later philosophy, but I also examine his philosophy in the Tractatus and the transitional texts of the 1930s. ;In chapter one I present the Tractatus' general theory of representation through an examination of Wittgenstein's work on the ancient problem of the unity of a proposition. I argue that Wittgenstein's solution draws upon Frege's philosophy of logic, despite the fact that Wittgenstein's interest in the problem is part of Russell's patrimony. In chapter two I present my interpretation of Wittgenstein's early treatment of logic and arithmetic. A major achievement of Wittgenstein's early work is the progress he makes towards showing that the logic of language is implicit in the concept of a proposition and that arithmetic is implicit in the concept of iteration. In chapter two I also discuss Wittgenstein's criticisms of Frege and Russell, on the one hand, and his relation to conventionalism of the logical empiricists, on the other. ;In the second half of the thesis I argue that the Tractarian idea that logic and mathematics belong to an ahistorical logico-metaphysical scaffolding surrounding language, thought, and the world evolves into the view that logical and mathematical truths are constitutive rules of the multifarious "language-games" that make up our human lives. Logical and mathematical truths are necessary because they are implicit in our concept of thinking or speaking about anything at all, as Wittgenstein had held in the Tractatus; but they enjoy this status on account of their position relative to our practices and not because they exhibit the prior constitution of the world. This treatment of these issues and my reading of Wittgenstein avoids both Kripke's interpretation of Wittgenstein, which collapses truth into assertability based on the agreement of a community and makes mathematical necessity a function of the affective side of human nature; and conventionalism, which is another standard reading of Wittgenstein and so-called "Wittgensteinian" account of mathematics.
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