OBJECTIVITY AND CONSISTENCY IN MATHEMATICS: A CRITICAL ANALYSIS OF TWO OBJECTIONS TO WITTGENSTEIN\u27S PRAGMATIC CONVENTIONALISM

Wittgenstein\u27s views on mathematics are radically original. He criticizes most of the traditional philosophies of mathematics. His views have been subject to harsh criticisms. In this dissertation, I attempt to defend Wittgenstein\u27s philosophy of mathematics from two objections: the objectivity objection and the consistency objection. The first claims that Wittgenstein\u27s account of mathematics is not sufficient for the objectivity of mathematics; the second claims that it is only a partial account of mathematics because it cannot explain the semantic properties of mathematical systems. The first chapter outlines Wittgenstein\u27s Philosophy of Mathematics by stressing the differences and similarities with more traditional accounts. The second chapter discusses the objectivity objection. I distinguish epistemic from non-epistemic objectivity and discuss their relation with the issue of realism and anti-realism. I conclude first that either notion of objectivity is insufficient for disparaging anti-realist accounts of mathematics and, second, that Wittgenstein\u27s account is sufficient for both. The third chapter is a detailed discussion of the consistency objection. Some unsuccessful replies are discussed. In the fourth chapter, I reformulate the objection by means of a rather simple axiomatic system that I call theory T. This allows us to see more clearly why the replies considered before are not successful and, furthermore, which view Wittgenstein is actually required to hold in order to reject the objection. In the fifth chapter, I offer a brief outline of Wittgenstein\u27s account of logic and, finally, show how it provides the required answer to the consistency objection