Is logic distinct from mathematics? This is the overarching question of this seminar. Specifically, we will discuss two contrasting types of answers to this question. One, which can be traced to Leibniz, holds that the answer is no, that the truths of mathematics, in part or altogether, are logical truths. The other, which can be traced to Kant, answers yes, because knowledge of mathematics requires some non-logical form(s) of intuition, perhaps of space and time. We will look closely at versions of the Leibnizian answer advanced by two originators of analytic philosophy: Gottlob Frege and Bertrand Russell. Both, but Frege in particular, have deep philosophical reasons for their positive answer, which is nowadays known as logicism. We will discuss some of those philosophical reasons. More importantly, both insist that their answers would not be truly compelling unless they identified precisely the fundamental truths of logic, and gave rigorous proofs of axioms of mathematical theories from the truths of logic. Part of our work is to understand Frege's proofs of the Dedekind-Peano axioms of arithmetic from his formulation of logic. We will discuss how Russell, through the Paradox that bears his name, demonstrates the inconsistency of Frege's logic. We will then investigate two programs of salvaging logicism. One, called neologicism, attempts to show how arithmetic is derivable from a consistent variant of Frege's logic. The other, due to Russell, is to circumvent his Paradox by a conception of logic as governing a hierarchy of types of entities, and then deriving arithmetic from this type-theoretic logic. We will next turn to a version of the Kantian answer advanced by David Hilbert. This answer, called (Hilbertian) formalism, holds that genuinely contentful mathematics rests on something like our intuitive knowledge of the possibilities of operations with signs, but that all of classical infinitary mathematics may be justified as a conservative extension of this content. If time permits, we will examine Kurt Gödel's theorems of the incompleteness of arithmetic and the trouble they raise for Hilbert's formalism. |