is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that (...) standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

We discuss the development of metamathematics in the Hilbert school, and Hilbert’s proof-theoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.

to constructive systems is significant for contemporary metaphysics. However, many are surprised by these results, having learned that the Axiom of Choice (AC) is constructively valid. Indeed, even among specialists there were, until recently, reasons for puzzlement-rival versions of Intuitionistic Type Theory, one where (AC) is valid, another where it implies classical logic. This paper accessibly explains the situation, puts the issues in a broader setting by considering other choice principles, and draws philosophical morals for the understanding of quantification, choice (...) principles, and the prospects for constructivism. (shrink)