 
Summary: Number theory and elementary arithmetic
Jeremy Avigad
1. Introduction
The relationship between mathematical logic and the philosophy of mathe
matics has long been a rocky one. To some, the precision of a formal logical
analysis represents the philosophical ideal, the paradigm of clarity and rigor;
for others, it is just the point at which philosophy becomes uninteresting
and sterile. But, of course, both formal and more broadly philosophical ap
proaches can yield insight, and each is enriched by a continuing interaction:
philosophical reflection can inspire mathematical questions and research
programs, which, in turn, inform and illuminate philosophical discussion.
My goal here is to encourage this type of interaction. I will start by
describing an informal attitude that is commonly held by metamathemati
cal proof theorists, and survey some of the formal results that support this
point of view. Then I will try to place these formal developments in a more
general philosophical context and clarify some related issues.
In the philosophy of mathematics, a good deal of attention is directed
towards the axioms of ZermeloFraenkel set theory. Over the course of
the twentieth century, we have come to learn that ZFC offers an extraor
dinarily robust foundation for mathematics, providing a uniform language
