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Number theory and elementary arithmetic Jeremy Avigad

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 Zermelo-Fraenkel 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


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics