 
Summary: Understanding, formal verification, and
the philosophy of mathematics
Jeremy Avigad
August 23, 2010
Abstract
The philosophy of mathematics has long been concerned with deter
mining the means that are appropriate for justifying claims of mathemat
ical knowledge, and the metaphysical considerations that render them so.
But, as of late, many philosophers have called attention to the fact that
a much broader range of normative judgments arise in ordinary math
ematical practice; for example, questions can be interesting, theorems
important, proofs explanatory, concepts powerful, and so on. The as
sociated values are often loosely classified as aspects of "mathematical
understanding."
Meanwhile, in a branch of computer science known as "formal ver
ification," the practice of interactive theorem proving has given rise to
software tools and systems designed to support the development of com
plex formal axiomatic proofs. Such efforts require one to develop models
of mathematical language and inference that are more robust than the
the simple foundational models of the last century. This essay explores
