 
Summary: Philosophy of Mathematics
Jeremy Avigad
Final version will appear in Boundas, Constantin editor, The Edinburgh Companion to
the 20th Century Philosophies, Edinburgh University Press. The annotations in the
bibliography will not appear in the published version.
1. Introduction
The philosophy of mathematics plays an important role in analytic philosophy, both as a
subject of inquiry in its own right, and as an important landmark in the broader
philosophical landscape. Mathematical knowledge has long been regarded as a paradigm
of human knowledge with truths that are both necessary and certain, so giving an account
of mathematical knowledge is an important part of epistemology. Mathematical objects
like numbers and sets are archetypical examples of abstracta, since we treat such objects
in our discourse as though they are independent of time and space; finding a place for
such objects in a broader framework of thought is a central task of ontology, or
metaphysics. The rigor and precision of mathematical language depends on the fact that
it is based on a limited vocabulary and very structured grammar, and semantic accounts
of mathematical discourse often serve as a starting point for the philosophy of language.
Although mathematical thought has exhibited a strong degree of stability through history,
the practice has also evolved over time, and some developments have evoked
controversy and debate; clarifying the basic goals of the practice and the methods that are
