Abstract: Many mathematicians understand their work as an effort to describe the denizens and features of an abstract mathematical world or worlds. Most philosophers of mathematics consider views of this sort highly problematic, largely due to two stark difficulties laid out by Benacerraf: first, if mathematical things are abstract, and thus not to be found in space and time, how can we come to know anything about them? Second, how can mathematics be the study of certain particular things, when all that seems to matter mathematically are various structural features and relations? The goal of this book is to develop a philosophically defensible version of the mathematician's pre-theoretic realism (sometimes called ‘Platonism’) about mathematical things. Beginning from an analysis of the strengths and weaknesses of Quine's and Gödel's versions of mathematical realism, I propose an alternative called ‘set theoretic realism’ and argue that it avoids both of Benacerraf's problems. In their place, I raise a new problem: given that some open questions of mathematics (like Cantor's Continuum Hypothesis) cannot be settled on the basis of the standard axioms, how are we rationally to evaluate new candidates for axiomatic status (such as Gödel's Axiom of Constructibility or various large cardinal axioms)? Set theoretic realism and its realistic cousins are not the only positions that face this important new challenge—various popular versions of nominalism and structuralism do as well—which suggests that it taps into a fundamental issue.

Keywords: Benacerraf, Cantor, Continuum Hypothesis, Gödel, Penelope Maddy, philosophy, philosophy of mathematics, Platonism, Quine, realism, set theoretic axioms, set theoretic realism