Realism in Mathematics (RM) by Penelope Maddy (1990) is a delightful, thought-provoking book which contains interesting ideas on almost every page. Maddy attempts to develop and defend a novel form of “naturalized” set-theoretic realism, which she portrays as “compromise platonism.” The compromise is supposed to be between Quine/Putnam platonism, on the one hand, and Gödelian platonism on the other, and the focus of the book, to make the project manageable, is on set theory in particular. In this chapter I shall discuss the arguments of her book in some detail.

Chapter 1 of RM, entitled “Realism”, leads the reader through a variety of positions in the philosophy of mathematics on the way to a characterization of compromise platonism. A number of antirealist positions are briefly evaluated – intuitionism, formalism, if-thenism, the logicism of the logical positivists, and conventionalism – and some issues concerning different theories of truth are canvassed. Traditional platonism about mathematics is then characterized as the view that mathematical entities are abstract, outside physical space, eternal, unchanging, and acausal. Knowledge of such entities is supposed to be a priori and certain, and mathematical truths are supposed to be necessary truths. Like traditional platonism, Gödelian platonism holds that mathematical entities are abstract, and it takes its lead from the actual experience of doing mathematics. Maddy says that, unlike Quine/Putnam platonism, it recognizes a form of evidence intrinsic to mathematics.