Abstract: The early sections describe the pre-theoretic realism of the mathematician, survey the basic forms of realism in philosophy, and attempt to disentangle the issues of realism from debates over the nature of truth. The final section begins by laying out traditional Platonism, intuitionism, and formalism. Quine's famous critique of Carnap's conventionalism then leads to Quine's realism, and Putnam's developments thereof, while a different sort of mathematical realism is found in Gödel's writings. Set theoretic realism emerges as an effort to build on the strengths of Quine's and Gödel's realisms, while avoiding their weaknesses and filling in their lacunae.

Keywords: Carnap, conventionalism, formalism, Gödel, intuitionism, Platonism, Putnam, Quine, realism, set theoretic realism,