Though extra-mathematical philosophy is irrelevant to the methodological decisions of mathematics, a second-philosophical understanding of the human practice of mathematics requires answers to traditionally philosophical questions of metaphysics and epistemology. This chapter explores three broad schools of thought on these questions: Robust Realism, Thin Realism, and Arealism. One familiar version of Robust Realism is Gödel's platonism, which raises an equally familiar epistemological objection associated with Benacerraf. Thin Realism reorients the epistemological discussion by beginning from the idea that the way to find out about mathematical things is to do mathematics; the Second Philosopher argues that traditional characterizations (objective, nonspatiotemporal, acausal) are retained while the proper methods are just those ratified by mathematical goals and values (as in §IV.3). Arealism embraces the very same mathematical methods as Thin Realism, the very same choices and decisions in practice, but includes no added claims of truth or existence. The Second Philosopher argues that Robust Realism is problematic (though not for strictly Benacerrafian reasons), that Thin Realism and Arealism are viable, and indeed that Thin Realism and Arealism are cosmetic variants of the same view.
Keywords: Arealism, Benacerraf, epistemology, existence, Gödel, metaphysics, methodology, platonism, Robust Realism, Thin Realism