Gödel, Wittgenstein and the Nature of Mathematical Knowledge

Abstract

The nature of mathematical knowledge can be understood only by locating the knowing mathematician in an epistemic community. This claim is defended by extending Kripke's version of the Private Language Argument to include informal rules and using Gödelian results to argue that such rules rules necessary in mathematics. A committed formalist might evade Kripke's original argument by positing internal mechanisms that determine rule-governed behavior. However, in the presence of informal rules, the formalist position collapses into the extreme skepticism that the Private Language Argument works against. The existence of a community of rule followers provides the only viable alternative to such skepticism.