ABSTRACT
This chapter briefly surveys three kinds of arguments sometimes alleged to make problems for platonism. These arguments divide into the following categories; arguments from paradox, which hold that the paradoxes generated by platonism require that rejects abstract entities; razor arguments, which appeal to the theoretical virtue of parsimony in defending nominalism; regress arguments, which hold that objectionable explanatory or metaphysical regresses result from a commitment to platonism. The contradiction underlying Russell's paradox thwarted early efforts to provide a set-theoretic foundation for mathematics. It also spawned a litany of proposals for recasting set theory in order to avoid this and other closely related paradoxes. The idea that parsimony or simplicity serves as a legitimate guide to theory choice is commonly glossed as a commitment to Ockham's razor, the methodological injunction to avoid positing entities beyond necessity. The history of the nominalist–platonist debate is peppered with regress arguments.
