ABSTRACT
This chapter introduces the non-uniqueness problem, which arises for the reduction of mathematics to set theory and for other platonist views. Platonist options were briefly canvassed and include plenitudinism, structuralism, pragmatism, and a thesis of epistemic humility regarding the species of abstract entities. The chapter sets outs Paul Benacerraf's version of the non-uniqueness problem, discusses some additional instances of the non-uniqueness problem that are not distinctively mathematical, and then explores the connection between non-uniqueness and broader worries about under determination and theory choice. In Benacerraf's initial presentation, the source of the non-uniqueness problem is the theoretical richness of set theory. Resolving the non-uniqueness problem forces the platonist into some substantial theoretical commitments. The non-uniqueness problem concerns competing proposals for reducing numbers or propositions to sets, this broader problem arises when platonists are forced to choose between many competing yet equally viable theories about the entirety of abstract reality.
