Abstract Objects and Modality

Abstract

In the field of philosophy of mathematics, various programs have been proposed in order to resolve Benacerraf's dilemma. Among them I would like to focus on the debate between Field's Fictionalism and Hale-Wright's Fregean Platonism. It is concerned with the modal status of the existence of mathematical objects. Through examining this debate, I will reveal some backgrounds of recent philosophy of mathematics. It will be showed that the difference between Fictionalism and Fregean Platonism is based on how they should take the notion of logical necessity, i.e. that while Field's program inevitably restricts the notion of logical necessity to physicalistically permissible one, Fregean Platonists must somewhat extend it to that of conceptual necessity and such an extension opens up the possibility that the existences of mathematical objects are derived from some kind of existencial statements.