Dr. Sam Cowling
Mathematical discourse about numbers, sets, and functions raises difficult questions about the nature and existence of mathematical entities.
According to Platonists, the primary subject matter of mathematics is a realm of abstract entities that exist outside of space-time. According to nominalists, there are no abstract entities, so mathematical discourse must be carefully reinterpreted to preserve its intelligibility and explain its profound usefulness. Modal discourse, which invokes concepts like necessity and possibility, parallels mathematical discourse by purporting to describe another range of abstract entities: possible worlds.
After surveying the on-going debate between Platonists and nominalists over the existence of mathematical entities, I consider the consequences of Platonism and nominalism for the status of possible worlds as well as the apparent necessity of mathematical propositions.