ZFC set-theory is often seen as a foundation for all mathematics. Yet many philosophers worry that at its core there is a ‘Deep-Problem’: to avoid the paradoxes that plagued naïve set-theory, the axiomatic versions have been developed ad-hoc. They do not offer a philosophically reasonable account of what it is about the nature of sets and mathematics that allow some entities to be collected but not others. The ‘Iterative Conception’ of the set is often thought to provide such an explanation. Sets are formed at stages. A set can only collect that which exists at a ‘lower’ stage, but not that which only exists at a ‘higher’ stage.
For a growing number of philosophers (especially Linnebo, Shapiro and Studd) this motivates Modal-Potentialism, which says that at no stage is all of mathematical reality present. Thus the paradoxical sets never form. It is not that they cannot be collected; it is that they can never be there all at once to be collected. At face value this sounds like a potentially infinite universe where all mathematical facts cannot be true all at once. Such an approach could have profound philosophical implications, not just for mathematics.
Soysal (2020) has offered an important critique of this (and related) positions. Arguing that the ‘Deep-Problem’ is not quite so deep after all, and that the very definition of set that is central to any understanding of the iterative conception can suffice to diffuse the problem. Modal-Potentialism, on this account, is not well-motivated, and there is no need to take the iterative metaphor quite so literally.
I want to argue that there is an important way to view the ‘Deep-Problem’ in which Soysal’s answer does not suffice. I argue that not only does Modal-Potentialism offer a compelling answer to the problem, but that Soysal’s own answer can in fact be reinterpreted as something akin to the Modal-Potentialists answer. I also argue that Modal-Potentialism has the resources to offer a different philosophical way of thinking about mathematics in general and numbers in particular that could be deeply illuminating and philosophically very exciting.
See the DPhil Seminar website for details.
DPhil Seminar Convenor: Mariona Miyata - Sturm
