Chair: Lewis Williams
Modal potentialism has recently received a lot of attention in philosophy of mathematics and philosophy of logic. Modal potentialism is the view that mathematical objects are generated successively, and thus those not yet generated are in some sense merely possible existents. Most modal potentialists take a modal object language as primitive for mathematics, and most deny that the primitive modality is metaphysical. Proponents of modal potentialism include Øystein Linnebo, Kit Fine, and James Studd. In this talk I show that Linnebo’s approach to modal potentialism in his book Thin Objects (2018) is susceptible to inconsistency.
I start by presenting Linnebo’s framework, in particular the core notion of Fregean abstraction. Second, I elucidate the meta-linguistic nature of his intended interpretation of the primitive modality. Third, I show that Linnebo is thus committed to the legitimacy of introducing a primitive modal predicate of formulae in his object language. Fourth, in a strikingly similar fashion to the semantic paradoxes, I show that natural principles for this modal predicate are inconsistent. I conclude that Linnebo’s intended interpretation of the primitive modality and his formal approach do not match up. I also provide some follow-up options available to Linnebo, requiring that his framework change or further develop.
Further work is needed to show whether other approaches to modal potentialism are meta-linguistic, and thus likely susceptible to the challenges posed in this talk. At the very least, modal potentialists must have a response to these challenges.
See the DPhil Seminar website for details.
DPhil Seminar Convenors: Lewis Williams and Kyle van Oosterum
