Boaz Laan

CHALLENGE THE INFINITE SPEAKERS

Boaz Laan is a third-year PhD student in philosophy at the University of Oxford.

His research interests lie in the overlap between mathematics and philosophy; asking philosophical questions about mathematical methods; and applying mathematical methods to philosophical questions.

Boaz is currently working on the semantics of modal predicates.

Title: A Paradox for Modal Potentialism

Abstract

Modal potentialism has recently received a lot of attention in philosophy of mathematics and philosophy of logic. Potentialism is the view that mathematical objects are generated successively, and that certain generative processes are incompletable. Modal potentialism explicates potentialism using quantified modal logic, taking a modal object language as primitive for mathematics. Most modal potentialists deny that the primitive modality is metaphysical or temporal. 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.