This will be a talk in the Centre for Philosophy and the Sciences Lunch Forum. Handout available here.

In this talk, I want to introduce two themes from my research in the foundations of mathematics, and discuss with the group relationships with the philosophy of science more broadly. My focus will be on generating discussion rather than presenting an argument per se.

The first theme is the idea that there are kinds of inference similar to those found in the philosophy of science at play in the justifications of axioms. In this regard we’ll discuss the idea that there is something like prediction and confirmation involved in certain justifications. I’ll ask what the data might be here, and how mathematical “data” might be similar to or different from our normal scientific data. We’ll also discuss the idea that when working with mathematics we ofteninfer to the best conception’. I’ll raise some questions for the group regarding pluralism and open texture in this context.

The second theme is that we often use extensions of a given universe of discourse in set-theoretic mathematics. This has implications for debates on absolute generality—if we can always transcend a given universe of discourse then absolute generality fails. The interesting fact in the context of set theory is that often we use an `external’ perspective in order to prove facts about the universe. This raises interesting questions about how seriously we should take the use of these resources. I’ll ask the group whether there are analogous methodologies and debates in the philosophy of science more widely (for example regarding multiverse theory and quantum mechanics).

This will be a talk on 21 June 2022 at the conference Engineering The Concept of Collection. The handout can be found here.

Abstract: In this talk I’ll present a main argument of a short book I’m working on entitled Engineering Set-Theoretic Concepts (I’m interested in comments on the draft, so please get in touch if you’d like to see it once ready). I’ll first note that conceptual engineering has formed a part of set-theoretic activity since its inception as a mainstream area of mathematical research, and that the development of the iterative (and other) conceptions of set was in part responding to inconsistency in the naive set-concept. I’ll then argue that whilst the iterative conception can be taken to be a consistent concept in its own right, it is deficient in various ways (in particular, it fails to tell us enough about the nature of infinite sets). Contemporary set theory, I’ll argue, has now moved to a maximal iterative conception of set, and this conception is inconsistent. Many contemporary accounts of the ontology underlying set-theoretic practice should be conceived of as attempts to engineer consistent conceptions of the maximal iterative concept of set. I’ll explain two such conceptions, the directed and schematic iterative conceptions of set. I’ll tentatively conclude that discussion in the philosophy of set theory should focus less on the vexed and seemingly intractable issue of ontology, and instead concern itself more with the (nonetheless difficult) question of the relative theoretical virtues of alternative conceptions.

This will be a talk on 09 March 2022 in the Oslo Infinity and Intensionality seminar on 09 March 2022. Handout here.

We explain and explore class-theoretic potentialism—the view that one can always individuate more classes over a set-theoretic universe. We examine some motivations for class-theoretic potentialism, before proving some results concerning the relevant potentialist systems (in particular exhibiting failures of the .2 and .3 axioms). We then discuss the significance of these results for the different kinds of class-theoretic potentialist.

This will be a talk in the Oslo Mathematical Logic Seminar on 17 February 2022. Handout here.

Often the view that every set is countable (call this position `countabilism’) is regarded as a viewpoint that is `restrictive’ as compared to uncountabilism (the view that there are uncountable sets). After all, doesn’t the uncountabilist hold that there are *more* sets than the countabilist? In this short talk I’ll present some results that challenge this assumption. In particular, we’ll examine a mathematical analysis of `restrictiveness’ proposed by Penelope Maddy. We’ll consider a version of Maddy-restrictiveness modified to weaker base theories that allows for countabilism, and we’ll see that in this context it is the uncountabilist, rather than the countabilist, who is making the restrictive claims. I’ll close with some philosophical remarks about (a) what we want out of set-theoretic foundations, and (b) what the role of analyses of `restrictiveness’ in this framework might be.

This will be a talk in the Oslo Infinity and Intensionality project seminar on 24th November 2021. Handout available here.

Title: Potentialist Sets, Intensions, and Non-Classicality

Abstract: There’s lots of different kinds of set-theoretic potentialism. We could add ranks, or subsets, or do something with a little of both. At the same time, we could be liberal potentialists (thinking that the modal truths are fixed) or strict potentialist (thinking that the modal truths get `made true’ as more sets come into being). The situation with sets is now a lot better understood on all these combinations, we have decent semantics for both strict and liberal potentialists and accounts of mirroring theorems. But what about classes (conceived of as intensional entities)? These are collection-like entities that can change members as more sets come to exist. Examples include the class of all (self-identical) sets, the class of all countable sets, and many more besides. So how do we handle talk of these intensional entities on each kind of potentialism?

In this talk I’ll do the following:

1. Give a very rough description of some in progress work on how to handle intensions, with truth values being collections of worlds capturing *when* a set gets into an intensional class.

2. Consider some philosophical questions regarding the approach (this will be the bulk of the talk). In particular we’ll discuss:

2.i. Is this kind of project anti-potentialist?

2.ii. What intensions exist?

2.iii What motivations might there be for strict potentialism in this context?