This was a talk at the Oslo Logic Seminar. Slides here.

This paper addresses the issue of the kinds of mathematical structure that exist, and the theories we use to talk about them. Often when philosophers discuss different varieties of structure and theories, they focus on the following distinction—particular structures and non-algebraic theories are those that are instantiated only by pairwise isomorphic structures (e.g. the axioms for second-order Dedekind-Peano arithmetic and the natural number structure), and general structures and algebraic theories are those that can have non-pairwise-isomorphic exemplars (e.g. the axioms for groups and the group structure).

In this paper I want to point to some results from model theory that show that the distinction between particular and general structures and non-algebraic and algebraic theories requires further refinement. General structures and algebraic theories can be subdivided into a further two philosophically salient different kinds, that I shall call LEGO-like and non-LEGO-like theories/structures. Roughly speaking a theory or structure is LEGO-like iff it is algebraic but any model/instantiation of a certain cardinality can be built from an initial template using a well-defined and determinate construction process. I’ll argue that the notions of model-theoretic geometry and strongly minimal set provide a nice way of capturing this idea. In this way, model theory provides philosophy with a rich variety of examples that have been neglected in virtue of our focus on the `canonical’ cases of arithmetic and basic abstract algebra.

This will be a talk at the conference Understanding Langauge. Slides here.

Abstract: In this talk, I’ll explain one implication of Engineering Set-Theoretic Concepts, namely that even in mathematics formalisation can fail to fix meaning. I’ll look at some different conceptions of set, and point out that if there are multiple competing conceptions at play in mathematical discourse, then there is a failure of certain internalist epistemological principles regarding set-theoretic knowledge.

This will be a talk on 14th November 2023 at the Konstanz Logic Colloquium. Handout here.

Abstract: On one view of set theory, the paradoxes precipitated a radical clarification of our concept of set, culminating with the isolation of the iterative conception of set. In this talk, I’ll present some work from a book I’ve been working on (also entitled Engineering Set-Theoretic Concepts). I’ll argue that in fact the iterative conception admits of further splitting into multiple conceptions of set, and that we are ourselves at a conceptual crossroads motivated by a kind of paradox. In one direction, we are pushed to the standard picture of ZFC. In the other, we are pushed towards a conception of set on which every set is countable. I’ll also (for the logic folks) present some of the mathematics behind this latter less familiar picture, and situate ZFC-based set theory within it. 

This (joint work with Ethan Russo and Chris Scambler) will be a talk at the Infinity and Intensionality project seminar. Slides available here.

Abstract: This article articulates and assesses a Fine-inspired approach to the foundations of mathematics. After Fine, we call the program `procedural postulationism’. The core idea for the program is that mathematical domains of interest can fruitfully be viewed as the outputs of construction procedures. Fine argues that this viewpoint has various potential epistemic and ontological benefits in philosophy of mathematics; and in any case we think the guiding idea is interesting enough to warrant close inspection.

To this date little formal flesh has been put on the program’s conceptual bones, at least in the published literature. This article seeks to remedy that deficit. We offer a logic for `creative’ imperatives — imperatives that command the introduction of new objects into the domain — and show how to treat the concept of indefinite iteration in that logic.

We then give consistency proofs for arithmetic and set theory starting from the formalized hypothesis that certain commands statable in the resulting logic are executable.

We also *begin* an assessment of Fine’s philosophical goals for the program in light of our formal framework.

This (joint work with Dan Waxman) will be an online talk at the conference Masterclass of Hamkins’ `The Set-Theoretic Multiverse’: 10 Years After. Handout here.

Abstract: Hamkins has argued for a very radical version of set-theoretic multiversism, on which every model of first-order ZFC is on an ontological par with every other. Moreover, his responses to the categoricity arguments and attitude towards the natural number structure indicate that he is committed to a principle we call Bridging: Only those structures and statements that are agreed upon by all models of first-order ZFC are determinate. In this paper, following arguments of Barton and Koellner, we suggest that this view is not coherent since the notion of a model of first-order ZFC is itself not determinate within the ZFC-multiverse. We consider a response given by Barton—that the multiverse view should be regarded as proposing an algebraic framework for set-theoretic practice—and argue that it is unworkable in the details in virtue of its dependence on another indeterminate notion, namely truth. We consider two responses available to the radical multiversist: (1.) Accept that the natural numbers are determinate, but otherwise hold on to the framework, or (2.) Plump for a further radicalisation of the view to only considering `feasible’ fragments of ZFC. Whilst both are coherent, we argue that each gives up a substantial part of the view. (1.) gives up Bridging, whereas (2.) gives up the naturalism that is often seen as a virtue of the position.