Talk: On Procedural Postulationism

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.

Talk: Is the Radical Multiverse view coherent?

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,┬ánamely 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.

Talk: Fusing Foundations: How similar are foundational debates in mathematics and science?

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).

Talk: Engineering Set-Theoretic Concepts

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.

Talk: Varieties of Class-Theoretic Potentialism

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.