Slides here. This will be a talk at the UCI LPS Colloquium.

Abstract: In this talk I’ll argue that we’re now at a conceptual crossroads regarding the iterative conception of set. To do this I’ll appeal to work on conceptual engineering. I’ll argue 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, and tentatively conclude that discussion 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 Juliette Kennedy’s book Gödel, Tarski and the Lure of Natural Language, to be held at the Pacific APA. Handout here. I’ll raise some questions concerning the relationship between the notion of formalism freeness and that of mathematical structure.

This will be a very short talk to the intro logic students at the University of Oslo. Slides here.

This will be a talk in the Oslo Logic Seminar. Handout here.

Abstract:

In this talk, I will present some early-stage work in progress on the iterative conceptions of set. In a recent booklet (itself entitled Iterative Conceptions of Set) I’ve argued that we can think more generally about the iterative conception as a way of generating sets, without committing to the Powerset operation. The booklet is rather “programmatic”, however, and leaves a lot of mathematical questions open. In this talk I’ll try to accomplish the following things:

(1.) I’ll present the rough idea I gave in the booklet.

(2.) I’ll outline one application I’m working on (a version of Steel’s multiverse programme), and the challenges I’m facing there.

(3.) I’ll present some thoughts about how one might try to extract a general notion of an “iterative conception of set”, at least model-theoretically.

This will be a talk at the workshop Generative Metaphysics and the Philosophy of Mathematics. Slides here.

Abstract: This article articulates and assesses an imperatival 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. 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. Using this framework, we will assess whether the view can claim to have epistemic and ontological benefits over standard `declarative’ approaches.