Talk: Philosophy and the Iterative Conceptions of Set

This will be a one-off talk at the University of California, Davis. Slides available here.

The iterative conception of set is often thought to provide a (if not “the”) mainstream resolution of the set-theoretic paradoxes. Whilst many philosophical questions remain (e.g. about the underlying axioms and metaphysics of the iterative conception), it is taken to largely be clear what the conception states—the universe of sets is formed by forming all possible subsets at additional stages and continuing this into the transfinite. In this talk, I’ll present some work suggesting that there are different kinds of iterative conception. The “default” version just mentioned is the “strong” version, but there is a “weaker” version too—we might not insist that we get all possible subsets of a given set at additional stages. Rather, we could instead have operations of set formation that are more fine-grained, and allow us to slowly “feed in” more and more subsets. Taking this approach, I argue, allows us more flexibility with the foundational theory we adopt, and reveals some interesting possibilities for the advancement of the philosophy of mathematics. Towards the end of the talk, I’ll mention some relationships with other areas of philosophy, including negative epistemology, conceptual engineering, and relationships with pluralism and perspectivism.

Talk: Philosophy and the Iterative Conceptions of Set

This will be a one-off talk at the University of Oslo. Slides here.

Abstract: The iterative conception of set is standardly taken to provide a (if not “the”) mainstream resolution of the set-theoretic paradoxes. Whilst many philosophical questions remain (e.g. about the underlying axioms and metaphysics of the iterative conception), it is taken to largely be clear what the conception states—the universe of sets is formed by forming all possible subsets at additional stages and continuing this into the transfinite. In this talk, I’ll present some work suggesting that there are different kinds of iterative conception. The “default” version just mentioned is the “strong” version, but there is a “weaker” version too—we might not insist that we get all possible subsets of a given set at additional stages, but we could instead have operations of set formation that are more fine-grained. Taking this approach, I argue, allows us more flexibility with the foundational theory we adopt, and reveals some interesting possibilities for the advancement of the philosophy of mathematics. Towards the end of the talk, I’ll mention some relationships with other areas of philosophy, including epistemology and the philosophy of science.

Talk: Algebraic Levels in Mathematical Structuralism

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.

Talk: To What Extent Does Formalisation Fix Meaning?

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.

Talk: Engineering Set-Theoretic Concepts

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.