Philosophy Department Colloquium

A Public Lecture by John Cardinal O'Hara Professor of Logic

'A Potentialist Perspective on Ultrafinitism' with Dr. Joel David Hamkins, Thursday, April 30, 2026 4pm to 6pm Scripps Hall Rm 111

Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist. I shall give an account of the deep model-theoretic connections between two otherwise very different-seeming approaches to ultrafinitism, which differ on the question of whether the feasible numbers are closed under successor. These connections are revealed and strengthened by adopting a potentialist outlook on the nature of arithmetic, where one's realm of feasibility can be successively enlarged and enlarged again. This approach opens the door to a modal perspective on arithmetic and the idea of expressing core ultrafinitist principles in a modal vocabulary. Ultimately, this is an actualist modal model theory of ultrafinitist potentialism, which I take to shed light on the nature of ultrafinitism.

Professor Hamkins is a mathematician and philosopher who undertakes research on the mathematics and philosophy of the infinite, working on a broad spectrum of topics in logic and the philosophy of mathematics, including mathematical and philosophical logic, modal logic, set theory and the philosophy of set theory, forcing and large cardinals, computability theory, infinitary computability, infinitary game theory, and infinitary utilitarianism. He earned his PhD in mathematics from the University of California at Berkeley and comes to Notre Dame from the University of Oxford, where he was Professor of Logic in the Faculty of Philosophy and the Sir Peter Strawson Fellow of Philosophy at University College, Oxford. Prior to that, he held longstanding positions in mathematics, philosophy, and computer science at the City University of New York, as well as diverse visiting positions at New York University, Carnegie Mellon University, University of California at Berkely, University of Amsterdam, University of Münster, and elsewhere. Professor Hamkins is the top user on the advanced mathematics Q&A site MathOverflow. He is serializing several new books on his substack Infinitely More, releasing the latest chapters and material each week from the books-in-progress The Book of Infinity and A Panorama of Logic.  His recent books include Lectures on the Philosophy of Mathematics. MIT Press, 2021, and Proof and the Art of Mathematics. MIT Press, 2020.