This will be a talk for the conference on Ultrafinitism: Physics, Mathematics, and Philosophy at Columbia University in New York, April 11-13, 2025.
Abstract. I shall argue in various respects that ultrafinitism is fruitfully understood from a potentialist perspective, an approach to the topic that enables certain formal treatments of ultrafinitist ideas, which otherwise often struggle to find satisfactory formalization.
This will be an online talk for the Infinity & Intentionality project of Øystein Linnebo in Oslo, 25 April 2023. Zoom link available from the organizers.
Abstract: I shall survey the surprisingly enormous variety of potentialist conceptions, even in the case of arithmetic potentialism, spanning a spectrum from linear inevitabilism and other convergent potentialist conceptions to more radical nonamalgamable branching-possibility potentialist conceptions. Underlying the universe-fragment framework for potentialism, one finds a natural modal vocabulary capable of expressing fine distinctions between the various potentialist ideas, as well as sweeping potentialist principles. Similarly diverse conceptions of ultrafinitism grow out of the analysis. Ultimately, the various convergent potentialist conceptions, I shall argue, are implicitly actualist, reducing to and interpreting actualism via the potentialist translation, whereas the radical-branching nonamalgamable potentialist conception admits no such reduction.
I was interviewed 26 August 2021 by mathematician Daniel Rubin on his show, and we had a lively, wideranging discussion spanning mathematics, infinity, and the philosophy of mathematics. Please enjoy!
Contents
0:00 Intro
2:11 Joel’s background. Interaction between math and philosophy
9:04 Joel’s work; infinite chess.
14:45 Infinite ordinals
22:27 The Cantor-Bendixson process
29:41 Uncountable ordinals
32:10 First order vs. second order theories
41:16 Non-standard analysis
46:57 The ZFC axioms and well-ordering of the reals
58:11 Showing independence of statements. Models and forcing.
1:04:38 Sets, classes, and categories
1:19:22 Is there one true set theory? Are projective sets Lebesgue measurable?
1:30:20 What does set theory look like if certain axioms are rejected?
1:36:06 How to judge philosophical positions about math
1:42:01 Concrete math where set theory becomes relevant. Tarski-Seidenberg on positive polynomials.
1:48:48 Goodstein sequences and the use of infinite ordinals
1:58:43 The state of set theory today
2:01:41 Joel’s recent books
Go check out the other episodes on Daniel’s channel!