Ultrafinitism as arithmetic potentialism
We may fruitfully view the philosophy of ultrafinitism in a potentialist light, helping to illuminate its philosophical commitments.
In recent weeks I have been writing a series of essays on the philosophy of ultrafinitism—find them in the ultrafinitism tag—and we come now finally into the main theme.
Namely, in this final essay I should like to discuss and defend what I see as an underlying potentialist nature to ultrafinitism. Specifically, I propose that we may fruitfully view the ultrafinitist perspective in a potentialist light, which will help illuminate its philosophical commitments, whilst also enabling a formal treatment of various ultrafinitist theories. What is more, I believe that the potentialist perspective brings to light certain fundamental issues on the nature of mathematical existence on which differing ultrafinitists might disagree, but which are most naturally discussed and adjudicated in a potentialist setting.
At the 2025 conference on ultrafinitism at Columbia University, Sam Buss mentioned that Ed Nelson had expressed ideas having a certain affinity with a potentialist outlook, in particular, the idea that things become true in arithmetic as you develop the theory—perhaps the twin primes conjecture could become true or the negation, depending on how the theory develops.
From the point of view of this essay, however, my entry into potentialism arises instead from a semantical perspective regarding the models of arithmetic, including models of the weak or the ultrafinitist theories. In a previous post, we saw, for example, how every model of finite arithmetic M ⊨ FA extends to taller models M+ and M++ and so forth, with which it is bi-interpretable, and in another post, we saw that M extends ultimately to the limit model M* ⊨IΔ0 of bounded induction. My view on this is to take it directly as a form of potentialism. Even if a finite-arithmetic ultrafinitist does not agree with the M* limit construction, nevertheless it seems to be in accordance with ultrafinitism to allow the move from M to M+, and this kind of move already leads exactly to the potentialist picture for arithmetic that I would like to paint.
