Potentialism and implicit actualism in the foundations of mathematics, Jowett Society lecture, Oxford, February 2019

This will be a talk for the Jowett Society on 8 February, 2019. The talk will take place in the Oxford Faculty of Philosophy, 3:30 – 5:30pm, in the Lecture Room of the Radcliffe Humanities building.

Abstract. Potentialism is the view, originating in the classical dispute between actual and potential infinity, that one’s mathematical universe is never fully completed, but rather unfolds gradually as new parts of it increasingly come into existence or become accessible or known to us. Recent work emphasizes the modal aspect of potentialism, while decoupling it from arithmetic and from infinity: the essence of potentialism is about approximating a larger universe by means of universe fragments, an idea that applies to set-theoretic as well as arithmetic foundations. The modal language and perspective allows one precisely to distinguish various natural potentialist conceptions in the foundations of mathematics, whose exact modal validities are now known. Ultimately, this analysis suggests a refocusing of potentialism on the issue of convergent inevitability in comparison with radical branching. I shall defend the theses, first, that convergent potentialism is implicitly actualist, and second, that we should understand ultrafinitism in modal terms as a form of potentialism, one with surprising parallels to the case of arithmetic potentialism.

Jowett Society talk entry | my posts on potentialism | Slides

2 thoughts on “Potentialism and implicit actualism in the foundations of mathematics, Jowett Society lecture, Oxford, February 2019

  1. Very nice slide presentation. I do have a question regarding a set-theoretic potentialist’s view of $V$, the universe of all sets and its relation to the Kunen Inconsistency in $ZFC$. As is known, what Kunen proved in his paper, “Elementary embeddings and Infinitary Combinatorics”, is the following, for $j$: $V$ $\rightarrow$ $M$, assuming $AC$:

    $\mathscr P$($\lambda$) $nsubseteq$ $M$

    where $\lambda$ = { $j^n$($\kappa$): $n$ $\lt$ $\omega$}.

    My question is simply this: If, for a set-theoretic potentialist, $V$ is the limit structure of all models of $ZFC$ (and it would seem that it would be, given your comments in the slide presentation regarding the limit structure), does the Kunen Inconsistency prove the set-theoretic potentialist’s point of view (since, according to what is written on the slide presentation, “the actual limit structure does not exist”) ?

  2. Pingback: Philosophy of Mathematics, graduate lecture seminar, Oxford, Trinity term 2020 | Joel David Hamkins

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