In, "Multiversism and Concepts of Set: How much relativism is acceptable?" Neil Barton distinguishes between an ontological interpretation of set-theoretic multiverses as referents and an algebraic interpretation that is very much like, or is just a version of, if-thenism, which countenances mathematical abstractions as true (or at least nontrivially justifiable) conditional sentences:
Under the algebraic interpretation, we do not take set theory as the sort of enterprise that is concerned with referring to objects. Rather it is seen as providing an intuitive framework that underlies an algebraic method of thinking. This then allows us to understand what will be possible on a given structure with certain properties. Thus the problem of reference dissolves: we are not even making the appropriate kinds of claims to be assessed for reference.
What struck me about the gist of the essay as a whole was how much it reminded me of the actualist-possibilist debate, esp. the question of the existential Barcan formula:
- ◊∃νφ → ∃ν◊φ?
Maybe this issue is addressed somewhere in one of the essays about set-theoretic potentialism that I've sifted through (e.g. in denying either (1) or its converse), though my memory isn't telling me so right now. So my question is whether the algebraic multiverse is ontologically noncommittal in the sense of being, for set-theoretic possible-worlds talk, what actualism is for basic possible-worlds talk? I.e., talk of set-theoretic multiverses can use "(relatively) maximal consistent sets of propositions" as individual universes formulated over only one "actual" universe (a core model? set-theoretic geological mantle? I still don't understand those notions very well, if at all), whereas the ontological multiverse involves a set-theoretic Barcan formula that describes its intended referential value.