Immanent or Transcendent Structuralism?
Happy new year everybody. I'm still alive, and I still have questions and comments on the metaphysics of David Lewis. This one is about Lewis' philosophy of mathematics.
In "Mathematics is Megethology", Lewis argues for structuralism in set theory: There is no particular relation of membership, connecting particular things with particular classes. Instead, there are just two sides of Reality, ordinary individuals on the one side, proper-class many mereological atoms (called 'singletons') on the other. Set theory is about all relations on this Reality that satisfy certain constraints, like 'every individual stands in that relation to a singleton'.
When Lewis wrote this, he believed that there are only few possible individuals, beth-3 perhaps or beth-omega. Later he was persuaded by Nolan that there are in fact proper-class many possibilia. Now if this is true -- contrary to what I've said recently --, an obvious move would be to jettison the entire realm of mathematical entities and let structuralism range over the possibilia themselves. I can see three advantages and one drawback for this 'immanent' structuralism.
The first advantage is of course a simpler ontology. There are the possibilia and that's it.
The second concerns modality. Lewis' Counterpart Theory doesn't work well for mathematical statements: It doesn't seem plausible that every single world contains all those proper-class many mathematical things. Why couldn't some of them fail to exist? And if they aren't parts of worlds, why then do they necessarily exist? What does it even mean to say that something exists necessarily if it doesn't have a counterpart in every possible world? By reducing classes to possibilia mathematical truths would be quantifications over possibilia. And there is a good reason why such statemants are necessary: "Necessarily, every thing at every world is F" means "At every world, every thing at every world is F". The second quantifier renders the first vacuous.
The third advantage concerns set theory. As Lewis and Nolan note, proper-class many individuals are incompatible with the ordinary image of the set-theoretical hierarchy. According to this image, the first stage of the hierarchy contains all individuals, the second every subset of these individuals. But if the individuals are a proper class, some of these 'subsets' are proper classes as well, and we doon't want proper classes at stage two. If on the other hand we adopt immanent structuralism, we can just stipulate that only those singleton relations are eligible with respect to which there are less than proper-class many individuals ('few' in Lewis' terminology).
The drawback is that "Julius Caesar is not a class" appears to be true, but wouldn't be true on immanent structuralism. (It will be either false or indeterminate depending on what kind of structuralism we chose. "Julius Caesar is a class" won't be true either.) This is why Lewis rules out singleton relations that treat ordinary things as classes in "Mathematics is Megethology".
Question: Does anyone know whether Lewis has changed his mind and favoured immanent structuralism after having been persuaded that there are proper-class many possibilia?
(I have a very tiny piece of evidence for such a change: In "Tensing the Copula", Lewis doesn't mention the restriction of classes to extra-ordinary things, and he talks about quantification over set-theoretical hierarchies, which would be slightly misleading if the iterative hierarchy didn't match his reconstrued set theory.)