## Magic, worlds, numbers, and sets

In On the Plurality or Worlds, Lewis argues that any account of what possible worlds are should explain why possible worlds represent what they represent. I am never quite sure what to make of this point. On the one hand, I have sympathy for the response that possible worlds are ways things might be; they are not things that somehow need to encode or represent how things might be. On the other hand, I can (dimly) see Lewis's point: if we have in our ontology an entity called 'the possibility that there are talking donkeys', surely the entity must have certain features that make it deserve that name. In other words, there should be an answer to the question why this particular entity X, rather than that other entity Y, is the possibility that there are talking donkeys.

It might be useful to consider parallel questions about mathematical entities.

Mars has two moons, Phobos and Deimos. So here is a fact about the number 2: it is the number of moons of Mars. Following Lewis, one might argue that any account of numbers should explain in virtue of what the number 2 has this property. If we have numbers in our ontology, surely it can't be a brute fact that precisely this one is the number of moons of Mars.

The von Neumann construction of numbers gives a plausible answer to the Lewisian challenge. Here the number 2 is identified with the set { {}, { {} } }. This set has two members. The set of moons of Mars also has two members. And that is why 2, i.e. { {}, { {} } }, is the number of moons of Mars. In general, a von Neumann cardinal n is the number of Xs iff there is a one-one map between the members of n and the Xs.

By contrast, consider a primitive platonism about numbers on which the numbers are irreducible extra entities, distinct from sets, sticks, Roman emperors, and everything else. I do think the Lewisian objection has some bite here. One of the Platonic entities, call it X, is supposed to be the number 2. But what makes it the case that X, rather than Y, is the number 2, and thereby the successor of 1, and the number of moons of Mars? How come our label '2' picks out X rather than Y?

There seems to be an argument here for reducing numbers to sets. However, the Lewisian challenge can also be raised for sets.

Consider the set { Phobos, Deimos }. In virtue of what, we can ask, is Phobos a member of that set? If the sets are simply a bunch of intrinsically unstructed abstract blobs (so to speak), it is hard to see what could single out any one of them as being { Phobos, Deimos }.

The problem is easy to miss because we are so used to picking out sets and numbers by certain names. In a sense it is of course trivial that { Phobos, Deimos } has Phobos as a member. By calling a set '{ Phobos, Deimos }' we have settled that the set has Phobos as a member. But think of the relationship between the sets themselves and our names for sets. Why is it that this particular set is properly called '{ Phobos, Deimos }'?

Lewis certainly took this question seriously. His 1991 book Parts of Classes is largely concerned with the present problem about sets.

Lewis only finds a partial answer. According to Lewis, the set { Phobos, Deimos } is not an unstructred primitive blob, but a mereological fusion of two other sets, { Phobos } and { Deimos }. This explains why { Phobos } is a subset of { Phobos, Deimos }: the subset relation is the relation of mereological parthood. But there is a residual question: in virtue of what is Phobos a member of { Phobos }? That is, what makes { Phobos } the singleton set of Phobos? Lewis surveys and rejects various possibilities and finally admits that he has no answer.

And so I have to say, gritting my teeth, that somehow, I know not how, we do understand what it means to speak of singletons. (p.59)

So Lewis (very reluctantly) accepts an analogue of magical ersatzism for singletons.

Incidentally, this illustrates that he plausibly didn't regard the explanatory deficiency of magical ersatzism as decisive. It's just that in the case of worlds he found an account (modal realism) that can explain how worlds deserve their names while in the case of singletons he didn't.

Well, not in 1991. By 1993 ("Mathematics is Megethology"), he changed his mind and no longer accepts singletonhood as primitive. His ontology remains the same. According to Lewis, fundamental reality divides into (a) the "concrete" building blocks of possible worlds and (b) "abstract" singletones. Sets are still fusions of singletons. But there is no longer a special, designated relation between Phobos in (a) and its singleton in (b), a relation that we inexplicably and magically pick out by our concept of a singleton. That's because none of the singletons on the abstract side of reality is determinately, intrinsically, the singleton of Phobos. The singletons are all on a par. They are unstructred blobs. None of them is objectively privileged to be the singleton of any particular object. When we speak of sets and singletons, we effectively quantify over all eligible relations between objects and singletons, where "eligibility" is a matter of satisfying the formal conditions on the singleton relation: nothing has more than one singleton, and so on. (The tricky part of the story is to explain away this "quantification over relations", but I won't go into that here.)

Lewis's 1993 position on sets is a form of "eliminative structuralism". Could one defend a similar position on possible worlds? What would that look like?

We would begin by postulating a bunch of primitive entities, none of which is intrinsically designated as a world "at which" donkeys talk. When we talk about worlds where donkeys talk, we implicitly quantify over all eligible representation relations: over all eligible assignments of representational content to the postulated entities.

But what's on the other side of that relation? Contents? Ways things could be? Do we have to believe in those things as further entities? That's not appealing, and we would run into the same problem again to explain what gives those entities their representational power -- what makes a particular "content" deserve the name 'that there are talking donkeys'.

We need to get clearer here about the original challenge. Lewis asks in virtue of what a candidate world w represents that there are talking donkeys. On the face of it, the question seems to concern a purported relation ("represents") between the candidate world w and a certain entity, that there are talking donkeys. But what is that entity? Couldn't the magical ersatzer follow Lewis and identify the possibility that there are talking donkeys with a set of worlds? And then isn't there a simple answer to the challenge: w represents that p in virtue of being a member of p?

That would clearly miss the point of Lewis's challenge. The ersatzist ontology now has some primitive blobs as well as sets of those blobs. Lewis asks: why is that blob here a world where donkeys talk? The ersatzer says: oh, because it's a member of this set of blobs here. OK, but why is that set of blobs the possibility that there are talking donkeys? Why is that set, rather than any of those others, picked out by 'there are talking donkeys'?

Can we construe the other side of the representation relation not as contents but as sentences, individuated by their use in a given community? On Lewis's view, the sentence 'there are talking donkeys' (as used in the English speaking community) picks out a determinate class of worlds (ignoring vagueness etc.). Our structuralist ersatzism denies that. It holds that all classes of worlds of appropriate size are equally well-suited to be picked out by 'there are talking donkeys'. None of them is intrinsically designated as the class of worlds with talking donkeys. When we speak of worlds with talking donkeys, we implicitly quantify over all eligible mappings from sentences to classes of worlds.

But now what makes a mapping eligible? One could suggest that the patterns of use in the relevant community reveal that certain sentences A entail other sentences B, and that an eligible mapping to classes of worlds must then assign to A a subset of B. Similarly, our usage could reveal that other sentences A are contraries of B, in which case the assigned sets must be disjoint. And so on.

But that seems too weak. The point of assigning sets of worlds to sentences is not just to capture facts about entailment, contrariness, etc. Possible-worlds semantics is not a form of conceptual-role semantics. Moreover, possible worlds are not only used in semantics. What if we want worlds to do something like the work Lewis outlines in chapter 1 of Plurality? Here we seem to need classes of worlds that go far beyond what can be picked out in English.

So I suspect the structuralist representation relation can't be understood as a relation to sentences. What else could it be? Can we somehow understand it as a relation to ordinary objects and sets, such as donkeys and things that talk? I don't know, but I think it would be worth exploring the options.

But perhaps the analogy to eliminative structuralism in set theory should not be pressed too hard. In fact, I'm tempted to say that we don't even need to postulate primitive extra blobs to make sense of possible-worlds talk. The general strategy illustrated by eliminative structuralism is that one might give an adequate (metaphysical) interpretation of number talk or possible worlds talk without believing in primitive entities of the relevant kind.

Suppose we can give analytically necessary and sufficient conditions for any statement involving possible worlds (or propositions, or properties, etc.) -- conditions that do not mention worlds or propositions or properties etc. It seems plausible to me that this is in principle possible. Why wouldn't that suffice as an interpretation of possible worlds talk?