Many people think that talking about possible worlds is useful in philosophy. A good number of those people think that talking about impossible worlds is also useful. In most cases, talking about impossible worlds as well as possible worlds is innocuous. On most of our views about what worlds are, impossible worlds are no more ontologically problematic than possible worlds: sets of propositions all of which can’t be true together are no more mysterious than sets of propositions all of which can be true together; if talk of possible worlds is a merely pragmatically useful fiction, talk of impossible worlds can be such without any further mystery. And so if you hold some such view about worlds, the question as to whether we should talk about impossible worlds depends solely on whether to do so is useful – there is no metaphysical problem in doing so.
Not so, seemingly, if you are a Lewisian realist about worlds. For Lewis, a world at which there are blue swans is a world with blue swans as parts, and so a world with round squares is a world with round squares as parts. And so, to believe in the latter world is to believe in round squares. And this is to raise a metaphysical problem, for now one must admit into one’s ontology objects which could not exist. In brief, impossible worlds for Lewis are problematic because of how he thinks worlds represent: they represent something being the case by being that way, whereas his opponents think worlds represent in some indirect manner, by describing things to be that way, or picturing them to be that way, or etc. Impossible worlds are not metaphysically mysterious on the latter views because there is no metaphysical puzzle in there being a description of something that couldn’t exist, or a picture of something that couldn’t exist; but they are a metaphysical puzzle for Lewis, because there is a metaphysical puzzle in there being something that couldn’t exist.
Nonetheless, some think this is a price worth paying: they like Lewis’s account of possibilia but are impressed by the arguments for the need for impossibilia, so want to extend Lewis’s ontology to include impossible worlds. I’ve heard this move a few times in conversation, but the one person I know of who has defended it in print is Ira Kiourti. (Yagisawa defends a view similar to Lewisian realism with impossible worlds, but with some crucial differences.) Now, there are some big, and familiar, problems with believing in genuine impossible worlds that they each try to deal with, but I am yet to be convinced can be solved. (See my critical study of Yagisawa.) But I want to raise a problem for Lewisian realism with impossible worlds that I haven’t seen discussed and which I don’t even know how one would start to answer.
I can see how Lewisian realism with impossible worlds is supposed to deal with impossibilities like ‘There is a round square’ or ‘Frank is taller than Jim and Jim is taller than Frank’. I just need to believe in impossible objects – a round square and a man that is both taller and shorter than some other man – and then I can believe in worlds composed in part of such objects. Now, personally I can’t conceive of such objects – but so what? If I’ve got good reason to believe in them, I can postulate them. But I don’t see how we are meant to account for an impossibility like ‘2+2=5’. For Lewis, ‘2+2=4’ is necessary not because there’s a number system that is a part of each world and which behaves the same way at each world; rather it’s necessary that 2+2=4 because the numbers are not part of any world – they stand beyond the realm of the concreta, and so varying what happens from one portion of concrete reality to another cannot result in variation as to whether 2+2 is 4. And so, since contingency just is, for Lewis, variation across certain portions of concrete reality – namely, the worlds, which are just big concrete objects – there is simply no room for contingency with respect to the mathematical truths. Necessary truths about the realm of concreta are necessary because each relevant portion of concrete reality is a certain way: that is, no matter what variation you get across these concrete portions of reality, things are that way. But necessary truths about the realm of pure abstracta are necessary because they have nothing to do with how concrete reality is: so irrespective of what variation you get across these concrete portions of reality, things are that way. (See my paper on Lewis and reduction, esp. footnotes 3-5, and the corresponding discussion in the text.)
In that case, while I can add to my ontology round squares if I wish, and hence believe in a big concrete object with a round square as a part, and thereby have a world that represents the impossible circumstance of there being a round square, I don’t see even what weird metaphysical move to make to get a Lewisian world that represents 2+2 as being 5. Worlds don’t have numbers as parts: they are sums of concrete individuals; and if we give that up, we don’t have something worth calling an extension of Lewisian realism (Yagisawa gives this up, which is why I’m ignoring his view here). A Lewisian world nonetheless represents that there are numbers, because numbers exist from the standpoint of that world. And while the concrete objects that exist in one Lewisian world are never the same as the concrete objects that exist at another world, the numbers that exist from the standpoint of each world are just the same. And that’s why mathematical truths are necessary, for Lewis: because a world represents some mathematical claim as being the case just because the numbers represented as existing from the standpoint of that world are as the claim says they are. And since the same numbers exist from the standpoint of any two worlds, no two worlds differ in what mathematical claims they represent as being the case. Given Lewis’s account of what a world is and how they represent something to be the case, there is simply no room for variation across worlds in what mathematical claims are represented as true: hence there is no room for contingency in mathematical claims, but nor is there room for a world that represents some impossible mathematical claim as true, no matter what we think about the extent of what concrete worlds there are. So the Lewisian simply can’t extend her ontology to admit worlds that account for every impossible situation.