In Sec. 16, “Modalism”, Parsons considers the stratgegy of rescuing eliminativist structuralism from the vacuity problem by going modal.
To recap, we’re considering the schematic idea that an ordinary arithmetical statement is elliptical for something along the lines of
For any N, 0, S, if Ω(N, 0, S) then A(N, 0, S),
where Ω(N, 0, S) lays down the conditions for a set N (equipped with a distinguished element 0, and a mapping S: N -> N – {0}) to be “simply infinite”, and A(N, 0, S) is appropriately correlated with the ordinary statement. And the idea is, of course, that the quantifications here are restricted to kosher, unproblematic, collections of physical objects N, and mappings on them (such as arrays of space-time points, and a ‘go to the next point’ map): in this way, problematic purely abstract entities drop out of the picture. And the vacuity problem is: what if, as a matter of physical fact, ours is a finite, granular, world and there is no infinite physical collection and physically realized mapping for which the condition Ω is true? In that case, the quantified conditional holds vacuously, and all arithmetical statements come out as indiscriminately true.
Now, the obvious modal gambit is to respond: Ah, we should require the quantified conditional to hold necessarily. For, even if in this world there are no physically realized simply infinite systems, there could be other possible worlds (maybe where the physical laws are very different) which do realize simply infinite systems. Why not? So even if the plain unmodalized conditional is vacuously true for any A, the modalized version won’t be.
Note, though, that the natural way of construing the modality here is surely in that sense of necessity which is spelt out as truth in all possible worlds (or at last, truth in all worlds that are in some sense physical worlds lacking abstract denizens) — as some kind of metaphysical necessity, in other words. So I’m a bit stumped as to why Parsons says “I will assume that the modal operators are understood either in the sense of mathematical modality or of metaphysical modality.” I’m quite unclear what the notion of mathematical modality would do for us here.
Anyway, Parsons considers two objections to modal structuralism. The first, however, is not an objection to the modal version in particular: “It is falsifying the sense of discourse about natural numbers [to take] arithmetical statements to be really about every simply infinite system.” And surely a structuralist of any sort will think this a pretty feeble objection: even if the structuralist account seems prima facie to do some violence to our intuitive model of what we mean, its defender will just say that that shows we are gripped by a bad philosophical picture of what we are really up to in doing mathematics.
Parsons considers another objection, by pressing “whether the modalist’s apparatus really does offer an elimination of mathematical objects” in Sec. 17, “Difficulties of modalism”. But in fact the way he develops the point seems not to tell against a modal structuralist account of arithmetic or indeed applicable analysis (given we can construct applicable analysis in such weak systems of second-order arithmetic). Rather the worry seems to be whether there could be structures realized in some sort of alternative physical world which have anything like the richness to be a model of higher set theory. Well, maybe not (though I’m not sure how one goes about settling the issue!). But then why shouldn’t the structuralist just say, so much the worse for higher set theory — it’s just a fiction, or a jeux d’esprit?