Weir, to summarize once more, wants to develop a position that allows him to say sincerely, speaking with the vulgar mathematicians (and not having to cross his fingers behind his back, or do that little dance with the fingers that signals scare-quotes, or do some radical reconstrual of what they “really” mean), “there are infinitely many prime numbers”, even though while amongst the learned, or at least amongst the metaphysicians, he consistently asserts “THERE ARE no numbers”. He hopes to have softened us up for the idea that there is room for such a have-your-cake-and-eat-it position by considering (i) how projectivism (supposedly) allows us to agree sincerely with the vulgar that “X is G” (for certain G) while also agreeing with metaphysicians who say “THERE IS no such property as being G”, and considering (ii) how a certain line on fiction (supposedly) allows us to agree with the vulgar reader of the stories that “Sherlock lived in London” while agreeing with metaphysicians who say “Sherlock never EXISTED“. Not that Weir want to be a projectivist or a fictionalist about maths: but the idea is that the prima facie tenability of those accounts elsewhere indicates that there is perhaps room for a similar ontological anti-realism about mathematics, one which rests on the key idea that in making mathematical assertions (as when making fictional assertions or “projective” assertions) we are playing a different game from when we are in the business of representing the world.
But Weir, as he now emphasizes again, wants more. He wants to combine ontological anti-realism about mathematical entities with “metaphysical realism” in the Putnamian sense of allowing for the possibility of evidence-transcendent truth in maths. Of course, this isn’t exactly a novel combination. The modal structuralist is similarly concerned to eliminate commitment to a distinctive ontology of mathematical abstracta, which he does by translating away mathematical claims into modal quantified truths, and he can allow that it is evidence-transcendent what the modal truths are. However, unlike the modal structuralist, Weir wants to take mathematical talk at face value (he doesn’t want to go in for telling mathematicians what they “really” mean by translating away their ostensible commitments). So he wants a brand of ontological anti-realism for mathematics akin to projectivism or his sort of account of fictional discourse — we again aren’t in the representational business — while allowing evidence-transcendent truth.
But it can’t be said that we’ve been softened up for that combination. Certainly, it is difficult to see how there could be e.g. evidence-transcendent truths about what is tasty! Maybe a projectivism about probability could be developed in such a way as to allow for evidence-transcendent truths in this case: but Weir doesn’t say anything about such a case — and, in sum, I think we get no illumination on the ontological-anti-realism/metaphysical-realism combo from anything he says about projectivism (have I missed something?). However, Weir does think his account of fiction gives us something to go on:
There is no incoherence in holding to this anti-realism [about fiction] while viewing truth in general as evidence-transcendent — perhaps even fictional truth, if the fact that S follows in the right way from the text, and thus is true, can be evidence-transcendent.
But what does Weir have in mind here?
Earlier, he talked about S following in the right way — “flowing from” the text — if “experienced readers would, on reflective consideration, judge [that S] must form part of the story if it is to make overall sense.” But that notion of flowing from the text, where what flows depends on our best judgements, would hardly make room for evidence transcendence! But perhaps the idea is that things may follow logically or indeed mathematically from the text, but in an evidence-transcending way. Thus suppose “2 is the least number such that P” is an evidence-transcendent mathematical truth. Then I guess we have “The number of Dmitri Karamazov’s half-brothers is at least as large as the least number such that P” as a truth about the fiction which would be evidence transcendent. But then the evidence transcendence of the fictional truth would be dependent on the evidence transcendence of the mathematical truth (and so we couldn’t use the possibility of former fact as illustrating how the latter could be possible). Well, maybe there are other cases we could think about here: but that’s enough to suggest that Weir’s one-sentence jab at persuading us that his story about fiction gives us a useful illustration of the desired ontological-anti-realism/metaphysical-realism combo is just too quick.
But let that pass. We now have some sense of where Weir wants to end up about mathematics: ontological anti-realism without radically reconstruing maths (we continue to take it “at face value”), to be achieved by seeing assertion in maths as playing a different role to representational assertion, BUT also “metaphysical realism”, in the sense of allowing for evidence-transcendent truth. The work of spelling out his attempted “neo-formalist” articulation of such a position starts in the next chapter.