When I started out as a graduate student writing a thesis defending mathematical nominalism, my naïve view was that mathematical claims (the correct ones) were true in a different way to empirical claims (the correct ones). Before long though, under the influence of Tarski, the basic model theory I was teaching in Logic 1, etc. I came to reject that view. Truth had to do with satisfaction in the Tarskian sense which required a domain of objects to do the satisfying. Recently though I came across this passage from Huw Price:
[W]e need to distinguish the notion of keeping track as something that we do within the assertoric language game – a notion constituted, within the game, by the fact that the normative structures always hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player – from a notion that we might employ from outside the game, in saying that in at least some of its versions its function is to aid the players in keeping track of their physical environment. … [T]here’s a temptation to call both kinds of external constraint ‘truth’, but we shouldn’t make the mistake of thinking we’re dealing with two aspects of sub-species of a single notion of truth. Both notions may be useful, for various theoretical purposes, but we shouldn’t confuse them’ (Expressivism, Pragmatism and Representationalism, 191)I’m beginning to think that my naïve view might just be the right one. Any assertoric discourse [Could there be a non-assertoric discourse? I’m not sure.] with standards of correctness and incorrectness will need a truth predicate for the sorts of expressive purposes deflationists get excited about. (We need a truth predicate to express commitments without having to state them explicitly. E.g. if a theory \(\Gamma\) entails infinitely many things I can say ‘Everything \(\Gamma\) entails is true’ but I can’t possibly explicitly assert everything entailed by \(\Gamma\) Hence truth predicates are indispensable for certain expressive purposes.) Mathematical discourse seems like a good candidate for a discourse that’s governed by internal rather than external standards. In empirical matters the world gets to answer back: we bump up against the world, probe it, test it experimentally. But mathematical investigations don’t involve interactions with mathematical objects, they involve proofs. Of course the results of these investigations can surprise us or be counterintuitive. But none of that requires an external world of mathematical objects; only that the normative structures constituted within the game ‘hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player’. Objectivity doesn’t require objects. Moreover, as I’ve argued before there’s good reason to think that the norms of correctness and incorrectness of mathematical discourse are internal, rather than external, in the senses above.
