I’ve just read through Steven Yablo’s nice paper “A Paradox of Existence”, which further advances his position that mathematical talk may all be metaphorical, and that metaphorical talk shouldn’t be construed as ontologically committing. His position strikes me in many ways as similar to Jody Azzouni’s, in that both suggest that no rigorization of our language will bring it in line with the Quinean picture where it’s in a first-order language and the ontological commitments can be read off from the quantifiers. The difference is that Azzouni thinks we can get it into that regimented form but the quantifiers aren’t committing, while Yablo suggests that some metaphors may be essential to our descriptions of the world, so any first-order expression of our theories contains some non-literal language, so there is no Quinean rigorization at all, much less one whose commitments can be read off the quantifiers.
I was very tempted by the position when reading this paper – more so than when reading “Does Ontology Rest on a Mistake?”, although the tendency it tempts me towards is more in keeping with the title of the latter.
Yablo points to Davidson’s analysis of adverbial constructions as being properties of events as an example of a theory that is useful and may not be eliminable, but may still be regarded as metaphorical. Davidson’s analysis allows a speaker to learn the language by learning only finitely many symbols, rather than learning every combination of a verb and an adverbial construction separately. But however useful this procedure is, it’s equally useful even if there turn out not to be events and properties thereof, and it’s just a pretense. It can be a purely semantic thesis rather than an ontological one.
But then I analogize this to science. Caloric (a reified notion of heat) is a very useful way of talking about the world for explaining how heat flows from one object to an adjacent one. In fact, it’s useful even if the world doesn’t have any of it. Similarly with electrons and their usefulness for science.
But now, if we’re going to repudiate our ontological commitments just because the world can be described treating this language as purely metaphorical, then it seems we would sink into complete scientific instrumentalism. I assume this is what Yablo means in footnote 53, where he says he wants to “provide a natural brake on the creeping metaphoricalism that might otherwise threaten.” However, I think such a brake isn’t possible. If this is the right account, then we can treat all talk of objects in general as metaphorical (which I think I’m happy with). But as long as we want to talk about objects of any sort, it seems that the constraint on our language is to accept the talk of objects that we can’t paraphrase away – so whether or not we’re committed to mathematical entities will turn again on the indispensability argument that Yablo and Azzouni want to get away from.