I was discussing indispensability arguments at the bar this evening with some of the philosophers in Canberra, and an interesting question came up regarding Penelope Maddy’s position on them. I unfortunately don’t have access to a copy of her book Naturalism in Mathematics right now, so I’m half doing this as a reminder to myself to check it out when I get a chance (or in case someone who knows better than me reads this and decides to comment to clear things up).
Anyway, my recollection of her position (in her naturalist phase) is roughly as follows. Quine has pointed out that natural science is a powerful and progressing body of knowledge that has helped us build a tremendous amount of understanding. Therefore, we should adopt the methods of its practitioners (or at least, the methods they follow when doing their best work, not necessarily the methods they say they adopt) when we want to find out what’s really going on fundamentally in the world. Maddy points out that mathematics is also such a body of knowledge, and that when Quine applied the methods of the natural sciences, he ended up with a much weaker theory than mathematicians (or at least, set theorists) want. Therefore, she suggests that when we talk about mathematics, we should adopt the methods of mathematicians – the needs of scientists are neither necessary nor sufficient (nor, perhaps, even relevant) for answering questions about whether various mathematical claims are true.
Of course, the methods of mathematics are fairly restrictive and straightforward, so we can’t even say anything about many supposed ontological questions about mathematics (like whether numbers really exist), and about basically all epistemic questions about mathematics (like how we come to have knowledge about numbers). As a result, these questions are effectively meaningless, because there is no way to answer them. So Maddy’s naturalism is a sort of third way, distinct from both realism and nominalism.
There’s also something misleading, it seems to me, about calling it “naturalism”. She develops it on analogy with Quinean naturalism, but it has important differences. In particular, it says that there is a body of knowledge that is not continuous with the natural sciences, namely mathematics! On at least some ways of putting Quinean naturalism, this is exactly what he wants to reject! (Of course, the alternative bodies of knowledge he was thinking about were things like “first philosophy”, rather than mathematics.)
But now I wonder – since Maddy (as I understand her) accepts something very much like Quinean naturalism about the physical sciences (when considered separately from mathematics), what does she have to say about traditional indispensability arguments? She obviously doesn’t think that they give one reason to believe that numbers and sets really exist (at one point she says something like, “if science can’t criticize, it also can’t support mathematical claims”). However, it seems that if entities of these sorts really are indispensable in doing natural science, then don’t we have scientific reason to say they actually exist, even if not mathematical reason to say so? Just as science makes us say there are electrons and quarks and genes and ions, it also seems to make us say that there are numbers and functions and the like, because all of these entities appear in our best theories. Maybe this is no reason from the mathematical point of view, but don’t we end up in effect with a reason to believe in physical objects with all the properties of numbers and functions and sets and the like? (Of course, these are quite unusual physical objects that have no spatiotemporal location and no causal properties, but science has already told us about strange particles that have no identity conditions and multiple positions (like electrons) or strange causal isolations (like black holes), so the “mathematical” entities mentioned in the theories could be seen as just even stranger physical objects, if Maddy won’t accept them as mathematical objects.)
I’m actually fairly sympathetic to this position – if I believed in the actual indispensability of mathematics, then I would grant mathematical entities exactly this kind of physical existence. But I’m also inclined to think that most people would regard this as a reductio of any position if it made one say that mathematical entities had such existence. Especially if the point of the theory was to remove mathematical existence claims from special philosophical consideration.
But maybe I’m just misreading one or more parts of the theory.