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.

Aidan(12:56:10) :I’m also thousands of miles away from my copy, but my recollection about Maddy on indispensibility arguments is that she’s pretty focused on set theory. What I mean by that is realism is generally taken as an ontological claim, or a claim about the objectivity of mathematical truths. I seem to remember Maddy’s realist isn’t straight-forwardly of this sort; isn’t her realist is a particular character in the discussion about how to justify axioms?

Indispensibility arguments then are assessed against their potency in the realist campaign to justify not-V=L, which somewhat different from the canonical indispensibility arguments (which invoke Quinean-style criteria of ontological committment).

So, assuming I’m not misremembering or misrepresenting, it may be that we need to distangle a cluster of issues obscured by the labels ‘realism’ and ‘nominalism’ before we can assess what Maddy’s committing herself to.

01010001(11:05:25) :Maddy’s hardly less consistent than Herr Doktor Quine, who starts off as nominalist, then switches to a somewhat platonic view of mathematical entities as indispensable, yet at the same time arguing against analyticity (wouldn’t that include supposed logic/math. identities as well?). He’s not too keen on set theory (rightfully so); later he affirms epistemological naturalism. No Stanford or Cal logician are we, but really, I think Quine was a naturalist (and nominalist?) the entire time (not to say behaviorist), and the hints of platonism (from Frege and Russell really) merely a pragmatic convenience. He’s following in the footsteps of William James and CS Peirce as much as he was doing Frege and Carnap; yet for all the naturalism, it’s a peculiarly anti-humanist naturalism. Economics or history, for example, rarely if ever appear in Quine’s austere meditations (or in Maddy’s for that matter).

Kenny(23:45:53) :I agree that Maddy’s inconsistency here is no worse than Quine’s – Quine accepted that some mathematics was indispensable, and therefore described actually existing things. But he somehow thought that “higher set theory”, since it’s only indispensable for mathematics, and not for physics, didn’t have any need to talk about actually existing things. But this means that he’s drawing an anti-naturalist distinction between mathematics and physics.

However, I don’t think Quine’s views about indispensability are either platonistic or conflict with his views on analyticity. Platonism requires that the entities be necessary, and that we have a priori access to them. Quine is skeptical of both ideas of necessity and a priority. So he’s really just a naturalist, without any platonistm. But I agree that he is too focused on physics, and not enough on higher-level sciences, much less social sciences. I guess it’s his “desert landscape” preferences coming through…

Anne Newstead(20:50:16) :I think the thing to remember about Maddy’s critique of Quinean indispensability arguments is that she aims to respect the disciplinary autonomy and integrity of mathematics. Her main point is that mathematicians don’t accept the existence of mathematical objects merely because those objects figure in applied science. Set theorists don’t look over their shoulders to see whether anything in nature has an inaccessible cardinality. Rather, there are norms internal to mathematical practice itself that are brought to bear in assessing whether to accept a mathematical statement as true. Quine’s indispensability arguments involved falsely assimilating mathematics to rest of the natural sciences. The result, as I argue in (forthcoming) is a distortion of the epistemology of mathematics.

What does Maddy say about indispensability arguments altogether (not just for the mathematical case)? I think she’s suspicious of them for the following good reasons:

(a) we can use terms for instrumental purposes in our best theories, without assuming real entities correspond to those terms;

(b) our best scientific theories may be literally false and thus block the inference to the conclusion that what they describe exists.

The primary illustration Maddy uses is the continuum. Probably, there is no physical space-time continuum. But it’s a useful fiction and it is indispensable to our best science. Nonetheless, we cannot conclude that a continuum really exists ‘in nature’ just because it figures in our best science. This does sound a lot like instrumentalism (an anti-realist position in philosophy of science). It’s unclear that Maddy wants to head there, of course.

The main idea in the naturalist programme is the Wittgensteinian methodological aim of describing existing mathematical practice, rather than seeking to revise or falsely assimilate mathematical practice to some other practice. Maddy was right to point out that Quine had led philosophers of mathematics pretty far away from focusing on mathematical practice.