As we can see from our initial specification of his position, to get Weir’s philosophy of mathematics to fly will involve accepting some substantial and potentially controversial claims in the philosophy of language and metaphysics. The first two chapters of TTP fill in some of the needed background. Weir starts by talking a bit about realism(s). Given that, in the Introduction at p. 6, he has already characterized himself as aiming for “an anti-realist … reading of mathematics”, we should get clear about what kind of realism he is anti.
However, I didn’t find the ensuing discussion altogether clear (is it perhaps extracted from something longer?). So in what follows, I’m reconstructing a little, but hopefully in a broadly sympathetic way, for I do at least want to end up pretty much where Weir does.
Traditional realisms, he says, “affirm the mind-independent existence of some sort of entity”. But what does ‘mind-independent’ mean here? The problems are immediate. For a start, which kinds of minds count? On the one hand, if it’s just finite sublunary minds, then Berkeley comes out a realist, which isn’t what we want (Weir himself contrasts realism with idealism). On the other hand — Weir might have noted — if we agree with Berkeley and count God as among the minds, then any traditional theist who believes that the physical world is dependent for its existence on God would ipso facto count as an non-realist about sticks and stones, which is also surely not what we want. Then there are other problems with the traditional formulation: on a crude reading, it seems to define away the very possibility of being a realist about minds.
Let’s put those worries on hold just for a moment, and turn to consider the modern theme that realism should instead best be characterized in epistemic terms. Thus Dummett (quoted by Weir): ‘Realism I characterise as the belief that statements … possess an objective truth value, independently of our means of knowing it.’ Of course, others such as Devitt have emphatically insisted contra Dummett that realism about Xs, properly understood, is an ontological doctrine about what there is, and is not to be confused with any epistemic or semantic doctrine. Where does Weir stand on this?
Well, he spends some time discussing the idea that realism is a species of fallibilism. We can present this sort of realism about a region of discourse R schematically as saying
For every (or some?) R-sentence s, it is possible (what kind of possibility?) for speakers (which speakers? even in optimal conditions?) to believe s though it is not true, or disbelieve s though it not false.
There are four dimensions along which versions of realism-as-fallibilism can vary, corresponding to the four queries. And Weir doesn’t hold out much hope that there is any way of setting the variables to give us a thesis which is substantive enough to be interesting but also sufficiently captures what a realist is after. He offers a number of considerations. Embroidering a bit, we could perhaps put one of them like this. Suppose we keep fixed our view about the nature of Xs but change our mind about the quality of our epistemic access to Xs. Suppose we become very optimistic — perhaps implausibly over-optimistic — and now think that, at least when we are optimally placed and exercising our cognitive faculties in the optimal way, there are (enough) claims about Xs which (in the relevant sense of ‘possible’) it is not possible for us to go wrong about. Then, the thought goes, surely changing our view like this about our fallibility with respect to claims about Xs doesn’t in itself entail changing our view as to whether Xs are really there, independently of us, etc. Coming to think we are more or less infallible about Xs can involve wildly upgrading our estimate of our epistemic powers, rather than downgrading our realism about Xs. So we shouldn’t tie realism to fallibilism too tightly.
Weir’s arguments here do go pretty quickly (too quickly to be likely to sway a Dummett or a Putnam, for example); but I won’t pause over the details as in fact I rather agree with his interim conclusion:
I find myself in sympathy with Devitt in wishing to return to a traditional ‘ontological’ characterization of realism as mind-independent existence. (p. 22)
Or at least, I agree that realism about Xs should be construed as an ontological claim, not an epistemic or semantic claim. But Weir’s version takes us back to those puzzles about how best to spell out ‘mind-independent’. And here, it seems to me, he takes a wrong turn. For having just explained why he thinks that realism-as-fallibilism won’t do, he now suggests that we can “effect a compromise” and proposes
a Devitt-style ‘ontological’ characterization of realism with respect to a given set of entities as constituted by a belief in their mind-independent existence, where mind-independence is, in turn, chararacterized in fallibilist terms à la Putnam and Dummett.
But will this do, even by Weir’s own lights? Isn’t this compromise package vulnerable to (some of) the same objections as pure realism-as-fallibilism? In particular, doesn’t it again implausibly imply that inflating our estimate of ourselves and supposing we have the relevant kind of infallibility with regard to claims about Xs would entail thereby rejecting realism about Xs themselves?
I’m not sure how Weir would respond to that jab, nor how he would fix those variables left dangling in a schematic statement of mind-independence as fallibilism. Instead he goes off on another — and more promising — tack, noting that
Someone who holds to evidence-transcendent truth and affirms that Xs exist should not count as a realist about Xs if the affirmation of the existence of Xs, though sincere, should not be taken at face value or else should not be read in a straight representational fashion.
That’s surely right: to be a realist about Xs involves affirming the existence of X without crossing your fingers as you say it, or proposing to ‘decode’ such an affirmation as in some way not being about what it at surface level seems to be about (or treating it as not in the business of representing how things are at all). Thus, to take Weir’s example, the modal structuralist might take at least some arithmetical claims to be true in an evidence-transcending way: but that hardly makes her a realist about numbers if she parses the claims — including apparently existence-affirming claims like ‘there is a prime number between 25 and 30’ — as really claims about what happens in concretely realized structures across possible worlds. Or to go back to Berkeley, the good bishop might allow some claims about the physical world to true independently of our human ability to discover them to be so, but that hardly makes him a realist about physical things, given the decoding he offers for such claims when thinking with the learned.
OK, suppose we say — taking the core of Weir’s line — that you are a realist about Xs if you affirm that there are Xs, where that is to be taken in a “straight representational fashion” and is to be “taken at face value” (not reconstrued, or decoded). You can immediately see why, quite trivially, Weir’s philosophy of mathematics will count for him as anti-realist, given that he has announced that on his view mathematical talk is non-representational. But of course, all the work remains to be done in explaining what it is to mean something as representational and intend it to be taken at face value.
Though here’s a concluding thought. We might suggest that it is a condition of talk of Xs being apt to be taken “at face value” that it involves continuing to respect enough everyday platitudes about the kind of things Xs are. And in some case — e.g. where X’s are everyday things like sticks and stones — those platitudes will involve ideas of ‘mind- independence’ (the sticks and stones are the sort of thing that will still be there even if no one is seeing them, thinking of them, etc.). So taking talk of sticks and stones at face value will involve taking it as respecting the ‘mind-independence’ of such things. Which suggest that perhaps that realism about X’s (meaning just representational face-value affirmation of the existence of Xs) will already bring with it as much ‘mind-independence’ as is appropriate to Xs — more or less independence , varying with the Xs in question. If that’s right, we needn’t build mind-independence into the general characterization of realism: it will just fall out for realism about Xs if and when appropriate.