One of the books which I blogged about at length here was Alan Weir’s Truth Through Proof (OUP, 2010). I found this difficult and puzzling, but also enjoyable to battle with (not least because Alan responded in comments at length). Things soon got intricate, but here is a (revised version) of a very early post in the series, where I try to initially locate Alan’s position on the map. So this is perhaps of stand-alone interest, and there are themes here connecting to issues raised in the previous Encore on Maddy.
TTP, 2. Introduction: Options and Weir’s way forward (May 30, 2011)
Our conception of ourselves as natural agents without God-like powers “imposes a non-trivial test of internal stability” (as Weir puts it) when combined with platonism. As Benacerraf frames the problem in his classic paper, ‘a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have’. Faced with this challenge, what are the options? Weir mentions a few; but he doesn’t give anything like a systematic map of the various possible ways forward. It might be helpful if I do something to fill the gap — not a complete map, of course, but noting various choice points, and the way Weir goes at each.
Start with this question:
- Can we say — without qualification, without our fingers crossed behind our backs! — that yes, 3 is prime and, yes, the Klein four-group is the smallest non-cyclic group?
The platonism will answer ‘yes’. A gung-ho fictionalist of a certain stripe, for instance, will say ‘no’, our ordinary talk of numbers or groups commits us to a platonist ontology of abstracta that a sensible naturalist with a coherent epistemology has no business believing in: the mathematical claims, as they stand, unqualified, are false.
Platonists, however, aren’t the only people who answer ‘yes’ to 1. Move on, then, to a second question:
- Are ‘3 is prime’ and ‘the Klein four-group is the smallest non-cyclic group’, for example, to be construed — as far as their ‘logical grammar’ is concerned — at face value, as the surface form suggests (on the same plan as e.g. ‘Alan is clever’ and ‘the tallest student is the smartest philosopher’)?
A more conciliatory stripe of fictionalist can answer ‘yes’ to (1) but ‘no’ to (2), since she doesn’t take ‘3 is prime’ as wearing its logical form on its face but re-construes it as short for ‘in the arithmetic fiction, 3 is prime’ or some such. Likewise, for a certain brand of naive (or naive-ish) formalist who takes ‘3 is prime’ as not attributing a property to a number but as saying that a certain sentence can be derived in a formal game.
Eliminative and modal structuralists will also answer ‘yes’ to (1) and ‘no’ to (2), this time construing the mathematical claims as quantified conditional claims about non-mathematical things (schematically: anything, or anything in any possible world, that satisfies certain structural conditions will satisfy some other conditions). It is actually none too clear how structuralism helps us epistemologically, and when given a modal twist it’s not clear either how it helps us ontologically. But that’s quite another story.
Suppose, however, we answer ‘yes’ to (1) and (2). Then we are committed to agreeing that there are prime numbers and there are non-cyclic groups, etc. (for it is true that 3 is prime and that the the Klein four-group is the smallest non-cyclic group, and — construed as surface form suggests — that requires there to be prime numbers and non-cyclic groups). Next question:
- Is there a distinction to be drawn between saying there are prime numbers (as an unqualified truth of mathematics, construed at face value) and saying THERE ARE prime numbers? – where ‘THERE ARE’ indicates a metaphysically committing existence-claim, one which aims to represent how things stand with ‘denizens of the mind-independent, discourse-independent world’ (following Weir in borrowing Terence Horgan’s words and Putnam’s typographical device)
According to one central tradition, there is no such distinction to be drawn: thus Quine on the univocality of ‘exists’.
The Wright/Hale brand of neo-Fregean logicism likewise rejects the alleged distinction. Their opponents are sometimes puzzled by the Wright/Hale argument for platonism on the cheap. For the idea is that, once we answer (1) and (2) positively (and just a little more), i.e. once we agree that ‘3 is prime’ is true in some anodyne, minimalist, sense, and that ‘3’ walks, swims and quacks like a singular term, then we are committed to ‘3’ being a successfully referring expression, and so committed to its referent, which (on modest and plausible further assumptions) has to be an abstract object; so there indeed exists a first odd prime which is an abstract object. Opponents think this is too quick as an argument for full-blooded platonism because they think there is a gap to negotiate between the likes of ‘there exists a first odd prime number’ as an anodyne mathematical truth and ‘THERE EXISTS a first odd prime number’. Drawing on inter alia early Dummettian themes (which have Fregean and Wittgensteinian roots), the neo-logicist platonist denies there is a gap to be bridged.
Much recent metaphysics, however, sides with Wright and Hale’s opponents (wrong-headedly maybe, but that’s where the troops are marching). Thus Ted Sider can write ‘There is a growing consensus that taking ontology seriously requires making some sort of distinction between ordinary and ontological understandings of existential claims’ (that’s from his paper ‘Against Parthood’). From this perspective, the claim would be that we must indeed distinguish granting the unqualified truth of mathematics, construed at face value, from being committed to a full-blooded PLATONISM which makes genuinely ontological claims. It is one thing to claim that prime numbers exists, speaking with the mathematicians, and another thing to claim that THEY EXIST ‘in the fundamental sense’ (as Sider likes to say) when speaking with the ontologists.
Now, we can think of Sider et al. as mounting an attack from the right wing on the Quine/neo-Fregean rejection of a special kind of philosophical discourse about what exists: the troops are mustered under the banner ‘bring back old-style metaphysics!’ (Sider: ‘I think that fundamental ontology is what ontologists have been after all along’). But there is a line of attack from the left wing too. Consider, for example, Simon Blackburn’s quasi-realism about morals, modalities, laws and chances. Blackburn is no friend of heavy-duty metaphysics. But the thought is that certain kinds of discourse aren’t representational but serve quite different purposes, e.g. to project our moral attitudes or subjective degrees on belief onto the world (and a story is then told about why a discourse apt for doing that should to a large extent retain the same logical shape of representational discourse). So, speaking inside our moral discourse, there indeed are virtues (courage is one): but as far as the make-up of the world on which we are projecting our attitudes goes, virtues do not EXIST out there. From the left, then, it might be suggested that perhaps mathematics is like morals, at least in this respect: talking inside mathematical discourse, we can truly say e.g. that there are infinitely many primes; but mathematical discourse is not representational, and as far as the make-up of the world goes – and here we are switching to representational discourse – THERE ARE NO prime numbers.
To put it crudely, then, we can discern two routes to distinguishing ‘there are prime numbers’ as a mathematical claim and ‘THERE ARE prime numbers’ as a claim about what there really is. From the right, we drive a wedge by treating ‘THERE ARE’ as special, to be glossed ‘there are in the fundamental, ontological, sense’ (whatever that exactly is). From the left, we drive a wedge by treating mathematical discourse as special, as not in the ordinary business of making claims purporting to represent what there is.
And now we’ve joined up with Weir’s discussion. He answers ‘yes’ to all three of our questions. A fourth then remains outstanding:
- Given there is a distinction between saying that there are prime numbers and saying THERE ARE prime numbers, is the latter stronger claim also true?
If you say ‘yes’ to that, then you are buying into a version of platonism that does indeed look epistemically particularly troubling (in a worse shape, at any rate, than for the gap-denying neo-logicist position; for what can get us over the claimed gap between the ordinary mathematical claim and the ontologically committing claim)? Weir thinks this position is hopeless. Hence he answers ‘no’ to (4). Hence he endorses claims like this: There are infinitely many primes but THERE ARE no prime numbers. (p. 8 )
But this isn’t because he is, as it were, coming from the right, deploying a special ‘ontological understanding of existence claims’. Rather, he is coming more from the Blackburnian left: his ‘THERE ARE’ is ordinary existence talk in ordinary representational discourse, and the claim is that ‘there are infinitely many primes’, as a mathematical claim, belongs to a different kind of discourse.
OK, what kind of discourse is that? “The mode of assertion of such judgements, I will say, is formal, not representational”. And what does ‘formal’ mean here? Well, part of the story is hinted at by the claim that the formal, inside-mathematics, assertion that there are infinitely many primes is made true by “the existence of proofs of strings which express the infinitude of the primes” (p. 7). Of course, that raises at least as many questions as it answers. There are hints in the rest of the Introduction about how this initially somewhat startling claim is to be rounded out and defended in the rest of the book. But they are much too quick to be usefully commented on here; so I think it will be better to say no more here but take them up as the full story unfolds.
Still, we now have an initial specification of Weir’s location in the space of possible positions. His line is going to be that, as a mathematical claim, it is true that are an infinite number of primes: and this truth isn’t to be secured by reconstruing the claim in some fictionalist, structuralist or other way. But a mathematical claim is one thing, and a representational claim about how things are in the world is another thing. And the gap is to be opened up, not by inflating talk of what EXISTS into a special kind of ontological talk, but by seeing mathematical discourse (like moral discourse) as playing a non-representational role (or dare I say: as making moves in a different language game?). That much indeed sounds not unattractive. The question is going to be whether the nature of this non-representational game can be illuminatingly glossed in formalist terms. Now read on …