Oops! A blunder in GWT

David Makinson has emailed to point out a foul-up in Episode 2, §9, of Gödel Without Tears. (Actually, he very kindly called it an “anomaly”. This suggests one of those irregular conjugations: “I made a little slip, what you wrote involves an anomaly, he or she made a terrible blunder”.)

Since I picked up the email on a train to London I had a few hours to worry about whether the same foul-up occurs in my Gödel book (in the corresponding §7.1). Phew. It doesn’t. It was indeed something too unthinkingly said as an aside in lectures, and afterwards plonked down in the notes. Which is a relief.

I’ll rewrite the relevant passage in Gödel Without Tears a.s.a.p. But here’s the basic story. Theorem 5 says: A consistent, sufficiently strong, axiomatized formal theory cannot be negation complete. Here, ‘sufficient strength’ you will recall is a matter of being able to capture or represent decidable properties of numbers, and ‘axiomatized’ of course means ‘decidably axiomatized’.

Now neither this theorem  nor the proof I gave of it  actually  delivers us a formally undecidable sentence. So it is weaker than a full Gödelian result. But I said more:

So suppose we start off with a consistent ‘sufficiently strong’ theory T couched in some language which just talks about arithmetic matters: then this theory T is incomplete, and will have arithmetical formally undecidable sentences. But now imagine that we extend T ’s language (perhaps it now talks about sets of numbers as well as about numbers), and we give it richer axioms, to arrive at an expanded consistent theory U. U will still be sufficiently strong if T is, and so Theorem 5 will still apply if it is properly axiomatized. Note, however, that as far as Theorem 5 is concerned, it could be that U repairs the gaps in T and proves every truth statable in T’s language, while the incompleteness has now ‘moved outwards’, so to speak, to claims involving U’s new vocabulary.

If that were right, it would be another way Theorem 5 is weaker than Gödel, for he shows that some incompleteness will always remain even in the theory’s arithmetical core. But it isn’t right. Sigh. Here’s David’s counter, only slightly edited:

Let T* be the set of all theorems of U in the language of T. Since T is included in T* which is included in U, we know that T* is sufficiently strong but consistent. And suppose for reductio that T* is negation complete in the language of T; we get a contradiction.

If T* is negation complete then T* must be decidable [by the same kind of argument used in proving Theorem 3]. Thus, given any sentence A in the language of T, just grind out theorems of U until you get either A or its negation (which you must, since T* is included in U and by supposition is negation complete in its language). But if T* is decidable, then it may serve as the axiom set for itself as a axiomatized formal theory. Thus T* is a consistent, sufficiently strong, axiomatized formal theory and so by Theorem 5 is not negation complete after all, giving us a contradiction.

Indeed. Oh dear, I’ve been leading the youth astray again …

TTP, CUP, and a shiny new MBA

I suppose it was mildly daft to plunge into blogging about Alan Weir’s  TTP just as the beginning of term looms. There’s now a flurry of other things which I really need to be thinking about, just as I’m getting into the book, and puzzling through the next chapter.  There’s admin as Chair of Examiners for Tripos to be done, plus putting  together some handouts for my last lectures on Gödel’s Theorems (the last for this academic year, at any rate), thinking about the response I’m down to give to one of the papers at the Phil. Logic & Maths conference here (about Brandom of all people), and that’s not to mention sorting out the techie logic seminar and preparing an initial talk to that. So the discussion of Alan’s book will stutter a bit for the next couple of weeks. Sorry about that.

For light relief, it is time for trips to the CUP Bookshop sale again. This is a great annual institution which I’ve mentioned before. The Press damage some books by stamping “damaged” across the title page, and then flog them (this year) at £3 for any paperback and £7 for any hardback. And during the week or ten days of the sale they keep putting out new stock in a random way, so you have to keep slipping back, just in case … It is amazing what turns up.

However, having badly run out of book shelving space, and then some, I really really do have to restrain myself. But I couldn’t resist the ‘Cambridge Companions’ to Haydn and Schubert, and David Crystal’s fun-if-you-like-that-kind-of-thing book on Shakespeare’s language.

And, erm, another category theory tome. Despite all the empirical evidence, I think I must subconsciously believe in a magical theory of learning-by-osmosis. Put the book on your shelves and the knowledge slowly seeps in … doesn’t it?

As for the MBA, that is, of course, a MacBook Air to you. Let me just say that the new version is awesome. I had an original version MBA, which was lovely but s-l-o-w and had a pretty poor battery life. But I’ve been given one of the new models, and the difference is impressive (it is the machine the original one almost promised to be, but fell quite a bit short of). Subjectively very fast, wonderful screen, and ludicrously long-lasting battery for academic writing/reading/surfing/mailing use. (For fellow Appleheads: I have the 13″, as I find the visual proportions of the 11.6″ unhappy — I can’t shake the sense of peering through a letter box — and I want big-enough side-by-side LaTeX windows. The base configuration with 2gb memory is more than just fine if you are not doing anything very fancy with it. If you have been wavering, treat yourself.)

TTP, 4. §1.III: Sense, circumstance, world

In the present section, Weir says something about the kind of semantic framework he favours, and in particular about issues of context-sensitivity. Here I do little more than summarize.

The basic idea is very familiar. “Utterances of declarative sentences are typically true or false, and what makes them one or the other is, in general, a triple product of firstly the Sinn or informational content they express, secondly the circumstances of the utterance, and finally the way the world is”. So this is the usual modern twist on the ur-Fregean story: it isn’t just sense, but sense plus context (broadly construed), that determines reference and so fixes truth-conditions. This basic picture is widely endorsed, and Weir doesn’t aim to develop a detailed account of how the three layers of story interrelate. Some general remarks are enough for his purposes.

Suppose we aim for a systematic story about how a certain class of sentences gets its truth conditions, for example those involving a demonstrative ‘that’. The systematic story will, perhaps, use a notion like salience, so for example it tells us that ‘that man is clever’ is true when the most salient man in the context is clever. Now, for this to be part of a semantic theory that is explanantory of speech-behaviour, speakers will have to reveal appropriate sensitivity to what we theorists would call considerations of salience. But those we are interpreting needn’t themselves have the concept of salience. And the explanatory statement of truth-conditions is not synonymous with ‘that man is clever’. We thus need to distinguish the literal content of the sentence as speakers understand it (what is shared by literal translation, for example) from the explanatory truth-conditions delivered by our systematic semantic theory.

Note though, semantics is one thing, metaphysics something else. It might be that what it takes (according to semantic theory) for ‘that man is clever’ to be true is that the most salient man in the context is clever. But what has to exist for that to be the case? — does it require the existence, for example, of a truth-making fact? Semantics is silent on the issue: so, for example, fans and foes of truth-makers can alike accept the same semantic story about explanatory truth-conditions.

Now, ‘literal content’ vs ‘explanatory truth-conditions’ was, Weir tells us (fn. 28) his own originally preferred terminology here. He now thinks ‘informational content’ or ‘sense’ vs ‘metaphysical content’ is less misleading. Really? Does dubbing something ‘metaphysical’ ever make things clearer?? Especially when you’ve just used ‘metaphysical’ in a significantly different way in talking of metaphysical realism, and also insisted on downplaying the metaphysical loading of the semantic story??? But let’s not get fractious! — if we are in the business of a traditional kind of semantic theory, there is a distinction to be made, whatever we call it. Though let’s also be on the watch for occasions where the possibly tendentious labelling is allowed to carry argumentative weight.

As Weir says, not everyone endorses the sense/circumstances/world (SCW) picture. For a start, there are radical contextualists who don’t like the idea of a given sense or meaning making a fixed contribution to determining truth-conditions. But Weir “side[s] with those who hold that radical contextualism makes language grasp a mystery”.

However, even if we go along with the basic SCW picture, there is room for debate about how much work circumstances do, just how much context-relativity we need to recognize. Cappelen and Lepore, for example, have argued that there is only a rather confined Basic Set of context-sensitive expressions in language, contra those who seek philosophical illumination by claiming to discern hidden context sensitivity. Weir hints that he is going to need to take a more generous line than Cappelen and Lepore (but without falling back into radical contextualism). But we’ll have to wait to see how this works out.

So far, as I said, that’s mostly summary. But let me add two final comments. (a) Weir’s anti-realism about mathematics, as we saw, is to be a built on a distinction between representational and non-representational modes of discourse. And on the face of it, you would expect that issues about different modes of discourse would be orthogonal to the issues about kinds of context-sensitivity most highlighted in this section. Again, we’ll have to wait to see just see what connections get forged. True, there is a murky hint on p. 38; but it didn’t at all help this reader. (b) A more thorough-going pragmatist, perhaps of Wittgensteinian disposition, will also emphasize the different roles of different discourses, but will resist the thought that one kind is more basic or more central than others. She might start to worry that Weir’s semantic framework — at least so far — is looking too traditionally biased towards privileging the representational discourse for which the SCW picture seems tailor-made.  More on this anon.

TTP, 3. §§1.I–II: Realisms

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.

Cambridge Conference on the Philosophy of Logic and Mathematics

The fourth in a series of now annual conferences takes place in Cambridge over the weekend of 22nd–23rd January 2011. The previous conferences have been excellent fun, so why not come along? Here is the line-up for this year’s event.

Keynote speakers:

  • Graham Priest (Melbourne/CUNY/St. Andrews): Dialetheism, Concepts, and the World
  • Rosanna Keefe (Sheffield): Modelling vagueness: what can we ignore?

Graduate papers:

  • Sharon Berry (Harvard): Solving the Access Problem for the Nominalist
  • James Collin (Edinburgh): Exists
  • Dmitri Gallow (Michigan): Frege Defining Functions
  • Jon Litland (Harvard): The Barcan Formulae for Determinacy
  • Jonathan Payne (Sheffield): Syntactic Priority Generalised
  • Giacomo Turbanti (Scuola Normale Superiore di Pisa): Modality in Brandom’s Semantics

If you want to come along you can register via the conference website. (Note that this is a fix-your-own-Saturday-night-accommodation-with-friends affair, which keeps registration very cheap.)

These you have loved …

To be honest, I am as bad as the rest of you! I almost daily visit someone’s website or blog, read their words of wisdom, download papers, talks, or overhead slides, and learn a huge amount this way. Yet I almost never post a comment or drop an e-mail to say “thanks”. Which doesn’t at all reflect how appreciative I am, or acknowledge what a difference all this web activity has made to my logical life. I am genuinely grateful.

And so I can hardly feel aggrieved when, in its turn, this blog gets about 900 visits a day, some of the papers and handouts posted here have been downloaded thousands of times, and then I get about one email every three months. (Hold on … maybe that’s welcome! What if even one in fifty visitors mailed ..?). Still, it is good to check out last year’s summary stats and find that the number of visitors steadily drifts upwards, and stuff indeed does get read.

It is  surprising too to discover what gets particularly picked up. For example, leaving aside stuff related to my logic book and the much-downloaded Gödel Without Tears series (yes, I’ll be finishing the updates on that over the coming weeks), who would have guessed that over 3000 people would have been interested enough to fetch the handout on Galois connections I dashed off for some grad students a few months back. It was very nice to see too that quite a few people downloaded my two page handout on Kleene’s very pretty proof of the first incompleteness theorem which deserves to be better known, and a similar number downloaded a longer expository piece on how to read Kreisel’s squeezing argument, something else that should be better understood (a later version of the latter has now been published in Analysis). As to other webpages, the LaTeX for Logicians section gets steadily visited (as one might have expected, though I’ve not been doing much to maintain it recently — I wonder if there is anyone out there who would sooner or later like to pick up the baton and run with it?). And I was amused to see that the special page I dashed off in the first flush of excitement as a proud owner of an iPad got visited over 8000 times in just five months (though I guess I should update that page: I’d now mention some limitations as well as the things it does well …).

Anyway, as I said, it is rather nice to know I’m not just talking to myself here. So very best wishes for 2011 to all of you out there. And thanks for reading!

Slides for more introductory logic lectures

I’ve somewhat belatedly put online slides for the last six of last term’s intro logic lectures. Lectures 11–13 introduce propositional trees, and lectures 14–16 introduce the language QL. Frankly, you would do much better to read my book: but since slides for previous lectures have been downloaded a surprising number of times, here’s the remainder.

TTP, 2. Introduction: Options and Weir’s way forward

Faced with the Benacerrafian 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.

One way of beginning to organize (some) positions in the philosophy of mathematics is to consider how they answer the following sequence of questions. Start with these two:

  1. Are ‘3 is prime’ and ‘the Klein four-group is the smallest non-cyclic group’, for example, (unqualifiedly) true?
  2. 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 — as the surface form suggests (on the same plan as e.g. ‘Alan is clever’ and ‘the tallest student is the smartest philosopher’)?

The platonist answers ‘yes’ to both. A naive formalist and one stripe of fictionalist, will get off the bus at the first stop and answer ‘no’ to (1) —  the former because there is no genuine content to be true, the latter because the content is (supposedly) a platonist fantasy. Another, more conciliatory stripe of fictionalist can answer ‘yes’ to (1) but ‘no’ to (2), since she doesn’t take ‘3 is prime’ at face value but re-construes it as short for ‘in the arithmetic fiction, 3 is prime’ or some such.

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 saying there are prime numbers and there are non-cyclic groups, etc. (for it is true that 3 is prime, and — construed as surface form suggests — that implies there are prime numbers). Next question:

  1. 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 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 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’ and ‘THERE EXISTS a first odd prime number’. Drawing on 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:

  1. 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.

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