Peter Smith's Blog, page 129

Beginning maths, beginning philosophy

Tim Gowers has begun what he plans to be a long series of posts on his blog giving advice for beginning Cambridge mathmos. But the posts should of course be of great interest for new undergraduate mathematicians anywhere. So spread the word.

And already some of his advice seems pretty applicable to beginning philosophers too. That shouldn't be too surprising: in both cases, what we are trying to teach isn't a body of facts so much as an understanding of argumentative moves. In both cases, students need to learn how to see how to fillet out the key ideas (and tell what's just joining-up-the-dots).

So I might be inspired to do what I have meant to do for a while and put together some how-to-get-started-thinking-philosophically and how-to-approach-your-work notes for new philosophers. It shouldn't be made a mysterious business, whether it's maths or philosophy. But meanwhile, read Gowers!

TTP, 14. Worries about excluded middle

Weir's formalist account of arithmetic in headline form comes to this: the arithmetical claim P is correct just in case that there is (or in practice could be) a concrete proof of P. (We'll stick to considering the arithmetical case.)

Weir needs proofs to be concrete: treating proofs as abstract objects wouldn't at all chime with his naturalist worries about abstracta. He needs to allow the practical possibility of a concrete proof to suffice for truth, or else — for all he knows — there could be random gaps even among the simple computable truths about small numbers because no one has yet bothered to write down a computation. But the modality had better be not be an extravagant one ("in some possible world, however remote, where our counterparts can e.g. handle proofs with more steps than particles in the actual world, …"), or again we could be taking on modal commitments that are inimicable to Weir's strong naturalism: so as we said, we need some notion of practical possibility here.

Are these concrete proofs particular tokens? If so, you and I always give different proofs. That wouldn't tally with our ordinary ways of talking of proofs (when we talk of Euclid's proof of the infinitude of primes, we don't mean just some ancient token; when we say there is only one known proof of P we don't mean that it has only been written down once). But Weir wants to avoid proof-types (or he is back with the unwanted abstracta); so he makes play with with a notion of a proof-tipe, where a tipe is a mereological sum of equiform tokens. Thus understood, a proof tipe is a scattered concrete thing, and you and I giving the same proof — to speak with the vulgar — is a matter of you and I producing tokens that are part of the same proof tipe. (Weir thinks that for his purposes he can get away with a lightweight mereological theory that doesn't get tangled with issue about e.g. absolutely restricted principles of composition. Let's give him that assumption. It will be the least of our worries.)

Now, concrete numerals and concrete proofs are few (even counting in practically possible ones as well). The obvious challenge to Weir's position is that his formalism will therefore lead to some kind of finitist revisionism rather than conserving the arithmetic we know and love.

To press the point, take a concretely statable claim P that some very large number n is prime. Then it could well be that there is no practically possible concrete proof of either P or not-P in your favourite formal system S (the system whose concrete proofs, according to Weir, make for the correctness of arithmetical claims on your lips). Yet it is an elementary arithmetical claim that either P or not-P, a truth now seemingly without a formalist-approved truth-maker (given that Weir endorses a standard truth-functional account of the connectives). What to do?

Well at this point I struggle. The idea seems to be this. Here is our practice of producing concrete formal proofs in the system S. Like other bits of the world, we can  theorize about this, doing some applied mathematics M which — like other bits of applied mathematics — purports to talk about some idealized mathematical model, in this case a model of the real-world practice of proving things in S. And now, we do have a concrete M-proof that, in the idealized model, the model's representative for the concrete claim that either P or not-P. So,

the EXISTENCE [remember: small caps indicate talk-in-the-metaphysics room] of concrete indeterminables [like P] should not inhibit reasoners from applying excluded middle to them so long as THERE IS a concrete proof that, in a legimitate idealization, the image of the indeterminable is decidable in the formal sense. (TTP, p. 205)

But hold on! Setting aside worries about 'legitimate', what makes an arithmetical claim correct, on your lips, is by hypothesis the practical availability of a concrete proof in a formal game S [maybe different speakers are working in different games with appropriate mappings between them, but let's not worry about that now]. So by hypothesis, a concrete proof in a different bit of mathematics M — a bit of applied mathematics about an idealized model of our arithmetical practice — just isn't the sort of thing that can make an arithmetical claim correct. If we allow moves external to S to make-true arithmetical claims, then S isn't after all the formal system proofs within which proofs provide correctness conditions for arithmetic.

So on the face of it, it looks as if Weir has simply cheated here by changing the rules of the formalist game midstream! And I'm evidently not alone in thinking this: John Burgess in his Philosophia Mathematica review finds himself baffled by the same passage. Weir himself recognizes that his readers might find the move here sticky. In one of his comments in the thread following my last blog post on his book, he writes

If we want to know whether a particular area is bivalent … we don't ask whether for every token there is a token proof or disproof, we ask if there is a concrete proof of a formal negation-completeness result for the idealised theory. If so, we are allowed to lay down all instances of LEM for that sub-language as axioms … even if that means that some concretely provable theorems — t is prime or t is not prime say — have disjuncts which are neither concretely provable nor refutable. I suspect this get out of jail manoeuvre will prove one of the biggest sticking points in the reception of the book, but I'm still confident about its legitimacy!

Like Burgess, I can't share Weir's confidence.

Log xy = Log x + Log y

In their (rich, original, ground-breaking) writings on plurals, Alex Oliver and Timothy Smiley more than once say that "mathematical practice" shows that addition is allowed to take plural terms as arguments. Thus, with a trivial change of variables, in their  'A modest logic of plurals' (JPL, 2006) they write

One might think that ['+'] can only take singular terms as arguments, … Mathematical practice shows that this is quite wrong, as witness the right-hand side of this equation (Hardy, 1925, p. 408)

Log xy = Log x + Log y

bearing in mind that Log z is an infinitely many-valued function.

But this is surely a mis-step. In fact Hardy himself realises that — given that the Log terms are many-valued — he'd better explain the extended use of the equation notation here (that is the novelty). So he does. Hardy tells us that we are to read this as "Every value of either side is one of the values of the other side". So already, the surface equation is frankly offered as syntactic sugar for something more structured. Fair enough: we do this sort of thing all the time. And how are to read Hardy's snappy explication? Obviously, if you want the gory details, like this — what else?

For any (particular!) a such that a is a value of Log xy, there is a b such that b is a value of Log x and a c such that c is a value of Log y, and a = b + c. And for any b such that b is a value of Log x and c such that c is a value of Log y, there is an a such that a is a value of Log xy and a = b + c.

Which reveals that, when the wraps are off, the addition functor here is  straightforwardly applied to singular terms, just as you'd expect. Hardy's usage is a handy extension of equation notation to plural terms produced by many-valued functions: but we don't need to discern any accompanying novelty in the use of addition. No?

Abortion, again

It has been depressing to see the issue of abortion being politicised again in the UK, when a hard-won, decent and humane settlement has been in place. And it is particularly depressing for a philosopher to see that usual dreadful arguments (on both sides, it has to be said) trotted out again. Here's one way of thinking about the issue which hopefully sheds some moral light.

Consider the following "gradualist" view: As the human zygote/embryo/foetus slowly develops, its death slowly becomes a more serious matter. At the very beginning, its death is of little consequence; as time goes on, its death is a matter it becomes appropriate to be gradually more concerned about.

Now, this view seems to be the one that almost all of us in fact do take about the natural death of human zygotes/embryos/foetuses. After all, very few of us are worried by the fact that a very high proportion of conceptions quite spontaneously abort. We don't campaign for medical research to reduce that rate (nor do opponents of abortion campaign for all women to take drugs to suppress natural early abortion). Compare: we do think it is a matter for moral concern that there are high levels of infant mortality in some countries, and campaign and give money to help reduce that rate.

Again, very few of us are scandalized if a woman who finds she is pregnant by mistake in a test one week after conception is then mightily pleased when she discovers that the pregnancy has naturally terminated some days later (and even has a drink with a girl friend to celebrate her lucky escape). Compare: we would find it morally very inappropriate, in almost all circumstances, for a woman in comfortable circumstances to celebrate the death of an unwanted young baby.

Similarly for accidental death. Suppose a woman finds she is a week or two pregnant, goes horse riding, falls badly at a jump, and as a result spontaneously aborts. That might be regrettable, but we wouldn't think she'd done something terrible by going riding and running the risk. Compare: we would be morally disapproving of someone badly risking the life of new born by carrying it while going in for some potentially very dangerous activity.

So: our very widely shared attitudes to the natural or accidental death of the products of conception do suggest that we do in fact regard them as of relatively lowly moral status at the beginning of their lives, and of greater moral standing as time passes. We are all (or nearly all) gradualists in these cases.

It is then quite consistent with such a view to take a similar line about unnatural deaths. For example, it would be consistent to think that using the morning-after pill is of no moral significance, while bringing about the death of an eight month foetus is getting on for as serious as killing a neonate, with a gradual increase in the seriousness of the killing in between.

Some, at any rate, of those of us who are pro (early) choice are moved by this sort of gradualist view. The line of thought in sum is: the killing of an early foetus has a moral weight commensurate with the moral significance of the natural or accidental death of an early foetus. And on a very widely shared view, that's not very much significance. So from this point of view, early abortion is of not very much significance either. But abortion gradually gets a more significant matter as time goes on.

You might disagree. But then it seems that you either need (a) an argument for departing from the very widely shared view about the moral significance of the natural or accidental miscarriage of the early products of conception. Or (b) you need to have an argument for the view that while the natural death of a zygote a few days old is of little significance, the unnatural death is of major significance. Neither line is easy to argue. To put it mildly.

We can agree however that killing a neonate is, in general, a very bad thing. So the remaining question is how to scale the cases in between. That's something that serious and thoughtful and decent people can disagree about to some extent. But note it is a disagreement about matters of degree (even if any legal arrangements have to draw artificial sharp lines). All-or-nothing views, on either side of the debate, have nothing to recommend them.

 

Music to sort books by

Who knew retirement could be so busy? Two jointly written papers just done; a draft book to comment on in detail; four book reviews to do; a growing pile of things I want to read (Huw Price's collected papers arrived today). Not to mention the books I'm supposed to be working on.

Though strictly speaking, I'm not retired till the end of next month. But apart from wrapping up a couple of reports as chairman of examiners, job duties are all done. And I'm on track for getting out of my office in the next couple of weeks and making room in my study at home for what I want to move there. I guess I could now just pack up the remaining office books, perhaps four medium boxes worth of them, and stash them in the garage. But that would be a Bad Move. Best not to keep postponing decisions. The trouble is, however, that I keep getting sidetracked into actually dipping in and reading the things.

And there a few box files of old lecture notes to dispose of. Gosh: did I really once know about all that stuff? For instance, 30 year old intro moral philosophy lectures anyone? Am I really going to type them up and put them on the web? Would enough people be much interested if I did? No, and no. So into the bin with them. Easier said than done of course. That's part of me that's gone. But needs must.

All that sorting out takes time, too. So, as I said, a busy time. But in good way. And some of the sorting out at home has been accompanied in the most enjoyable way, listening to Haydn. At my retirement drinks party, I was presented inter alia with the boxed set of his complete symphonies (the Adam Fischer/Austro-Hungarian Haydn Orchestra recordings — a mere 33 CDs to explore). Which is just perfect while fossicking about trying to decide  what books to keep, what to give away. So thanks again, everyone who contributed so generously. I started with Symphony no. 1 and am slowly working my way through. From the outset, a delight and very rewarding.

TTP, 13. Formalism and "pluralism"

In TTP 11, I emphasized that Weir's position interweaves two separable strands. One strand I called "formalism about arithmetical correctness": at a first approximation, what makes an arithmetical claim correct is something about what can be done in some formal game(s) played with uninterpreted tokens. The other strand proposes, as I put it, that the content of arithmetical claims is not "transparently representational". So far, in these blog posts, I've been talking mostly about the way the second strand gets developed. It has been Hamlet with only brief appearances of the Prince of Denmark.

Weir's story about content is intended to serve as a kind of protective wrapping around the formalist core (so he can say e.g. that although arithmetical claims have formalist correctness conditions we aren't actually talking about synactic whatnots when we make common-or-garden arithmetical claims — thus avoiding at least some incredulous stares). But there is no getting away from it: when the wraps are off, the story about those correctness conditions are indeed very starkly formalistic. What makes arithmetical claims correct, when they are correct, is facts about plays with concrete tokens in some rule-governed practice of token-shuffling (actual plays, or practically possible ones).

Well, here I am, making arithmetical claims. These are supposedly made true by facts about concrete moves in some formal practice. Which formal practice? Look again at the toy models I offered in TTP 10. There was a (1) a game with an abacus, with facts about this making true tokens like '68 + 57 = 125' (whose content is tied to such facts, but non-representationally). Then there was (2) something like school-room practice where we write down "long additions" etc. on our slates, with facts about what can be written down in this practice making true equiform tokens '68 + 57 = 125' (which therefore have a different content from before). Then there was (3) a formal proof-system for quantifier-free arithmetic, and tokens such as '68 + 57 = 125' are now tied to facts about what can be derived inside the formal system. So when I say '68 + 57 = 125' which formal game is my utterance tied to? What content does my utterance have? What's to choose?

Weir's response is: don't choose.

The neo-formalist position is pluralist rather than relativist. The truth-value of '68 + 57 = 125' is not relative to a formal system. Rather there is a plurality of systems, and the sentence expresses different senses in each, whilst it is made true (or false) in the context of a given one iff it is provable therein. (p. 108)

Which is the line he has to take (irrespective of the details of his story about non-representational content). Take the child brought up "bilingually" to play the abacus game and comment on that, and play the school-room game and comment on that. By mishap the child could come to believe 68 + 57 = 125 in the comment mode in one context and disbelieve in the other: so the contents had better be different.

OK: two children taught two different games and two different commenting practices will mean something by equiform comments. Yet it would be mighty odd, wouldn't it, to say that two real children taught arithmetic by different methods mean something different by '68 + 57 = 125'? The natural thing to say, most of us will think, is that if the kids end up as practical arithmeticians counting the world in the same ways, using addition when they want to put together the counts of two different piles to give a count of their combination, then whether they get to '68 + 57 = 125' by abacus, school-room sum, doing a formal proof (heaven help us!) or using a calculator, they mean the same. For the sense of what they say is essentially grounded in their applied practice, in how they use arithmetic in the world.

From this natural perspective, Weir's story look upside down. He first talks of unapplied formal games (as it might be with an abacus, or writing down sums as in the school room, or operating with an uninterpreted proof system); assigns various contents to '68 + 57 = 125′ treated as comments on moves in those various as-yet-unapplied games. And only afterwards, with the various informational contents fixed by the liaisons to the uninterpreted formal games, does Weir talk about applying the 'arithmetic' to the world. But at least as far as arithmetic is concerned, this looks topsy turvy: it's the embedding in the applied practice of counting and adding and multiplying that gives sense to arithmetic, properly so called. (Cf. Wittgenstein in the Big Typescript discussing what makes arithmetic different from a game.)

Later (p. 218) Weir modifies his pluralism. He says two speaker S and S* using e.g. of '68 + 57 = 125' "express the same sense" if there is an "admissible mapping" between theie linguistic practices L and L*. The details don't matter (which is good, because they are not clear). The trouble is that I don't see what entitles Weir to play fast and loose with the notion of sense like this. He's spent chapters earlier on the in book trying to give us a story about sense or "informational content" that allows him to distingish the thin sense of '68 + 57 = 125' from the rich sense of a statement of the correctness conditions of the claim in terms of moves in a formal game. He can do this because he cuts sense fine. And his criterion of difference in sense is that you can believe 68 + 57 = 125 (as a comment on the game) without being able to frame the concepts necessary to state the metaphysical correctness conditions. But then, by the same Fregean criterion of difference, the bilingual L/L* speaker like the abacus user who also does sums on paper could still express two different senses by different uses of '68 + 57 = 125'. Weir seems, later, just to be changing the sense of "sense".

So I think Weir is landed with the radical pluralism he cheerily embraced earier, and his later attempt to soften it is a mis-step by his own lights. Many will count this as a strike against neo-formalism.

To be continued

Weir on colours, concepts, fiction, numbers, and more!

Alan has written a long reply to my reply to his comments on my last post.  This should be of particular interest to anyone who is reading his book, so I thought I'd flag it up at the top level here!

TTP, 12. 'The formal mode of assertion'

Weir himself distinguishes three model cases where a claim's content is not transparently representational — to use my jargon for his idea — and I added a fourth case. (We are assuming, for the sake of argument, that the general idea of having NTR content is in good order.) The question left hanging at the end of the last post was this: Which of the models on the table, if any, is the appropriate one when it comes to elucidating Weir's idea that arithmetical claims have NTR content?

Well, we have some idea from the opening chapters what Weir wants — see the preceding blog posts in this series! For a start, (1) he wants to treat arithmetical claims 'at face value' in the sense that he doesn't construe them as being misleading in their suface form and as requiring unmasking as really representing some subject-matter not obviously revealed at the surface level. But it isn't that he thinks that arithmetical claims do actually represent numbers and their properties; rather (2) he wants them to be treated as belonging to a fundamentally non-representational mode of discourse.

So the sort of NTR content which is illustrated by cases with demonstratives — cases which are still fundamentally representational — can't provide us with a model of what's going on in arithmetic. Nor will my example of talk of colours be helpful: for Weir, it isn't that arithmetical talk deploys "confused ideas" and represents but in a way that calls for an unmasking story to lift the fog. Rather, "The mode of assertion of [arithmetical claims] … is formal, not representational".

The models to look at, then, for illuminating the story about the NTR content of arithmetic must indeed be the non-representational ones that Weir himself emphasizes, i.e. the cases of projectivist discourse and of fictional discourse.

What will make it plausible (I'm not saying right!) to tell a projectivist, non-representational, story about (say) moral discourse? It must look reasonably natural to tell a story about the mental states of speakers according to which moral assertions are keyed not to kosher beliefs representing the world but to evaluative attitudes. What will make it plausible to tell a projectivist story about probabilities? Again, we need to tell a natural story about how assignments of probability are keyed not to having a belief with a certain content but to the strength of another belief. Such projectivism about claims that p gets off the ground, then, when such claims can be seen as suitably keyed to mental states other than believing-that-p — and for this to be plausible, we'd need already have reason to discern such states. For example, we already have reason to think of agents as having attitudes pro and con various actions, and as having a desire that such attitudes be shared: it's not so surprising, therefore, if we should have acquired ways of talking whose purposes is to express such attitudes and facilitate their coordination. Similarly, we already have reason to think of beliefs as coming in degrees: no surprise, either, that we should have ready ways of expressing degrees of belief.

It's similar to the case of talking within a fiction, at least in the key respect that the claim that p (e.g. that Sherlock lived within five miles of Westminister) is not keyed to a common-or-garden belief but to something else, a pretence to be representing.

But now compare the "just so" stories in my post TTP 10. Take the extended abacus game, for example (the same applies to the other games, mutatis mutandis). We imagined children playing with an abacus, and then learning to "comment", first by learning to write '74 + 46 = 120′ when they have just achieved a certain configuration in a correct play of the abacus game. Now ask: at this stage, what mental states are those tokenings keyed to? Surely beliefs, common-or-garden beliefs about what has just happened in a correct play of the abacus game. There's no call for any story yet about a special non-belief state of mind behind such tokens as '74 + 46 = 120′. To be sure, the children may well lack the resources to frame a transparent representation of the correctness condition for their tokens and may not yet fully conceptualize the business of getting to an arrangement of beads by correct play. Fine. But that in itself would only make their claim a "confused" representation, in Leibniz's sense, not make it non-representational. It is surely beliefs, representational states, that are being expressed. (What alternative, non-belief, state that we already have reason to discern would their tokenings be keyed to?)

What about the end stage of the language game where the rules are relaxed to allow the children to write '74 + 46 = 120′ even when they haven't in fact just executed a (correct) play of the abacus game, so long as they could in (correct) practice achieve the configuration?

Well, there's a modality here, and if you are a projectivist, or other non-representationalist, about modality you could now take a non-representationalist line about the arithmetical tokenings in the developed language game. But that's seemingly not Weir's line. For a start he only fleetingly mentions the possibility of being a projectivist about modalities, which would be very puzzling if he intended to lean on such a view. But more tellingly, his own neo-formalist account of the correctness conditions for arithmetical claims comes to this: such a claim is true just if a proof of it (a concrete proof-token) actually exists or is practically possible. So Weir seemingly likes facts about practical possibility, and takes them to be available as inputs to his explanatory metaphysical-cum-semantic story about arithmetic. So again we might ask: why isn't it confused representations of such facts that are being expressed by the children's tokenings of '74 + 46 = 120′ at the final stage of the developed abacus game?

Here's the worry. Non-representational claims that p in other cases are non-representational because keyed to non-beliefs. In the abacus game, the children's tokenings by contrast do seem to be keyed to beliefs, albeit ones that may only foggily represent the structure of the facts that make them true when they are true. So it needs argument to show that, despite such appearances, the children's tokenings in fact aren't expressions of belief but belong to a different kind of non-representational mode of utterance, Weir's so-called 'formal' one? And then what type of non-belief state are such 'formal utterances keyed to? I'm not seeing that Weir offers us the necessary account here.

In sum: maybe arithmetical claims e.g. in the abacus game are not transparently representational. But it doesn't follow that they are outright non-representational, involving a different mode of assertion keyed to some new class of non-beliefs. What's the argument that they are so?

TTP, 11. Disentangling neo-formalism

The introductory sketch in the last post reveals at least this much about Weir's neo-formalism: it is the marriage of two independent lines of thought.

One idea — call it "formalism about arithmetical correctness" — is that, at a first approximation, what makes an arithmetical claim correct (and we'll stick for the moment to the example of arithmetic) is something about what can be done in a some formal game(s) played with uninterpreted tokens.

To introduce the other idea, let's first say that the content of a claim C is transparently representational when what we grasp in grasping C is just its correctness condition according to our best explanatory metaphysical-cum-semantic theory. Thus, plausibly, in grasping "cats are mammals" we grasp what best theory surely will say is its correctness condition, i.e. just that cats are mammals. But it isn't like this, according to Weir, for arithmetical claims. Here, the suggestion goes, the content of a claim, as grasped by an ordinarily competent speaker, falls short of the thought that its correctness conditions are satisfied (which the formalist takes to mean: falls short of a thought about what can be done in the relevant formal game). Here, then, the content is not transparently representational; so call this the idea that arithmetic has  "NTR content" for short

Both ideas need to be embroidered in various ways to give us a decently worked through position, and Weir offers his versions, about which more anon. But the first point to make is that these plainly are two quite different basic ideas and their further developments will be very largely independently of each other. You could buy Weir's line on one without buying his story about the other.

For example, if you are a modal structuralist, who gives a quite different account of the correctness conditions for arithmetical assertions, you might — very reasonably — also   want to say that the content of the ordinary arithmetician's claims doesn't involve concepts of possible worlds or whatever. In other words, although no formalist, you might very tempted to embrace some story about how arithmetic content is not transparently representational (again holding that what is grasped in our ordinary understanding of an arithmetical claim falls short of grasping what makes the claim correct, according to your preferred story). So you might be interested to see whether you could borrow themes from Weir's story about NTR content. Alternatively, of course, you could embrace something like Weir's formalism about correctness while not liking the way he spins his account of NTR content.

So we'd better clearly disentangle the two themes. We'll turn to "formalism about arithmetical correctness" in later posts. Here let's begin to consider how Weir handles the idea of NTR content.

Weir's first two chapters are supposed to have softened us up for the viability of this idea. For recall, he has given three examples of cases where, it seems, the idea of NTR content looks appealing. First, and least controversially, there is the case of assertions involving demonstratives: "that animal is a mammal". Here the story about correctness (in such a case, correctness is plain truth) will talk about the most salient animal given contextual indications (pointings etc.) But what the ordinary conversationalist grasps surely isn't a thought involving the concept of salience, etc. Second, and much more controversially, there's the case of assertions supposedly inviting a Blackburnian projectivist treatment, e.g. moral claims. Here the story about correctness (and perhaps correctness can be thought of a matter of truth again, if we are sufficiently minimalist about truth) will talk about appropriateness of attitudes: but the content of a moral claim is not a thought about human attitudes. Third, Weir considered claims made in elaborating a fiction: "Holmes lived less than five miles from the Houses of Parliament". Here correctness ('truth in the fiction', perhaps) is keyed to what experienced readers would, on reflective consideration, judge must belong to an elaboration of the Holmes stories if everything is to make good enough overall sense. Again the content of the claim about Holmes, on the lips of a casual conversationalist, is not plausibly to be said to involve ideas about the coherence of a fictional corpus.

OK. But let's note again, as I've noted before, that these are three very different stories about instances of NTR content. We might say that, in the demonstrative case, the content — though not fully transparently representational — is still partially representational: the claim "that animal is a mammal", in context, aims to represent the world as it is. But a projectivist will say that moral judgements, by contrast, are in a different game from representation, they get their content from the practical business of encouraging and coordinating attitudes. As for fictional claims, they aren't representational either, but are articulating a make-belief.

So: a claim can fail to be (fully) transparently representational because it is representational but part of the representation has to be supplied by context; it can fail because it actually isn't primarily in the game of representation at all; or it can be fail because it is only pretending to represent. But that's not the end of it. Here's another sort of case which Weir doesn't mention. On a plausible metaphysical view, it is correct to say of something that it is green just if it is disposed, in normal viewing conditions for things of the relevant kind, to produce a certain characteristic response in normal viewers. But again, it seems wrong to say that the ordinary speaker, in grasping the content of "grass is green" is grasping a thought about dispositions or normal viewers. (To use a favourite style of argument of Weir's, Alan can believe that grass is green without believing that grass is disposed, in normal viewing conditions, etc. etc.). So, "grass is green" also has NTR content. But it isn't that "grass is green" is non-representational or is only pretending to represent: rather, it represents, but in a foggy way (that doesn't transparently yield correctness conditions). It seems apt to echo here Leibniz's talk of "confused ideas".

The question, then, for Weir is this. Let's grant him the idea that (in our terminology) some claims are not transparently representational: their sense peels apart from their explanatory correctness conditions. But the divorce can arise in various ways. Weir himself distinguishes three model cases; we've just added a fourth. So which of these models, if any, is the appropriate one when it comes to elucidating the idea that arithmetical claims have NTR content?

To be continued …

TTP, 10. Neo-formalism introduced

As you might remember, I'm supposed to be writing a review of Alan Weir's Truth through Proof. I started blogging here about the book some time ago, intermittently discussing the first couple of chapters at length, and then I'm afraid I stopped. That was partly through pressure of other things. But also I got to a point where I was finding it quite difficult to be sure I was getting the picture. (A seminar discussion at Birkbeck was comforting, as it revealed that I'm not the only one to find Weir's exposition of his "neo-formalism" rather opaque.)

Ok: the review is now overdue, and I don't have the time or energy to carry on blogging in such detail. (Who knew retirement could be so busy?) But let me try in this post to give the headline news about the  content of Weir's neo-formalism as first sketched in Chapter 3. Reading the chapter again, I think I'm following the plot better. But rather than try to summarize his own presentation, let me start by telling a story, one that I hope works up to a toy model of the sort of position which Weir wants to develop. The story involves developing three games, and each game we develop in three steps.

The abacus game Step one. We teach children to play with an abacus — let's imagine one with seven horizontal wires, six with nine beads on them, with the last wire having just a single bead. In the game, the inital state of the abacus has the beads in the first four rows each distributed into two (possibly empty) groups, a left one and a right one, and the beads on the final three rows are all shuffled to the right. Then allowed moves are made until a position is reached where all the beads on the top four rows are shuffled to the right, and there's some new distribution on the bottom three rows. In fact the rules are such that the game tracks the addition of two numbers under 99 (represented in the initial state of the abacus by units and then tens of one number on the top two rows, then units and tens of the other number) with the result represented by the state of the final three rows at the end of the game. But the children don't know this: they are taught allowed moves, which they apply (not by counting but) by pattern-recognition.

Step two. We now teach the children another step, augmenting the abacus game. Initially they learn to write in a ledger e.g. the symbols '74 + 46 = 120′ when they have just got from a certain initial configuration (the configuration that we would describe as follows: the beads on the first 'units' row separated four to the left, five to the right; on the second 'tens' row, seven to the left, two to the right; etc.) to a certain end configuration (on the first four rows, the beads all to the right; the same on the next row represent the null 'units' in the conclusion; then two beads on the left of the next row, and the final row the single bead is on the left).

Step three. The children now get rewards when they write down (what we would regard as) a 'correct' addition, and there are disincentives for incorrect ones. Moreover, they quickly learn that the goodies will be forthcoming whether or not the 'additions' are cued to actually playing with the abacus (behind a screen, perhaps) — it is only the written 'additions' in the ledger which are inspected. They learn that if an 'addition' is challenged, the challenge can met by actually going through the abacus game and getting the result cued to the addition, and the challenge is lost if they get a different result.

Now, in the original abacus game (at the G level for short), the tokens the children are moving around — the beads on wires — have no significance. But, at the level of the written tokens, where the children are giving a kind-of-commentary (call this the C level for short), the tokens in the language game can be thought of as having a certain significance, in that there are now correctness conditions for the issuing of a token '74 + 46 = 120′.

However, although the correctness condition for issuing '74 + 46 = 120′ is that you can get from a certain initial state of the abacus to a certain final state via moves according to the rules of the abacus game, it would arguably be over-interpreting to suppose that this is what the 'equation' means for the childish player, who after all need have no reflective grasp of e.g. the concept of a rule-of-the-game, and indeed no descriptive concepts for initial and final states either. If we tie the idea of having content to the idea of having correctness conditions in the language game, then the token '74 + 46 = 120′ can be said to have content: but the content falls short of the explicit thought that the correctness conditions obtain.

In sum, this gives us a toy model for a language-game in which (i) there are correctness conditions for the issuing of 'equation'-like tokens, (ii) these correctness conditions for tokens are given in terms of the availability of moves in an abacus-shuffling game, but (iii) it would be over-interpreting to suppose that the players, in issuing such an 'equation'-like token mean that the correctness conditions obtain.

Ok, so far let's suppose, so good. Now let's move on to imagine another scenario, again with a G-level formal game which involves shuffling items around, and with a second C-level of linguistic tokens keyed to the availability of moves at the G-level.

The school-room addition sums game This time, take the G-level to be decimal arithmetic, as taught to children as a package of routines. So at step one they write down, as it might be, '74′ above '46′, draw a horizontal line underneath, and then — following what they are taught as an uninterpreted syntactic game — they 'add the units', carry one, 'add the tens' and write down '120′.

After being taught these symbol-shuffling routines (so far empty of significance), the children are then taught a new step, and learn to write down '74 + 46 = 120′ after their addition routine is executed. And then thirdly, as before ,the game is expanded so that they are allowed to write down such a token even if they haven't done the routine, so long as they could respond to a challenge by 'doing the sum'.

Here, unlike the abacus cases where we had beads-on-a-wire at one level and symbols to play with at another, the same token '74′ can appear at the G-level as what an uninterpreted symbol-shuffling game operates on, and at the C-level as part of a language game keyed to the G-level game. In the first case it is empty of significance, in the second case part of a move with content. For as before, the thought goes, (i) there are correctness conditions for the issuing of 'equation'-like tokens at the C-level, (ii) these correctness conditions are to be given in terms of the availability of moves in a formal symbol-shuffling game, but (iii) it would again be over-interpreting to suppose that the players, in issuing such an 'equation'-like token, mean that the correctness conditions obtain. So the content of the token '74 + 46 = 120′, such as it is, falls short of the explicit thought that you can get from '74′ and '46′ to '120′ by legitimate moves in the adding game. Still, even if '74 + 46 = 120′ doesn't explicitly represent a fact about the formal game, its correctness conditions might be said to be, in an obvious sense, formal.

Now, a smallish tweak takes us to …

The DA game Again, we are dealing at the G-level with decimal arithmetic again, but we this time imagine decimal arithmetic presented as a formal quantifier-free system of equations, with axioms and rules of infererence, giving a formal theory which we'll call DA (on the model of our old friend PA). Again we go through three steps. First we imagine the neophyte learning to play with DA — and being taught by the 'direct method' to recognize legimate DA manipulations by pattern-recognition and training, not by explicit instruction that mentions 'axioms' and 'rules' etc. This time the items being shuffled in the formal game are equation-like, but as yet — at the G-level — they have no content, any more than the arrangement of beads in the abacus game.

At the next step, however, the game is expanded: a player is taught to enter one of those equation-like tokens on a ledger if they are produced at the end of a DA game. Then, thirdly, the practice is expanded to allow a player to write down such a token even if they haven't done the DA 'derivation' routine, so long as they could respond to a challenge by 'doing the proof'.

So now the very same kind of equation-like token '74 + 46 = 120′ gets into the story twice over. Firstly, a token can appear in the G-level DA symbol-shuffling game, which is again as empty of content as bead-arrangements in the abacus game. And second, a token can appear again at the C-level in a kind-of-commentary on the DA game, making a move in a language game which has the correctness condition that you can derive that sort of token inside DA.

As before, then, at the C-level, (i) there are correctness conditions for the issuing of equation-like tokens, (ii) these correctness conditions are given in terms of the availability of moves in a formal game, but (iii) it would be over-interpreting to suppose that the players, in issuing such an equation-like token mean that the correctness conditions obtain.

Neo-formalism Now here, at last, comes the punch. The neo-formalist claims that the content of our arithmetical claims is, or is like, that of the tokens at the C-level cued to the DA game. So: unlike the classic formalist who avers that arithmetic is an empty game with signs, the neo-formalist allows that arithmetical claims do have content. But he's a formalist because the correctness conditions for such claims are given in formal terms, in terms of moves in a formal game. However, the claim goes, the content is not as rich as the thought that the correctness conditions obtain. So it would be wrong to say, as a cruder formalist might, that an arithmetical claim is about the formal game facts about which supply the correctness conditions.

That, then, is the basic story about arithmetic in introductory form. And similarly, it is hoped, for other claims in other areas of mathematics. But does this sort of account work? Watch this space!