TTP

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.

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 is 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 (one of these or another)? 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 real 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 begins to 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; claims, 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. ‘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 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 want to construe them 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 give 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 do already have reason to think of agents as having attitudes pro and con various actions, and as having desires 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 with 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 still beliefs, albeit foggily representational states, that are being expressed. (And what alternative, non-belief, state that we already have reason to discern would their tokenings be keyed to?)

How 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, then. Outright non-representational claims that p in other model 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?

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 in almost anybody’s book) 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 …

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). Note, however, it isn’t enough just to have ended up with the appropriate configuration of beads: for a legitimate entry in the ledger, the configuration must have been achieved by correct play. A child’s entry on the ledger can be challenged by getting them to repeat the shuffling and showing a misplay, and a challenge met by a correct re-play.

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!

Weir, to summarize once more, wants to develop a position that allows him to say sincerely, speaking with the vulgar mathematicians (and not having to cross his fingers behind his back, or do that little dance with the fingers that signals scare-quotes, or do some radical reconstrual of what they “really” mean), “there are infinitely many prime numbers”, even though while amongst the learned, or at least amongst the metaphysicians, he consistently asserts “THERE ARE no numbers”. He hopes to have softened us up for the idea that there is room for such a have-your-cake-and-eat-it position by considering (i) how projectivism (supposedly) allows us to agree sincerely with the vulgar that “X is G” (for certain G) while also agreeing with metaphysicians who say “THERE IS no such property as being G”, and considering (ii) how a certain line on fiction (supposedly) allows us to agree with the vulgar reader of the stories that “Sherlock lived in London” while agreeing with metaphysicians who say “Sherlock never EXISTED“. Not that Weir want to be a projectivist or a fictionalist about maths: but the idea is that the prima facie tenability of those accounts elsewhere indicates that there is perhaps room for a similar ontological anti-realism about mathematics, one which rests on the key idea that in making mathematical assertions (as when making fictional assertions or “projective” assertions) we are playing a different game from when we are in the business of representing the world.

But Weir, as he now emphasizes again, wants more. He wants to combine ontological anti-realism about mathematical entities with “metaphysical realism” in the Putnamian sense of allowing for the possibility of evidence-transcendent truth in maths. Of course, this isn’t exactly a novel combination. The modal structuralist is similarly concerned to eliminate commitment to a distinctive ontology of mathematical abstracta, which he does by translating away mathematical claims into modal quantified truths, and he can allow that it is evidence-transcendent what the modal truths are. However, unlike the modal structuralist, Weir wants to take mathematical talk at face value (he doesn’t want to go in for telling mathematicians what they “really” mean by translating away their ostensible commitments). So he wants a brand of ontological anti-realism for mathematics akin to projectivism or his sort of account of fictional discourse — we again aren’t in the representational business — while allowing evidence-transcendent truth.

But it can’t be said that we’ve been softened up for that combination. Certainly, it is difficult to see how there could be e.g. evidence-transcendent truths about what is tasty! Maybe a projectivism about probability could be developed in such a way as to allow for evidence-transcendent truths in this case: but Weir doesn’t say anything about such a case — and, in sum, I think we get no illumination on the ontological-anti-realism/metaphysical-realism combo from anything he says about projectivism (have I missed something?). However, Weir does think his account of fiction gives us something to go on:

There is no incoherence in holding to this anti-realism [about fiction] while viewing truth in general as evidence-transcendent — perhaps even fictional truth, if the fact that S follows in the right way from the text, and thus is true, can be evidence-transcendent.

But what does Weir have in mind here?

Earlier, he talked about S following in the right way — “flowing from” the text — if “experienced readers would, on reflective consideration, judge [that S] must form part of the story if it is to make overall sense.” But that notion of flowing from the text, where what flows depends on our best judgements, would hardly make room for evidence transcendence! But perhaps the idea is that things may follow logically or indeed mathematically from the text, but in an evidence-transcending way. Thus suppose “2 is the least number such that P” is an evidence-transcendent mathematical truth. Then I guess we have “The number of Dmitri Karamazov’s half-brothers is at least as large as the least number such that P” as a truth about the fiction which would be evidence transcendent. But then the evidence transcendence of the fictional truth would be dependent on the evidence transcendence of the mathematical truth (and so we couldn’t use the possibility of former fact as illustrating how the latter could be possible). Well, maybe there are other cases we could think about here: but that’s enough to suggest that Weir’s one-sentence jab at persuading us that his story about fiction gives us a useful illustration of the desired ontological-anti-realism/metaphysical-realism combo is just too quick.

But let that pass. We now have some sense of where Weir wants to end up about mathematics: ontological anti-realism without radically reconstruing maths (we continue to take it “at face value”), to be achieved by seeing assertion in maths as playing a different role to representational assertion, BUT also “metaphysical realism”, in the sense of allowing for evidence-transcendent truth. The work of spelling out his attempted “neo-formalist” articulation of such a position starts in the next chapter.

The projectivist about e.g. judgements of tastiness explains how “X is tasty” (as an ordinary judgement made in the restaurant, not the philosophy class) is an assertion that can be correct or incorrect even though there is no such property-out-there as tastiness, so the assertion isn’t representationally-true (or correspondence-true, if your prefer). Or so the story goes.

The theory about fiction that Weir sketches explains how “Sherlock Holmes lived in Baker Street” (as an ordinary judgement made in discussing the stories, not the in history class) is an assertion that can be correct or incorrect even though there is no such person-out-there as Sherlock (or Meinongian substitute), so again the assertion isn’t representationally-true. Or so the story goes.

The projectivist line about tastiness or goodness or beauty, the theory about fiction, allow us to speak with the vulgar but think with the learned (assuming the learned have a naturalistic bent). We can legitimately talk as if there is a kosher property of tastiness or as if there are fictional beings such as Sherlock, while not being really ontologically committed to such things. If we use small caps to signal when we are making assertions in full-on, stick-by-it-even-in-the-metaphysics-classroom, genuine-representation mode, then we can say (ordinary conversation) “Marmite is tasty” even though (when in Sunday metaphysical mode) we can agree “Marmite IS NOT TASTY“; and likewise we can say (conversationally) “Holmes lived in London” while (on Sundays) agreeing that “Holmes NEVER EXISTED“. Hence the projectivist story and the story about fiction allow us to eliminate some of our ostensible ONTOLOGICAL commitments in talking with the vulgar (Weir calls this “ontological reductionism”, but I’ve grumbled before about that label). So the story goes.

What does Weir add to the story in this section, to further set the scene before trying to paint a comparable picture of mathematics as non-representational? (1) Some remarks about what makes for the difference between a representational mode of assertion and a non-representational one. (2) Some remarks about why the difference between “Holmes existed” and “Holmes EXISTED” shouldn’t be confused with a lexical or structural ambiguity. (3) Some remarks about what a projectivist should say about the likes of “If sentient beings had never existed, there would still have been beautiful sunsets”.

Concerning (1), I’d have thought the way to go is to illustrate the kind of basic semantic story that applies to canonical examples of “representational” discourse, and then say that non-representational discourse is whatever needs some different kind of semantic story (I’m not saying that’s easy to do! — but Weir’s p. 59 seems to go off in a slightly skew direction.)

Concerning (2), I agree. I’m not sure it is helpful then to go on to talk about ‘metaphysical ambiguity’ (but maybe that’s just complaining about Weir’s taste in labels again).

Concerning (3), Weir discerns a wrinkle, but also thinks that it doesn’t carry over his promised non-projectivist but analogously anti-realistic account of mathematics. So we needn’t pause over this.

The projectivist’s root idea is that a judgement that “X is G”, for a predicate G apt for projectivist treatment, is keyed not to a belief that represents X as having a special property but to an appropriate non-cognitive attitude to X. But what does being “keyed” to an attitude amount to? Well, for a start, there should be a basic preparedness to affirm X is G when one has the right attitude. But, as we noted, the projectivist wants to put clear water between his position and that of the crude (strawman?) expressivist for whom the judgement is no more than a “snapshot” evocation of the speaker’s current attitudes. So the projectivist will want to complicate the story to allow what Weir calls “correctional practices”, where snap judgments are allowed to be corrected in the light of thoughts about the judgements of others and oneself at other times, thoughts about how attitudes might be improved, etc.

Weir is pretty unspecific about how the story about correctional practices is to work out in detail, even in the case of “tasty”, which is rather oddly his favourite replacement for “G”. Maybe his reticence about the details is not so surprising given his choice of example: for I rather doubt that there are enough by way of correctional practices canonically associated talking about what’s “tasty” to makes ideas of “correct judgement” robustly applicable here. But still, I’m willing to go along with Weir’s general hope that there are might be other cases where projectivism works, and so (i) can illustrate how anchorage in “snapshot-plus-correctional” practices can be meaning-constituting for “X is G”, (ii) without giving the judgement realist truth-conditions, while (iii) imposing enough discipline to make it appropriate to talk of such a judgement being correct/true (at least in a thin enough, non-correspondence sense).

As I said before, I doubt that Weir’s discussions will do enough to really help out those philosophers of maths to whom the idea of projectivism is (relatively) new. But in this section he goes on to offer another purported illustration of how we might get a (i)/(ii)/(iii) story to fly, this time in a context which will probably be a lot more familiar to logicians, i.e. the treatment of discourse about fiction.

Thus consider Weir’s example ‘Dimitry Karamazov has at least two half-brothers’ in the context of discussion of Dostoyevsky’s book. He suggests (as a first shot at describing the relevant “snapshot dispositions”)

It is constitutive of grasp of ‘Dimitry Karamazov has at least two half-brothers’, in the context of discussion of a given English translation of The Brothers Karamazov, that one sincerely assent (if only ‘privately’) to the sentence iff one believes that the sentence ‘flows from’ the translated text.

Here ‘flows from’ is to be elucidated in turn roughly (again, as a first shot) along the lines of “what experienced readers would, on reflective consideration, judge must form part of the story if it is to make overall sense”, and this gives us a role for “correctional practices”.

I’m not sure why Weir relativizes to a particular translation, which seems unnecessary; but let that pass. And “must form part [sic] of the story” must mean something like “must belong to any sensible/natural filling out of the story text”, which raises more problems which we’ll let pass too. But the root idea, at any rate, is that (i) the sketched “snapshot-plus-correctional” story means that that (ii) when we say ‘Dimitry Karamazov has at least two half-brothers’ we are not representing D.K. or expressing truths about the real world (not even truths about what is written in a certain book), nor indeed expressing truths about some other world (whatever that quite means) but are going in for a different kind of speech-performance, as it were a going-along-with a bit of story-telling. But the framework in which we do this is not subject merely to our creative whim (after all we are not Dostoevsky, who is more entitled to carry on just as he wants!) but is constrained enough for us to be able to talk of (iii) correct and incorrect ways of going along with the story-telling.

I don’t myself have decided views about discourse about fiction, and don’t know whether this line is a “best buy” (indeed Weir himself raises some issues). But it does serve, I think, to give us a case where it seems that the (i)/(ii)/(iii) schema can be filled out in a prima facie plausible way, without tangling with the special problems of projectivisms. So that’s a plus point. The attending minus point, I suppose, is that the more you like this account of the semantics of discourse about fiction, the more tempted you might be to recycle it to serve the ends of a fictionalist account of mathematics. So why does Weir after all prefer “neo-formalism” to a brand of fictionalism? We’ll have to see …!

Suppose we want to claim that some class of sentences that are grammatically like those of straightforwardly fact-stating, representational, belief-expressing discourse actually  have a quite different semantic function (and remember, this is going to be Weir’s line about mathematical sentences: where a fictionalist error-theorist sees a failed representation, or a kinder fictionalist sees a pseudo-representation made in a fictional mode, Weir is going to argue that mathematics isn’t in the business of representation at all). How then might we further explicate this idea of superficially representational claims which in fact have a different role?

One context in which such an idea has been developed and put to work is in the neo-Humean “projectivist” account of morals, modals, and the like, as nowadays particularly associated with Simon Blackburn. The root idea is that a judgement like ‘X is good’ doesn’t express a belief about how the world is with respect to some special property of goodness, but rather a sincere such judgement is keyed to the utterer’s attitude of approval of X. NB, it isn’t that the judgement is about the attitude; rather that it is semantically appropriate, other things being equal, to assertorically utter the judgment when you have the right attitude. Likewise, ‘E is highly probable’ doesn’t express a belief about the occurrence in the world of a special property of objective chance, but rather a sincere judgement is keyed to the utterer’s having a high degree of belief in the occurrence of event E. And so it goes.

But of course, the devil is in the details! The root idea here is equally available to the crudest expressivist: the hard work for the Blackburnian projectivist comes in explaining (a) why, despite the anchoring of the judgements in non-cognitive attitudes, it is still appropriate that they have the logical “look and feel” of cognitive judgements — i.e. can be negated, embedded in conditionals, and the like — and there’s related work to be done in explaining (b) why it makes sense to reflect “In my view, X is good, but I could be wrong” and the like. What distinguishes the projectivist from the crude expressivist is the sophisticated way in which he tries to explain (a) and (b).

Weir’s §2.I touches on the projectivist’s treatment of these matters  — but in a way that I expect is going to be far too quick for those philosophers of mathematics (surely most of them!) who aren’t already familiar with a particular strand of contemporary debate that’s mostly conducted remote from home, in meta-ethics. In particular, Weir’s constrast between earlier and later Blackburn, and the role of the idea of non-correspondence truth in his later work, will probably mystify (well, I can’t say I found it at all clear or helpful, and I start probably knowing a bit more than many logicians about these things, having Blackburn as a colleague!).

And as well as the discussion going too quickly, Weir’s discussion of projectivism is oddly framed. The full title of the section is “Projectivism in the SCW framework”, and you’ll recall that in his §1.III, the so-called sense/circumstances/world picture is exemplified in the treatment of demonstratives and the story about how the situation represented by an utterance involving “that” is co-determined by the literal meaning (or sense) of the utterance and the relevant circumstances of utterance which make a particular thing appropriately salient. But that was a story about context sensitivity in fixing what state of affairs was being represented (it is still good old-fashioned representation that is going on). The new issues raised by projectivist stories about non-representational content seem, then, to be quite orthogonal to the issues about how we need to tweak Fregean semantics to cope with demonstratives.

OK: we have a story about what is happening in the use of sentences with demonstratives and another story about sentences with “good” or “probable” (or whatever else invites projectivist treatment), and in each case the story deploys concepts (salience, pro-attitudes, degrees of belief) which are not part of the thought expressed in the circumstances. But there the similarity surely ends. Needing circumstances to help fix what is being represented is one thing; going in for non-representational thinking is surely something else, about which we need a quite different sort of story than is provided within the confines of the SCW framework as introduced in §1.III.

Still, let’s agree that Weir’s (over?) brisk remarks serve to point up that there is possibly space for, though also problems attending, treatments of areas of statement-making discourse as non-representational. And that’s perhaps all we really need for now, given that Weir has already announced in his Introduction, p. 7, that he doesn’t want to offer a projectivist account of mathematical discourse. So let’s not get unnecessarily bogged down in worries about how best to develop projectivisms.

Though let me end this instalment with a very small protest about calling projectivism a species of reductionism (rarely a helpful label, of course). Projectivism “populates the world … with certain naturalistically unproblematic attitudes or relations between humans and objects”, and in so doing does away with the need to postulate problematic properties of goodness, chance or whatever. So, to be sure, projectivism reduces ontology — but, if we want one word to describe what is happening, we are eliminating the need for the supposedly troublesome non-natural properties.

So at long last, it’s back to discussing Alan Weir’s Truth Through Proof (henceforth, TTP). And apologies to Alan, and anyone else, who has been eagerly waiting for further  instalments.

Let’s quickly, in this post, review where we’ve got to (cutting-and-pasting a few snippets from previous posts which you’ve now forgotten!). In his short Introduction, Weir sketches out the ground he wants to occupy. He wants to say that, as a mathematical claim, it is true that are an infinite number of primes. And this common-or-garden mathematical truth isn’t to be reconstrued in some fictionalist, structuralist or other way. However, he wants to say, a mathematical claim is one thing, and a claim about how things are in the world is another thing. Speaking mathematically, there are an infinite number of primes; but there is also a good sense in which THERE ARE NO primes at all.

How is the gap here to be opened up? Not by construing talk of what really EXISTS as a special level of ontological talk, distinct from other talk that aims to represent the world. Rather, the small caps just signal that straightforwardly representational discourse is in play, and the key idea is going to be that mathematical discourse (like e.g. moral discourse) plays a non-representational role — if you like, mathematics makes moves in a different language game.

If Weir is going to be able to develop this line, we’ll need to hear more in general about styles of discourse, representational vs non-representational. It’s the business, inter alia, of Chap. 2 to provide some of this background. And in the next posts I’ll start discussing this chapter. But some semantic groundwork, and some terminology, has already been provided in Chap. 1.

Suppose we aim for a systematic story about how sentences of a certain class get to convey the messages that they do: take, for example, sentences involving a demonstrative ‘that’. The systematic story will, perhaps, use a notion like salience, so for example the story tells us that ‘that man is clever’ expresses a message which is true when the most salient man in the context is clever. Now, for this to be part of a semantic theory that is suitably explanantory of speech-behaviour, speakers will have to reveal appropriate sensitivity to what we theorists would call considerations of salience. But note: those we are interpreting needn’t themselves have the concept of salience. So the explanatory account given in our theoretical story doesn’t supply a synonym for ‘that man is clever’: we need to distinguish the literal content of the demonstrative sentence as speakers understand it (what is shared by literal translation, for example) from what we might call the explanatory conditions as delivered by our systematic semantic theory.

It is a familiar and not-too-contentious point that such a distinction needs to be made, and made not just in the case of the semantics of demonstratives. Somewhat unhappily, I’d say, Weir has chosen different terminology to mark it: in particular, he talks not of ‘explanatory (truth)-conditions’ — which indeed was his initially preferred term — but of metaphysical content. And he says “metaphysical content specifies what makes true and makes false a sentence in a circumstance”. This talk of truth-making might suggest that the business of metaphysical content is to specify truth-makers in the sense favoured by some metaphysicians. But not so! Weir in fact is quite sceptical about truth-makers, so understood. Hence we mustn’t read more into Weir’s terminology than he really intends to put into it: to repeat, so-called metaphysical content is just a specification of the situation where an utterance of the sentence in question would be correct or appropriate or disquotationally-true.

The thought is going to be then that, when it comes to giving the ‘metaphysical content’ of mathematical claims, the story about what makes a mathematical sentence correct or appropriate or disquotationally-true doesn’t mention mathematical entities of a platonist kind but runs on quite different lines. But how? Back in his Introduction, Weir says “The mode of assertion of [mathematical claims] … is formal, not representational”. And what does this mean? Well, part of the story is hinted at by the claim that the formal, inside-mathematics, assertion that there are infinitely many primes is rendered correct by “the existence of proofs of strings which express the infinitude of the primes”. Hence Weir’s “neo-formalism”. Our task is going to be that of making sense of this surprising claim, and evaluating it.

Now read on …