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?
Perhaps I was right to avoid colour, whether my reasons were good ones or not! It’s too complicated a topic and I haven’t thought about it enough myself, though I’d like to work on this more and move beyond a hazy recollection of Hardin and a few bits and bobs of more recent philosophical stuff. I did enjoy the PASSV article, albeit I could do with thinking on it a bit more. (It was nice, also, to be reminded of Greg McCulloch your respondent and also of course sad to think he has been lost to us for some years now.)
One more general point you make, though, I think we agree on, and that is on the importance of the Fregean sense/reference distinction, applied, as Frege applied it, to predicates as well as to singular terms. If I have made some sort of conflation there, I’ll be aghast. I agree with you that two predicate terms can have the same referent, but different senses. Can express different concepts, as we would say, but stand for the same property, as ‘Hesperus’ and ‘Phosphorus’ can stand for the same object but have different senses. Alas Frege, (before he made the sense/reference distinction I think) crazily and tragically (from the perspective of a realist about properties) chose ‘concept’- ‘begriff’ for the referent, not the sense! Aside from Frege studies, I think we should avoid that dreadful choice entirely. And of course his referents aren’t properties as most of us understand them, but the mysterious ‘unsaturated’ functions with truth values for outputs.
Certainly then a realist about properties should distinguish the sense or concept a predicate expresses from its referent (if it has one, on an austere view of properties, not all do, comprehension for predicates is false) and the referent, the property, in turn from its extension, the set of instances of the property. Where does that get us? Are there examples of non-synonymous predicates which are not just co-extensive or even necessarily co-extensive but pick out the same property? (When you are clear that properties aren’t senses/concepts, are on the other side of the world/language-mind divide, there’s no reason to think that necessary co-extensiveness entails identity. That’s an unFregean line of course.)
Perhaps a dispositional term like ‘soluble’ might provide an example. Although I don’t, the chemist might grasp a complex chemical predicate which is not just co-extensive with, but picks out the same property as, ‘is soluble’. (Though as I understand it, it will be messy, dissolving in water working in a rather different way for electrolytes than from non-electrolytes and so forth). By the way, when you talk of ‘dispositional properties’ do you think dispositionality pertains to the property, rather than to (some) expressions which pick it out (but not others)? In the PASSV article it looks as if not to the property, the talk there is of dispositional specifications and concepts. As is perhaps evident, I agree. I think properties are no more dispositional, or negative, or conjunctive than people are, (I can uniquely specify a given person by a complex predicate with conjunction the main operator, likewise disjunction and so on). These notions of dispositional/categorical or conjunctive, disjunctive, positive, negative, apply to expressions and concepts, not the referents.
And dispositionality is another reason why colour introduces a complexity which is perhaps not relevant here. I agree ‘red’ is a disposition term, even though like many it lacks an ‘ble’ ending. But I’d be more inclined to go with your formulation II on p. 248 and say that the disposition term it links to is something like ‘disposed to transfer radiation conforming to such and such a pattern’ (despite the ‘unperspicuous and apparently arbitrary mess’ you note in specifying the latter). The occurrent property of transferring light in this way is not one which involves a relation to the neural responses of sentient creatures as in your preferred formulation (III). Perhaps then sound provides a less tangled example for our purposes in separating out irrelevant dispositions. I think ‘sounding middle C’ is an occurrent property which does not involve a relation to typical neural responses. Presumably you’d take the same sort of tack as with colour, and think that it does? But for me the Fregean point may well apply. There are predicates of acoustic science, some to do with frequency profiles vis a vis amplitudes, others to do with amplitudes across time, each of which picks out the same property as the much less theoretical ‘sounding middle C’.
On pp. 254ff. you consider a distinction between intrinsic and non-intrinsic accounts of colour, I assume the same would be true of sound, and develop this in terms of ‘rigidified’ versus non-rigidified dispositions. You argue that it doesn’t matter much philosophically which way we go here but that if we ‘rigidify’ then intrinsic accounts and those which relate to the neural response come to the same thing. So on the intrinsic account, blue is the property which causes the specific effects on us given the neural systems we actually have. (It seems unlikely the non-rigid disposition picks out any actual property at all.) Here I accept the Fregean point: one and the same objective property- the property which Weir, as actually constituted, likes in a drink, say- can be picked out by distinct terms with distinct ‘modes of presentation’, one via chemistry say, the other via subjective matters- (the ‘Weir likes’ term co-referential, it might cruelly be said, with a rather simple chemical property, ‘containing lots of ethanol’). This term ‘x has the property which Weir likes in a drink’, though, does not have the same referent according to me as ‘is a good drink’ even on my lips- for the latter picks out no property at all on the projectivist view, the terms work semantically in very different ways. I say something about why the latter isn’t representational, even though there will be objective properties which ground it, at pp. 58-60. At any rate, if you think ‘blue’ or ‘manifesting blue’ picks out, albeit with a different sense, the actual property in common to all objects which transfer (emit, reflect, diffuse etc.) light waves in certain complex patterns then we are very close on colour.
Even if one were to plump for a non-intrinsic analysis of ‘manifesting blue’ or ‘playing a loud C major chord’ as involving a relation to observers that still might, I agree, count as ‘thumpingly realist’. Humans (and other animals) have properties and stand in relations. As you say, being poisonous-for-rats is an objective property or feature of some substances. But a term which picks out a particular neural response, (whatever its sense) at the sensory input stage does not pick out redness or even ‘experiencing redness’ (what goes on further back in the optic nerve and brain matters). Whilst an account which takes ‘red’ (and any co-referential term whatever its sense ) to pick out some intrinsic mental state, a functionalist one perhaps on a physicalist reading, a sense datum quality on a non-physicalist one, that is clearly a subjectivist account.
The psychology of colour, sound and the like is so Heath-Robinson-esque (the systems evolved blindly after all, they weren’t designed!) that I think this tempts people away from realism about secondary qualities (as well as a strangely persisting tendency to be suckered by the same fallacies- of division, composition etc- that Locke was prey to). The concepts don’t cut nature up at the joints and all that. Certainly on an austere account of properties, which I would favour, that a complex heterogeneous ‘unperspicuous’ disjunction is true of some things does not mean it picks out a property. But it does not mean it does not: any property, in fact, will be picked out by arbitrarily many formulae, including long, mangled ones. What is at work in some varieties of anti-realism about colour is perhaps an extreme physicalism: a predicate which is complex in the language of physics/chemistry can’t refer to a property. Why ever think that? All the physicalist has to say is that every object, property and relation can be picked out by a term (perhaps too long for us to digest) of physics, or physics and chemistry or some such.
To be sure, there are reasons to think ‘green’, ‘blue’ and so on do not pick out properties: the grouping of shades in simple colour terms is language-relative, varying even across European languages (e.g. English versus Gaelic). But there is reason to think ‘that shade on that part of the tomato’ picks out a property: I see the property. (But herein lies another complication, the relation between perception and thought, particularly intrinsically linguistic thought. This probably makes comparing our accounts even harder as I’m a naïve realist (if it’s naïve, I’m going to like it- witness naïve set theory!) and so think that when I see the red tomato, the tomato and the physical property which is that colour shade are literal constituents of my mind.) The property of that patch on the tomato almost certainly isn’t a spectral property to be sure, it’s more like a chord in which the notes really are physically separate. And you looking at the self-same bit of the tomato might well see a different, but equally objective, property, just as you might hear harmonics in a note I don’t hear. Objects have more than one property at a given time, and they can even have more than one colour at a given time even when to each individual they look to have a single colour (I think Mark Kalderon has a view something like this).
What’s all this got to do, though, with the anti-platonism I’m defending about maths, simple arithmetic being in focus for the moment? You worried whether I have good grounds for saying that arithmetical sentences express beliefs, in a relatively minimal sense, but beliefs which are made true or false by the existence of concrete proofs, disproofs, calculations and the like, whilst at the same time denying that these beliefs are representational beliefs, beliefs* about those concrete truth-makers.
A very pertinent question, as the passage you quote admits. What is common ground, I think, underneath the complex issues involving colour etc. is firstly that we agree that two co-referential predicates can have different senses. I also agree, of course, that we will have incomplete knowledge of properties as of objects, and that this varies among humans; our ideas are rarely ‘adequate’ to move from Leibniz to a more Cartesian terminology, if I remember it right. I refer to the same object and property as the scientists using ‘the moon’ and ‘has spin one-half’ but the scientists know vastly more truths about that object and that property than me.
Where does a problem for me arise in this? There seem two lines of thought here. The first is the worry that my account of arithmetic will make it ‘thinly’ representational, not a non-representational discourse, as my anti-platonism requires. I don’t see this one, it certainly doesn’t follow just because the metatheoretical account is couched in more theoretical language than that of the target discourse. Matters specific to semantics and psychology aside, it needn’t even be so. Suppose I’m trying to give a metatheoretic account of our grasp of the relatively low-grade, non-theoretical ‘soluble’ which tries to offer some explanation of what it is about us which constitutes such grasp and what makes our utterances involving the term true or false. A very difficult task indeed. Will it fail utterly, if I know nothing of the chemistry of solubility, if I don’t know some scientifically deep identity of the form ‘solubility = the chemical property C’? I don’t see why that must be the case. Even if so, it might well be that though there is no limit to the amount of chemical facts we could find out about solubility after a while advances in this knowledge aren’t very important in semantic accounts. Conversely (and I think we are agreed on this) if I do use much richer, scientifically speaking, concepts in the metatheory in accounting for our grasp of some relatively observational term (like ‘soluble’), this does not mean that I’m proffering an anti-realist account of the target object language term. But it also doesn’t mean I’m *not* giving a reductionist account in the arithmetic case, we need to look at the account.
I agree that even on my account, if the arithmetic sentences have different senses, express different thoughts, from those giving their metaphysical contents, that on its own does not make them representational nor (more importantly still for me, reductionist). But I think the account is clearly not representational (numerals don’t work by picking out objects etc.) and is clearly reductionist (arithmetic truths are made true by concrete proof structures, not abstract structures).
It is true that if I am after a reductionist, in my sense, account of ‘solubility’ or ‘sounding the note middle C’ it will not suffice to state the account in a metatheory which entails no existential consequences involving those expressions (I could do that by moving to French in the metalanguage) or even an account which involves no synonymous expression. If my metatheory entails that the property P exists, and the expression P picks out the same property as ‘sounding middle C’ then I’ve failed in anti-realist reduction. But my goal, at any rate, is to have no commitment to abstract objects of any type, arithmetical, syntactical, in the metatheory, regardless of the terms which pick them out, or their senses. And I can’t see how these considerations- the sense/reference distinction, the possibility of acquiring ever deeper knowledge of the nature of some object or property- cause problems for me in achieving that goal.
A second worry I think is on these lines. Projectivism ties belief in the projective utterances to non-doxastic attitudes and that gives us a handle on how it can be reductionist. But if it is beliefs* which we tie our response to arithmetical sentences to, why am I not committed to an implausible realist representational theory of arithmetic, which makes it a body of representations about concrete proofs and calculations?
Well go back to the fiction case. I want to say what makes ‘Sherlock uses cocaine’ true, ‘Sherlock smokes dope’ false (if that’s right- I’m not a reader of the books!) are facts about Conan Doyle’s text, including facts about ink marks in books, or other concrete realisations. But these concrete things don’t magically make the sentences true and false on my lips, there have to be other facts about language use, and about my language use in particular and indeed my beliefs*, if I actually am to connect with Conan Doyle’s text in uttering ‘Sherlock’. What makes the concrete marks relevant to the truth of my assertions about fiction (and not the beliefs I happen to have about the texts) is that it is what I, or my linguistic community, would TRULY believe* about the texts which matters and this tracks the actual corpus (via ‘correctional dispositions’) even if I in actuality do not. If a ready implication of one text, not contradicted anywhere else, is that Sherlock has property P- wasn’t in London at the time of the murder say- but that gets missed and the view that he was takes root now and forever more, that belief about Sherlock is false. It’s the text which makes it false, and it is the content *we* grasp which is made false because, at least in the simpler cases, we could have come to believe*- indeed know*- what the text actually said and implied.
But this essential role of representational beliefs about texts in the account of the metaphysical content of fictional utterances does not imply that ‘Sherlock uses cocaine’ means ‘reflective readers of the texts would add that sentence to a body of sentences filling out the stories in a coherent way’. Just as well, as it seems obvious they mean something different. You might say, ‘yes that only shows why you shouldn’t be committed to that view but it doesn’t show how your theory can avoid that implication’. Well, I avoid it because I explicitly situated the theory in a view of language which holds that sense is usually augmented by something else (context in simple cases like demonstratives) before generating a truth-apt proposition. And that gives me space to hold that the explanatory correctness conditions, the metaphysical content, can differ from the sense (and not just by ‘saturating’ indexical parameters or the like, witness projectivism). Which means it is consistent for me to hold that the metaphysical content, though involving or having connections with representational beliefs, is not the informational content of the fiction statement. And that is further reinforced when one looks at intentional contexts. ‘Sherlock would have smoked dope if he’d been a late 20th century detective’ does not mean “reflective readers filling out the stories but situating them later, would have added ‘Holmes is a dope head’ ” to the body of assertions about him’. And then when one looks at the metaphysical contents one sees they are not only different as senses from the senses of the target sentences, but they don’t carry commitment to detectives with the properites Conan Doyle gives Holmes, it’s reductionist.
Turn then to arithmetic and to the most favourable case for an anti-platonist specifically a ‘nominalist’ in the modern sense, one who rejects abstract objects (including not just formalists but fictionalists, at any rate those who think 2+2=5 ‘incorrect’ but ‘2+2=4’ correct). These is the case of utterances for which there are concrete feasible proofs/disproofs lying around (even if I haven’t come across them) whether in books, in the (not too lengthy) output of some automated theorem-prover or whatever. I agree with you entirely that we anti-platonists are under an obligation to explain how our theories are meant to account for *arithmetic*, the discourse which we all got introduced to at school and professional mathematicians take to dazzling, for most of us incomprehensible, lengths and depths; and not something else. So we need an account of the sense of the arithmetical claims which i) it is plausible we creatures could grasp; ii) more than that, is plausible as an account of the senses we actually grasp and iii) an account of the meaning as a whole which gets the truth or correctness conditions right not only in the sense of putting the ‘true’ ticks against the theorems, false crosses against the anti-theorems but in not making these truth values depend on anything abstract.
I agree that it is crucial for the anti-platonist to face the challenge to meet all three criteria, and that it is not easy to do so even in this simplest case. My claim that I did so in the book I’d defend on similar lines to the fiction example above. What, on my proposal, makes the arithmetical claims true or false are concrete proofs, disproofs, calculations … so (iii) is met. In these simple cases, it is plausible that we do have the ability to make our yeas or nays to arithmetical claims (fallibly) sensitive to the posited truth-making conditions, the proofs or disproofs. Not even the hard-core platonist will deny that so (i) is met. (If I can’t grasp even the simplest proofs of some rather tricky theorem, then I don’t have full understanding, but reliable tendencies to defer to the right folk might entitle me to partial understanding).
What gets my yeas and nays linked to the concrete proofs and disproofs are dispositions (those dispositions again) which may not be actualised, dispositions to say yes when confronted with a proof, no with a disproof. This will, in anyone moderately reflective, involve him or her having representational beliefs* about concrete proofs and disproofs. That doesn’t right away require equating his or her belief in the theorem with such a representational belief* (even a world-expert on salience doesn’t mean ‘the salient animal is a dog’ when uttering ‘that animal is a dog’). And if I had thus equated the belief that there are infinitely many primes with the belief* that concrete tokens of proofs of a string expressing that in a suitable system exist then I’d have gotten (ii) wrong and not have an account of arithmetic. That I’d get (ii) wrong is a) intuitively obvious and b) evident from intensional contexts: ‘necessarily there are infinitely many primes’ does not mean “necessarily there exist proofs establishing ‘there are infinitely many primes’ ”. So that’s why I didn’t thus equate them! Coherently and plausibly, in the context of the overall framework of sense/circumstance and world in which I was working. But maybe the most important thing is not whether arithmetic is representational or not on my account- I think it isn’t but I readily agreed the right account is not projectivist- but whether it is reductionist in my sense. That is, whether it, if successful, allows us to say consistently in the metaphysics room that there are no numbers, no abstract objects, because that’s not what makes ‘there are infinitely many primes’ true.
Overall, I deny my theory is committed to the false conclusion that arithmetic sentences express propositions about concrete proofs and the like. That’s a virtue, not a vice of my theory. And the (fairly orthodox) position on Sense versus Circumstance/Context versus ‘World’- what makes things true- which I adopt makes it relatively easy to show, or anyway sketch, how those categories of proposition- arithmetic on the one hand, ‘concretist proof theory’ on the other, have different senses yet how the truth-makers for the former can be given by the latter. You might say, well you’ve created a framework in which informational content and metaphysical content can come apart in such a way that arithmetic beliefs aren’t representational beliefs* about concrete proofs. But you haven’t filled in enough detail about these types of content to establish that they really do come apart, if your account is right. Well I admitted in the book I don’t have a comprehensive account of metaphysical content even less of informational content- pp. 26-7 fn 30. Considerations of space forbid etc. Ok we all know such pleas mean: ‘I haven’t actually figured one out in detail yet’! But even if I had a fully comprehensive theory of language and mind which fleshed out the different types of content as much as one would wish, if I’d put it in the book it would never have been reviewed for it would never have been published, it would have been too big. Moreover I don’t think any other anti-platonists have shown at all how to meet the challenges of i), ii) and iii) simultaneously, even sketchily and even in the simplest cases of ‘feasible’ arithmetic and the like.
Sorry, went on at some length (secondary qualities is a pretty absorbing topic)! But I’ll forbear commenting on TTP11 as penance!)
i) A minor matter: you say fictional claims are ‘not keyed to a common-or-garden belief but to something else, a pretence to be representing’. That’s not my position; in the very rough sketch of a view of fiction, e.g. on p.54, I contrast my account of fictional truth with pretence theories such as Walton and Currie’s. However I do allow (footnote 24) that one might be able to ally the two ideas. I’m not committed though to there being any pretence involved either in reading fiction or in critically discussing it.
ii) ‘Which model, if any, is the appropriate one for arithmetic’. None is the answer, none of the models I give in chapter two, nor your colour one, fit the mathematical case. The idea is that the mathematical case is a distinctive subspecies of a more general species many varieties of which I hope will already be found plausible, softening up, as you say, those philosophers who do so find them, for the mathematical case.
iii) I don’t think I’m committed to anything like arithmetic claims being ‘confused’ or ‘foggy’ representations, for the reasons given in my response to TTP 12.
iv) Modality, and my liking for practical possibility. Yes, indeed, I don’t find anything problematic about claims like: ‘I could write out a full proof of this theorem of propositional calculus inside of about 20 minutes if I could be bothered’. And I don’t of course want to say only these theses which have been proved are true. I want to steer clear completely of modal realism, though. So even in the practical possibility case, rather than thinking of there being a ‘possible proof’, I prefer to think in terms of actual structures, which could be dimples in a computer disk or strings of polymers for example, structures which could be, or could have been, (in an everyday sense not involving superhero powers), interpreted, parsed (though not usually in one intellectual glance) as proofs (pp. 168-170).
v) The worry that for me arithmetical claims are keyed to beliefs, not non-beliefs (compare the non-doxastic attitudes of the projectivist): the worry I take it is: aren’t these then beliefs which foggily represent the formal ‘game’?
Well firstly, not foggily, see above (iii). Moreover for me arithmetic assertions, also projective ones, can, like other assertions be used to represent beliefs contra:
it’s just that content of the beliefs is non-representational. (One could stipulate that ‘beliefs’ have to have representational content but then disagreement threatens to be merely terminological.)
Thirdly, what makes the arithmetical claims true is not a belief but a token proof, whether of ink marks, a diagram, a series of moves with the abacus or whatever. Of course we have to relate such things to the competent speaker’s practice if they are to ‘get into’ the metaphysical content of that speaker’s language. I talk of a crude grasp being given by assenting to an utterance whenever a comprehensible concrete proof is ‘manifest to the speaker’ (p. 74) and this manifesting need not involve belief.
However a fuller grasp, I go on to say 74-5 (using brackets for subscripting here):
So the metaphysical content of the arithmetical assertion is being linked to speaker’s beliefs (as also in the fiction case — there beliefs about what is said in novels etc.). This, though, is part of the ‘snapshot’ dispositions which form part of understanding. For there to be objectivity, the speaker must have ‘correctional’ dispositions keyed to proof tokens not beliefs. Still there is a linkage to fairly representational notions of belief in provability. I go on in p.75 to add
My answer, pp. 75-6, (including fn 11 p. 76) is of the type you allude to in TTP 11 ‘To use a favourite style of argument of Weir’s’. The transparent informational content of ‘7+5=12’ contains nothing to do with proofs, concrete proofs in particular. Why: a) (most brutally) that seems intuitively right, b) ‘Hamish said that 7+5 = 12’ is not synonymous with “Hamish said that Hamish [or Weir] believes ‘7+5=12’ is provable in decimal arithmetic”.
So with respect to your final
I’d say that grasp of such claims involves ‘sensitivity’ to concrete proofs, e.g. assenting when presented with such. This sensitivity needn’t involve beliefs. And it is the concrete proofs which make mathematical claims true for me, these are the non-beliefs to which truth is keyed. Such grasp, to be sure, also will involve beliefs: if I believe that p though unaware of a proof it maybe because I believe it provable, though I might have a rather unsophisticated notion of ‘provable’. But the latter belief is not part of the content of p. (Of course rampant platonists might believe p and p is unprovable’, I say a little about this case e.g. at p.107.).
I don’t pretend this position is obviously right, of course, even for what might seem the easiest case for the anti-platonist, simple, decidable (better still ‘short’, ‘feasible’, perhaps ‘polynomially-bounded-derivable’) maths such as (short, feasible ) decimal arithmetic. The platonist will agree that truth in these cases is extensionally equivalent to provability and will, surely, if reasonable, allow that appropriate abilities to spot proofs are part of grasp. The emphasis on this as part of grasp, at least for arithmetic of the above limited sort, is surely not an implausible aspect of neoformalism.
But the platonist will quite rightly ask after a more general account of meaning which can explain how such abilities could constitute all of grasp, even in those cases, and how, even if so, truth is given by proof when it seems clearly not to be part of the meaning of such claims. I’ve tried to answer this by applying a fairly widespread picture about how propositions/truth-conditions are determined (usually) not solely by sense alone and arguing this idea can play out in a number of different ways. So the anti-platonist, I agree, should not complacently think all is easy in these simple foothills of mathematics. And that’s before we get on to what I think are the really difficult cases for anti-platonists, utterances which lack any feasible proof or disproof, then beyond that negation-incomplete theories.
Thanks again for the further comments!
i) Oops, I didn’t meant to foist a particular view about fiction onto you: I just needed the observation that, whatever we say about the claim “Sherlock Holmes lived in Baker Street”, its utterance by those chatting “within the fiction” doesn’t go with a straight historical belief about the denizens of that street.
ii) Yep. My worry though, in headline form, is this. The “softening up” examples seem to fall into two classes. (a) Demonstratives/colours where there is representation, but not fully transparent representation (to use that terminology) and (b) projectivist/fiction where some non-representational performance is in question. I thought that your “formal” mode of assertion — by your own lights — was supposed not to be an (a)-case; but to make it plausibly a (b) case, we need a story about what mental states are behind it which aren’t kosher doxastic states. But in the abacus model, for example, there seems no need to postulate such states. But if “formal” assertion is neither an (a) case nor a (b) case, just saying it is a “distinctive subspecies” doesn’t help me!
iii) See my comment on your replies to TTP 11!
iv) I’m not quite sure what the claim is here. But the question really is: are you happy to allow practical possibilities to feature in your story of explanatory truth conditions at the metaphysical level? It seems so.
v) Two points are perhaps enough for now. (a) About beliefs. Sure, we can use the idea in a wide, all-inclusive way: so where we assert (broad inclusive sense) p there we express the belief (broad inclusive sense) that p. So, on this usage there are aesthetic beliefs, fictional beliefs, arithmetical beliefs, whatever — all quite uncontroversially. Even the projectivist can say, in this sense, that e.g. moral judgements express moral beliefs. But of course she, in her explanatory story about correctness conditions, must also want to say that such judgements express attitudes* rather than beliefs* (where now the starred notions carry a bit of theoretical weight in the explanatory theory, and belong to a philosophy of mind that discriminates beliefs* as a subclass of beliefs-in-the-broad-sense as the states that play a certain role in carrying information rather than in setting behavioural goals, etc).
Now, since you are minded to use projectivism (about taste, if not morals) as an illustrative case of how sense can peel apart from explanatory truth-conditions, you seem to be committed to the thought that there is a discrimination to be made, and to allow that some assertions do not express beliefs*. I was going along with that notion of belief*. So when I meant in the quoted passage from me is that it needs argument to show that, despite such appearances, the children’s tokenings in fact aren’t expressions of belief*. (It doesn’t need argument, I agree, that they can be expressions of belief in the undiscriminating sense).
(b) Thanks for highlighting the quoted passage from your pp. 74–75 (and I feel embarrassed not to have picked up on this as I should have done). But it makes my case for me! Now, it seems, you agree that in the DA game, and no doubt the abacus game too, the players comments will be keyed to common-or-garden belief states (indeed belief* states — why not?). So, as you say “why not say that the informational content … is representational”?
But your answer is what the colour example (in my view) shows is a bad argument. To believe the tomato is red is, we agree, not to believe that it is disposed to produce a certain response in normal perceivers etc. etc. But it doesn’t follow that “the tomato is red” isn’t a representational belief. Think Frege: the senses red and disposed to blah blah blah are different modes of representation of the same physical property. Or if you don’t like that example there will be others. It’s just the point that senses cut finer than what they represent.