Truth Through Proof, 0. Preamble

I am eventually going to be writing a (short) review for  Mind of Alan Weir’s new book  Truth Through Proof: A Formalist Foundation for Mathematics (OUP, 2010). The blurb on the publisher’s website gives  you an idea what of what the book is about. The clue is in the subtitle — but note, this is a philosophy book, not a technical book in foundational studies.

To help fix my ideas,  I’ll be posting a (much longer) series of discussion notes here, as I sporadically work through the book. Given other commitments, however, I’ll have to take things pretty slowly over the coming weeks.

As with similar series of postings on other books, I expect that these notes will weave around and about Weir’s book (henceforth  TTP) in a more free-ranging way than would be appropriate in a review: but I’ll try to make it clear when I’m summarizing TTP, when I’m directly commenting on Weir’s views, and when I am striking out more on my own account. And more generally, I’ll try to keep things accessible to students, even if that sometimes means including more background explanations than some other readers here will feel they need.  All comments as we go along will be very gratefully received!

After the Introduction, Weir’s chapters are divided into sections numbered off with roman numerals: so here ‘3.III’ means Chapter 3, §III. Double quotation marks are reserved to signal quotations from TTP (and otherwise unattributed page numbers of course refer to the book too).

OK. With that by way of brisk preamble, let’s dive in … tomorrow!

TTP, 1. Introduction: Platonism vs ‘naturalized epistemology’

Let me begin by setting the scene, embroidering only a little on Weir’s opening pages.

Consider then the following claims, ordinarily regarded as mathematical truths:

  1. 3 is prime.
  2. The Klein four-group is the smallest non-cyclic group.
  3. There is an uncountably infinite set of nested subsets of Q, the set of rationals.

On the surface, (1) looks structurally very similar to ‘Alan is clever’. The latter is surely about some entity, namely Alan, and is true because that entity has the property attributed. Likewise, we might initially be pretty tempted to say, (1) also is about something, namely the number three, and is true because  that thing has the property attributed. Similarly (2) is about something else, the Klein four-group. And (3) is about the set of rationals, and claims there is a further thing, an uncountably infinite family of sets of rationals nested inside each other.

So what  kind of things are three, the Klein group, the set of rationals? Not things we can see or kick, but non-concrete things, surely — i.e. abstract objects. And given the standard view that (1), (2) and (3) don’t just happen to be true but are necessarily true, it would seem that these abstract objects must be necessary existents.

But how can we possibly know about such things? Once upon a time (Weir might have reminded us), the thought seemed attractive that we are made in God’s image, and — albeit to a limited extent — partake in his rational nature (for Spinoza, indeed, ‘the human mind is part of the infinite intellect of God’). And God, the story went, can just rationally see all the truths of mathematics: sharing something of his nature, in a small way we can come to do that too. Thus Salviato, speaking for Galileo in his Dialogue, says that in grasping some parts of arithmetic and geometry, the human intellect ‘equals the divine in objective certainty, for here it succeeds in understanding necessity’. And Leibniz writes `minds are … the closest likenesses of the first Being, for they distinctly perceive necessary truths’. (For more on this theme, see Ch. 1 of Edward Craig’s wonderful The Mind of God and the Works of Man (OUP, 1987), from which the quotations are taken.)

However, this conception of ourselves as approximating to the God-like has lost its grip on us. So, getting back to Weir, given the sort of beings we now think we are, with the sorts of limited and ramshackle cognitive powers with which evolution has provided us to enable us to survive in our small corner of the universe, the question becomes pressing: how come that we can possibly get to know anything about supposedly necessarily existent abstract entities (entities in Plato’s heaven, as they say)? How are we supposed to cognitively ‘lock on’ to such things? Indeed, we might wonder, how do we ever manage even to frame concepts of things apparently so remote from quotidian experience?

Now, note that to find this sort of question pressing it isn’t that we already have to bought into the idea that epistemology should be ‘naturalized’ in some strong sense, or that a Quinean ‘naturalized epistemology’ exhausts the legitimate parts of what used to be epistemology. And though Weir does talk about mathematical platonism being “put to the test by naturalized epistemology”, he officially means no more than that our conception of ourselves as natural agents without God-like powers “imposes a non-trivial test of internal stability” (p. 3) when combined with views like platonism. The problem-setting issue, then, is an entirely familiar one: as Benacerraf frames it in his classic paper, ‘a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have’.

So far, then, so good. We have a familiar but still pressing question, and in the next post, I’ll say something about various lines we might take in response and indicate how, in rest of his Introduction, Weir situates on the map the position he wants to defend.

But first just a word or two more about Weir’s enthusiasm for naturalism. He goes on to write “I accept, as a general methodological maxim, the prescription that one should push a naturalistic approach as far as it will go.” (p. 5) And what does that involve? “The methodological naturalist … prescribes that one ought to follow scientific method, at a level of sophistication appropriate to the problem at hand, whenever attempting to find out the truth about anything.” Really? If using the methods of science is construed more narrowly as involving the careful rational weighing of evidence to test specific, antecedently formulated, empirical conjectures, etc., then of course this isn’t the only way to discover truths. Evolution has thankfully provided us with other quick-and-dirty ways of fast-tracking to the truth reliably enough to avoid fleet-footed predators often enough! While if the methods of science are understood in a more relaxed and embracing way, as whatever goes into the mix as we develop our best overall theory of nature, then (a familiar old-Quinean point) these methods would seem to subsume the methods of (much) mathematics which seem so entangled, and ‘methodological naturalism’ in itself has no special bite again platonism (over and above Benacerraf’s problem, which doesn’t depend on the naturalism).

But we really don’t want to start off on that debate again, about the appropriate formulation of a possibly-defensible ‘naturalism’. So let’s re-emphasize the key point that I think Weir would make, which his rhetoric hereabouts could possibly obscure: we don’t need to endorse any strong form of naturalism, or have any commitment to naturalized epistemology as the one legitimate residuary legatee of the epistemological tradition, to find troubling the combination of platonism with our conception of ourselves as limited creatures epistemically geared to the sublunary world. That‘s enough of a problem to get Weir’s project going.

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

Faced with the Benacerrafian challenge, what are the options? Weir mentions a few; but he doesn’t give anything like a systematic map of the various possible ways forward. It might be helpful if I do something to fill the gap.

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

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

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

Eliminative and modal structuralists will also answer ‘yes’ to (1) and ‘no’ to (2), this time construing the mathematical claims as quantified conditional claims about non-mathematical things (schematically: anything, or anything in any possible world, that satisfies certain structural conditions will satisfy some other conditions). It is actually none too clear how structuralism helps us epistemologically, and when given a modal twist it’s not clear either how it helps us ontologically. But that’s quite another story.

Suppose, however, we answer ‘yes’ to (1) and (2). Then we are committed to saying there are prime numbers and there are non-cyclic groups, etc. (for it is true that 3 is prime, and — construed as surface form suggests — that implies there are prime numbers). Next question:

  1. Is there a distinction to be drawn between saying there are prime numbers (as an unqualified truth of mathematics, construed at face value) and saying THERE ARE prime numbers? – where ‘THERE ARE’ indicates a metaphysically committing existence-claim, one which aims to represent how things stand with ‘denizens of the mind-independent, discourse-independent world’ (following Weir in borrowing Terence Horgan’s words and Putnam’s typographical device)

According to one central tradition, there is no such distinction to be drawn: thus Quine on the univocality of ‘exists’.

The Wright/Hale brand of neo-Fregean logicism likewise rejects the alleged distinction. Their opponents are puzzled by the Wright/Hale argument for platonism on the cheap. For the idea is that, once we answer (1) and (2) positively (and just a little more), i.e. once we agree that ‘3 is prime’ is true and that ‘3’ walks, swims and quacks like a singular term, then we are committed to ‘3’ being a successfully referring expression, and so committed to its referent, which (on modest and plausible further assumptions) has to be an abstract object; so there indeed exists a first odd prime which is an abstract object. Opponents think this is too quick as an argument for full-blooded platonism because they think there is a gap to negotiate between the likes of ‘there exists a first odd prime number’ and ‘THERE EXISTS a first odd prime number’. Drawing on early Dummettian themes (which have Fregean and Wittgensteinian roots), the neo-logicist platonist denies there is a gap to be bridged.

Much recent metaphysics, however, sides with Wright and Hale’s opponents (wrong-headedly maybe, but that’s where the troops are marching). Thus Ted Sider can write ‘There is a growing consensus that taking ontology seriously requires making some sort of distinction between ordinary and ontological understandings of existential claims’ (that’s from his paper ‘Against Parthood’). From this perspective, the claim would be that we must indeed distinguish granting the unqualified truth of mathematics, construed at face value, from being committed to a full-blooded PLATONISM which makes genuinely ontological claims. It is one thing to claim that prime numbers exists, speaking with the mathematicians, and another thing to claim that THEY EXIST ‘in the fundamental sense’ (as Sider likes to say) when speaking with the ontologists.

Now, we can think of Sider et al. as mounting an attack from the right wing on the Quine/neo-Fregean rejection of a special kind of philosophical discourse about what exists: the troops are mustered under the banner ‘bring back old-style metaphysics!’ (Sider: ‘I think that fundamental ontology is what ontologists have been after all along’). But there is a line of attack from the left wing too. Consider, for example, Simon Blackburn’s quasi-realism about morals, modalities, laws and chances. Blackburn is no friend of heavy-duty metaphysics. But the thought is that certain kinds of discourse aren’t representational but serve quite different purposes, e.g. to project our moral attitudes or subjective degrees on belief onto the world (and a story is then told about why a discourse apt for doing that should to a large extent retain the same logical shape of representational discourse). So, speaking inside our moral discourse, there indeed are virtues (courage is one): but as far as the make-up of the world on which we are projecting our attitudes goes, virtues do not EXIST out there. From the left, then, it might be suggested that perhaps mathematics is like morals, at least in this respect: talking inside mathematical discourse, we can truly say e.g. that there are infinitely many primes; but mathematical discourse is not representational, and as far as the make-up of the world goes – and here we are switching to representational discourse – THERE ARE NO prime numbers.

To put it crudely, then, we can discern two routes to distinguishing ‘there are prime numbers’ as a mathematical claim and ‘THERE ARE prime numbers’ as a claim about what there really is. From the right, we drive a wedge by treating ‘THERE ARE’ as special, to be glossed ‘there are in the fundamental, ontological, sense’ (whatever that exactly is). From the left, we drive a wedge by treating mathematical discourse as special, as not in the ordinary business of making claims purporting to represent what there is.

And now we’ve joined up with Weir’s discussion. He answers ‘yes’ to all three of our questions. A fourth then remains outstanding:

  1. Given there is a distinction between saying that there are prime numbers and saying THERE ARE prime numbers, is the latter stronger claim also true?

If you say ‘yes’ to that, then you are buying into a version of platonism that does indeed look epistemically particularly troubling (in a worse shape, at any rate, than for the gap-denying neo-logicist position; for what can get us over the claimed gap between the ordinary mathematical claim and the ontologically committing claim)? Weir thinks this position is hopeless. Hence he answers ‘no’ to (4). Hence he endorses claims like this: There are infinitely many primes but THERE ARE no prime numbers. (p. 8 )

But this isn’t because he is, as it were, coming from the right, deploying a special ‘ontological understanding of existence claims’. Rather, he is coming more from the Blackburnian left: his ‘THERE ARE’ is ordinary existence talk in ordinary representational discourse, and the claim is that ‘there are infinitely many primes’, as a mathematical claim, belongs to a different kind of discourse.

OK, what kind of discourse is that? “The mode of assertion of such judgements, I will say, is formal, not representational”. And what does ‘formal’ mean here? Well, part of the story is hinted at by the claim that the formal, inside-mathematics, assertion that there are infinitely many primes is made true by “the existence of proofs of strings which express the infinitude of the primes” (p. 7). Of course, that raises at least as many questions as it answers. There are hints in the rest of the Introduction about how this initially somewhat startling claim is to be rounded out and defended in the rest of the book. But they are much too quick to be usefully commented on here; so I think it will be better to say no more here but take them up as the full story unfolds.

Still, we now have an initial specification of Weir’s location in the space of possible positions. His line is going to be that, as a mathematical claim, it is true that are an infinite number of primes: and this truth isn’t to be secured by reconstruing the claim in some fictionalist, structuralist or other way. But a mathematical claim is one thing, and a representational claim about how things are in the world is another thing. And the gap is to be opened up, not by inflating talk of what EXISTS into a special kind of ontological talk, but by seeing mathematical discourse (like moral discourse) as playing a non-representational role (or dare I say: as making moves in a different language game?). That much indeed sounds not unattractive. The question is going to be whether the nature of this non-representational game can be illuminatingly glossed in formalist terms.

TTP, 3. §§1.I–II: Realisms

As we can see from our initial specification of his position, to get Weir’s philosophy of mathematics to fly will involve accepting some substantial and potentially controversial claims in the philosophy of language and metaphysics. The first two chapters of TTP fill in some of the needed background. Weir starts by talking a bit about realism(s). Given that, in the Introduction at p. 6, he has already characterized himself as aiming for “an anti-realist … reading of mathematics”, we should get clear about what kind of realism he is anti.

However, I didn’t find the ensuing discussion altogether clear (is it perhaps extracted from something longer?). So in what follows, I’m reconstructing a little, but hopefully in a broadly sympathetic way, for I do at least want to end up pretty much where Weir does.

Traditional realisms, he says, “affirm the mind-independent existence of some sort of entity”. But what does ‘mind-independent’ mean here? The problems are immediate. For a start, which kinds of minds count? On the one hand, if it’s just finite sublunary minds, then Berkeley comes out a realist, which isn’t what we want (Weir himself contrasts realism with idealism). On the other hand — Weir might have noted — if we agree with Berkeley and count God as among the minds, then any traditional theist who believes that the physical world is dependent for its existence on God would ipso facto count as an non-realist about sticks and stones, which is also surely not what we want. Then there are other problems with the traditional formulation: on a crude reading, it seems to define away the very possibility of being a realist about minds.

Let’s put those worries on hold just for a moment, and turn to consider the modern theme that realism should instead best be characterized in epistemic terms. Thus Dummett (quoted by Weir): ‘Realism I characterise as the belief that statements … possess an objective truth value, independently of our means of knowing it.’ Of course, others such as Devitt have emphatically insisted contra Dummett that realism about Xs, properly understood, is an ontological doctrine about what there is, and is not to be confused with any epistemic or semantic doctrine. Where does Weir stand on this?

Well, he spends some time discussing the idea that realism is a species of fallibilism. We can present this sort of realism about a region of discourse R schematically as saying

For every (or some?) R-sentence s, it is possible (what kind of possibility?) for speakers (which speakers? even in optimal conditions?) to believe s though it is not true, or disbelieve s though it not false.

There are four dimensions along which versions of realism-as-fallibilism can vary, corresponding to the four queries. And Weir doesn’t hold out much hope that there is any way of setting the variables to give us a thesis which is substantive enough to be interesting but also sufficiently captures what a realist is after. He offers a number of considerations. Embroidering a bit, we could perhaps put one of them like this. Suppose we keep fixed our view about the nature of Xs but change our mind about the quality of our epistemic access to Xs. Suppose we become very optimistic — perhaps implausibly over-optimistic — and now think that, at least when we are optimally placed and exercising our cognitive faculties in the optimal way, there are (enough) claims about Xs which (in the relevant sense of ‘possible’) it is not possible for us to go wrong about. Then, the thought goes, surely changing our view like this about our fallibility with respect to claims about Xs doesn’t in itself entail changing our view as to whether Xs are really there, independently of us, etc. Coming to think we are more or less infallible about Xs can involve wildly upgrading our estimate of our epistemic powers, rather than downgrading our realism about Xs. So we shouldn’t tie realism to fallibilism too tightly.

Weir’s arguments here do go pretty quickly (too quickly to be likely to sway a Dummett or a Putnam, for example); but I won’t pause over the details as in fact I rather agree with his interim conclusion:

I find myself in sympathy with Devitt in wishing to return to a traditional ‘ontological’ characterization of realism as mind-independent existence. (p. 22)

Or at least, I agree that realism about Xs should be construed as an ontological claim, not an epistemic or semantic claim. But Weir’s version takes us back to those puzzles about how best to spell out ‘mind-independent’. And here, it seems to me, he takes a wrong turn. For having just explained why he thinks that realism-as-fallibilism won’t do, he now suggests that we can “effect a compromise” and proposes

a Devitt-style ‘ontological’ characterization of realism with respect to a given set of entities as constituted by a belief in their mind-independent existence, where mind-independence is, in turn, chararacterized in fallibilist terms à la Putnam and Dummett.

But will this do, even by Weir’s own lights? Isn’t this compromise package vulnerable to (some of) the same objections as pure realism-as-fallibilism? In particular, doesn’t it again implausibly imply that inflating our estimate of ourselves and supposing we have the relevant kind of infallibility with regard to claims about Xs would entail thereby rejecting realism about Xs themselves?

I’m not sure how Weir would respond to that jab, nor how he would fix those variables left dangling in a schematic statement of mind-independence as fallibilism. Instead he goes off on another — and more promising — tack, noting that

Someone who holds to evidence-transcendent truth and affirms that Xs exist should not count as a realist about Xs if the affirmation of the existence of Xs, though sincere, should not be taken at face value or else should not be read in a straight representational fashion.

That’s surely right: to be a realist about Xs involves affirming the existence of X without crossing your fingers as you say it, or proposing to ‘decode’ such an affirmation as in some way not being about what it at surface level seems to be about (or treating it as not in the business of representing how things are at all). Thus, to take Weir’s example, the modal structuralist might take at least some arithmetical claims to be true in an evidence-transcending way: but that hardly makes her a realist about numbers if she parses the claims — including apparently existence-affirming claims like ‘there is a prime number between 25 and 30’ — as really claims about what happens in concretely realized structures across possible worlds. Or to go back to Berkeley, the good bishop might allow some claims about the physical world to true independently of our human ability to discover them to be so, but that hardly makes him a realist about physical things, given the decoding he offers for such claims when thinking with the learned.

OK, suppose we say — taking the core of Weir’s line — that you are a realist about Xs if you affirm that there are Xs, where that is to be taken in a “straight representational fashion” and is to be “taken at face value” (not reconstrued, or decoded). You can immediately see why, quite trivially, Weir’s philosophy of mathematics will count for him as anti-realist, given that he has announced that on his view mathematical talk is non-representational. But of course, all the work remains to be done in explaining what it is to mean something as representational and intend it to be taken at face value.

Though here’s a concluding thought. We might suggest that it is a condition of talk of Xs being apt to be taken “at face value” that it involves continuing to respect enough everyday platitudes about the kind of things Xs are. And in some case — e.g. where X’s are everyday things like sticks and stones — those platitudes will involve ideas of ‘mind- independence’ (the sticks and stones are the sort of thing that will still be there even if no one is seeing them, thinking of them, etc.). So taking talk of sticks and stones at face value will involve taking it as respecting the ‘mind-independence’ of such things. Which suggest that perhaps that realism about X’s (meaning just representational face-value affirmation of the existence of Xs) will already bring with it as much ‘mind-independence’ as is appropriate to Xs — more or less independence , varying with the Xs in question. If that’s right, we needn’t build mind-independence into the general characterization of realism: it will just fall out for  realism about Xs if and when appropriate.

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

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

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

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

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

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

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

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

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

TTP, 5. New readers start here …

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 …

TTP, 6. §2.I Projectivism

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.

TTP, 7. §2.II Snapshot dispositions, correction, fiction

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 …!

TTP, 8. §2.III Reduction

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.

TTP, 9. §2.IV A map of the terrain

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.

Scroll to Top