# 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.

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.

This entry was posted in Phil. of maths, Truth Through Proof. Bookmark the permalink.

### 10 Responses to TTP, 3. §§1.I–II: Realisms

1. Alan Weir says:

First many thanks to Peter for taking the time and trouble to go through my book in such a detailed way. It’s very gratifying (and of course very daunting too) to have such attention paid to one’s work, in particular to be given the opportunity to respond, try to correct misunderstandings and answer, if I can, objections.

Now on the treatment of realism(s) in §§I.I-1.II, one thing I was trying to do was to mark out the corner of conceptual space I want to occupy and defend; this is an anti-realist one, even though I consider myself to be something of a hard line realist, for example in philosophy of science. The solution was to distinguish two different broad types of realism, ‘metaphysical’- the epistemic/fallibilist construal of realism- and ‘ontological’, where the realist makes existential claims. As I note, (pp. 24 fn 26, 69 fn. 39) this is a widely accepted distinction.

The resolution then is that whilst I think that the truth about many questions outruns the ability of humans reliably to determine that truth (even in non-trivially specifiable optimal circumstances)- so I’m a metaphysical realist- and even though I’m an ontological realist in lots of areas, e.g. with respect to muons or black holes, I reject ontological realism in mathematics. I deny that mathematical entities exist (but with a complication you note). As well as trying to make my own position conceptually clear, I also have some strategic argumentative goals here. I note (p. 67, chapter two- the stage setting work spreads over two chapters) that my ontological anti-realism in mathematics is compatible, as far as I can see, with metaphysical anti-realism; but if I can make plausible the claim that it is also compatible with (a fairly strong brand of) metaphysical realism, which happens to be my own view, all the better for persuasive purposes, I haven’t presupposed at the outset the falsity of lots of metaphysical positions.

Where you say, ‘Weir’s arguments here do go pretty quickly (too quickly to be likely to sway a Dummett or a Putnam, for example)’ do you mean to sway them on the right way to characterize realism? Or to persuade them to abandon their verificationist (metaphysical) anti-realism? I’d have a bit more hope in the former case than the latter where I’m pretty sure they’d remain unswerved, as I think Quine would put it. I disavowed any claim to produce a compelling argument against such verificationism but I suppose I couldn’t resist having a bit of a pop at it by highlighting some well-known difficulties. But the main thrust §1.II.3 is to argue that the Dummett/Putnam fallibilist framework doesn’t capture one prominent type of realism.

You aren’t convinced, moreover, by my attempt to go for a ‘compromise’ combination of metaphysical Putnam/Dummett conceptions and ontological, Devitt-style ones. My reasoning was that, though I’m more interested in the latter, nonetheless merely making sincere existence claims: ‘there were three witches who appeared before Macbeth’ isn’t enough to count as an ontological realist. Some sort of ‘mind-independence’ clause seems needed, as Devitt recognizes (p. 23) and the fallibilist idea does seem to me to capture the idea that our minds and the universe aren’t so snugly fitted to one another that we can, operating at our best limits, figure everything out.

Your concluding suggestion is that the notion of mind-independence might be built into avowals that there are Xs for certain types of X, because the platitudes we accept about the sort of things Xs are require them to be mind-independent- stones are still there when we don’t look at them or think of them. My worry here would be that this is too weak a notion of mind-independence, that Berkeley would come out as realist on this account (if we restrict to human minds, cf. p. 12).

My idea was that if someone says that in general the truths about what things there are outruns our ability to reliably detect such truth (perhaps even to conceptualize it, as Nagel suggests) then that is enough to credit them with having a mind-independent notion of existence, so that when they go on to say Xs exist, this can be interpreted as a mind-independent existence claim even if they think that there is no reasonable chance of us being wrong about the existence of Xs- stones say; or more generally if they think that at the optimal limit we are guaranteed to get things right about Xs, even if not for Ys.

Of course that’s just a ‘can be interpreted’ for as you note, I actually do affirm existence claims in mathematics- that there are eight complex roots of x^8 = 256 and so on. That’s the qualification needed above. At the same time I also say there aren’t any complex numbers (or any other sort of mathematical entity), and I’m not a dialetheist nor do I think there is any ambiguity in these cases in ‘exist’. I try to have my cake and eat it here by distinguishing representational existence claims- ‘THERE ARE NO complex numbers’ from non-representational, non-mind-independent existence claims, of which ‘there are eight complex roots of x^8 = 256’ is one. Your ‘initial specification’ in ‘Truth through Proof’ post January 1st (posting on January 1st!!) captures my position very well. The final section of chapter one and a good bit of chapter two involve dipping far enough into philosophy of language to try to make the representational/non-representational distinction plausible.

2. Rowsety Moid says:

I think Berkeley *should* come out as a realist. His idealism is in contrast to materialism and to (substance) dualism, not to realism.

• Alan Weir says:

Rowsety: certainly a Berkeleyan can be a metaphysical realist in the sense of holding that the truth value of propositions about the external world is independent of finite, human minds (and this might be cashed out in fallibilist ways).
But is Berkeley an *ontological* realist about material objects? Certainly he thought he was with the common man against Lockean scientific realists, whose views he thought threatened atheism. It seems to me, though, that a contemporary Berkeleyan attempting a metatheoretic account of everyday empirical talk, attempting to give a systematic account of what conditions render ‘the table in the empty room is brown’ true, should specify the truth conditions in terms of ideas, or bundles of ideas, not tables. If they aren’t prepared to go beyond disquotational conditions in this direction, how can they be Berkeleyans? And that, I would say, is a form of ontological reductionism, a reduction of the physical to ideas. Idealism in other words, an idealism which contrasts with ontological realism about the physical world.

• Rowsety Moid says:

But to Berkeley, tables *are* ideas or bundles of ideas. The account of truth conditions would involve tables; it just wouldn’t say tables were material — and surely it’s not necessary to buy into materialism or physicalism in order to be a realist. After all, Berkeley can’t really be refuted by kicking a rock. Rocks remain as toe-crushing as ever; it’s just that people are mistaken if they think that means rocks must be material.

Also, Berkeley’s idealism is nicely in tune with the epistemic / fallibilist version of realism, in a way. Tables are … whatever they are, regardless of whether we can discover their true nature. So perhaps we’re wrong about tables, and perhaps one of the ways we’ve been wrong is in thinking they’re material (in thinking that anything is material).

• Alan Weir says:

True Berkeley does think tables are ‘bundles’ of ideas (including of course the archetypal ideas in God’s mind). So maybe you are right, we can see him as a metaphysical realist and an ontological realist about everyday things just one who espouses a sort of identity theory of tables and bundles or collections of ideas. Similarly both the dualist and the materialist are ontological realists about minds (leaving aside the tricky questions about how realism applies to the mental), it’s just that one identifies them with disembodied souls the other with brains.

I’m uneasy about that, though, since the materialist thesis might not be read as a semantic one whereas the Berkeleyan one might. Or rather, if I can leave Berkeley to the scholars for now, there is a sort of textbook phenomenalism which ‘analyses’ everyday talk in terms of sets of sense data. We have an object language in which we talk of tables and trees, sense data and sets. But we explain what makes true or false the sentences of this language in a metalanguage which only talks of sets and sense data. That, in the framework of my second chapter, is anti-realism about tables, it’s an ontological reductionism reducing tables to sense data (and sets) even though the account renders true object language sentences of the form ‘there are tables’.

Similarly someone who claims (most implausibly) that a correct metatheoretic account of mathematicians’ talk of numbers, functions, groups, categories and sets should be given in a metatheory which speaks solely of sets is, according to me, an ontological realist about sets, but not about numbers, functions etc. Although the theory might explain the truth of ‘there are eight complex eighth-roots of 256’ it’s not thereby a realism about complex numbers, rather it reduces them to sets.

This is crucial to my position of course. Though I too say there are complex numbers, I attempt to give an account of what makes such claims true in a metatheory which talks only of concrete proof tokens, and that I think is anti-realist, it’s saying that in mind-independent reality there are concrete proof tokens, but no numbers, functions, sets etc. as conceived of by the platonist.

To be sure, these anti-realisms add a negative component. The phenomenalist says there are tables *but* there aren’t really any in mind-independent reality; and then I say that murky idea should be explained in terms of reductionism as above. I say there are numbers, *but* not in mind-independent reality. So going back to Berkeley there may be a disanalogy in that he doesn’t, it might be said, want to say that although there are tables there aren’t any. But he does deny there are tables in a Lockean sense, and if you thought that this actually captures part of our ordinary conception you might take that to be a denial of mind-independent existence. On the other hand the role of the divine archetypes means you can say he thinks tables exist independently of human minds. Maybe Berkeley can’t be determinately fitted into my scheme, it would be anachronistic to do so, but phenomenalists, ‘textbook’ ones anyway, perhaps fit it better.

3. Peter Smith says:

1. Yes, I’d picked up that two brands of realism are in play, traditional vs modern, ontological vs epistemic/fallibilist. Alan now says that “the main thrust of §1.II.3 is to argue that the Dummett/Putnam fallibilist framework doesn’t capture one prominent type of realism” (my emphasis). But in TTP, he spends some pages taking potshots at the very idea that a strong form of epistemic/fallibilist anti-realism (which Alan pointedly calls “verificationism”) is going to be plausible. That suggested to me the stronger claim — which I’m sympathetic to — that the Dummett/Putnam approach doesn’t successfully capture any position that could reasonably (with even half an eye to tradition) be called realism. But ok, let’s take it that Alan doesn’t need the stronger claim.

(By the way, am I the only person who finds that it still really grates to call a version of realism defined in epistemic terms metaphysical? I guess we can blame Putnam for the terminological mishap here. My discomfort with it is why I didn’t follow Alan in using “metaphysical” this way.)

2. Alan still says that a Devitt-style, ontological, realism about Xs needs to fleshed out with some story about ‘mind-independence’. His suggested reason now is “that merely making sincere existence claims: ‘there were three witches who appeared before Macbeth’ isn’t enough to count as an ontological realist”. Well yes, of course it isn’t. The claim needs not merely to be sincere, but made representationally, not in the scope of a suppressed ‘In the fiction’ operator, not otherwise to be parsed away, etc. etc. But by my lights, it isn’t issues about fallibilism that come in to explaining what is involved in discourse in a fiction’, or Blackburnian projective discourse — or (I bet!) Alan’s mathematical discourse in the formal mode. We’ll see in the next chapters whether Alan really thinks differently.

3. “The fallibilist idea does seem to me to capture the idea that our minds and the universe aren’t so snugly fitted to one another that we can, operating at our best limits, figure everything out.” Absolutely. Typically, ontological realism about Xs + a becoming modest about cognitive grip on the universe –> fallibilism about Xs. Dropping the modesty may indeed mean dropping the fallibilism. But that doesn’t ipso facto cancel the realism about Xs. That’s why I said realism about Xs and fallibilism about Xs can peel apart.

4. But actually Alan agrees with that (and this is what I wasn’t sufficiently clear about before — and I leave it to others to judge whether that was me being rather dim, or Alan being less that ideally clear!). His idea, he now explains, is that “if someone says that in general the truths about what things there are outruns our ability to reliably detect such truth … then that is enough to credit them with having a mind-independent notion of existence” and then, given they have this general “mind-independent notion of existence”, they might credit Xs with existence in this sense too even though they think we happen to be pretty infallible about them.

Two points about this. (A) On the face of it, everyone sane — including, mostly, Dummett and Putnam! — will agree that there are at least some (disquotational) truths about what things there are which outrun our ability to reliably detect them. It’s just part of our well-evidenced overall theory of nature that there are things in remote parts of the universe beyond our ken. If accepting that is all it takes to have a “mind-independent notion of existence” then it is a notion shared by hard-core “metaphysical” realists and sensible Dummett/Putnam-style anti-realists alike. So the debate between the latter is not to the present point. (Perhaps, though, Alan has in mind something meatier than the anodyne claim about our cognitive shortcomings that everyone can agree on. But what, exactly?) (B) Is there an implication that there is also a contrasting non-mind-independent notion of existence someone might have? Or that someone might play with both notions? I’m Quinean enough to be unhappy with this suggestion: existence is just what the existential quantifier expresses in a topic neutral way. And certainly, I took it that Alan doesn’t need to suppose that there is some special ontologically committing notion of existence to develop his overall position — that’s why I had him in my first post as in the “left-wing” camp (counting as an anti-realist about numbers because he discerns a special mode of assertion in there is a prime number between seven and twelve’ rather than a special non-ontological but still representational use of `there is’). And that chimes still with the concluding remarks in his first comment.

5. I suggested 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 that could — as it were — get us mind-independence where appropriate for free. I meant necessary condition, not sufficient. The case of Berkeley illustrates why. He respects platitudes about sticks and stones still being there when not in our minds. So far so good; so far he is treating them realistically as mind-independent. But then he offers a further decoding, a further gloss …

6. But note, all this minor skirmishing is intended in a spirit of friendly amendment, aiming to highlight Alan’s core proposal by downplaying what seems to me to be unnecessarily distracting diversions. As he says at the end of his first comment, the leading idea is to distinguish “representational existence claims” as in ‘THERE ARE NO complex numbers’ from “non-representational” existence claims, of which ‘there are eight complex roots of x^8 = 256’ is one. What has been bugging me a bit is the idea that “representational” is helpfully glossed in terms of “mind-independent” which is in turn to be glossed in fallibilist term. But let’s now put such worries on hold. The suggestion that a distinction between representational and non-representational modes of discourse can be applied in the philosophy of maths as much as in the philosophy of morals, say, looks intriguing and not unpromising. Great. Now let’s see — in future installments — if Alan can get it to fly.

• Alan Weir says:

Re Comment 2, and relatedly Comment 6 ‘What has been bugging me a bit is the idea that “representational” is helpfully glossed in terms of “mind-independent” which is in turn to be glossed in fallibilist term.’

Yes I agree representational vs non-representational is distinct from mind-independence issues (and whether they should be construed in fallibilist terms). One way, at least, of giving a representationalist account of maths would be to produce a metatheoretic account of the truth conditions which was roughly speaking homophonic, modulo any issues of simple indexicality (rare in maths anyway where tense, personal pronouns etc. don’t feature in any serious sense) and with no attempt at reduction in the sense of chapter two. I allow (p. 67), someone who rejects mind-independence in my sense could proffer such a metatheoretic account so, in my terminology, be an ontological realist, a platonist in maths in fact, but a metaphysical anti-realist. My position is also available to metaphysical realists (like me!) though. (I should have said ‘complex solutions to x^8 – 256’ of course, or spoken of ‘complex eight-roots of 256’).

Comment 3. Yes I agree realism about Xs and fallibilism, about Xs can come apart, surely that’s a good thing about a specification of ‘realism’. One could aver that trees exist and that causally isolated miniverses exist, no ambiguity in ‘exist’, adopt metaphysical realism as a general viewpoint, but have no doubt about the former, whilst being unsure and hesitant about the latter. It would be weird to say one was thereby realist about miniverses but not trees.

Re >’That’s why I said realism about Xs and fallibilism about Xs can peel apart. … But actually Alan agrees with that (and this is what I wasn’t sufficiently clear about before — and I leave it to others to judge whether that was me being rather dim, or Alan being less that ideally clear!).’ I did write (p. 22) “Clearly we can be in error as to whether Sean is taller than Iain or not. But in optimal circumstances? Insofar as one can make sense of optimality here, this might be disputed by many philosophers. But do they ipso facto reject the reality of the properties and relations of shape, distance, and so forth which obtain among the everyday objects we deal with? This does not seem right and this in turn tells against fallibilistic formulations as a way of making sense of the traditional
realism debate.”

Comment 4 (A): Putnam, because of his refusal to define truth in terms of assertibility, presumably can consistently believe in evidence-transcendence truths- I take truths which outrun our ability reliably to detect them to be a species of evidence-transcendence. I don't want to impugn the sanity of Dummett (and of course it may be that Dummett is not an anti-realist at all but a theist realist who thinks atheists can't but be anti-realists) but it seems to me his anti-realist does deny there are any determinate truths which outrun our ability to detect them. Presumably also someone like Neil Tennant thinks this too.

So the question as to whether there are hydrogen atoms inside a sphere 1 micrometre in diameter exactly n light years distant from this point in this direction (add perhaps 'at a time exactly m billion years ago'), that question (with suitably choice of distant and direction to some dark, unnoticed and unremarkable region of space) presumably has no determinate answer for them, is neither determinately true nor determinately false. Likewise 'did an even number of dinosaurs break a bone on this day exactly 70 millions of years ago?' (prescinding from vagueness) and so on and so on. I agree it is preposterously anthropocentric to say this sort of thing (but this is no argument, more of an incredulous stare). But if the anti-realists aren't saying that, what is their position?