Truth Pluralism and Mathematics

When I started out as a graduate student writing a thesis defending mathematical nominalism, my naïve view was that mathematical claims (the correct ones) were true in a different way to empirical claims (the correct ones). Before long though, under the influence of Tarski, the basic model theory I was teaching in Logic 1, etc. I came to reject that view. Truth had to do with satisfaction in the Tarskian sense which required a domain of objects to do the satisfying. Recently though I came across this passage from Huw Price:

[W]e need to distinguish the notion of keeping track as something that we do within the assertoric language game – a notion constituted, within the game, by the fact that the normative structures always hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player – from a notion that we might employ from outside the game, in saying that in at least some of its versions its function is to aid the players in keeping track of their physical environment. … [T]here’s a temptation to call both kinds of external constraint ‘truth’, but we shouldn’t make the mistake of thinking we’re dealing with two aspects of sub-species of a single notion of truth. Both notions may be useful, for various theoretical purposes, but we shouldn’t confuse them’ (Expressivism, Pragmatism and Representationalism, 191)

I’m beginning to think that my naïve view might just be the right one. Any assertoric discourse [Could there be a non-assertoric discourse? I’m not sure.] with standards of correctness and incorrectness will need a truth predicate for the sorts of expressive purposes deflationists get excited about. (We need a truth predicate to express commitments without having to state them explicitly. E.g. if a theory \(\Gamma\) entails infinitely many things I can say ‘Everything \(\Gamma\) entails is true’ but I can’t possibly explicitly assert everything entailed by \(\Gamma\)  Hence truth predicates are indispensable for certain expressive purposes.) Mathematical discourse seems like a good candidate for a discourse that’s governed by internal rather than external standards. In empirical matters the world gets to answer back: we bump up against the world, probe it, test it experimentally. But mathematical investigations don’t involve interactions with mathematical objects, they involve proofs. Of course the results of these investigations can surprise us or be counterintuitive. But none of that requires an external world of mathematical objects; only that the normative structures constituted within the game ‘hold open the possibility that one’s present commitments will be challenged, so that ‘correctness’ is always in principle beyond the reach of any individual player’. Objectivity doesn’t require objects. Moreover, as I’ve argued before there’s good reason to think that the norms of correctness and incorrectness of mathematical discourse are internal, rather than external, in the senses above.

Horwich and Wittgenstein's Metaphilosophy

In a recent book, Wittgenstein’s Metaphilosophy, Paul Horwich defends (as the title might lead one to expect) a Wittgensteinian metaphilosophy which opposes the idea that philosophical thinking can lead us to surprising, substantive metaphysical results, and accords good philosophy the more deflationary role of dissipating confusions. This is a view I find attractive and plausible. According to Horwich, Wittgenstein’s metaphilosophy involves the claim that within the domain of philosophy:

There are no surprising discoveries to be made of facts, inaccessible through the methods of science, yet discoverable ‘from the armchair’ by means of some blend of pure thought, contemplation, and conceptual analysis. (Wittgenstein’s Metaphilosophy, 1–2)

Philosophical problems arise from 'bewitchment of our intelligence by means of language’. This typically involves overgeneralisation: applying some principle, which is locally valid in some domains, where it doesn’t belong:

[A] common way for us to be mislead by language, according to Wittgenstein, is that when we see a noun appearing as the subject of a true sentence, we expect there to be a thing to which the noun refers. This expectation comes from reflection on countless statements such as

- Neptune is a planet

- Boston has subways

- Plato taught philosophy

which suggest a universal underlying semantic structure for all sentences of that simple syntactic type: the subject, a noun phrase, picks out a particular object in the world, and the rest of the sentence denotes some property or characteristic attributed to that object. (ibid. 11)

Now, all this sounds great for the nominalist. Here is a nice way to diagnose how philosophers erroneously arrive at platonism: we expect, by overgeneralising on some paradigmatic cases, that whenever nouns feature in true sentences there are things to which the nouns refer. But mathematical language isn’t like this—that isn’t it purpose, mathematical standards of correctness and incorrectness haven’t to do with accurately depicting an abstract mathematical realm, etc.—we have simply overgeneralised from the paradigmatic representational use of language. Having diagnosed this overgeneralisation we are free to see that platonism is unmotivated, and adopt nominalism accordingly.

This isn’t what Horwich does though—he’s a staunch anti-nominalist. Why? I think the answer lies not in Horwich’s philosophy, but in the “pre-philosophical” convictions he brings to it. Horwich talks about this deflationary metaphilosophy allowing us to retain the ‘naïve idea that numbers are abstract objects [and] our naïve aspiration to discover what is true about them’ (ibid. 16). Horwich was a pre-philosophical platonist, so his belief that philosophy could not produce surprising, substantive metaphysical results leads him to believe that it cannot overturn platonism. I, on the other hand, was a pre-philosophical nominalist; so the self-same belief that philosophy cannot produce surprising, substantive metaphysical results leads me to think that the existence of mathematical objects can’t be discovered by philosophical means.

I wonder if the lesson here is that the platonism-nominalist debate is irresolvable by philosophical means. I wonder, but I’m not sure; and that’s because even some of our pre-philosophical convictions can arise from the sorts of confusions that good philosophy can dissolve. In this case, I think that attention to mathematical-linguistic practice could plausibly show that it’s not in the business of depicting an extant mathematical realm; but that’s a story for another day.

The norms of mathematical discourse and inquiry

David Lewis once (influentially) commented that it would be ludicrous to expect mathematicians to change their ways on the basis of philosophical arguments that mathematical objects don’t exist. Why he thought mathematical practices would have to be emended in the light of ontological facts about the existence of mathematical objects, I’m not sure.

Here’s a thought experiment to make explicit your own implicit commitments about this. Imagine that, instead of a philosopher, an infallible oracle told the world that mathematical objects don’t exist. Would mathematics professors be obliged to hand in their resignations? Would their discipline have been exposed as a sham?

I think the answer to these questions is a, very obvious, “no”, and I suspect that almost everyone would agree. But notice what that means. If we don’t accept that mathematical practices ought to change in light of word from an infallible oracle that mathematical objects don’t exist, then we must also accept that the norms governing mathematical discourse are not representational, in the robust sense of that word as pertaining to mapping, tracking or picturing how things stand with a domain of mathematical objects. The standards of correctness and incorrectness in mathematics do not derive from mathematical objects, but from standards internal to the game (or perhaps “game”) of mathematics itself.

Call this view normative nominalism. But if one is committed to normative nominalism (as a “no” answer to the above questions would reveal), then what could possibly be the motivation for platonism?

Wright on Deflationism

I’m reading Crispin Wright’s Truth and Objectivity for a reading group at the moment, where he sums up an argument against deflationism about truth in the following way (I quote at length):

The deflationist holds that “true”, although gramatically a predicate, denotes no substantial quality of statements, or thoughts, but is merely a device of assertoric endorsement, of use to us only because we sometimes wish so to endorse a single statement, referred to in a way which doesn’t specify it’s content, or batches of statements all at once. Apart from applications of those two kinds, it is, for the deflationist, a complete explanation of the truth predicate that it satisfies the Disquotational Schema. It is a consequence of this general conception of the role of the truth predicate that it can register no norm governing assertoric discourse distinct from warranted assertibility. Yet the central place assigned to the Disquotational Schema—and thereby to the Negation Equivalence—actually clashes with that consequence, for it follows that, while normative of assertoric discourse, and indeed coincident in (positive prescriptive) normative force with warranted assertibility, “true” is nevertheless potentially extensionally divergent from warranted assertibility—and hence has to be accounted as registering a distinct such norm. Since it’s compliance or non-compliance with a norm distinct from assertoric warrant can hardly be an “insubstantial”property of a statement, and since a uniform account is possible of what it is for any particular statement so to comply, deflationism collapses. (pp. 71–2)

For reference, the Disquotational Schema is:

(DS) “P” is T is and only if P

One can’t, without engaging in a kind of doublethink, say or believe things like ‘P and it is not warrantedly assertible that P’ for some proposition P, since you can rationally assert P if and only if it is warrantedly assertible (for you) that P. But truth can be used to contrast with warranted assertibility. Take (DS) and substitute ‘It is not the case that P’ for P:

(i) ‘It is not the case that P’ is T is and only if it is not the case that P.

From (i) and (DS) one can infer:

(ii) It is not the case that P if and only if it is not the case that ‘P’ is T.

And from (i) and (ii) you get:

(iii) ‘It is not the case that P’ is T if and only if it is not the case that ‘P’ is T.

But (iii) cannot be right if T means warrantedly assertible, so ‘true’ registers a norm distinct from warranted assertibility. Deflationism is the view that ‘true’ just is a devise of assertoric endorsement, and Wright thinks that a mere devise of assertoric endorsement couldn’t register a norm distinct from warranted assertibility, so deflationism must be false.

But there is a way to register the truth norm without using the truth predicate. One can say ‘P and it is not warrantedly assertible for S that P’ or ‘¬P and it is warrantedly assertible for S that P’ where S is some person other than yourself, and in doing so can register the truth norm without using the truth predicate. What is required is that one contrasts one’s own perspective with that of another. Registering the truth norm requires some kind of I-Thou contrast. If that’s the case then ‘true’ is not what, fundamentally, allows one to register the truth norm contrasting with the norm of warranted assertibility, and deflationism is off the hook.

I think this might tap into something deep about objectivity—more specifically, our ability to see the world as being objective or to conceptualise there being objective facts that outstrip our ability to know them—as it coheres with something Robert Brandom says about objectivity. Brandom (I won’t spell out the details here) also argues that conceptualising objectivity requires I-Thou relationships. This is made explicit in paradigmatically referential of-statements like ‘He believes of this criminal that he is an innocent man.’ Understanding “of” requires navigating between one’s own perspective and that of another. Since “of” is how we refer objectively to the world, talking (and hence thinking) about the world objectively requires navigating between one’s own perspective and that of another.

Why I really hope we vote no on Thursday

Maybe I’m odd this way, but I love Britain. I love Britain because it has a kind of contrapuntal brilliance—the way the craggy summits of the Munroes perfectly compliment the gentle pastures of Cambridgeshire, and Edinburgh stands like a poised and dignified sister to exuberant, thrumming London. And I love Britain because the rest of the UK isn’t Westminster—it’s J.R.R. Tolkein, and The Wind in the Willows, and Radiohead, and Wallace and Gromit, and Stewart Lee, and Viz, and Bertrand Russell, and Ant and Dec, and my cherubic little nephew.

I believe in many of the things that have made breaking away from the UK seem attractive to a lot of people: essentially the benefits of political localism—a political class that is close to, and so responsive to, the needs and concerns of the constituents they serve. The thing is, that all these things could be achieved, without the damage involved in breaking up the union, through devo max. Not only that, devo max is the democratically mandated option; it’s what most Scottish people actually want. Far better that than the division we’ll have within Scotland if we permanently break away from the UK on the basis of a tiny majority of separatists. Moreover, it’s an option that’s on the table if and only if we vote no in the upcoming election.

This isn’t primarily why I’m voting no, because I would vote no even if devo max wasn’t on the table. It’s not primarily for economic reasons either, although I would by no means dismiss these as somehow crass or “not what’s really important”. Questions about the economy just are questions about how the most vulnerable in our society will fare. They’re also questions about how my family will fare. If the dire warnings about recession, the flight of business, the disaster of shared currency without political union and mortgage rates skyrocketing are even close to true, breaking up the union might mean losing our home.

But the fundamental reason I’m voting no isn’t economic, because I would vote no even if the economic consequences of separation weren’t so grim. It’s not economic because even if, by some miracle, we were sitting on an oil bonanza, I wouldn’t, for one moment, resent it paying for someone’s medical treatment in Yorkshire, or Liverpool, or wherever. The reason I'm voting no has to do with the value of unity itself. People within a nation state have differing political, religious and ethical convictions. A nation state is held together by bonds of mutual trust, solidarity and coöperation that somehow transcend these things. Just look to Iraq, Syria or any other truly dysfunctional states to see that these coöperative bonds are not natural necessities but deeply contingent, and deeply valuable. A united kingdom is a precious and hard-won achievement, and it would be a terrible waste to throw that away by choice. Salmond’s convinced many of us that the SNP are somehow progressive visionaries. That’s what his well-greased rhetoric is designed to suggest anyway, though the facts don’t match the bluster. Unionists are voting for unity, separatists are, in practice, voting for the opposite.  Scots have always been cosmopolitan and internationalist in outlook, and have always used Britain to make our mark in the world.  But if we break up the UK and sever the unique bonds of coöperation that link compatriots, we will, in a very real, very concrete way, be making Scotland a less inclusive, less open and more parochial place. That’s the politics of division, and that’s why I really hope we say no to it on Thursday.

Is nominalism self-defeating?

Here's an objection to nominalism I've heard a few times.  Sometimes the 'access problem' to abstract objects is motivated by the idea that embodied creatures adapted to a particular environment, such as ourselves, need to interact with the world in order to gain knowledge of what it's like.  If we're to learn something about a given domain of objects, at some point we will require some kind of causal interaction with at least some members of that domain of objects.  Abstract objects, such as mathematical objects, are not like this: there is no method by which we could interact with anything in a domain of abstract objects at any time.  As a result, even if abstract objects exist, there is no means by which we could come to gain knowledge of which abstract objects exist or what properties they have.  This kind of minimal causal condition for knowledge is sometimes called the (or a) "eleatic principle".  But it's sometimes said that this eleatic principle is self-defeating.  Here's Sorin Bangu in his nice book The Applicability of Mathematics in Science (pp.18-9):

My naturalist’s reaction to the reformulated [eleatic principle] challenge is to point out that it is ultimately self-defeating.  That is, the naturalist notes that one cannot even formulate the challenge without actually making appeal to mathematics: one simply can’t grasp what the new naturalized [eleatic principle] actually says unless one understands the physical theories describing the abovementioned types of interactions.  But these theories are, of course, thoroughly mathematical!  So, anyone attempting to advance a challenge of the [eleatic principle] type in naturalistically acceptable terms finds herself engaged in the self-undermining enterprise of rejecting the very (mathematical) terms which allow the (acceptable naturalistic version of the) challenge to be meaningfully formulated in the first place.

This criticism seems wrong to me, on two counts.  Firstly, it ignores responses to the indispensability argument.  The nominalist will need some response to the indispensability argument.  If this response doesn't work then the nominalist is in trouble anyway.  If it does work—whether it involves doing without reference to or quantification over mathematical objects in scientific theories, like Field, Chihara etc., or offering some account of why it's acceptable for the nominalist to continue to refer to or quantify over mathematical objects in scientific theories, like Leng—then it will work here too: that we give mathematical models of how we (concrete) creatures interact with (concrete) parts of the world will pose no special problems.  Secondly, even in lieu of a response to the indispensability argument, the eleatic principle can be used to give a sort of reductio of mathematical platonism: (i) assume mathematical platonism is true, (ii) motivate the eleatic principle, (iii) our own mathematicized theories which describe how we interact with the world show that we cannot have knowledge of mathematical objects. So the assumption we began with is unknowable and rationally self-defeating.

What I talk about when I talk about numbers

Here is a valid argument:

(1) The number of Front national MEPS is worrying. 

(2) The number of Front national MEPS is 24. 

(3) 24 is worrying.

At least it’s valid if you think, as almost all philosophers who think about mathematical language seem to, that (2) refers to a number.  Contrast (2) with

(2*) There are 24 Front national MEPS.

(2*) is a statement about Front national MEPS, but (2) and (2*) are treated as being equivalent; not in the sense that they have the same meaning (one refers only to a political party, the other refers to a number) but in the sense that given (2) we can always infer (2*) and given (2*) we can always infer (2).  We can do this because we accept the abstraction principle:

(*) There are n Fs if and only if the number of Fs is n.

I’m not sure what to make of this.  I used to think that claims like (2*) were true because they predicate a property of something real, whereas claims like (2) were literally false because they make reference to something that doesn’t really exist—a number.  Making inferences using (literally false) claims like (2) was, I thought, fine, because doing so wouldn’t lead us astray with respect to how things stood with what really existed.  Similarly we could accept (*), not as being literally true, but as being “nominalistically adequate”, i.e. unable to lead us astray with respect to how things stand with what really existed.  (Compare: we accept ‘There is a dent in the car’ not because dents really exist or because they are an extra bit of the furniture of reality over and above the car, but because saying this doesn’t lead us astray with respect to the topographical properties of the car.)  But here is a problem with this: (3) is absurd.  A convenient fiction that aids inference-making is one thing; an absurd convenient fiction that aids inference-making is something else.  This is disastrous for the platonist who thinks that numbers really exist.  For the platonist (3) is true.  But it’s also bad news for the fictionalist who accepts that (2) refers (or at least purports to refer) to a number, because although fictionalist take (3) to be false, they’re still left with a problem: (3) isn’t even nominalistically adequate.  (3) can be used to infer falsehoods about the concrete world:

(3) 24 is worrying. 

(4) The number of Tunnock’s Teacakes in a four-pack is 24. 

(5) The number of Tunnock’s Teacakes in a four-pack is worrying.

(5) is about the concrete world and is false.  Maybe the only option is to drop the claim that phrases of the form ‘The number of Fs is n refer to the number n.  In this case the ‘is’ can’t be the ‘is’ of identity; the phrase can’t mean ‘The number of Fs = n’.