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.

How to Eschew Metaphysics

Here’s how Blackburn describes pragmatism:

You will be a pragmatist about an area of discourse if you pose a Carnapian external question: how does it come about that we go in for this kind of discourse and thought?  What is the explanation of this bit of our language game?  And then you offer an account of what we are up to in going in for this discourse, and the account eschews any use of the referring expressions of the discourse; any appeal to anything that a Quinian would identify as the values of the bound variables if the discourse is regimented; or any semantic or ontological attempt to ‘interpret’ the discourse in a domain, to find referents for its terms, or truth-makers for its sentences … Instead the explanation proceeds by talking in different terms of what is done by so talking.  It offers a revelatory genealogy or anthropology or even a just-so story about how this mode of talking and thinking and practising might come about, given in terms of the functions it serves.  Notice that it does not offer a classical reduction, finding truth-makers in other terms.  It finds whatever plurality of functions it can lay its hands upon. [Simon Blackburn, Expressivism, Pragmatism and Representationalism: 75]

I'm interested in the claim often made by pragmatists, such as Simon Blackburn or Huw Price, that they are eschewing metaphysics, in contrast to platonists, fictionalists, error theorists and the like. Pragmatic accounts of a discourse provide a genealogy, or some consanguineous account, of why it is we go in for this way of talking and, as it may happen, this account may be metaphysically deflationary.  So it may be that the motivation for talking about, say, mathematical objects, does not involve representing how things stand with a domain of mathematical objects.  If there is some such story—if we can account for the uses of mathematical talk, without invoking mathematical objects—then we have an ontologically deflationary pragmatic account of mathematical discourse.

But so far, what’s been said about pragmatic accounts of mathematical discourse is open for the fictionalist to adopt.  The difference between the fictionalist (who is apparently engaged in metaphysics) and the pragmatist (who apparently eschews metaphysics) is that the fictionalist claims that mathematical talk is, strictly speaking, false, whereas the pragmatist does not.

Fictionalists and pragmatists then agree in methodology: provide an account of the usefulness of mathematical (or moral, or possible worlds) discourse that makes the existence of mathematical (or moral, or modal) objects orthogonal to the practice.  Their point of divergence is not methodological or ontological, but semantic: whether one opts for fictionalism or pragmatism depends on what one takes the meaning of existential quantification to be.  Here, the pragmatist reads the pragmatic purpose of quantification over mathematical objects back into the semantics of quantification over mathematical objects, and the fictionalist does not.  A truism: people can engage in ontological disputes.  There is something at stake between someone who claims that the Higgs boson exists and someone who claims that it does not, or between someone who claims that God exists and someone who claims that he does not.  The interlocutors in these debates are in disagreement over what the world is like.  So, sometimes at least, quantificational talk is used to express disagreements about what the world is like.  Ultimately then, the difference between the fictionalist and the pragmatist lies in what they take the meaning of existential quantification to be.  Fictionalists take existential quantification to be univocal: it always expresses claims about what the world is like.  Pragmatists (are committed to) taking existential quantification to be multivocal: within discourses whose purpose is to describe the world existential quantification expresses claims about what the world is like; within discourses whose purpose is not to describe the world, existential quantification does not express claims about what the world is like. (Note that the point of divergence is not, or need not be, over semantic minimalism. The person engaged in metaphysics need not couch what he is doing in terms of finding truth-makers or referents to be relata in substantive relations of truth or reference to given sentences; he can simply couch what she is doing in terms of whether such and such objects exist. Hartry Field is a case in point.)  

The take-away claim: whether one gets to eschew metaphysics depends on whether existential quantification is univocal or multivocal.

Is naturalism coherent?

Here is how Huw Price characterises naturalism in a recent book, although I think it’s a characterisation that many philosophers would endorse:

What is philosophical naturalism?  Most fundamentally, presumably, it is the view that natural science properly constrains philosophy, in the following sense.  The concerns of the two disciplines are not simply disjointed, and science takes the lead where the two overlap.  At the very least, then, to be a philosophical naturalist is to believe that philosophy is not simply a different enterprise from science, and that philosophy should defer to science, where the concerns of the two disciplines coincide. [Expressivism, Pragmatism and Representationalism: 3]

But in what sense is it possible for philosophy to defer to science?  One way we might think this could go is in the following scenario: we have in our possession, say, both a successful scientific theory which posits backwards causation, and an a priori philosophical argument that backwards causation is impossible.  Deferring to science—which is an essential trait of naturalism, as understood above—involves accepting the scientific theory and rejecting the philosophical argument as (somehow) unsound.  But there is a problem with thinking of this as philosophical deference to science (as opposed to some other kind of deference to science).  Consider the maxim: When a claim of a successful scientific theory conflicts with the conclusion of an a priori argument, reject the conclusion of the a priori argument in favour of the claim of the successful scientific theory.  This is, on any reasonable measure, a philosophical dictum rather than the claim of a scientific theory.  In which case, someone who follows the maxim is being guided by a philosophical dictum rather than the claim of a scientific theory.  Moreover—although I’m not really arguing for this latter claim here—it is plausible that any adjudicative maxim of this kind would be philosophical rather than scientific per se; and in that case, it wouldn’t make sense to say that philosophy could defer to science.

Horwich vs. Nominalism

In a recent exchange with Huw Price (Expressivism, Pragmatism and Representationalism) Paul Horwich makes an 11-point case against nominalism (he calls this a case against naturalism, but it seems to be nominalism in particular that is being targeted).  The argument is quite condensed, but it provides a nice summary of why (I think) so many philosophers embrace some form of platonism (or 'anti-nominalism' for those who prefer the apophatic formulation) as well as why (I think) the case against nominalism goes awry.

Firstly, there is the purported motivation for nominalism (for 'naturalism' read 'nominalism'):

1. Naturalism rests on the impression that non-natural facts would be intolerably weird. 

2. That impression has three sources: first, the singular practical and explanatory importance of naturalistic facts; second the very broad scope of the naturalistic order – the striking range and diversity of the facts that it demonstrably encompasses; and, third, the feeling that reality must 'surely' be fundamentally uniform – so all facts must be naturalistic. [EPR: 124-5]

This theme is continued:

6. The committed naturalist will not be greatly perturbed by the accusation that [his defence of naturalism is] ad hoc, contrived and intrinsically implausible.  For he will reason that although such defects may indeed be present, and are indeed unwelcome in themselves, they are a price well worth paying for the wonderfully simple metaphysics that naturalism provides. [EPR: 125]

Now, perhaps some nominalists really are motivated by metaphysical simplicity, and for some reason take simple pictures of reality to be intrinsically more plausible than complicated ones; but nominalists don't, as a matter of habit, primarily motivate their view  by appealing to any metaphysical claims of this kind.  On the contrary, the most prominent  defenders of nominalism, such as Field or Leng, object to platonism on epistemological grounds.  Knowledge of abstract (and hence acausal) objects is problematic because the existence or non-existence of mind-independent abstract objects can make no difference to any grounds one might have for believing in them.  Since the existence of abstract objects has no consequences for anything could possibly take place it seems impossible in principle to (i) provide justificatory grounds for belief in abstract objects, or even (ii) provide some hardcore externalist model of belief in abstract objects that could explain how these beliefs could count as knowledge, even in the absence of justification.  (Note that this applies to indispensability arguments: the indispensability of quantification over mathematical objects in scientific theories does not depend on the existence of a domain of abstract mathematical objects—it is a function of more mundane things, such as the complexity of the concrete systems being modelled and the expressive resources available to the agents doing the modelling.)

The crux of the matter however seems to lie in the pro-case for platonism:

4. Note, to start with, that it's prima facie extremely implausible that amongst the facts we recognise, some are non-natural – for example, that there are numbers […]  An unbiased consideration of such facts will indicate that they aren't naturalistic.  For it's as plain as day (to anyone not 'in the grip of a theory') that they aren't spatio-temporally located, aren't engendered by facts of physics and don't enter into causal/explanatory relations with other facts. 

7. But this apology for naturalism is inaccurate in two related respects.  In the first place, what is given up for its sake is not justly described as 'local theoretical simplicity'.  For what must be denied are data – epistemologically basic convictions.  It is blindingly obvious to us … that Julius Caesar wan't a number. […] And no less obviously false are certain implications of every one of the sceptical 'error theories' (i.e. denials of existence) and strained reductive analyses aimed at safeguarding naturalism from the threats posed by numbers…  

8. And, in the second place, the norm of simplicity, as it is deployed in science, is not in fact a licence to reject recalcitrant data … A scientist is obliged to respect all relevant data, and when they don't conform to a simple pattern, that reality must be accepted. [EPR: 125-6]

The thought that facts about an abstract domain of numbers are simply data, I would hazard a guess, is an important motivation for contemporary platonists, and explains why arguments for nominalism are often simply written off on the grounds that they entail an unacceptable conclusion.  But are we right to see these as data?  Horwich seems to hold that everyone is (or was, at some point) a pre-theoretical platonist, and adopted nominalism for the sake of metaphysical simplicity.  I suggested before that the second part of this claim is false, but the first part is also, at least to some extent, inaccurate.  I for one was a pre-theoretical nominalist: I didn't realise that anyone believed in the existence of numbers until I took a class in metaphysics, and when I made mathematical claims I didn't take the purpose of this practice to reside in describing a domain of abstract objects.  It would be interesting to see some stats on the topic, but at the very least, a number of people don't take facts about an abstract domain of numbers to be data (I've met a few).  So, not everyone has epistemologically basic convictions about the existence of mathematical objects, and to treat these as data is question-begging within the context of this debate.

Besides being question-begging in the current context, there is a kind of hard-line stance against the appeal to "epistemologically basic convictions" of this sort—on the grounds that it is unduly conservative: insulating views from criticism—as articulated, with characteristic understatement, by Kant:

To appeal to ordinary common sense when insight and science run short, and not before, is one of the subtle discoveries of recent times, whereby the dullest windbag can confidently take on the most profound thinker and hold his own with him.  So long as a small residue of insight remains, however, one would do well to avoid resorting to this emergency help.  And seen in the light of day, this appeal is nothing other than a call to the judgement of the multitude; applause at which the philosopher blushes, but which the popular wag becomes triumphant and defiant. [Prolegmonena: 4:259]

Now, Horwich isn't exactly a dull windbag or a popular wag (nor, for that matter, was Reid); and, in any given debate, something will be treated as data, if only temporally, since possessing some common commitments is a precondition of debate in the first place.  So what sorts of things ought one allow as data in the debate between platonists and nominalists?  A plausible supposition is that the division between platonists and nominalists coincides with a division over the kind of things that one allows as data.  Platonists will take the data to be true sentences about mathematical objects, whilst nominalists will take the data to be mathematical practices.  I take it that the latter view is the correct one (and usefully non-question-begging in the context of this debate): what must be accounted for are facts about how (e.g.) solving differential equations and carrying out measurement procedures allows us to track features of and make predictions about concrete systems. This would explain why nominalists (like me) are drawn to nominalism: it is very hard to see how the existence of abstract objects would be required to explain any practice. The take-home claim (that I've not really defended here with any rigour): there are deep connections between nominalism and pragmatist methodology.