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.

Two Kinds of Indispensability Argument

Continuing on the theme of nominalism and pragmatism…

The Putnam of yore took it that mathematical objects exist and is credited along with Quine as being an early proponent of the indispensability argument.  There are though two very different kinds of indispensability argument that Putnam made.  The first runs like this:

[Q]uantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.  This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes. [Philosophy of Logic: 347]

Why be a platonist?  Because, according to the argument, nominalism is inconsistent with physics.  One big problem with the argument is that there is plenty that gets quantified over in the sciences that we don’t take to exist; especially idealised versions of physical systems, the stock examples being frictionless surfaces, continuous fluids and the like.  So nominalism’s being “inconsistent with physics” in this sense isn’t a big deal, since the (clearly true) claims that fluids are not continuous, that there are no frictionless planes etc. are also “inconsistent with physics”.  (Penelope Maddy in Naturalism in Mathematics and Mary Leng in Mathematics and Reality both make this kind of point.)

Putnam also made a very different kind of indispensability argument, often conflated with the first, that goes like this:

I believe that the positive argument for realism has an analogue in the case of mathematical realism.  Here too, I believe, realism is the only philosophy that doesn’t make a success of science a miracle.  [Philosophy of Logic: 73]

There is an important shift from looking flatly to what entities are quantified over in our best scientific theories to looking at what quantification over these entities can be used to achieve.  This pragmatic spin is in fact necessary because quantification over mathematical objects is not indispensable simpliciter (if such a notion even makes sense), but indispensable for certain ends.  We could do without quantification over mathematical objects; we might just also have to do without iPhones, air travel and so on, if we did.  As Sellars famously said (in ‘A Semantical Solution to the Mind-Body Problem’) ‘[c]learly human beings could dispense with all discourse, though only at the expense of having nothing to say’.  So, if the indispensability of quantification over mathematical objects is supposed to be a problem for nominalism, it must be because talk of mathematical objects must be made use of to achieve certain ends; in which case what is at issue are mathematical practices.  The best explanation for the success of mathematical practices must involve the existence of mathematical objects, or so the thought goes.  But here’s the kicker: mathematical objects, because they are acausal, changeless and not subject to any events, cannot be invoked to explain any practices.  So the best explanation of the success of science needn’t invoke mathematical objects.

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.