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?

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’.

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.