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.