Thursday, January 08, 2009

How mysterious is Platonism?

Arithmetical Platonism is supposed to be prima facie suspect because how, it’s asked, could we have arithmetical knowledge if the objects of that knowledge are in a world apart from us, a timeless world of Platonic objects, with which we cannot interact causally? I reply by wondering, how strange are abstract objects? You have just been reading this, for the obvious example. What have you been reading? You have been reading sentences. You look at the physical instances of these words, but you see the words, you read the words, and as you do so you are (hopefully) thinking about the thoughts expressed by means of them. So, there are, in the physical world around you, those physical instances of the shapes of (written modern English) words, and there are in your mind those thoughts; so, where are the sentences? What are the sentences? Sentences are made of words, and words are parts of a language (i.e. modern British English). They can be spoken or written, and can sometimes be spelt in different ways. Furthermore they have a meaning, a sense, and they have it essentially. The mere shape of a word is not a word, no more than meaningless strings of letters are words. Words, it seems, are abstract objects (I’m not entirely sure about that, or about what abstract objects are, so I’d welcome corrections) and you’ve just been reading some words of mine (and I’ll add that words can be true, insofar as they describe the world sufficiently accurately, or not, in case anyone wants to argue that thoughts and not words are truth-bearers:)


Anonymous said...

Here's your argument:

3 3 3

How many numbers are presented? Three or one?

Or, three tokens of a type?

As I've seen it put before: "If I've read the Penguin Classics edition of "Pride and Prejudice" and the Oxford World Classics edition of "Pride and Prejudice", have I read the same book twice, or two different books?"

This is a common argument and I think what your argument actually attempts to show, if anything, is the existence of universals by way of multiple instantiation/realization. Certainly not the truth of platonism.

The existence of universals is a necessary condition for the truth of platonism, but not vice versa.

Note that your argument, if true, could be accommodated by a restricted form of Aristotelian realism about universals; probably of the timid, moderate Armstrongian variety.

Your argument probably backfires: the argument that mathematics is like a language is often (and powerfully) wielded by anti-realists about abstract objects; viz. nominalists (google: "resemblance nominalism"). Chomskians (and other platonists) have been arguing about the Piraha tribe's apparently numberless language.

A fortiori, I think "gavagai" means that this argument is not worth pursuing.

Enigman said...

Hi Anonymous. You say "Here's your argument" but what follows is no argument, just a question. The answer is, I think, the same book twice, although 'book' does equivocate. I might be carrying three copies of a certain book, and if asked how many books I'd been carrying the correct answer might be "Three." So other answers can also be correct. Anyway, that not an argument, but a question (and perhaps, you may have thought, an implicitly correct answer), as indicated by how you speculated about what it might be purporting to show.

Your speculation about my intention was false (and if you reread my post, you'll see that I wasn't too unclear). My argument was to do with Platonism, but not its truth, rather its supposed mystery. My motivation is that some philosophers go from Platonism being mysterious (they think) to it not being worthy of consideration. They often then consider some alternative to be empirically indicated (usually without examining their own metaphysical presumptions as they interpret the evidence), and so accept that as true, and so end up having implicitly argued (badly) for the falsity of Platonism. But my argument was aiming to undermine their first step, which is usually unchallenged.

It was an argument that arithmetical Platonism is no more mysterious than the fact than you can read these words. Of course, both are in many ways mysterious (there is also, for example, the mystery of perception), but we don't conclude that reading (or seeing) is impossible; not, that is, unless we're going a bit Eleatic. Incidentally, the charge of being Eleatic is often made against those who argue for the indefinite extensibility of arithmetic; it's better made against those who argue for standard set theory on the grounds that the natural philosophy of arithmetic is hopelessly mysterious, and standard set theory is adequate for our current needs.

But anyway, maths is like and unlike a language, in different ways (although that is little to do with my argument, except superficially). Since there is considerable disagreement (much of it confused) about even what a language is, I don't think that much follows from that fuzzy fact. (The examples you give are all very interesting ones though:)

Enigman said...

E.g. "gavagai" actually makes my point quite well, I think (upon reflection). I'm not sure what that word means, but perhaps that's the point of it? My point is that we can have a pretty good idea of what all these other words mean. The idea that we cannot (e.g. because of words like "gavagai") is pretty incoherent, and pretty obviously fallacious.

Similarly, if I met you I wouldn't know everything about you, and there'd be all sorts of sceptical scenarios that I couldn't rule out, which follow from my ignorance of how I'd know what little I did know. But none of those mysteries makes the idea that you're another person, like me in many ways, about whom I could find things out upon meeting, an idea not worth considering, which is the form of the argument against Platonism I was considering. I can't be sure of facts about even the people I meet, and perceptual mechanisms remain a bit mysterious, but I can be quite sure that an argument from that to the absurdity of supposing that there could be (and are probably) people who I could learn about by meeting is a fallacious argument.

There are lots of analogies for this fallacy in analytic philosophy (sadly), e.g. some say that colours are abstract objects. Another is the case of natural laws, or nomological necessity. David Lewis thinks it mysterious how there could be a sort of necessity that isn't logical necessity, and so regards natural laws as mere regularities, following Hume. But his reasoning there is fallacious. If there weren't natural laws the world would very probably not be as it seems to be, for all that the scientific discovery of their forms is an uncertain affair, and for all the mysteriousness of their metaphysics. We can know about natural laws via science, even if they're more than mere regularities (and indeed, there're problems with them being mere regularities).

And similarly we could know about numbers even if they were abstract objects; and indeed, there are metaphysical problems with them being mere conventions. We ourselves are (or seem to be) logical individuals, so we can know (directly) that there's such an objective kind. And the objective properties of that kind are the arithmetic that we seek to know about. Our arithmetic may be uncertain, at least when it goes into the infinite, and we may be guessing even more about what we mean by "kind" there, but none of that makes it implausible that there are such objective facts to find out about and (most appositely) that we could discover them (with some confidence, insofar as we're good at maths) whether or not the whole numbers are logical objects in some Platonic realm. Whether or not they are is a different question; my point was that this argument that they aren't is fallacious (and obviously so).

Enigman said...

And finally (for more analogies), consider those fallacious arguments against Idealism that go via kicking stones and such (e.g. see The Reasoner 2(4)). If anything, those are arguments against Humean supervenience, badly expressed. Such Straw Man arguments seem to be very common in mainstream analytic philosophy (the response of Anonymous above is just another example). Presumably there's some interesting sociological reason for that.