Thursday, July 27, 2017

Gunky Pluralities

Philosophers have been squeezed out of most traditional areas of physics by fancy scientists with their “experiments” and their “research budgets”, but one thing they’ve managed to cling on to is mereology. We’ve even reclassified it as metaphysics so the physicists won’t accidentally stumble across it when looking for a book on something more important. Mereology is the theory of parts and wholes. Mereologists try to answer questions like these:
  • Is there a great big object which all the other objects are parts of?
  • When x is part of y, is there always another object z which is the rest of y?
  • Does anything even have proper parts? (A proper part of x is a part of x that isn’t x itself.)
  • Could there be two different objects made out of the same parts at the same time?
  • Could there be something all of whose parts had multiple parts?

The last question is about gunk. I wrote here once before about gunk, when describing an argument by David Hume against the possibility of gunk. You have to be a little bit careful when defining gunk, because what seem like equivalent definitions might not be equivalent if you accept some other exotic principles about parthood, or at least reject some mundane ones. One fairly careful definition of gunk would be “something all of whose parts have proper parts”, but that doesn’t really get at what we had in mind if we allow that things can be proper parts of each other.

Now, parts and wholes aren’t the only game in town when it comes to gathering objects together. Somebody once told me that Bolzano identified like fifteen different ways of gathering objects together. Maybe it wasn’t fifteen, but I’m pretty sure it was a lot. Although perhaps this person was pulling my leg.

Anyway, one alternative to making a whole out of some objects is to make a plurality out of them. A broom is an object made of a brush and a handle. A brush and a handle are a plurality whose members are a brush and a handle. (Sets are different again; two things form a set, but they are a plurality.) Now, you might think that there’s no distinction here: a broom just is a brush and a handle. If that’s what you think then I’m actually on your side, but most people who work on this stuff don’t think that’s right. (One person who works on this stuff and thinks it is right is Meg Wallace, of whose work on both this and other things I am a fan.) The mainstream view is that a broom can’t be a brush and a handle, because (apart from anything else) a broom is one thing and a brush and a handle are two things. Anyway, let’s make the distinction.

It turns out that a lot of the candidate principles governing wholes and their parts are analogous to candidate principles governing pluralities and their subpluralities. Let’s say that Tom and Harry are among Tom, Dick and Harry. Let’s also say that Tom is among Tom, Dick and Harry, and that Tom, Dick and Harry are among Tom, Dick and Harry, but not properly among them. And let's allow that "some things, the xs, are F" can be true even if only one thing is F, so some things are (each!) Buzz Aldrin. Now we have a language in which to ask similar questions about pluralities and subpluralities to the ones we asked about parts and wholes.
  • Are there some things, the xs, such that whenever there are some things they’re among the xs?
  • When the xs are among the ys, are there always some things, the zs, that are the rest of the ys?
  • Are any things properly among any other things?
  • Could there be some things, the xs and the ys, such that any things among the xs were also among the ys and vice versa, but the xs weren’t the ys?
  • Could there be some things, the xs, such that whenever some things the ys were among the xs there were some things, the zs, properly among the ys?

Call some xs that fit the definition in the last question a gunky plurality. Could you have gunky pluralities? Are they ridiculous? I asked Twitter if they were ridiculous, and the eight respondents were evenly split on the matter.

Gunky pluralities twitter poll results.png
Am I an experimental philosopher yet?

I was a little bit surprised. I think gunky pluralities are coherent, but in the past I’ve never detected much enthusiasm for them. While a Twitter poll with eight respondents doesn’t give much of an indication of the frequency of a position among any population apart from the people who responded to it, I was quite surprised to see that four people, not including me, saw the tweet who don’t think gunky pluralities are ridiculous. Maybe they didn’t understand the question. I did phrase it in terms of membership instead of in terms of amongness, but if anything I’d expect that to make the position seem more ridiculous, not less.

Now, I’m invested in gunky pluralities being coherent because I think composition’s identity and amongness is parthood, and so if gunky pluralities are incoherent then gunk is incoherent, and nobody wants to be committed to that. (Someone with a fancy research budget might come along and make a fool of you.) But even if you don’t think composition’s identity, and I suppose even if you don’t think merelogical gunk is possible, you can still make sense of the question about gunky pluralities. Are they ridiculous or aren’t they?

I think that Hume and Leibniz probably took the view that they were ridiculous. Hume may well have been thinking only about mereological gunk, and I’m pretty sure Leibniz was, but it would have been kind of weird to endorse their arguments for the mereological case and not the analogous arguments for the pluralities case. It’s possible of course that they didn’t really see a distinction, Bolzano not having arrived on the scene until the following century. Hume credited his argument to a Monsieur Malezieu, who I guess is probably this guy, although his English Wikipedia article could use some work and his French one doesn't mention Hume. Here are some quotes for you:

It is evident, that existence in itself belongs only to unity, and is never applicable to number, but on account of the unites, of which the number is composed. Twenty men may be said to exist; but it is only because one, two, three, four, &c. are existent, and if you deny the existence of the latter, that of the former falls of course. It is therefore utterly absurd to suppose any number to exist, and yet deny the existence of unites; [...]
But the unity, which can exist alone, and whose existence is necessary to that of all number, is of another kind, and must be perfectly indivisible, and incapable of being resolved into any lesser unity. (Hume, Treatise on Human Nature, 1.2.2)

And there must be simple substances, since there are compounds; for a compound is nothing but a collection or aggregatum of simple things. (Leibniz, Monadology, translated by Robert Latta, proposition 2)

It’s also true that our language gets a bit strained when we try to talk in a way that never presupposes that the thing we’re referring to is just one thing. You’ll have noticed that when I was trying to do it earlier. On the other hand, there doesn’t seem to be any reason you can’t construct a logic that can accommodate gunky pluralities. You might struggle to get a model theory in terms of sets that didn’t make a certain kind of person a bit grumpy, but this kind of person is already grumpy about variable-domain model theory for modal logic, so you’ll be in good company. (It may be that their grumpiness is warranted, but my impression is that even if the objection in the case of modal logic succeeds, the analogous objection would be question-begging in the case of gunky pluralities. But I’m pretty open to being wrong about that.) One possibility is that the idea of gunky pluralities is one of those things that’s ridiculous without being incoherent. I don’t really get what the problem is supposed to be, though. If you think they’re ridiculous, and it seems at least four of you do, let me know why in the comments!

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