Speak with the vulgar.

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Getting in touch with the universe

with 7 comments

In my last post I argued that the set-theoretic problems with “absolutely everything” carry over even for those who don’t believe in sets, by appealing to the possibility of “supergunk”. There’s another route to the same conclusion by way of some principles about contact. I think it’s kind of neat.

Let’s take contact to be a two-place relation between objects; it is reflexive (we count overlap as contact), symmetric, and monotonic: if X touches a part of Y, then X touches Y. These are all standard so far.

The following additional principles seem jointly possible:

  1. A pretty weak separation principle: if X and Y don’t touch, then there is some further Z that doesn’t touch either of them. (Think of Z as being located between X and Y, keeping them apart.)

  2. A very strong distribution principle: if X touches the fusion of the \phi’s, then X touches some \phi. (Since the last post, I’ve switched from plural quantification to schemes, since I think it helps avoid some issues.) We might call this contact supervenience: what touches the whole touches some part.

The finite version of distribution is completely tame and standard: if X touches Y + Z, then X touches Y or X touches Z. It’s very hard to imagine the finite version failing. It turns out that the general version can fail, though. For instance, none of the intervals \left[\frac{1}{n}, 1\right] touches the interval \left[-1, 0\right]; but their fusion does (under ordinary topology). But this is pretty counterintuitive (John Hawthorne has written a whole paper about the principle’s failure). And so, even if it turns out that actually contact doesn’t supervene on parts, it still strikes me as a way things could have been.

But these two principles together give rise to another extensibility argument. Suppose that something doesn’t touch X. Given any \phi’s that don’t touch X, their fusion doesn’t touch X by (2), and so by (1) there is some further thing that doesn’t touch X. So the \phi’s, whatever they may be, don’t exhaust the things that don’t touch X: the non-X-touchers are indefinitely extensible. Thus, in a world where (1) and (2) hold, it doesn’t make sense to talk about absolutely everything there is.

To sum up: (1) and (2) are jointly possible; therefore, generality absolutism is possibly false. Since generality absolutism isn’t contingent, generality absolutism is actually false.

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Written by Jeff

December 6, 2008 at 7:59 pm

7 Responses

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  1. Hmm, I’m not sure I agree with your two principles.

    The second one I think I have a countexample to here (see the first picture): http://possiblyphilosophy.wordpress.com/2008/10/07/when-do-two-objects-touch/

    As for the first, what about (\infty, 0) and (0, \infty) in the reals? There doesn’t seem to be another object not touching either.

    Lastly, I’m not sure I agree that those two principles could be jointly true. Suppose X doesn’t touch something. Let Y be the fusion of the objects that don’t touch X. By 2, Y doesn’t touch X. By 1 there is an object Z which doesn’t touch either. But Z must be a part of, and hence touch Y, since Y is the fusion of the things that don’t touch X. #.

    Andrew

    December 8, 2008 at 5:41 am

  2. Sorry, I noticed you already gave a counterexample to 2 in your post. Also, that should have been (0, -\infty) not (0, \infty) .

    Andrew

    December 8, 2008 at 5:43 am

  3. You’re pretty much right about both counterexamples. I’m not so sure about the second one; the most natural condition for contact in the real line I know of is having overlapping closures, in which case the intervals (-\infty, 0) and (0, \infty) do touch.

    A more convincing counterexample to the separation principle is the case of wholly disconnected spaces: they don’t touch, but not by virtue of there being something else in between them.

    But I wasn’t claiming that the principles are necessarily or even actually true: just that they’re possibly true (and in particular, though I didn’t specify this, possibly true when the quantifiers are restricted to the parts of some object). So I can grant the counterexamples.

    As for your objection to the mere possibility: the upshot of the extensibility argument is precisely that we can’t speak of all the objects that don’t touch X. So of course there is no object that is the fusion of all of them. In fact, your argument basically restates the extensibility argument—doesn’t it?

    Jeff

    December 8, 2008 at 2:24 pm

  4. “In fact, your argument basically restates the extensibility argument—doesn’t it?”

    Yes it does. Sorry, I was being a bit slow catching on.

    So I guess my worry is this: when I saw your two principles I thought to my self, they’re inconsistent, so they can’t be jointly possible. But of course, the indefinite extensibilist will say they’re not inconsistent, your quantifiers are just stretchy: I can try to take the fusion of *all* the non-touchers, but I will always have missed something out. But where do you draw the line? What about the possibility that (1) everything is red, and (2) something is not red. Suppose you have some phi’s. They’re all red by (1). There’s a non red thing by (2), so it can’t be a phi… etc.

    “the most natural condition for contact in the real line I know of is having overlapping closures”

    Actually I disagree. I think the most natural condition is that x touches y iff x’s closure overlaps y, or y’s closure overlaps x. Given standard definitions, this coincides with the topologists notion of self connectedness (not being the fusion of two open objects.)

    But that aside, your definition has consequence that \mathbb{Q} is self-connected, when intuitively its very disconnected. (Also the four colour theorem turns out false :D.)

    Andrew

    December 8, 2008 at 3:02 pm

  5. 1. Yeah, as I said in my comment on the previous post, I’m no better off than any other indefinite extensibility guy. It’s tricky to figure out how to talk about “stretchy” quantifiers. I think Kit Fine’s way is formally pretty sharp (though one of the upshots of these arguments is that his way of glossing the formalism, in terms of “postulation”, looks like a mistake). And in both of my original posts, I was being quite sloppy about it.

    2. Yeah, that does sound like a better definition. So the separation principle probably is false in the real line. So much the worse for the real line! =)

    Jeff

    December 9, 2008 at 12:48 pm

  6. Yeah, I’m inclined to agree with you on Fine (on the benefits of going modal, and the mysteriousness of postulational modality.) Actually, Oystein Linnebo has an interesting modal version of indefinite extensibility (in terms of what sets you can construct.) I’m not sure if its been published yet, but I recommend having a look at that.

    Andrew Bacon

    December 9, 2008 at 2:23 pm

  7. […] interested, I’ve written a short paper on my nominalistic indefinite extensibility arguments. (This is also my way of making good on my offer in the comments to discuss a sort of consistency […]


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