Topless Models and Mereology

December 25, 2007

I’ve recently been thinking about a thesis once held by Whitehead about the structure of space. Toplessness. Say that the world is topless iff there is nothing which has everything as a part. There are two ways this might happen. 1) Antigunk: everything is a proper part of something. In this case no maximal object exists. 2) Branching: There are at least two things such that the only things they are a part of is themselves. In this case there is no unique maximal object. (Call an object, x, maximal iff for all y, if x is a part of y then x = y.)

Branching is a bit more esoteric. Perhaps someone might hold the universe is branching if they believe the abstract/concrete distinction is exhaustive, and take this to mean that you can’t fuse sets and cats together for fear of hybrid objects, even though objects from the same side of the abstract/concrete division may be fused. Or perhaps you believe in otherworldly objects, but don’t believe in transworld fusions. My main problem with branching is that it violates the principle that any two objects have a mereological sum. If x and y are maximal and distinct, they can’t have a sum otherwise they would both be part of something other than themselves. This aside, the case I am most interested in is that of antigunk. Antigunk is consistent with with the existence of finite sums (take the non-empty finite subsets of omega with subsethood as our model for standard mereology + the sum principle + toplessness – unrestricted fusion.) The real question is whether antigunk is a genuine metaphysical possibility?

I can think of two reasons to think not. The first, and probably weakest, appeals to a particular conception of possible worlds as maximal situations, propositions or something similar and the principle that, x exists at world w iff x is a part of w. For a clear example, on David Lewis’s view worlds are maximal fusions of spatiotemporally related objects. On this theory antigunk is impossible – by definition every world has a maximal part. If there was antigunk it could not be a part of, and hence would not exist in any world.

(To be continued…)



  1. I should have thought of posts with titles like “Topless Models” as a way to get my blog going 🙂

    Having said that, I wasn’t really sure why you thought either of these options aren’t actualised. It’s true that branching doesn’t allow you to have universal pairwise composition. But it’s also arguably true that we don’t have much by way of metaphysical argument that there is, say, a fusion of my desk and the number 7. A systematic ban on abstract/concrete fusions certainly wouldn’t violate Lewis’s strictures against vague composition.

    The possibility of unresricted pairwise composition, but restricted infinite composition is pretty interesting I think. If you think set union is fusion, then orthodox set theory gives you a place where this happens. But it’s certainly possible that there’s a good reason not to believe each of those things (or either of them!).

  2. Hi Brian,

    Thanks for your post – it’s great to have a veteran blogger christen the comments wall :-D.

    It’s true that you’re not going to find an argument for a fusion of your desk and 7 from Lewis style considerations. But I’m more inclined to hold that fusions exist in virtue of the meaning of “part”, and this way pairwise composition should hold unrestrictedly. Your second point is very interesting, and pre-empts some of my next post. I think it is still open for indefinite extensibilists, like Dummett, to retain infinite comprehension, and set fusion as union, due to their view that quantifiers are always restricted on pain of Russell’s paradox. In fact I think unrestricted composition with parthood understood as subsethood can be rewritten as naïve comprehension for sets, so it immediately falls out from the Dummettian position.

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