Here’s some thoughts I had, inspired by Jonathan Schaffer’s talk at the APA a month and a half ago. Basically, I point out that if gunk is a problem for the nihilist, then anti-gunk (if it makes sense) is a problem for the monist. But gunk might not be a problem for the nihilist in the end.

The paper is called “From Nihilism to Monism”, where he argued that any argument leading one to believe that there are no composite objects should in fact push one all the way to believing that there is only one object – the entire universe. Unfortunately, I didn’t stick around for the comments by Ted Sider and Ned Markosian, which I’m sure shed some light on very interesting issues. However, I’m wondering whether some of the arguments could be turned around. For instance, the seeming possibility of gunk (stuff such that every part of it has even smaller parts) can’t be paraphrased by the minimal nihilist (someone who thinks there are just lots of small simples), though it can by the monist (someone who thinks there’s just one big simple).

But what about the possibility of anti-gunk? Just as we used to have an unquestioned assumption that every object has atomic parts, don’t we also have an unquestioned assumption that there is a biggest thing, that is not a part of anything else? For instance, if everything that there is has a finite size, but there is unrestricted (finitary) composition, we could have bigger and bigger things without end. This possibility could not be paraphrased by the monist, but the minimal nihilist could deal with it just fine.

The standard representation of unrestricted (perhaps finitary) composition is with all the objects being elements in a boolean algebra (the bottom element is the only one that doesn’t represent an object). A relatively straightforward theorem points out that a dense subset of the algebra will suffice to represent every element of the algebra as a set of parts. If the algebra is atomic, then the set of atoms will be a dense set. But, as Ted Sider points out in “Van Inwagen and the Possibility of Gunk”, if it’s atomless (or has an atomless part) then any dense set will contain two elements, one of which is a part of the other. This is incompatible with the nihilist position, on which no object is a part of any other.

The dual worry for the monist is if we drop the top element of the boolean algebra, just as we drop the bottom element. Or perhaps if we consider just some distributive lattice, rather than a boolean algebra. There’s no a priori reason why the objects should form a boolean algebra rather than one of these other structures (at least, not obviously, not any more than that the algebra should be atomic).

There might be a paraphrase strategy, where we just talk about some fictional largest thing. But maybe we can do the same in the other direction – even if there are no atoms, we can talk as if there are some! Just as we can fictionally add a top element to the algebra, we can fictionally add elements at the bottom of chains – that is, instead of considering elements of the algebra, we can consider infinite descending chains of elements. Any element can then be represented as the set of all chains containing it. This is exactly analogous to the process by which we represent real numbers as Dedekind cuts or Cauchy sequences of rationals – we add ideal elements at the limits of chains, even though in the “actual” structure, there are no limits. Sider says, “A hunk of gunk does not even have atomic parts ‘at infinity’; *all* parts of such an object have proper parts.” However, for any boolean algebra in which there is gunk (ie, some non-atomic object), there is an atomic boolean algebra in which it can be embedded. Every object in the old algebra will be represented as some object in the new one containing continuum-many atoms. This might raise some concern, because the atomic algebra will have, in addition to the atoms, many new objects (like the finitary joins of atoms, and possibly some countable joins as well) – but the monist can say that the reason we don’t talk about those in ordinary language is that our grasp on the world only gets really large, crude chunks, rather than anything closer to the atoms – this is why the world looks gunky.

Thus, the possibility of gunk isn’t really much of a worry for the nihilist. Sider says, “Surely there are both atomistic possible worlds and gunk worlds, and for that matter in-between worlds with both atoms and gunk.” But I suggest that the nihilist could say that there are only atomistic possible worlds – the ones we might ordinarily call gunk worlds are really just ones in which all our ordinary predicates pick out continuum-sized sets of atoms with certain uniformity properties.

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