All things great and small

This is a blog-sized summary of a paper I’m working on.

For more than a century now, there’s been a problem with “everything”. Here’s a simple version: say you have all of the sets. Then there ought to be a set of just those things—a set X that contains all the sets. But in that case X is a member of itself, which no set can be. Paradox!

In 1906 Bertrand Russell writes,

[T]he contradiction results from the fact that…there are what we may call self-reproducing processes and classes. That is, there are some properties such that, given any class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property.

Michael Dummett (1993) calls properties like this indefinitely extensible—the main example is “set”, but related paradoxes also show up for “cardinal number”, “order-type”, “property”, and “proposition”. Because of this a lot of philosophers are driven to conclude that we can’t speak intelligibly of all the sets (cardinals, properties, etc.). Whenever we think we’ve caught them all, another pops up to defy us. And if we can’t talk about every set, then we also can’t talk about plain everything—since that would have to include all the sets.

This kind of argument leaves open an escape to somebody with enough nerve: one way out is to deny outright that there are any sets (cardinals, properties, etc.). This is kind of an attractive view anyway, since sets are a lot spookier than, say, tables and chairs and galaxies and electrons—even without the paradoxes. The strong-nerved people who deny the existence of such things are called nominalists (contrasted with platonists or realists).

I have a way to close of the nominalists’ escape route. What we need is a new indefinitely extensible property that isn’t “abstract” (like “set”, etc.): instead, it applies to concrete, material objects. (Even nominalists don’t want to deny those!) I don’t claim that there actually are any such things, though: instead I claim that there could be. This is enough, because it would be very odd if it turned out that “absolutely everything”-talk was intelligible just by luck. The people who think it makes sense to talk that way think that it necessarily makes sense to talk that way. If they’re right, then it shouldn’t even be possible for something to be the way I suggest.

Here’s the idea. Material things could be made of atoms: they might have smallest parts that cannot be divided any further. Alternatively, they could be made of “atomless gunk” (David Lewis’s term (1991)): any piece of it contains ever-smaller bits. Inside our “atoms” we find protons, in the protons we find quarks, and it never stops. Gunk has a long pedigree as a theory of how the world is—and even if it happens to be false about our world, it sure seems like a way a world could possibly be.

But gunk doesn’t by itself give us what we need: it could be that the parts of a gunky material object eventually run out. If you follow finite chains of decreasing objects, there is always something further down—but if you follow infinite chains, you may succeed in getting all the way to the bottom, with nothing smaller below. But also, (it seems) that might not happen. As you go further and further down to smaller and smaller parts, there are always smaller parts further on. An object with parts like this I’ll call supergunk.1

More precisely, an object X is hypergunk iff it satisfies the following condition:

• For any parts of X, the x’s, such that each x is a part of or has as a part each of the x’s, there is something that is a proper part of each of the x’s.

From this condition it follows that “part of X” is an indefinitely extensible property: X is indefinitely divisible. So if there’s trouble for the sets, there is just as much trouble for supergunk. And it sure seems like there could be supergunk (even if there isn’t any in the actual world). So the nominalist has a problem with “everything”, too.

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1. Daniel Nolan (2004) describes something he calls “hypergunk”, but unfortunately that’s a bit different. ↩