An example of a value-driven epistemological approach to metaphysics

1. Everything that exists is intrinsically valuable.

2. Shadows and holes are not intrinsically values.

3. So, neither shadows nor holes exist.

Fake counting

When someone’s walking speed is two miles per hour, there are not two things, “one mile per hour walkings”, that are present.

When we say that a sculpture has three dimensions, we are not saying there are exactly three things—dimensions?—that are present in it. But are there not width, height and depth? In a way. But rotate the sculpture by 45 degrees, and “width”, “height” and “depth” refer to measurement along three other axes. There are, it seems, infinitely many axes along which the sculpture can be non-trivially measured.

These are examples of what one might call “fake counting”. We speak as if there were n of something, but the following argument is invalid:

1. There are n Fs.

2. n ≥ 1.

3. So, there are some Fs.

And, similarly, this is invalid:

4. There are exactly two Fs.

5. So, ∃x∃y(F(x)&F(y)&∀z(F(z)→(z = x ∨ z = y))).

In fake counting of Fs, there is counting involved, but it is not counting of Fs. For instance, when we say that the sculpture has three dimensions, we mean something like this:

• there are three mutually perpendicular axes such that the sculpture has non-zero extent along each of them, but there are no four such axes.

So, there is a counting of axes, but it is not a counting of dimensions. If we were counting dimensions, we would have to have say what the first one is, what the second one is and what the third one is, and as the rotation thought experiment shows, that doesn’t work. And the counting of axes doesn’t involve counting axes overall, but rather axes in a particular set of them.

We need to beware of fake counting when making metaphysical arguments for the existence of entities of some sort. For instance, topologists have ways of “counting holes”. But topological properties are invariant under deformations. Now, imagine a pancake with, as we would say, “one hole in the middle”. Well, however we distort the pancake, it has one topological hole. But if we ask where that hole is, there is no topological answer to it (in the animation below, is the hole outlined in red or in blue?). So, topological hole counting is fake counting.

Shapes of holes

The ordinary notion of a hole is kind of dubious. Consider the hole in the thin wavy sheet of rubber on the right. What is the shape of that hole? How thick is it? Is it exactly as thick as the rubber sheet? But the rubber sheet varies in thickness, actually. How does it stretch from its wavy edges to the middle? Does it have a sinewave bump in the middle, to correspond to where there are sinewave bumps in the sheet elsewhere? Or does that depend on the history of its formation (e.g., maybe if the sheet used to have a bump there but then a hole was made--that's how my code generating this picture works--then the hole has a bump, but if the sheet was pre-made with a hole, then the hole is flatter)? I think there really are no good answers to these questions, and hence holes don't exist.

Holes and substantivalism

Suppose substantivalism about space is correct. Imagine now that the following happens to a slice of swiss cheese: the space where the holes were suddenly disappears. I don’t mean that the holes close up. I mean that the space disappears: all the points and regions that used to be in the hole are no longer there (and any air that used to be there is annihilated). The surfaces of the cheese that faced the hole now are at an edge of space itself.

The puzzle now is that in this story we have an inconsistent triad:

1. There is no intrinsic change in the cheese.
2. The slice of cheese no longer has holes.
3. Changing with respect to whether you have holes is intrinsic.

Here are my arguments for the three claims. There is no intrinsic change in the slice of cheese as something outside the cheese has changed—space has been annihilated. The slice of cheese no longer has holes, as it makes sense to talk of the size or shape or volume of a hole, but there is no size or shape or volume where there is no space. And changing with respect to whether you have holes is change of shape, and changes of shape are intrinsic.
It seems that the above story forces you to reject one of the following:

4. Substantivalism about space
5. Intrinsicness of shape.

But there is another way out. Deny (3). Whether you have holes is not intrinsic. What is intrinsic is your topological genus with respect to your internal space and similar topological properties.

Note, also, a lesson relevant to the famous Lewis and Lewis paper on holes: the counting of holes should not involve the counting of regions, but the computation of a numerical invariant, namely the genus.

Caves and holes

On the face of it, the ontology of caves is just like that of holes. But there do seem to be some significant differences. A hole is, as it were, pure nonbeing, albeit of a sort that depends on its walls for its existence. But a cave includes the hole and the walls. When we talk of cave walls, we think of them as parts of the cave. But what is the wall? Is it just a veneer of the rock? It seems to me that if you could take a cave and cut it out of the rock, keeping a thin veneer of rock, and put it in space (so the veneer wouldn't collapse under its own weight), the cave would survive. So only a veneer is essential. Is only a veneer of rock part of the cave? No—more is. For the physical properties of the cave—say, the material causes of stalactites—probably include more than a veneer. Besides, if only a veneer of wall is included in the cave, then only a veneer of stalactites and stalagmites would be included. But that's absurd. So, more than a veneer of wall is part of the cave. But how much? This is, surely, vague.

So caves seem to be more ontologically solid than holes, and it seems that we can answer a bunch of questions about them. But like sand sculptures, they depend ontologically on nonbeing—if you fill in the hole (or cover the sand sculpture with sand) with solid rock, it seems the cave perishes. This ontological dependence shows that caves (and sand sculptures) are not substances. If they are at all.