*holes*. I say there are holes, and I once spent a while thinking about how one would try to define "hole" adequately. The conclusion I came to was based on the notion of a hole in a manifold---roughly, a region has a hole in it if that region cannot be smoothly contracted to a point. I hadn't thought about it for ages, but pleasingly, this is the definition given by Eric Weisstein at Wolfram Mathworld:

A hole in a mathematical object is a topological structure which prevents the object from being continuously shrunk to a point.

Does this really give you a way of picking out holes, or a way of nominalistically paraphrasing them away? After all, anyone who denies the existence of holes will surely still accept the existence of regions that cannot be contracted (in whatever precise sense is settled upon), so on your proposed definition you can't be in disagreement with them.

ReplyDeleteAlso, here is a tricky case: I take a closed shoebox, which has a hole in a sense that roughly corresponds to our everyday notion of hollowness (if we are going to treat real objects topologically at all, we should consider this one a sphere, so it is simply connected, but not contractible), and I puncture it with a sharp object. We would normally call this 'making a hole', but the shoebox is now topologically a filled disc, which has no holes according to your definition. You could say that various parts of it contain a hole, but I don't think that is enough, since the same could have been said of the intact box.

"so on your proposed definition you can't be in disagreement with them."

ReplyDeleteThey might well just be confused, since we're not sure what a hole is.

"We would normally call this 'making a hole', but the shoebox is now topologically a filled disc, which has no holes according to your definition."

Yes, similarly, a sphere with a small hole is topologically a disk, which, intuitively, has no holes. The number of holes in a surface is not quite topologically invariant. Also, a sphere with two small holes is topologically an annulus, which has, intuitively, one hole (in the middle). So, a hole disappears under the deformation. A sphere with n small holes becomes a disk with n-1 holes. So, perhaps one might count the boundary as a degenerate hole. Or maybe the notion of a hole is not entirely topological, but requires metric notions.

But I do agree with your main point - this definition is still problematic. I think it is a step in the right direction, but it is not quite there.