[Edit: My interpretation of the thingspace text was probably colored by the context in which I was thinking of it. I still have criticisms of it, but it's not as useless as I was implying it to be. Read the comments for details or proceed with caution.]
Yvain of Less Wrong fame is presently investigating scholastic metaphysics and has posted a list of exploratory questions on his blog. The list is a bit hefty, so I basically referred him to a book. But now I'll cherry-pick a question I found interesting:
5. On the very remote chance that there's anyone here who is familiar both with Aristotelian forms and with the idea of cluster-structures in thingspace, does the latter totally remove the need for the former, do they address different questions, or what?
Do follow his link, because that's the argument I want to bash and that won't make much sense unless you know it.
On the philosophical side the easy answer would be no, because we also need forms to explain diachronic identity as well as creation and corruption. But right now I feel like arguing something more radical: Cluster-structures in thingspace remove the need for nothing because they are bunk. The idea is less than wrong, it is simply meaningless.
To see the meaninglessness consider this ladder of questions: What is a cluster? Basically a cloud of data points that each are close to the next member of the cluster but further away from data points not belonging to the cluster. What does "being close" mean? It means the distance is small. And here comes the main point: What does "distance in thingspace" mean?
The hand-wavy math metaphor of a space is a trap here, because it makes you think of a "nice" space like the real 3D space we (seem to) live in, where the normal laws of geometry apply.  If that was a correct analogy, you could just use the Pythagorean theorem to calculate the distance from the coordinate differences. For example, if I walk 1 meter to the right and then 1 meter forward, I'll end up meters away from where I started. So if I know the coordinates of two points I also know their distance.
But thingspace is not nearly that nice. You can see that from its different dimensions having different units. To take the sparrows example, you might measure volume in liters and mass in kilograms. But what is the sum of 1 liter and 3 kilograms? Is it more or less than the sum of 3 liters and 1 kilogram? Even if you assume some conversion ratio of liters and kilograms, this way of calculating distances will not coincide with your intuitive sense of similarity. For example, if another man is a kilogram heavier than I am, that probably isn't all that dissimilar. But if one amoeba is a kilogram heavier than another one, you'd better watch out for aliens planning to lay their eggs in you.
In fairness, Eliezer Yudkowsky knows it's not quite that simple and proposes
using a distance metric that corresponds as well as possible to perceived similarity in humans
Mathematically, this doesn't make much sense either, but at that point we get the meaning: "distance in thing-space" is just another formulation for "perceived dissimilarity". And then a "cluster in thingspace" is just a set of things that seem similar to other things in the cluster and not to things outside of it. The whole pseudo-math-talk is just borrowing plausibility from a prestigious domain of knowledge that turns out not to have much of a relation to the question at hand.
But dropping the math-stuff, doesn't the idea make sense? Can't the squirrality of a squirrel consist simply in being similar to other squirrels and not so much to dogs? Well, it depends on what you mean by "similar". If you apply it naively, it will turn whales into fishes, because a whale clearly looks a lot more similar to a shark than to a cow. So basically your standard of similarity will have to use a lot of your knowledge of the problem domain. But at that point it gets circular: Essentially similarity is supposed to impose a structure on reality while being defined by that structure. Different fish are all fish by virtue of being similar to each other, but that similarity pretty much consists in them all being fish. At the end of the day you still need an explanation of what all fish have in common. Which means the whole idea of phlogiston thingspace buys you nothing.
While I'm at it, I think the "clusters in thingspace" idea is an instance of a more general failure mode that is fairly common in Less Wrong style arguments. The steps to reproduce the problem on other questions are (1) hand-wavingly map your question to a mathematical structure that isn't well-defined, (2) use that mapping to transfer intuitions, and (3) pretend that settles it. Note that doing steps 1&2 without step 3 is a fine way to generate ideas. But those ideas can still be wrong. If you want them to be right, you either need to replace step 1 by something much more rigorous or restate the ideas without the mathematical analogy and check if they still make sense.
Major examples of the Less Wrong groupthink falling into this particular trap include their vulgar utilitarianism, where the individual utility functions and their sums turn out not to be well-definable, and their radical Bayesianism, which basically assumes a universal probability measure that has no sample space or σ-algebra to live on.
- In this context the technical meaning of "nice" would be that it's a Hilbert space and the given coordinates refer to an orthonormal basis. But don't worry if you don't know what that means, it's unimportant for the argument↵
- Actually that should be squared liters and kilograms, but let's not get pedantic↵
- because that metric would necessarily presuppose knowledge of the clusters, violate the triangle inequality, or result in highly counter-intuitive clusters↵