One interesting concept I read in Wittgenstein was the idea of decomposing a word into its constituent parts. I’ll use the term broom for it because that was the classic example and also the motivation for David Foster Wallace’s novel “Broom of the System.”

So you take “broom” and you could decompose it into “handle” and “bristles”. But then you could decompose it more, by recursively decomposing “handle” into “grains of wood” and “bristles” into “pieces of fiber” (or whatever).

You keep doing this ad infinitum, I guess on down to the summation of a bunch of quarks or whatever.

The question of interest to Wittgenstein was where does this process bottom out. What would it mean, either physically or semantically, to have a word identifying a concept that could not be broken down into further constituent parts.

Wittgenstein was interested in this for the philosophy of language. But I got interested in it by thinking about the decomposition as a mathematical operator,

D(“broom”) = {“handle”, “bristles”}

and then asking what it could mean if this operator D had an “eigenvector” with an “eigenvalue” of 1, so that Dx = x for some non-decomposeable word x.

In some ways, you can see how it could relate to things like word2vec and embedding representations if you could represent a decomposition operator, and define a hierarchical relationship of words as an ordering of how to more or less specifically decompose a word’s representation.