Many thanks for your thoughtful reply. I copied a transcript to my blog to take up first thing next year. Here’s hoping we all have a better one!
- I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between “dimensional variety” and “cardinal variety”. Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system where the are dimensions (something that can vary), typically cast as sets, so that here is Cartesian product. Here is the dimensional variety (number of dimensions, -adicity), while is the cardinal variety (cardinality of dimension -tomicity (-tonicity, actually?)). One might think of the two most classic examples:
- Multiadic diatom/nic: Maximal (finite) dimensionality, minimal non-trivial cardinality: The bit string where there are Boolean dimensions One can imagine an infinite bit string, even moreso.
- Diadic infini-omic: Minimal non-trivial dimensionality, maximal cardinality: The Cartesian plane where there are real dimensions.
- There’s another quantity you didn’t mention, which is the overall “variety” or size of the system, so which is itself a well-formed expression (only) if there are a finite number of finite dimensions.