## Sign Relations, Triadic Relations, Relation Theory • Discussion 4

Dear Cliff,

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!

Regards,

Jon

CJ:
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 $S = \times_{i=1}^k X_i,$ where the $X_i$ are dimensions (something that can vary), typically cast as sets, so that $\times$ here is Cartesian product.  Here $k$ is the dimensional variety (number of dimensions, $k$-adicity), while $n_i = |X_i|$ is the cardinal variety (cardinality of dimension $i,$ $n_i$-tomicity ($n_i$-tonicity, actually?)).  One might think of the two most classic examples:
• Multiadic diatom/nic:  Maximal (finite) dimensionality, minimal non-trivial cardinality:  The bit string $(b_1, b_2, \ldots, b_k)$ where there are $k$ Boolean dimensions $X_i = \{ 0, 1 \}.$  One can imagine $k \to \infty,$ an infinite bit string, even moreso.
• Diadic infini-omic:  Minimal non-trivial dimensionality, maximal cardinality:  The Cartesian plane $\mathbb{R}^2,$ where there are $2$ real dimensions.
There’s another quantity you didn’t mention, which is the overall “variety” or size of the system, so $\prod_{i=1}^k n_i,$ which is itself a well-formed expression (only) if there are a finite number of finite dimensions.

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cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

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