Sign Relations, Triadic Relations, Relation Theory • Discussion 4

Posted on December 31, 2020 by Jon Awbrey

Dyadic Versus Dichotomic

Re: Previous Post
Re: Cybernetics • Cliff Joslyn

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.

Resources

cc: Cybernetics • Ontolog • Peirce List (1) (2) • Structural Modeling • Systems Science

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