ONT Why Tri?
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Tri, Tri, Tri Again ...
The compositional irreducibility of all 3-adic relations and
the projective irreducibility of many 3-adic relations is
now beyond doubt for all people who grasp the definitions
of terms like "relation", "composition", "projective",
and, last but not least, "and", but of course there
is much more to say about 3-adic relations once
we get past these elementary rudiments of play.
That 3 is a prime is a fact of ordinary arithmetic --
and we have learned to recognize that people who say
e.g. "3 is not a prime because, behold! 3 = 1 + 1 + 1"
are people lacking cognizance of pertinent definitions.
Of course, the fact that 3 is a prime is not the only fact of mathematics,
but it is a fact of some major consequence in mathematics, which is one
good reason to care about the fact of such elementary facts in general.
So, returning from that analogically parallel world of rudimentary arithmetic,
we see that our next task is to investigate the consequences of the facts that
all 3-adic relations are irreducible under relational composition, that being a
matter of definition, and that some, indeed, many 3-adic relations are irreducible
to 2-adic relations plus the 3-adic relation that's invoked by the connective "and",
that being a matter of simple demonstrations and exhaustively concrete exemplitudes.
Many years ago, everyone I knew who had read even a little bit of C.S. Peirce
understood that this "irreducibility of triadics" (IOT) was the linkpin of his
semiotics (the theory of signs), his synechism (the philosophy of continuity),
his platonic or scholastic realism, and indeed his whole system of thinking.
I have been shocked of late to discover that this understanding is no longer
prevalent in some of the places that I would have thought it might have been,
but then "reading" and "understanding" are relative terms, so there is naught
to do but lay out the reasoning of these various connections.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o