ONT Re: Zeroth Order Ontology
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
ZOO. Note 7
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
In order to comply with recurrent requests, I have been
taking considerable pains to keep my gnosis as close to
the sawdust of this arena as I possibly can, but let me
now catch a breath, two, or three, and indicate several
of the grander and more generic themes that can be seen
stirring already at this level of form, albeit in their
most protean shapes and their most prototypical shifts.
I will try to give the relevant themes generic enough names that they
can be easily recognized again when we encounter them in full bloom,
but please understand that we are presently looking at some of the
most primitive evolutionary precursors and even the most formally
"degenerate" species that fall under the genera in question, and
so the invoking of these anachronistic titles is bound to sound
just a bit too grandiosely precocious as applied to the present
layers of eofauna and eoflora.
Generators and Relations.
To illustrate the algebraic theme of "generators and relations",
let's revert to an even more seedy case than that of bare trees,
the case of "tally marks <[/]>, mod //", also known as <[/|//]>.
This starts out with the formal language L that has a single "generator",
to wit, the tally mark "/". The generator bracket "<[...]>" is used to
indicate the language that is generated by whatever symbols are placed
between the brackets. The duplex bracket "<[...|...]>" manifests the
"generators" on the left side and the "relations" or the "relators"
on the right side of the central divider.
Each "relator" is implicitly considered to define a corresponding "relation",
which is an algebraic equation that sets the relator equal to the relevant
identity element of the algebraic system or formal language in question.
In the case of L/[//] = <[/|//]>, the appropriate identity element is the
empty string !e! = "" of the formal language L, and so the marking of "//"
as a relator means that the relation "//" = !e! = "" holds for the system,
in short, that any pair of tally marks in any string of the intended formal
language can be replaced by the empty string !e! = "", and vice versa, and
in this way producing an "equivalent" string of the language in question.
It is understood in all this that the identity element always belongs
to the generated language L, so <[/]> = {"", "/", "//", "///", ...},
otherwise known as /*. What does it mean to impose the relator "//"?
Just that the "quotient language" L/[//] = <[/|//]> partitions into
a couple of equivalence classes, namely, the even and odd tallies.
Like I said, a grandiose way of describing a very simple critter.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o