Re: Zeroth Order Ontology

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ZOO.  Note 12

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I will continue on the hopeful presumption that folks
are becoming more transversant with parsing my graphs,
after all -- "a man's reach must exceed his graph, or
what's a heaven for".

When last we convened we had just discovered 
the following two "formal equations" (FEQ's):

o-----------------------------------------------------------o
| Equation E_1                                              |
o-----------------------------------------------------------o
|                                                           |
|       o   o       o                                       |
|       |   |       |                                       |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                   =                   @         |
|                                                           |
o-----------------------------------------------------------o
|     ( (()) (()) ( (()) ))   =                             |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
| Equation E_2                                              |
o-----------------------------------------------------------o
|                                                           |
|     o       o   o   o                                     |
|      \     /     \ /                                      |
|     o o   o o   o o o                                     |
|      \|   |/     \|/                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                             =         @         |
|                                                           |
o-----------------------------------------------------------o
| ( (()(()))((())()) ( (()(()())()) ))  =                   |
o-----------------------------------------------------------o

Letting the respective expressions indited here be !e!, e_1, e_2,
the transitive property of "formal equivalence relations" (FER's)
yields the fact that [!e!] = [e_1] = [e_2], in short, the FEC of
one is the FEC of all.

But bringing the exprssions e_1 and e_2 together on the same page like this,
we may now see a certain analogy, or an "abstract family isomorphism" (AFI),
that affects the forms of the two expressions.  The AFI between e_1 and e_2
finds articulation in many ways, in particular, it may be given a so-called
"parametric expression" in the following fashion.

Let us consiliate the two expressions, e_1 and e_2, under a common pattern
by marking the places where e_1 and e_2 differ in the following way, where
the "x" signs mark in each spot where they appear in the composite drawing
of the expression e_3 an appearance of the empty constant !e! or the blank
constant " " in the expression e_1 and an appearance of the bound constant
"()" in the expression e_2.

o-----------------------------------------------------------o
| Equation E_3                                              |
o-----------------------------------------------------------o
|                                                           |
|     x o   o x   x o x                                     |
|       |   |       |                                       |
|     x o   o x   x o x                                     |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                       =    @    |
|                                                           |
o-----------------------------------------------------------o
| ( ( x ( x )) (( x ) x ) ( ( x ( x x ) x ) ))    =         |
o-----------------------------------------------------------o

This begins to appear like a full-fledge algebraic expression,
but try for a moment to forget that you ever heard of algebra,
a fantasy earnestly to be desired, and seek to view it afresh.
In this fresh light, all that we have done is to introduce an
ad hoc markup device to mark two variations on a single theme.

Jon Awbrey

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