Re: Zeroth Order Ontology

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ZOO.  Note 10

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Distinction and Coincidence.

I introduce here a slightly more general, slightly more formal heading
that subsumes the specific algebraic theme of generators and relations.

Let L_2 be the combinatorial species of rooted trees,
regarded modulo the arithmetic relations I_1 and I_2.

o-----------------------------------------------------------o
|                                                           |
|                 o   o                 o                   |
|                  \ /                  |                   |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                ( ) ( )      =        ( )                  |
|                                                           |
o-----------------------------------------------------------o
| Axiom I_1.   Distract <---- | ----> Condense              |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
|                                                           |
|                   o                                       |
|                   |                                       |
|                   o                                       |
|                   |                                       |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                 (( ))       =                             |
|                                                           |
o-----------------------------------------------------------o
| Axiom I_2.     Unfold <---- | ----> Refold                |
o-----------------------------------------------------------o

Let us run through an example of how to evaluate
a string or a tree expression modulo the initial
relations I_1 and I_2.

o-----------------------------------------------------------o
| Example E_1                                               |
o-----------------------------------------------------------o
|                                                           |
|       o   o       o                                       |
|       |   |       |                                       |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|           o       o                                       |
|           |       |                                       |
|           o       o                                       |
|          /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|                   o                                       |
|                   |                                       |
|                   o                                       |
|                   |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|         @                                                 |
|                                                           |
o=============================< QEI >=======================o

In this way, one discovers the formal equation recorded below:

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

Using the square bracket notation for a "formal equivalence class" (FEC),
one says that "( (()) (()) ( (()) ))" is in the FEC [""] = [!e!] of the
empty string, or since we are tacitly ignoring blank spaces, in the FEC
of the blank character.  Consequently, ["( (()) (()) ( (()) ))"] = [""].

Jon Awbrey

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