Re: Zeroth Order Ontology

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ZOO.  Note 6

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ZOOlogists,

Last time we looked at what certain "schools of thought" (SOT's)
in graph theory would call "bare, free, rooted trees".  They are
"bare" because they bear no colors, labels, or other distinctive
markers on the nodes, "free" because, even though they appear to
be drawn in the plane of the page, they are not really conceived
to be bound in 2 dimensions the way that so-called "plane trees"
and "planted plane trees" would be embedded, fixed, or tied down.
At this point, the language begins to waffle from SOT to SOT, so
I will just lay out the relevant distinctions roughly in the way
that I was taught, as they naturally come to bear on the account.

Taking the freeness and rootedness as understood for the rest of
this discussion, the bare trees correspond to what GSB called the
"arithmetic" of the corresponding logical "algebra".  He made the
all-important observation that every algebra has its arithmetic,
but that in logic we find ourselves in the peculiar situation of
having discovered the algebra first, before we had a good grasp
of what the proper arithmetic might be.  With this in hindsight,
or maybe hindmind, many of the graphical explorations that CSP
went through can be seen as a search for a fitting arithmetic.

In the formalisms that derive from CSP and GSB,
the logical arithmetic is defined by the first
two "laws of form", namely, the following pair
of "arithmetic initials":

o-----------------------------------------------------------o
|                                                           |
|                 o   o                 o                   |
|                  \ /                  |                   |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                ( ) ( )      =        ( )                  |
|                                                           |
o-----------------------------------------------------------o
| Axiom I1.    Distract <---- | ----> Condense              |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
|                                                           |
|                   o                                       |
|                   |                                       |
|                   o                                       |
|                   |                                       |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                 (( ))       =                             |
|                                                           |
o-----------------------------------------------------------o
| Axiom I2.      Unfold <---- | ----> Refold                |
o-----------------------------------------------------------o

I will go ahead and call these initial rules by the name of "axioms",
though strictly speaking they are purely formal transformations, and
formally precede any conception of truth or falsity.

Jon Awbrey

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