SUO: Re: Examples! Examples! Examples!
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EEE. Note 13
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Example 1. John Sowa's "Top Level Categories" (cont.)
Returning to the main attraction of the TLC example,
the thing that I would like to really understand as
thoroughly as possible is how exactly the adoption,
the assertion, or the assumption of a particular
sentence in the role of an axiom modulates the
space of models, that is, the space that is
via the medium of that act being conceived.
For ease of reference the details
of the formalization are repeated
here in their revised formulation.
| The Alphabet !TLC!
|
| a_1 = "Abstract"
| a_2 = "Actuality"
| a_3 = "Continuant"
| a_4 = "Description"
| a_5 = "Form"
| a_6 = "History"
| a_7 = "Independent"
| a_8 = "Intention"
| a_9 = "Juncture"
| a_10 = "Mediating"
| a_11 = "Nexus"
| a_12 = "Object"
| a_13 = "Occurrent"
| a_14 = "Participation"
| a_15 = "Physical"
| a_16 = "Prehension"
| a_17 = "Process"
| a_18 = "Proposition"
| a_19 = "Purpose"
| a_20 = "Reason"
| a_21 = "Relative"
| a_22 = "Schema"
| a_23 = "Script"
| a_24 = "Situation"
| a_25 = "Structure"
!A!(!TLC!) = !TLC! |_| !M!, where !M! = {" ", "(", ",", ")"}.
!C!(!TLC!) c !A!*, in accord with the applicable grammar.
$A$(!TLC!) = {!a!_1} = {the axiom inscribed in Table 1}.
Table 1. TLC in Cactus Language (Version 1)
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| |
| (( Object ),( Process ),( Schema ),( Script ), |
| ( Juncture ),( Participation ),( Description ),( History ), |
| ( Structure ),( Situation ),( Reason ),( Purpose )) |
| |
| ( Independent ,( Actuality ),( Form )) |
| ( Relative ,( Prehension ),( Proposition )) |
| ( Mediating ,( Nexus ),( Intention )) |
| |
| ( Physical ,( Actuality ),( Prehension ),( Nexus )) |
| ( Abstract ,( Form ),( Proposition ),( Intention )) |
| |
| ( Continuant ,( Object ),( Schema ),( Juncture ), |
| ( Description ),( Structure ),( Reason )) |
| |
| ( Occurrent ,( Process ),( Script ),( Participation ), |
| ( History ),( Situation ),( Purpose )) |
| |
| ( Actuality ,( Object ),( Process )) |
| ( Form ,( Schema ),( Script )) |
| ( Prehension ,( Juncture ),( Participation )) |
| ( Proposition ,( Description ),( History )) |
| ( Nexus ,( Structure ),( Situation )) |
| ( Intention ,( Reason ),( Purpose )) |
| |
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We are, in effect, back at the observation that I made at the beginning:
| Looked at this way, the universe of discourse for the TLC example
| potentially has 2^25 positions and 2^(2^25) propositions, but the
| axiomset $A$, which we may think of as singling out a single one
| of these propositions, eliminates all but a certain number of
| cells from consideration. This informational constraint can
| be thought of as a sort of quotient operation, whose result
| we can indicate as TLC%/$A$, commonly read as "A% mod $A$".
But what is most earnestly desired here is a way
of intuitively visualizing, and with a bit of luck
quickly computing, the indicated quotient operation.
The mind continues to boggle, and when that happens
I am usually reduced to looking at a reduced example
or else to drawing on experience with related objects.
If it were a mathematical "group" G, and I was trying to compute
or at least to intuit the "quotient group" G/K, which makes sense
"modulo" a "normal subgroup" K, then there are a couple of pictures
that would leap to mind, one of them being category-theoretic and the
other being by analogy geometric. The latter I would sketch this way:
G o
|\
| \
| \
K o \
|\ o G/K
| \ |
| \ |
| \|
1 o----o 1
This makes salient the analogy or the proportion
where G : K :: (G/K) : 1 and G : (G/K) :: K : 1.
It memoizes the notion that K being mapped onto
the trivial group {1}, that wholly consists of
a lone identity element 1, is what induces the
mapping of the big group G to its quotient G/K.
A structure that forms the inverse image of the
identity element under a transformation is called
the "kernel" of that transformation, which is why
the letter "K" came first to mind. According to
this paradigmatic example, if I were to try and
use the potential analogy with group theory,
I would be led to ask: What is the arrow,
the mapping, morphism, or transformation,
and what is the kernel in this case?
On that note, I think that I will call it a day,
and see if addresses the problem to sleep on it.
Jon Awbrey
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