[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

PLOI.  Note 11

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

The term "graph" in graph theory refers to an abstract formal object,
of which any scribble or sketch can be but a concrete representation.
Abstraction being a relative condition, one finds that there will be
"labeled graphs", "painted graphs", "plane embedded graphs" (PEG's),
"planar lattice embedded graphs" (PLEG's), and so on to the limits
of formal imagination or practical application, whichever runs out
first on a given day of the week.  Though still abstract, each in
its own peculiar but beautiful way, each example of one of these
more devolved species of graphs is naturally regarded as being
relatively more concrete than its own "underlying graph" (UG).
Note also here the handy strategy of using the longer names
to nomenclate the more concrete abstractions, pinning the
most winged niche names on the most abstract concretions.

One way to tell the abstracter thing from the concreter thing
is by counting up the relative multiplicities, the abstracter
counting as one, relatively speaking, to the concreters' many.

For instance, consider the graph that is pictured as follows:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/|\` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / | \ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` | `\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o ` o ` o ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

There is only one graph of this description.  It is in fact
our old friend K_2,3, as becomes manifest when the relevant
partition of points is disclosed.  There are, in comparison,
10 distinct labeled graphs that share this underlying graph:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` 1 ` ` 1 ` ` 1 ` ` 1 ` ` 2 ` ` 2 ` ` 2 ` ` 3 ` ` 3 ` ` 4 ` `
` `/|\` `/|\` `/|\` `/|\` `/|\` `/|\` `/|\` `/|\` `/|\` `/|\` `
` 3 4 5 2 4 5 2 3 5 2 3 4 1 4 5 1 3 5 1 3 4 1 2 5 1 2 4 1 2 3 `
` `\|/` `\|/` `\|/` `\|/` `\|/` `\|/` `\|/` `\|/` `\|/` `\|/` `
` ` 2 ` ` 3 ` ` 4 ` ` 5 ` ` 3 ` ` 4 ` ` 5 ` ` 4 ` ` 5 ` ` 5 ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o