[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 7

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Cybernetics List, Peirce List,

Two paces back I used the word "category" in a way that will turn out
to be not too remote a cousin of its present day mathematical bearing,
but also in way that's not unrelated to Peirce's theory of categories.

When I call to mind a "category of structured individuals" (COSI),
I get a picture of a certain form, with blanks to be filled in as
the thought of it develops, that can be sketched at first like so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` Category` ` ` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `
` Individuals ` ` ` ` ` ` o ` `...` ` o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` / ` \ ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` `/` ` `\` ` ` ` ` ` ` ` ` `
` Structures` ` ` ` ` o->-o->-o ` o->-o->-o ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The various glyphs of this picturesque hierarchy serve to remind us
that a COSI in general consists of many individuals, which in spite
of their calling as such may have specific structures involving the
ordering of their component parts.  Of course, this generic picture
may have degenerate realizations, as when we have a 1-adic relation,
that may be viewed in most settings as nothing different than a set:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` Category` ` ` ` ` ` ` ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `
` Individuals ` ` ` ` ` ` o ` `...` ` o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
` Structures` ` ` ` ` ` ` o ` `...` ` o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

The practical use of Peirce's categories is simply to organize
our thoughts about what sorts of formal models are demanded by
a material situation, for instance, a domain of phenomena from
atoms to biology to culture.  To say that "k-ness" is involved
in a phenomenon is simply to say that we need k-adic relations
to model it adequately, and that the phenomenon itself appears
to demand nothing less.  Aside from this, Peirce's realization
that k-ness for k = 1, 2, 3 affords us with a sufficient basis
for all that we need to model is a formal fact that depends on
a particular theorem in the logic of relatives.  If it weren't
for that, there would be hardly any reason to single out three.

Jon Awbrey

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