[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 36

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Cybernetics List, Peirce List,

The relation between the primary arithmetic and the primary algebra is
founded on the idea that a variable name in the algebra indicates the
contemplated absence or presence as an operand of any expression in
the arithmetic, with the understanding that each appearance of the
same variable name indicates the same state of contemplation with
respect to the same expression of the arithmetic.

For example, consider the following expression:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` a ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` o-----o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

We may regard this algebraic expression as a general expression
for an infinite set of arithmetic expressions, starting like so:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` o ` ` ` ` ` o ` o o ` o
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` `\`/` `\`/`
` ` ` ` ` ` o ` ` o o ` o o ` o o ` ` o o o o o o o o ` ` o `
` ` ` ` ` ` | ` ` | `\`/` `\`/` | ` ` | `\|/` `\|/` | ` ` | `
` o-----o ` o-----o ` o-----o ` o-----o ` o-----o ` o-----o `
` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `
` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `
` O ` ` ` ` O ` ` ` ` O ` ` ` ` O ` ` ` ` O ` ` ` ` O ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Now consider what this says about the following algebraic law:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` a ` ` a ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` o-----o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` O ` ` ` ` = ` ` ` ` O ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

It permits us to understand the algebraic law as saying,
in effect, that every one of the arithmetic expressions
of the contemplated pattern evaulates to the very same
canonical expression as the upshot of that evaluation.
This is, as far as I know, just about as close as we
can come to a conceptually and ontologically minimal
way of understanding the relation between an algebra
and its corresponding arithmetic.

Jon Awbrey

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