[Inquiry] Re: Sign Relations -- Commentary

[Jon Awbrey]

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SR.  Commentary Note 35

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Re: SR-COM 34.  http://stderr.org/pipermail/inquiry/2005-December/003339.html
In: SR-COM.     http://stderr.org/pipermail/inquiry/2005-December/thread.html#3325

Arisbe List, Inquiry List,

Well, it appears that some dawns take a longer time arriving than others.
But the question stays the night, no matter the number of live-long days.

We return to question of what sort of formal or informational process
makes a circumstance different from what it might have been otherwise,
analogous to the causal or material process in the Rain-Stone example.

To address this question let us think of the form of determination in
the "two points determine a line" example as proceeding by a stepwise
information process, as if we were to be given one point, asking what
sort of constraint it places on the lines that must be incident to it,
and then we are given a second point, which finally determines one of
the lines among the set of lines that are incident to the first point.
Then the circumstance that is determined to be different from what it
might have been otherwise is just our state of information before and
after the information is actually given about the first point and the
second point.

To make this concrete, but also to keep it simple,
let's focus on a small sample of points and lines
that we pick from a larger geometry, specifically,
the six points and the three lines in this Figure:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` 1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` 2 o ` ` ` o 6 ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o-------o-------o ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `3` ` ` ` 4 ` ` ` `5` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

    1.  The set of points is !P! = {p_1, p_2, p_3, p_4, p_5, p_6},
        it being usually enough to use the indices for the points.

    2.  The set of lines is !L! = {L_1, L_2, L_3}, with the data:

        a.  L_1  =  {1, 2, 3}

        b.  L_2  =  {3, 4, 5}

        c.  L_3  =  {5, 6, 1}

Earlier I defined a "dyad" as a pair of distinct points.
Notice that I did not say "ordered pair", so we have as
many dyads in the overall geometry as there are choices
of two different things from a collection of six things,
in short, (6 * 5)/2 = 15 dyads.  But only nine of these
dyads lie on the three lines of our sample, leaving the
following picture of the incidence relations among this
particular sample of geometric points, dyads, and lines:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1 ` 2 ` 2 ` 3 ` 3 ` 1 ` 3 ` 4 ` 4 ` 5 ` 5 ` 3 ` 5 ` 6 ` 6 ` 1 ` 1 ` 5
o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o ` o
`\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/` ` `\`/`
`D_1` ` `D_2` ` `D_3` ` `D_4` ` `D_5` ` `D_6` ` `D_7` ` `D_8` ` `D_9`
` `\` ` ` | ` ` `/` ` ` ` `\` ` ` | ` ` `/` ` ` ` `\` ` ` | ` ` `/` `
` ` \ ` ` | ` ` / ` ` ` ` ` \ ` ` | ` ` / ` ` ` ` ` \ ` ` | ` ` / ` `
` ` `\` ` | ` `/` ` ` ` ` ` `\` ` | ` `/` ` ` ` ` ` `\` ` | ` `/` ` `
` ` ` \ ` | ` / ` ` ` ` ` ` ` \ ` | ` / ` ` ` ` ` ` ` \ ` | ` / ` ` `
` ` ` `\` | `/` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` `\` | `/` ` ` `
` ` ` ` \ | / ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` \ | / ` ` ` `
` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` `
` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` `
` ` ` ` `L_1` ` ` ` ` ` ` ` ` ` `L_2` ` ` ` ` ` ` ` ` ` `L_3` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

Considered within the context of our geometric sample, then, we can
say that picking the point p_1 by itself would pin down the dyads to
the set {D_1, D_3, D_8, D_9}, and that picking the point p_3 by itself
would pin down the dyads to the set {D_2, D_3, D_4, D_6}, so picking the
points p_1 and p_3 determines the dyad D_3, which determines the line L_1.

Jon Awbrey

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