[Inquiry] Re: Dynamics And Logic

[Jon Awbrey]

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DAL.  Note 25

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Staying with the example pq : X -> B, Figure 25-1 shows
the enlargement or shift map E[pq] : EX -> B in the same
style of differential field picture that we drew for the
tacit extension !e![pq] : EX -> B.

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o` `o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` P ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` Q `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /`\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` ` ` ` ` `/```\` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /`````\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` ` ` ` ` `/```````\` ` ` ` ` ` `` ` ` ` ` \` ` ` |
| ` ` / ` ` ` ` ` ` ` ` ` ` ` /`````````\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o`(dp)`(dq)`o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|``o-->--o``|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|```\```/```|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` `(dp) dq` `|``` \`/````|` `dp (dq)` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` o----------------->o<-----------------o ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````^`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o`````|`````o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` ` \ ` ` ` ` ` ` ` ` ` ` ` \````|````/ ` ` ` ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` `\```|```/` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` \``|``/ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o | o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` dp | dq ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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Figure 25-1.  Enlargement E[pq] : EX -> B

A very important conceptual transition has just occurred here,
almost tacitly, as it were.  Generally speaking, having a set
of mathematical objects of compatible types, in this case the
two differential fields !e!f and Ef, both of the type EX -> B,
is very useful, because it allows us to consider these fields
as integral mathematical objects that can be operated on and
combined in the ways that we usually associate with algebras.

In this case one notices that the tacit extension !e!f and the
enlargement Ef are in a certain sense dual to each other, with
!e!f indicating all of the arrows out of the region where f is
true, and with Ef indicating all of the arrows into the region
where f is true.  The only arc that they have in common is the
no-change loop (dp)(dq) at pq.  If we add the two sets of arcs
mod 2, then the common loop drops out, leaving the 6 arrows of
D[pq] = !e![pq] + E[pq] that are illustrated in Figure 25-2.

o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o` `o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` / ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` ` |
| ` ` ` ` `/` P ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` Q `\` ` ` ` ` |
| ` ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /`\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` ` |
| ` ` ` `/` ` ` ` ` ` ` ` ` ` ` `/```\` ` ` ` ` ` ` ` ` ` ` `\` ` ` ` |
| ` ` ` / ` ` ` ` ` ` ` ` ` ` ` /`````\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` ` |
| ` ` `/` ` ` ` ` ` ` ` ` ` ` `/```````\` ` ` ` ` ` `` ` ` ` ` \` ` ` |
| ` ` / ` ` ` ` ` ` ` ` ` ` ` /`````````\ ` ` ` ` ` ` ` ` ` ` ` \ ` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o```````````o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|```````````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|```````````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` `(dp) dq` `|```````````|` `dp (dq)` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` o<---------------->o<---------------->o ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````^`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `|` ` ` ` ` ` ` ` ` ` ` `|`````|`````|` ` ` ` ` ` ` ` ` ` ` `|` ` |
| ` `o` ` ` ` ` ` ` ` ` ` ` `o`````|`````o` ` ` ` ` ` ` ` ` ` ` `o` ` |
| ` ` \ ` ` ` ` ` ` ` ` ` ` ` \````|````/ ` ` ` ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` ` ` ` `\```|```/` ` ` ` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` \``|``/ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `\`|`/` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` |
| ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` \|/ ` ` ` ` ` ` ` ` ` ` ` / ` ` ` ` |
| ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` ` |
| ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` /|\ ` ` ` ` ` ` ` ` ` ` / ` ` ` ` ` |
| ` ` ` ` ` `o-------------------o | o-------------------o` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` dp | dq ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `v` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
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Figure 25-2.  Difference Map D[pq] : EX -> B

Jon Awbrey

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