SUO: Re: Transformations Of Discourse
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[ Comment On Notation:
|
| I cannot apologize profusely enough for all of
| those rudely mechanical, syntacky, tendentious,
| and ugly devices and all of the rigors that it
| takes to mortise such bits and pieces together
| into any semblance of a suitably congeried fit.
| This has been, from my point of view, like one
| of those stories that gets to be a movie first
| and only a bit later, as an afterthought, gets
| turned into a book. As I write, my head swims
| with variegated pictures and sculptural shapes
| that I thrash about for a way to linearize, in
| all of the manifold connotations and senses of
| that undeservedly and unexpectedly polysemious
| term "linear", and therefore I, at least, have
| the consolation of my images and my ideals, if
| and when I fail to convey them in this printed
| way, but I know that the inverse metamorphosis,
| from string to parti-colored picture, and from
| all of that to solid bodies in glorious action,
| well, I know that I have left the harder parts
| to the interpreters and the performers thereof.
|
| After that doleful note of notational benediction,
| I suppose that it would be a good idea to refresh
| our memories and continue the series of pictorial
| images that are envisioned here to carry the data
| in its liveliest, succinctest, and most vivid way.
]
Extension from 1 to 2 Dimensions
Figure 18-a lays out the "angular form" of venn diagram for universes
of 1 and 2 dimensions, indicating the embedding map of type B^1 -> B^2,
and detailing the coordinates associated with individual cells. Because
all points, cells, or logical interpretations are represented as coherent
geometric areas, we can say that these pictures provide an "areal view" of
each universe of discourse.
o o
/ \ / \
/ \ / \
o o o o
/ \ / \
/ \ / \
o o o 1 1 o
/ / \ / \ / \
/ / \ / \ / \
o 1 o o o o o o
/ / \ / \ / \
/ / \ / \ / \
o o o >>>--->>> o 1 0 o 0 1 o
|\ / / |\ / \ /|
| \ / / | \ / \ / |
| o o 0 o | o o o o |
| \ / / | \ / \ / |
| u \ / / | u \ / \ / v |
o-----o o o-----o 0 0 o-----o
\ / \ /
\ / \ /
o o o o
\ / \ /
\ / \ /
o o
Figure 18-a. Extension from 1 to 2 Dimensions
Figure 18-b shows the differential extension from X° = [x] to EX° = [x, dx]
in a "bundle of boxes" form of venn diagram. As awkward as it may seem at
first, this type of picture is often the most natural and easily available
representation when we require to conceptualize the localized information
or the momentary knowledge of an intelligent dynamic system. It affords
a ready picture of a "proposition at a point", in the present instance,
an image of a proposition about changing states, altogether composing
a depiction of a dynamic situation that is itself associated with or
attached to a particular dynamic state of the system in question.
I think that it is easy to see how this pattern of application,
this style of "appliqué", might be extended to conceive of
more general types of instantaneous knowledge possessed
by the appropriate kind of intelligent dynamic system.
o---------o---------o---------o---------o o---------o---------o
| X | | dX |
| o | | o |
| / \ | | / \ |
| / \ | | / \ |
| / o---------------------->o o dx o o
| / \ | | \ / |
| / \ | | \ / |
| / \ | | o |
| / \ | | |
| / \ | o---------o---------o
| / \ |
o o x o o
| \ / |
| \ / | o---------o---------o
| \ / | | dX |
| \ / | | o |
| \ / | | / \ |
| \ / | | / \ |
| \ / o------------>o o dx o o
| \ / | | \ / |
| \ / | | \ / |
| o | | o |
| | | |
o---------o---------o---------o---------o o---------o---------o
Figure 18-b. Extension from 1 to 2 Dimensions
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Have to take a break here,
Hope to continue later on,
Jon Awbrey
[ "Differential Logic & Dynamic Systems"
|
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Created: Dec 16, 1993
| Revised: Oct 31, 1994
| Revised: Mar 15, 2000
| Revised: Mar 02, 2001
| Project: Engineering 690
| Advisor: M.A. Zohdy
| Setting: Oakland University
| Excerpt: Pages 28-29.
]
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