SUO: Re: Manifolds Of Sensuous Impressions (MOSI's)
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| This paper is based upon the theory already established, that the function of
| conceptions is to reduce the manifold of sensuous impressions to unity, and that
| the validity of a conception consists in the impossibility of reducing the content
| of consciousness to unity without the introduction of it. (CSP, CP 1.545, CE 2.49).
SUO Work Groupers,
I have been charged, if not tried and convicted,
of scaring people and horses, on account of the
the noise of this new-fangled contraption makes
and all of that outlandish, unheard-of rumbling
that emanates from under my bonnet and manifold.
And so I am sentenced to the punishment that my
grandparents once told me befit their times, of
sending a forerunner ahead of the car, you know,
the one whose tired new wheels are still yet to
get themselves invented, a harbinger as it were,
to wave a flag or ring a bell or cry the alarum,
but softly, very softly.
All kidding aside, I was getting to point of drawing you a picture, anyway,
since it is just the thing that called for in order to reduce the manifold
of symbolic ingressions to a unity of iconic complexion and due proportion.
Here is the typical picture of their subject to which manifold theorists
have become accustomed, that, were it to be drawn in a more fluid medium,
and not so badly quartered in this e-current style, would be e-mediately
recognizable as the "Planarian", more popularly, the "Flatworm Diagram".
Here, again, for ease of reference, is the definition of an atlas of class C^p:
| Let X be a set. An "atlas" of class C^p (p >= 0) on X is a collection
| of pairs (U<i>, q<i>) (i ranging in some indexing set), satisfying the
| following conditions:
|
| AT 1. Each U<i> is a subset of X and the U<i> cover X.
|
| AT 2. Each q<i> is a bijection of U<i> onto an open subset q<i>U<i>
| of some Banach space E<i> and for any i, j, [it is true that]
| q<i>(U<i> |^| U<j>) is open in E<i>.
|
| AT 3. The map
|
| q<j> o q<i>^-1 : q<i>(U<i> |^| U<j>) -> q<j>(U<i> |^| U<j>)
|
| is a 'C^p'-isomorphism for each pair of indices i, j.
|
| (Lang, DARM, page 20).
And here is (a squared-off version of) the paradigmatic picture,
capturing what is most of the essence in our manifold situation:
o---------------------------------------o o-------------------o
| X | | E<i> |
| | | |
| | | o |
| | | / \ |
| o | | / \ |
| / \ | | / \ |
| / \ | | / \ |
| / \ q<i> | | / q<i>U<i>\ |
| / o---------------------->| o o o |
| / \ | | \ / \ / |
| / \ | | \ / \ / |
| / U<i> \ | | o o |
| / \ | | \ / |
| / \ | | \ / |
| o o o | | o |
| \ / \ / | | |
| \ / \ / | | |
| \ / U<i>\ / | o---------|---------o
| \ / \ / | |
| o |^| o | q<j> o q<i>^-1
| / \ / \ | |
| / \ U<j>/ \ | o---------v---------o
| / \ / \ | | E<j> |
| / \ / \ | | |
| o o o | | o |
| \ / | | / \ |
| \ / | | / \ |
| \ U<j> / | | o o |
| \ / | | / \ / \ |
| \ / | | / \ / \ |
| \ o---------------------->| o o o |
| \ / q<j> | | \ q<j>U<j>/ |
| \ / | | \ / |
| \ / | | \ / |
| o | | \ / |
| | | \ / |
| | | o |
| | | |
| | | |
o---------------------------------------o o-------------------o
Figure 1. Manifold Of Sensuous Impressions
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Next time, I'll be transmitting to you
from the other hemisphere of the brain.
It won't be long till you long for the
days when all I did is read you poetry!
Until Then,
Jon Awbrey
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