ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
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Manifold SIG,
I am planning to parade before you only a few more squibs of Lang's snappy presentation of DARM's
before I lay into it with my own hamlet-fausted soliloquy on what it all means to me, but I think
that I can now safely vouchsafe to your long-suffering souls one key of importance to its imports.
I hope that this will serve to suggest at least a hint of a connection to the business of inquiry,
modeling, semiotics, and sign relations, especially with regard to many pressing questions about
change and diversity in our conceptual and symbolic systems, including the problems of designing
interoperable perspectives and mutually intelligible codes for the worlds we vorpally construe.
Let's view our archetype of a manifold, the Figure of a space X
and a couple of charts (U_i, q_i) and (U_j, q_j) from its atlas:
| o---------------------------------------o o-------------------o
| | X | | E_i |
| | | | |
| | | | o |
| | | | / \ |
| | o | | / \ |
| | / \ | | / \ |
| | / \ | | / \ |
| | / \ q_i | | / q_i U_i \ |
| | / o---------------------->| o o o |
| | / \ | | \ / \ / |
| | / \ | | \ / \ / |
| | / U_i \ | | o Eij o |
| | / \ | | \ / |
| | / \ | | \ / |
| | o o o | | o |
| | \ / \ / | | |
| | \ / \ / | | |
| | \ / \ / | o---------|---------o
| | \ / \ / | |
| | o Uij o | q_j o q_i^-1
| | / \ / \ | |
| | / \ / \ | o---------v---------o
| | / \ / \ | | E_j |
| | / \ / \ | | |
| | o o o | | o |
| | \ / | | / \ |
| | \ / | | / \ |
| | \ U_j / | | o Eji o |
| | \ / | | / \ / \ |
| | \ / | | / \ / \ |
| | \ o---------------------->| o o o |
| | \ / q_j | | \ q_j U_j / |
| | \ / | | \ / |
| | \ / | | \ / |
| | o | | \ / |
| | | | \ / |
| | | | o |
| | | | |
| | | | |
| o---------------------------------------o o-------------------o
|
| Figure 1. Manifold Of Sensuous Impressions
For ease of reverence, I resuscitate the revelant liturgy:
¤~~~~~~~~~¤~~~~~~~~~¤~DEFINITION~¤~~~~~~~~~¤~~~~~~~~~¤
| 2. Manifolds
|
| 2.1. Atlases, Charts, Morphisms
|
| 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 holds 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.
|
| Each pair (U_i, q_i) will be called a "chart" of the atlas.
| If a point x of X lies in U_i, then we say that (U_i, q_i)
| is a "chart at" x.
|
| Lang, DARM, pages 20-21.
¤~~~~~~~~~¤~~~~~~~~~¤~NOITINIFED~¤~~~~~~~~~¤~~~~~~~~~¤
Backing away from all of the pointy little pointillistic details a bit,
let us now take in a grandly more impressionistic view of this picture.
Regard all of that busy-ness about Banach this and C^p that as nothing
more than somebody or another's personal aesthetic with regard to what
they think it might be that makes a space "pretty" or a mapping "nice".
Now think of X as being the "object space", the "real" space in which
all of us are really the most interested, at least, if we know what's
good for us, and consider E_i and E_j to be the spaces of, let us say,
my impressions, measurements, nomenclature, senses, signs, symbology,
terminology, utterances, vocabulary, whatever, and yours, respectively.
Let us now focus on the subsets of X, E_i, E_j that are indicated as follows:
| 0. U_ij = U_i |^| U_j c X
|
| 1. E_ij = q_i (U_ij) c E_i
|
| 2. E_ji = q_j (U_ij) c E_j
The mapping of the form q_j o q_i^-1 is what does the work of partially translating
my code into yours, to the extent that it is possible to do so by flipping charts.
This is easier to see if one lays out the maps in a straight line presentation:
| q_i^-1 q_j
| E_ij ------------> U_ij ------------> E_ji
|
Hence, maps of the form q_j o q_i^-1 are called "transition" or "translation" maps.
A helpful hint in this regard is to read "(q_j o q_i^-1)(s)" in either one
of the following ways, according to which reading best suits the occasion:
| (q_j o q_i^-1)(s) = thy own name for what I usually call s.
|
| (q_j o q_i^-1)(s) = the new name for what I used to call s.
In other words, as one says, we are talking about an objective interpretive situation,
with the sign s and the interpretant sign t = (q_j o q_i^-1)(s) for the shared object
x = (q_i^-1)(s).
Next question: Does this manifold picture capture the
most generic brand of objective interpretive situation?
Exercise for the interpreter ...
Jon Awbrey
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