[Arisbe] SUO: Re: Manifolds Of Sensuous Impressions (MOSI's)
Jon Awbrey
arisbe@stderr.org
Thu, 08 Mar 2001 14:52:02 -0500
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
I owe you an explanation -- I knew this even before
a veritable Hitchcockian flock of little birds told
me so -- of why I'm afrighting the sensitive viewer
with "brief quotations of elementary definitions
from the introductions to mathematics texts".
So here is my first pass to e-mit,
hopefully someday to re-quit,
this promissory, all too
promissory note.
One of my intentions is intellectually historical,
to try to heal a certain breach that has ariven
in our "welt-hysterical-lack-of-consciousness".
I am quite serious about the suggestion that
is implied by my third-hand title, the one
that I borrow from Peirce's lift of Kant.
And I have not looked into it yet, but
I just suspect that these connotations
suffused the air that Riemann breathed.
Query. If there are any folks on the German scene
who have any knowledge germane to this speculation,
it would really save me the painstaking archeology
of having to dig it up out of some dustier bins of
the library, so I would be grateful to learn of it.
On this same historical note, I have of late vaunted a number of assertions,
addressed to no one in particular, in spite of how they chanced to be taken,
to the effect that many of these wheels were set spinning quite a ways back.
Just to refresh our intermediate term memories, I cite four recent examples:
---------------------------------------------------
| Mathematicians are as lacking in historical consciousness
| as most folks can be, but those brave souls who do manage
| to master their own recursèd horrors of drowning in their
| own "horror of history" (HOH) will most likely be able to
| inform you that the concepts of "category" and "manifold",
| as used in their current and continuing argosies of argot,
| come straight -- well, moderately straight! -- out of the
| precursory and prolegomenal problematics of Aristotle and
| Kant, and there are specific reasons for their provenance
| that we would do well to recognize in these fora, even as
| we strive to recreate ourselves here in re-creating a lot
| of what was developed in 19th Century mathematics, if but
| slightly transmuted with epistemological and ontological
| nuances. But that yarn will have to be the carding and
| the spinning and the weaving of yet another looming day.
|
| http://suo.ieee.org/email/msg02683.html
---------------------------------------------------
| Some folks I know say that our ontology, that is to say,
| "what ought to be our ontology", is related to the level
| of formal, logical, & mathematical description in the way
| that invariants are related to the transformations between
| representations over which they are in fact the invariants.
|
| This brings in a slightly novel perspective, in that,
| even though we tend to think of our ontology as being
| on the reality-side of our representational interface,
| it is by operations on the inner-face of this carapace
| that we come to acquire a re-constituted image of being.
| This is connected, I imagine, to that algebraic principle
| by which the "double dual" is closely akin to the original,
| all of which happens, sure enough, only under ceratin cases.
|
| http://suo.ieee.org/email/msg03627.html
---------------------------------------------------
| I am not sure, so I will have to let John Sowa correct
| me if I have guessed wrong, but I took him to speaking
| at least partly metaphorically -- you know how he is! --
| that is to say, making use of a certain analogy between
| "ontological frameworks" (OF's) and "coordinate systems".
| Myself, I often use the phrase "frame of reference" (FOR)
| in a broadly punning sense, as a bridge concept to cover
| what both varieties of conceptual systems have in common.
|
| Thus, if we follow this tentative analogy a little further,
| the next question that we ask is: How do we conceptualize
| and represent a "trans-FOR-mation" between two FOR's? And
| what we might have in mind, if we try to push the analogy
| to its ultimate breaking point, is this general question:
| How do we recognize the things, represented as functions
| of each interpreter's or observer's conceptual coordinates,
| that do happen in fact to remain invariant over very wide
| varieties of transformations among these particualr FOR's?
|
| All of this compasses a very wide topic area that I have had on my mind
| for a quite a while now, and so I will lean to say some things about it
| on a thread that I can stretch out in the general direction of that aim.
| I plan to entitle it, as I have before: "Transformations of Discourse".
|
| http://suo.ieee.org/email/msg03671.html
---------------------------------------------------
| Fortunately -- though it would be a shock to some people to think
| they might have missed the Boom! -- this particular sound barrier
| was pretty much conquered in the Nineteenth Century when Riemann
| worked out his theory of manifolds, stealing a goodly measure of
| the aweful thunder that those "manifolds of sensuous impressions"
| have enjoyed since the Days of Kant, and I think that it is safe
| to say that regular flights across this obstruction have been in
| full operation since the Company of Category Theory, Ltd. first
| began, in the late great Twentieth Century, to make it possible
| for even the most workaday mathematicians to escape the tedium
| of mindless calculation that they were once accustomed to call
| the "pile of numbers" (PON's) approach to mathematical objects.
|
| http://suo.ieee.org/email/msg03756.html
---------------------------------------------------
And just to assure you that I did not make up all
this stuff just the other day in order especially
to annoy you -- singular "you" here, and you know
who you are! -- here is a longer extract from the
essay that I submitted pursuant to my application
to enter my current Systems Engineering programme,
instancing my dedication to it for a decade or so:
¤~~~~~~~~~¤~~~~~~~~~¤~AUTOCITATION~¤~~~~~~~~~¤~~~~~~~~~¤
Intelligent Systems Project
Division 1, Version 4
September 1, 1992
1.1.2.3 The Trees & The Forest
A sticking point of the whole discussion has just been
reached. In the idyllic setting of a knowledge field the
question of systematic inquiry takes on the following form:
What piece of code should be followed in order to discover that code?
It is a classic catch, whose pattern was traced out long ago in the paradox
of Plato's 'Meno'. Discussion of this dialogue and of the task it sets for
AI, cognitive science, education, including the design of intelligent tutoring
systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, '72, '75, '80, '86),
(Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and in (Collins & Stevens, 1991).
Though it appears to mask a legion of diversions, this text will present itself at
least twice more in the current engagement, both on the horizon and at the gates
of the project to fathom and to build intelligent systems. Therefore, it is
worth recalling how this inquiry begins. The interlocutor Meno asks:
| Can you tell me, Socrates, whether virtue can be taught,
| or is acquired by practice, not teaching? Or if neither
| by practice nor by learning, whether it comes to mankind
| by nature or in some other way? (Plato, 'Meno', p. 265).
Whether the word "virtue" (arete) is interpreted to mean virtuosity
in a special skill or a more general excellence of conduct, it is
evidently easy, in the understandable rush to "knowledge", to forget
or to ignore what the primary subject of this dialogue is. Only when
the difficulties of the original question, whether virtue is teachable,
have been moderated by a tentative analysis does knowledge itself become
a topic of the conversation. This hypothetical mediation of the problem
takes the following tack: If virtue were a kind of knowledge, and if
every kind of knowledge could be taught, would it not follow that
virtue could be taught?
For the present purpose, it should be recognized that this "trial factorization"
of a problem space or phenomenal field is an important intellectual act in itself,
one that deserves attention in the effort to understand the competencies that
support intelligent functioning. It is a good question to ask just what sort
of reasoning processes might be involved in the ability to find such a middle
term, as is served by "knowledge" in the example at hand. Generally speaking,
interest will reside in a whole system of middle terms, which might be called
a "medium" of the problem domain or the field of phenomena. This usage makes
plain the circumstance that the very recognition and expression of a problem
or phenomenon is already contingent upon and complicit with a particular set
of hypotheses that will inform the direction of its resolution or explanation.
One of the chief theoretical difficulties that obstructs the unification of
logic and dynamics in the study of intelligent systems can be seen in relation
to this question of how an intelligent agent might generate tentative but plausible
analyses of problems that confront it. As described here, this requires a capacity
for identifying middle grounds that ameliorate or mollify a problem. This facile
ability does not render any kind of demonstrative argument to be trusted in the
end and for all time, but is a temporizing measure, a way of locating test media
and of trying cases in the media selected. It is easy to criticize such practices,
to say that every argument should be finally cast into a deductively canonized form,
harder to figure out how to live in the mean time without using such half-measures
of reasoning. There is a line of thinking, extending from this reference point
in Plato through a glancing remark by Aristotle to the notice of C.S. Peirce,
which holds that the form of reasoning required to accomplish this feat is
neither inductive nor deductive and reduces to no combination of the two,
but is an independent type.
Aristotle called this form of reasoning "apagogy" (Prior Analytics, 2.25)
and it was variously translated through the Middle Ages as "reduction" or
"abduction". The sense of "reduction" here is just that by which one question
or problem is said to reduce to another, as in the AI strategy of goal reduction.
Abductive reasoning is also involved in the initial creation or apt generation of
hypotheses, as in diagnostic reasoning. Thus, it is natural that abductive reasoning
has periodically become a topic of interest in AI and cognitive modeling, especially
in the effort to build expert systems that simulate and assist diagnosis, whether in
human medicine, auto mechanics, or electronic trouble-shooting. Recent explorations
in this vein are exemplified by (Peng & Reggia, 1990) and (O'Rorke, 1990).
But there is another reason why the factorization problem presents an especially
acute obstacle to progress in the system-theoretic approach to AI. When the states
of a system are viewed as a manifold it is usual to imagine that everything factors
nicely into a base manifold and a remainder. Smooth surfaces come to mind, a single
clear picture of a system that is immanently good for all time. But this is how an
outside observer might see it, not how it appears to the inquiring system that is
located in a single point and has to discover, starting from there, the most fitting
description of its own space. The proper division of a state vector into basic and
derivative factors is itself an item of knowledge to be discovered. It constitutes
a piece of interpretive knowledge that has a large part in determining exactly how
an agent behaves. The tentative hypotheses that an agent spins out with respect to
this issue will themselves need to be accommodated in a component of free space that
is well under control. Without a stable theater of action for entertaining hypotheses,
an agent finds it difficult to sustain interest in the kinds of speculative bets that
are required to fund a complex inquiry.
States of information with respect to the placement of this fret or fulcrum can
vary with time. Indeed, it is a goal of the knowledge directed system to leverage
this chordal node toward optimal possibilities, and this normally requires a continuing
interplay of experimental variations with attunement to the results. Therefore it seems
necessary to develop a view of manifolds in which the location or depth of the primary
division that is effective in explaining behavior can vary from moment to moment.
The total phenomenal state of a system is its most fundamental reality, but the
way in which these states are connected to make a space, with information that
metes out distances, portrays curvatures, and binds fibers into bundles --
all this is an illusion projected onto the mist of individual states
from items of code in the knowledge component of the current state.
The mathematical and computational tools needed to implement such a perspective
goes beyond the understanding of systems and their spaces that I currently have
in my command. It is considered bad form for a workman to blame his tools, but
in practical terms there continues to be room for better design. The languages
and media that are made available do, indeed, make some things easier to see,
to say, and to do than others, whether it is English, Pascal (Wirth, 1976),
or Hopi (Whorf, 1956) that is being spoken. A persistent attention to this
pragmatic factor in epistemology will be necessary to implement the brands
of knowledge-directed systems whose intelligence can function in real time.
To provide a computational language that can help to clarify these problems
is one of the chief theoretical tasks that I see for myself in the work ahead.
A system moving through a knowledge field would ideally be equipped with
a strategy for discovering the structure of that field to the greatest extent
possible. That ideal strategy is a piece of knowledge, a segment of code existing
in the knowledge space of every point that has this option within its potential.
Does discovery mark only a different awareness of something that already exists,
a changed attitude toward a piece of knowledge already possessed? Or can it be
something more substantial? Are genuine invention and proper extensions of the
shared code possible? Can intelligent systems acquire pieces of knowledge that
are not already in their possession, or in their potential to know?
If a piece of code is near at hand, within a small neighborhood of a system's place in
a knowledge field, then it is easy to see a relationship between adherence and discovery.
It is possible to picture how crumbs of code could be traced back, accumulated, and gradually
reassembled into whole slices of the desired program. But what if the required code is more
distant? If a system is observed in fact to drift toward increasing states of knowledge,
does its disposition toward knowledge as a goal need to be explained by some inherent
attraction of knowledge? Do potential fields and propagating influences have to be
imagined in order to explain the apparent action at a distance? Do massive bodies
of knowledge then naturally form, and eventually come to dominate whole knowledge
fields? Are some bodies of knowledge intrinsically more attractive than others?
Can inquiries get so serious that they start to radiate gravity?
Questions like these are only ways of probing the range of possible systems that
are implied by the definition of a knowledge field. What abstract possibility best
describes a given concrete system is a separate, empirical question. With luck, the
human situation will be found among the reasonably learnable universes, but before that
hope can be evaluated a lot remains to be discovered about what, in fact, may be learnable
and reasonable.
http://suo.ieee.org/email/msg02666.html
¤~~~~~~~~~¤~~~~~~~~~¤~NOITATICOTUA~¤~~~~~~~~~¤~~~~~~~~~¤
Now, it has occurred to me once or twice that nobody will believe
this just because I say it -- perhaps especially because I say it --
and so I think that it is my duty to try and depose myself of the
evidence that I fancy that I might have, if it bears on this case.
That is what I will attempt to do for my next trick, but another time!
I must split the scene right about here, and by a split to say I case
out the time of my dis-jointed address in this doubled wise, those to
whom I must beg off as it was too short, and those to whom I must bid
leave as it was too long, for the fragmentary character my assymboled
discourse is assembled to the measure for measure's sake of this rule:
that I esteem how weary my own dear reader is liable to become in the
reading of it from how bleary I become in my writing of it, so to end!
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~REFERENCE~MATERIALS~¤~~~~~~~~~¤~~~~~~~~~¤
| 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.
|
| (DARM, page 20).
|
| Serge Lang, 'Differential And Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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
¤~~~~~~~~~¤~~~~~~~~~¤~SLAIRETAM~ECNEREFER~¤~~~~~~~~~¤~~~~~~~~~¤