ONT Transformations of Discourse
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Transformations of Discourse
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Proem
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| Way out here they have a name
| For wind and rain and fire
| The rain is Tess, the fire's Joe,
| And they call the wind Mariah
|
| Mariah blows the stars around,
| Sets the clouds a'flyin'
| Mariah makes the mountain sound
| Like folks was up there dyin'
|
| Mariah, Mariah
| They call the wind Mariah.
|
| Alan Jay Lerner & Frederick Loewe
|
| http://persweb.direct.ca/fstringe/oz/w1792.html
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Episode 1
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| Differential Logic & Dynamic Systems
|
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Created: Dec 16, 1993
| Revised: Oct 31, 1994
| Revised: Mar 15, 2000
| Project: Engineering 690
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan
| Excerpt: Transformations of Discourse, pages 28-32
Transformations of Discourse
| It is understandable that an engineer should be completely absorbed
| in his speciality, instead of pouring himself out into the freedom
| and vastness of the world of thought, even though his machines are
| being sent off to the ends of the earth; for he no more needs to
| be capable of applying to his own personal soul what is daring and
| new in the soul of his subject than a machine is in fact capable of
| applying to itself the differential calculus on which it is based.
| The same thing cannot, however, be said about mathematics; for here
| we have the new method of thought, pure intellect, the very well-spring
| of the times, the 'fons et origo' of an unfathomable transformation.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 39]
In this section we take up the general study of logical transformations,
or maps which relate one universe of discourse to another. In many ways,
and especially as applied to the subject of intelligent dynamical systems,
our argument develops the antithesis of the statement just quoted. Along
the way, if incidental to our ends, we hope this essay can pose a fittingly
irenic epitaph to the frankly ironic epigraph we have placed at its head.
Our goal in this section is to answer a single question:
What is a propositional tangent functor? In other words,
our aim is to develop a clear conception of what kind of
thing would pass in the logical realm for a genuine analogue
of the tangent functor, an object conceived to generalize as
far as possible in the abstract terms of category theory the
ordinary notions of functional differentiation and the familiar
operations of taking derivatives.
As a first step we discuss the kinds of transformations
we already know as projections and extensions, and we use
these special cases to illustrate several different styles
of visual and logical representation that will figure heavily
in the sequel.
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Prologue
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| Notation: Terminological & Interminological --
|
| I will note here only the most pressing bits of notation
| that we need in order to read the present note, and save
| the running accumulation for a tabular form another time.
|
| Given $X$ = {x<1>, ..., x<n>} as an alphabet of logical features:
|
| Define X<j> = <(x<j>), x<j>> = <~x<j>, x<j>> as the "coordinate dimension" j,
| an ordered pair that makes up the oriented space of abstract type B = {0, 1}.
|
| Define X = <$X$>
| = {<x<1>, ..., x<n>> : x<j> in X<j>}
| = X<1> x ... x X<n>
| = Prod<j> X<j>,
|
| the set of interpretations, cells, points, vectors
| in the universe of discourse that is based on $X$,
| a space that is of the abstract type B^n.
|
| Define X* = {f : linear X -> B}
|
| as the space of "linear propositions" on X,
| also called the "algebraic dual space" of X,
| a space that is also of the abstract type B^n.
|
| Define X^ = {X -> B} = {f : X -> B}
|
| as the space of "boolean functions" on X,
| also called the "truth-valued functions" on X,
| and loosely described as the "propositions" on X,
| a space that enjoys the abstract type of B^n -> B.
|
| Define X° = [$X$]
| = [x<1>, ..., x<n>]
| = <X, X^>
| = {X +-> B}
| = <X, {X -> B}>
|
| as the "universe of discourse" that is based on the features in $X$,
| a space of the "complex type" <B^n, B^n -> B> = <B^n +-> B> = [B^n].
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Metalogue
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| Notational Note
|
| Yes, there's more notation. What I gave last time
| was basically just the minimum amount that we need
| to formalize our casual insights into such schemes
| of representation as truth tables or venn diagrams,
| and to paste the two layers of the latter style of
| depiction, the "point" layer and the "paint" layer,
| if you please, into a unified category-theoretical
| object, the kind of thing, once grasped, which can
| be turned over in the mind and contemplated in its
| manifold aspects, changes, facets, highlights, and
| shadows, but the time has come to enlarge the span
| of our formalism by just a bit, at least enough to
| encompass an order of differential features, those
| that we need to talk about change in a logical way.
|
| Given $X$ = {x<1>, ..., x<n>} as an alphabet of logical features:
|
| Define d$X$ = {dx<1>, ..., dx<n>} as an alphabet of differential features,
| more specifically, the "(first order) differential alphabet" based on $X$.
|
| Define dX<j> = <(dx<j>), dx<j>> = <~dx<j>, dx<j>>
| as the "differential dimension" j, a space of the
| abstract type B = {0, 1}. It is convenient, just
| as an informal way of keeping tabs on such spaces,
| to distinguish them as having an abstract type dB.
|
| Define dX = <d$X$>
| = {<dx<1>, ..., dx<n>> : dx<j> in dX<j>}
| = dX<1> x ... x dX<n>
| = Prod<j> dX<j>,
|
| as the generic "(co)tangent space" at a point of X,
| intuitively intended to be interpreted as the space of
| alterations, changes, departures, deviations, differences,
| motions, steps, options, the (co)tangent interpretations, or
| the (co)tangent vectors at a point of the underlying space X.
| It is convenient to mark dX as having the abstract type dB^n.
|
| Nota Bene. Some folks will be fussy about calling
| these "cotangent spaces", but I have not been able
| to decide whether the distinction matters all that
| much in these varieties of qualitative situations.
|
| Define dX* = {f : linear dX -> B}
|
| as the space of "linear propositions" on dX,
| also called the "algebraic dual space" of dX,
| a space that is also of the abstract type dB^n.
|
| Define dX^ = {dX -> B} = {f : dX -> B}
|
| as the space of "boolean functions" on dX,
| also called the "truth-valued functions" on dX,
| and loosely described as the "propositions" on dX,
| a space that enjoys the abstract type of dB^n -> B.
|
| Define dX° = [d$X$]
| = [dx<1>, ..., dx<n>]
| = <dX, dX^>
| = {dX +-> B}
| = <dX, {dX -> B}>
|
| as the "(co)tangent universe of discourse" at a point of X°,
| a space of the type <dB^n, dB^n -> B> = <dB^n +-> B> = [dB^n].
|
| Finally, we have now reached the point where we can proceed to define
| the "(first order) differential extension" of a universe of discourse.
|
| Define E$X$ = $X$ U d$X$
| = {x<1>, ..., x<n>, dx<1>, ..., dx<n>}
|
| as the "(first order) extended alphabet"
| or the "(first order) bundled alphabet" based on $X$.
|
| Define EX = X x dX
| = <E$X$>
| = <$X$ U d$X$>
| = {<x<1>, ..., x<n>, dx<1>, ..., dx<n>>}
|
| as the "(first order) differential extension" of X,
| or the "tangent bundle" formed on the base space X,
| a space that takes the abstract type of B^n x dB^n.
|
| At last!
|
| Define EX° = [E$X$]
| = [x<1>, ..., x<n>, dx<1>, ..., dx<n>]
|
| as the "(first order) differential extension" of the universe X°,
| or the "tangent bundle universe" that is based on the universe X°,
| in brief terms, the "extended universe" based on the features $X$,
| altogether constituting the totality of points and maps, the full
| set of interpretations and propositions, which are generated from
| the extended alphabet of features in E$X$. The extended universe
| has the type {B^n x dB^n +-> B} = <B^n x dB^n, B^n x dB^n -> B>.
|
| A proposition in the extended universe EX° = [E$X$] is called
| a "differential proposition" and amounts to a logical analogue
| of a system of differential equations, constraints, or relations
| in the ordinary calculus. With these constructions, namely, the
| differential extension EX and the differential proposition of the
| form F : EX -> B, we have arrived, in concept at least, at one of
| our major subgoals in this study. But that is just the beginning!
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Apologue
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| 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.
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Episode 2
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Foreshadowing Transformations: Extensions & Projections of Discourse
| And, despite the care which she took to look behind her at every moment,
| she failed to see a shadow which followed her like her own shadow, which
| stopped when she stopped, which started again when she did and which made
| no more noise than a well-conducted shadow should.
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 126]
Many times in our discussion we have occasion to place one universe of discourse
in the context of a larger universe of discourse. An embedding of the general
type [$X$] -> [$Y$] is implied any time we make use of one alphabet $X$ which
happens to be included in another alphabet $Y$. But when we are discussing
differential issues we usually have in mind that the extended alphabet $Y$
has a special construction or lexical relation with respect to the initial
alphabet $X$, one which is often marked by characteristic types of accents,
indices, or inflected forms.
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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Interlogue
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I would like to take a moment's rest from that
long litany of "te deums" that will need to be
canted and dinned and droned and intoned on my
present theme, before the time can arrive when
a poor creature such as I, so finitely endowed
with any capacities for control or information,
can approach to anything like doing it justice,
and I would like to point to a couple of links
that this topic has with matters that you have
of late raised here on several of your threads.
Let me then try to forge for you here an image
of the pattern of connections that I see among
these three questions:
1. The number of dimensions that we need to think about.
2. The character of the coordinate systems that we need.
3. All of this stuff about transformations of discourse.
Consider a type of system, prospectively having some bearing on the issue
of whether any systen can have both dynamics and intelligence all at once,
that is commonly described, at least by me, as a "brain in a cube" (BIAC),
whose life traverses a "discrete approximation to a brain state" (DATABS)
at each and every moment of its surprisingly variegated life, for a' that.
Each DATABS x is a bit-string or a bit-vector whose type may be specified
in the form x : B^N, inasmuch writing that x is a member of the space B^N,
and where N is somewhere in the vicinity of 10^10 the last time I checked.
Now this brain of which we speak resides in 3-space and develops over time,
at least for a time, or that is what I think that I may safely assume that
the likes of us brands of thinkers will agree on here, for the sake of the
present argument, at any rate. And so, yes, we 'could', if we insisted on
doing so, index every "discrete approximation to a neuron" (DATAN) in some
form of "prospective geometry" (PG), say, by addressing each and every one
according to its location in relation to some suitably distinguished point,
say, the center of levity in the "pineal gland" (PG), and that would maybe
suffice to relate what is going on in this amazingly complex system to our
familiar manners of assigning "a local habitation and a name" to the stuff
that dreams are made on, but I just think that it ought to be rather clear
how artificial is the aspect of simplicity that is thereby pulled over all.
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Episode 3
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| I return to the main development, illustrating the remaining two types
| of diagrams that I have found to be useful in representing differential
| situations in these sorts of discretely finite and qualitatively limited
| universes of discourse.
Extension from 1 to 2 Dimensions (Continued)
Figure 18-c shows the differential extension from X° = [x] to EX° = [x, dx]
in a "compact" style of venn diagram. Here, the differential features "at"
a point are represented by arrows that stretch from that point to cross the
corresponding feature boundaries.
o---------------------------------------o
| X |
| |
| |
| o |
| / \ |
| / \ |
| / \ |
| / /\ \ |
| / | | dx |
| / | o---------->o |
| / | ^ \ |
| / \/ \ |
| / (dx) \ |
| / \ |
| o x o |
| \ / |
| \ / |
| \ / /\ |
| \ / | | |
| \ o<----------o | |
| \ dx ^ | |
| \ / \/ |
| \ / (dx) |
| \ / |
| \ / |
| o |
| |
| (x) |
| |
o---------------------------------------o
Figure 18-c. Extension from 1 to 2 Dimensions
Figure 18-d compresses our picture of the differential extension from
X° = [x] to EX° = [x, dx] even further, yielding a "directed graph"
or a "digraph" form of representation. (Note that our definition
of a digraph allows for loops or "slings" at individual points,
in addition to arcs or "arrows" between pairs of points.)
o---------------------------------------o
| |
| |
| |
| dx |
| ------->------- |
| /\ / \ /\ |
| | |/ \| | |
| (dx) ^ o x (x) o ^ (dx) |
| | |\ /| | |
| \/ \ / \/ |
| -------<------- |
| dx |
| |
| |
| |
o---------------------------------------o
Figure 18-d. Extension from 1 to 2 Dimensions
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Epilogue
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| Point Of Order, Point Of Focus
|
| Once again, I find that there is a "form of mis-apprehension" (FOMA)
| that is likely to arise here on account of the fact that I am forced,
| by all sorts of rhetorical considerations that lie beyond my control,
| to present these materials in almost exactly the opposite order from
| that of the temporal evolution in which they were actually developed.
|
| Invoking the motif of my initial overture, I could say that I first
| came to know these elemental qualities as Tess, and Joe, and Mariah,
| only later to learn their more common names, "rain", "fire", "wind",
| and only much later in life, after a considerable effort to acquire
| the necessary level of abstraction, with all of the hard-won trials
| of incremental detachment and increased sophistication that it took
| to do so, finally got the requisite nerve to replace every trace of
| concrete sensation with the paltry qualities of x1, x2, x3, no more.
| And so, once again, I recognize that I have left what is really the
| tougher leg of the round and round trip of transformation to others.
|
| And when it comes to why I would even want to go through all of this,
| I will have to confess, in order to answer that, a quite ego-centric
| bias. I am mostly concerned with the changes of that special system
| that I call my self, though, of course, I realize that it is as much
| an act of self-deception to call it a "self" as it is an illusion of
| systematic perspective to call it a "system". But skip that for now.
| In either case, we are only talking about forms of address, forms of
| description, and we can put new shirts when these are due for a wash.
|
| So what we have here is three languages:
|
| 1. The language of "Way Out Here" (WOH).
| 2. The language of "Any War Else" (AWE).
| 3. The language of "Way Out Thar" (WOT).
|
| And while I have proceeded through the series 1, 2, 3,
| You are being asked to take them in the order 3, 2, 1.
| I hope that serves to alleviate some of the confusion!
Jon Awbrey
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