SUO: Re: Transformations Of Discourse
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[ Notation:
|
| 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!
]
E(nuff) for E(now)!
Jon Awbrey
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