ONT Re: Differential Logic
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DLOG. Note D45
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The Tangent Operator: $T$
| They take part in scenes of whose significance they have no inkling.
| They are merely tangent to curves of history the beginnings and ends
| and forms of which pass wholly beyond their ken. So we are tangent
| to the wider life of things.
|
| William James, 'Pragmatism', [Jam, 300]
The operator tagged as $d$^1 in the analytic diagram (Figure 33) is called
the "tangent operator", and is usually denoted in this text as $d$ or $T$.
Because it has the properties required to qualify as a functor, namely,
preserving the identity element of the composition operation and the
articulated form of every composure among transformations, it also
earns the title of a "tangent functor". According to the custom
adopted here, we dissect it as $T$ = $d$ = <!e!, d>, where d is
the operator that yields the first order differential dF when
applied to a transformation F, and whose name is legion.
Figure 34 illustrates a stage of analysis where we ignore everything but the
tangent functor $T$, and attend to it chiefly as it bears on the first order
differential dF in the analytic expansion of F. In this situation, we often
refer to the extended universes EU% and EX% under the equivalent designations
$T$U% and $T$X%, respectively. The purpose of the tangent functor $T$ is to
extract the tangent map $T$F at each point of U%, and the tangent map $T$F =
<!e!, d> F tells us not only what the transformation F is doing at each point
of the universe U% but also what F is doing to states in the neighborhood of
that point, approximately, linearly, and relatively speaking.
U% $T$ $T$U% $T$U%
o------------------>o============o
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| | |
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F | | $T$F = | <!e!, d> F
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| | |
v v v
o------------------>o============o
X% $T$ $T$X% $T$X%
Figure 34. Tangent Functor Diagram
NB. There is one aspect of the preceding construction that remains especially
problematic. Why did we define the operators W in {!h!, E, D, d, r} so that the
ranges of their resulting maps all fall within the realms of differential quality,
even fabricating a variant of the tacit extension operator to have that character?
Clearly, not all of the operator maps WF have equally good reasons for placing their
values in differential stocks. The only explanation I can devise at present is that,
without doing this, I cannot justify the comparison and combination of their values
in the various analytic steps. By default, only those values in the same functional
component can be brought into algebraic modes of interaction. Up till now, the only
mechanism provided for their broader association has been a purely logical one, their
common placement in a target universe of discourse, but the task of converting this
logical circumstance into algebraic forms of application has not yet been taken up.
Jon Awbrey
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