ONT Re: Differential Logic
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DLOG. Note D44
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The Chord Operator: $D$
| What difference would it practically make to any one if this
| notion rather than that notion were true? If no practical
| difference whatever can be traced, then the alternatives
| mean practically the same thing, and all dispute is idle.
|
| William James, 'Pragmatism', [Jam, 45]
Next I discuss an operator that is always immanent in this form of
analysis, and remains implicitly present in the entire proceeding.
It may appear once as a record: a relic or revenant that reprises
the reminders of an earlier stage of development. Or it may appear
always as a resource: a reserve or redoubt that caches in advance
an echo of what remains to be played out, cleared up, and requited
in full at a future stage. And all of this remains true whether or
not we recall the key at any time, and whether or not the subtending
theme is recited explicitly at any stage of play.
This is the operator that is referred to as $r$^0 in the initial stage
of analysis (Figure 33-i), and that is expanded as $d$^1 + $r$^1 in the
subsequent step (Figure 33-ii). In congruence, but not quite harmony,
with my allusions of analogy that are not quite geometry, I call this
the "chord operator" and denote it $D$. In the more casual terms that
are here introduced, $D$ is defined as the remainder of $E$ and $e$,
and it assigns a due measure to each undertone of accord or discord
that is struck between the note of enterprise $E$ and the bar of
exigency $e$.
The tension between these counterposed notions, in balance transient but
regular in stridence, may be refracted along familiar lines, though never
by any such fraction resolved. In this style we may write $D$ = <!e!, D>,
calling D the "difference operator" and noting that it plays a role in this
realm of mutable and diverse discourse that is analogous to the part taken
by the discrete difference operator in the ordinary difference calculus.
Finally, we should note that the chord $D$ is not one that need be lost
at any stage of development. At the m^th stage of play it can always
be reconstituted in the following form:
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| |
| $D$ = $E$ - $e$ |
| |
| = $r$^0 |
| |
| = $d$^1 + $r$^1 |
| |
| = Sum_(i = 1 to m) $d$^i + $r$^m |
| |
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Jon Awbrey
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