ONT Re: Differential Logic
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DLOG. Note D57
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Differential of Conjunction
| By deploying discourse throughout a calendar,
| and by giving a date to each of its elements,
| one does not obtain a definitive hierarchy of
| precessions and originalities; this hierarchy
| is never more than relative to the systems of
| discourse that it sets out to evaluate.
|
| Michel Foucault, 'The Archaeology of Knowledge', [Fou, 143]
Finally, at long last, the differential proposition dJ can be gleaned from the
difference proposition DJ by ranging over the cells of [u, v] and picking out
the linear proposition of [du, dv] that is "closest" to the portion of DJ that
touches on each point. The idea of distance that would give this definition
unequivocal sense has been referred to in cautionary quotes, the kind we use
to distance ourselves from taking a final position. There are obvious notions
of approximation that suggest themselves, but finding one that can be justified
as correct is not as straightforward as it seems.
| He had drifted into the very heart of the world.
| From him to the distant beloved was as far as to
| the next tree.
|
| Robert Musil, 'The Man Without Qualities', [Mus, 144]
Let us venture a guess about where these developments might be heading.
From the present vantage point, it appears that the ultimate answer to
the quandary of distances and the question of a fitting measure may be
that, rather than having the constitution of an analytic series depend
on our familiar notions of approach, proximity, and approximation, it
will be found preferable, and perhaps unavoidable, to turn the tables
and let the orders of approximation be defined in terms of our favored
and operative notions of formal analysis. Only the aftermath of this
conversion, if it does converge, could be hoped to prove whether this
hortatory form of analysis and the cohort idea of an analytic form --
the limitary concept of a self-corrective process and the coefficient
concept of a completable product -- are truly (in practical reality)
the more inceptive and persistent of principles and really (for all
practical purposes) the more effective and regulative of ideas.
Awaiting that determination, I proceed with what seems like
the obvious course, and compute dJ according to the pattern
in Table 45.
Table 45. Computation of dJ
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| |
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| => |
| |
| dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| |
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Jon Awbrey
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