ONT Re: Differential Logic
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DLOG. Note D25
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| 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]
Transformations of Discourse
In this section we take up the general study of logical transformations,
or maps that relate one universe of discourse to another. In many ways,
and especially as applied to the subject of intelligent dynamic systems,
my argument develops the antithesis of the statement just quoted. Along
the way, if incidental to my ends, I hope this essay can pose a fittingly
irenic epitaph to the frankly ironic epigram I have inscribed at its head.
My goal in this section is to answer a single question:
What is a propositional tangent functor? In other words,
my aim is to develop a clear conception of what manner 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 all
too familiar operations of taking derivatives.
As a first step I discuss the kinds of transformations that
we already know as "extensions" and "projections", and I use
these special cases to illustrate several different styles of
logical and visual representation that will figure heavily in
the sequel.
Jon Awbrey
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