Differential Propositional Calculus • 36
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https://inquiryintoinquiry.com/2024/12/30/differential-propositional-calcul…
Transformations of Discourse —
❝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
Here we take up the general study of “logical transformations”,
or maps relating one universe of discourse to another. In many
ways, and especially as applied to the subject of intelligent
dynamic systems, the argument will develop the antithesis of
the statement just quoted. Along the way, if incidental to
my ends, I hope the present essay can pose a fittingly irenic
epitaph to the frankly ironic epigraph inscribed at its head.
The goal is to answer a single question: “What is a propositional
tangent functor?” In other words, the 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 we examine the types of transformations we already know as
“extensions” and “projections” and we use their special cases to illustrate
several styles of logical and visual representation which figure in the sequel.
Resources —
Differential Logic and Dynamic Systems
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https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Differential Logic • Transformations of Discourse
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https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Regards,
Jon
cc:
https://www.academia.edu/community/lnY2aN
cc:
https://www.researchgate.net/post/Differential_Propositional_Calculus