Differential Propositional Calculus • 32
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https://inquiryintoinquiry.com/2024/12/28/differential-propositional-calcul…
❝I open my scuttle at night and see the far‑sprinkled systems,
And all I see, multiplied as high as I can cipher, edge but
the rim of the farther systems.❞
— Walt Whitman • Leaves of Grass
Example 2. Drives and Their Vicissitudes —
Before we leave the one‑feature case let's look at a more substantial example,
one which illustrates a general class of curves through the extended feature
spaces and affords an opportunity to discuss important themes concerning
their structure and dynamics.
As before let †X† = {x₁} = {A}. The discussion to follow considers
a class of trajectories having the property that dⁱA = 0 for all i
greater than a fixed value n and indulges in the use of a picturesque
vocabulary to describe salient classes of those curves.
Given the above finite order condition, there is a highest order
non‑zero difference dⁿA exhibited at each point of any trajectory
one may consider. With respect to any point of the corresponding
curve let us call that highest order differential feature dⁿA the
“drive” at that point. Curves of constant drive dⁿA are then
referred to as “n‑th‑gear curves”.
Note. The fact that a difference calculus can be developed for boolean functions
is well known and was probably familiar to Boole, who was an expert in difference
equations before he turned to logic. And of course there is the strange but true
story of how the Turin machines of the 1840s prefigured the Turing machines of the
1940s. At the very outset of general purpose mechanized computing we find the motive
power driving the Analytical Engine of Babbage, the kernel of an idea behind all his
wheels, was exactly his notion that difference operations, suitably trained, can serve
as universal joints for any conceivable computation.
Resources —
Differential Logic and Dynamic Systems
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Differential Logic • Drives and Their Vicissitudes
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Regards,
Jon
cc:
https://www.academia.edu/community/VBqpez
cc:
https://www.researchgate.net/post/Differential_Propositional_Calculus