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   The discussion to follow considers a class of trajectories having the property that for all greater than a fixed value 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 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 the drive at that point.  Curves of constant drive are then referred to as ‑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 of 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
• Differential Logic • Drives and Their Vicissitudes

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