Cf: All Process, No Paradox • 8
http://inquiryintoinquiry.com/2021/03/16/allprocessnoparadox8/
 These are the forms of time, which imitates eternity
 and revolves according to a law of number.

 Plato • Timaeus 38 A
 Benjamin Jowett (trans.)
Re: Laws of Form (
https://groups.io/g/lawsofform/topic/81284216 )
Dear Seth, James, Lyle, All ...
Nothing about calling time an abstraction makes it a nullity.
I'm too much a realist about mathematical objects to ever mean
that. As a rule, on the other hand, I try to avoid letting
abstractions leave us so absentminded as to forget the concrete
realities from which they are abstracted. Keeping time linked to
process, especially the orders of standard process we call “clocks”,
is just part and parcel of that practice.
Synchronicity being what it is, this very issue came up just last night in
a very amusing Facebook discussion about “windshield wipers slappin' time …”
(
https://www.youtube.com/watch?v=Mc7qmE5CiuY )
At any rate, this thread is already moving too fast for the pace
I keep these days but maybe I can resolve remaining confusions about
the game afoot by recycling a post I shared to the old Laws of Form list.
This was originally a comment on Lou Kauffman's blog back when he first
started it. Sadly, he wrote only a few more entries there in the time since.
Re: Lou Kauffman
https://homepages.math.uic.edu/~kauffman/
::: Iterants, Imaginaries, Matrices
http://kauffman2013.wordpress.com/2013/12/27/iterantsimaginariesandmatrices/
As serendipity would have it, Lou Kauffman, who knows a lot about
the lines of inquiry Charles Sanders Peirce and George Spencer Brown
pursued into graphical syntaxes for logic, just last month opened a blog
and his very first post touched on perennial questions of logic and time —
Logos and Chronos — puzzling the wits of everyone who has thought about
them for as long as anyone can remember. Just locally and recently
these questions have arisen in the following contexts:
[Links omitted here. Please see the blog post linked above for the list.]
Kauffman's treatment of logic, paradox, time, and imaginary truth values
led me to make the following comments I think are very close to what I'd
been struggling to say before.
Let me get some notational matters out of the way before continuing.
I use B for a generic 2point set, usually {0, 1} and typically but
not always interpreted for logic so that 0 = false and 1 = true.
I use “teletype” parentheses (...) for negation, so that (x) = ¬x
for x in B. Later on I’ll be using teletype format lists
(x_1, ..., x_k) for minimal negation operators.
[ See
https://oeis.org/wiki/Minimal_negation_operator ]
As long as we’re reading x as a boolean variable x in B
the equation x = (x) is not paradoxical but simply false.
As an algebraic structure B can be extended in many ways
but it remains a separate question what sort of application,
if any, such extensions might have to the normative science
of logic.
On the other hand, the assignment statement x := (x) makes perfect sense
in computational contexts. The effect of the assignment operation on the
value of the variable x is commonly expressed in time series notation as
x' = (x) and the same change is expressed even more succinctly by defining
dx = x' − x and writing dx = 1.
Now suppose we are observing the time evolution of a system X
with a boolean state variable x : X → B and what we observe is
the following time series.
Table. Time Series 1 (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/allprocessnoparadoxe280a22e280a2timeseries1.png
Computing the first differences we get:
Table. Time Series 2 (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/allprocessnoparadoxe280a22e280a2timeseries2.png
Computing the second differences we get:
Table. Time Series 3 (also attached)
https://inquiryintoinquiry.files.wordpress.com/2021/03/allprocessnoparadoxe280a22e280a2timeseries3.png
This leads to thinking of the system X as having an extended state
(x, dx, d²x, ...), and this additional language gives us the facility
of describing state transitions in terms of the various orders of
differences. For example, the rule x' = (x) can now be expressed
by the rule dx = 1.
The following article has a few more examples along these lines.
Differential Analytic Turing Automata (DATA)
https://oeis.org/wiki/Differential_Analytic_Turing_Automata_%E2%80%A2_Overview
Resources
=========
Differential Logic and Dynamic Systems
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview
Regards,
Jon