Re: All Process, No Paradox

Post   : All Process, No Paradox : 2
URL    : http://inquiryintoinquiry.com/2014/01/15/all-process-no-paradox-2/
Posted : January 15, 2014 at 1:00 pm
Author : Jon Awbrey

Note.  The text below is much better formatted in the blog post linked above.

Here's the continuation of another attempt to get at
a host of recurring and rather long-running issues.

| These are the forms of time,
| which imitates eternity and
| revolves according to a law
| of number.
|
| Plato • Timaeus

Re: Lou Kauffman • Iterants, Imaginaries, and Matrices
At: http://kauffman2013.wordpress.com/2013/12/27/iterants-imaginaries-and-matrices/

As serendipity would have it, Lou Kauffman, who knows a lot about the lines of work
that Charles Sanders Peirce and George Spencer Brown carried out on graphical syntaxes
for logic, just last month opened a new blog and his very first post touched on perennial
questions of logic and time — Logos and Chronos — that have puzzled 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:

• Alpha Now, Omega Later
http://inquiryintoinquiry.com/2013/11/16/alpha-now-omega-later-1/

• Objects, Models, Theories
http://inquiryintoinquiry.com/2013/09/10/objects-models-theories-1/

• Discussion with Helmut Raulien
http://stderr.org/pipermail/inquiry/2013-November/004123.html

Kauffman's treatment of logic, paradox, time, and imaginary truth values led me to make the 
following comments that 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 2-point set, usually {0, 1} and usually 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 http://intersci.ss.uci.edu/wiki/index.php/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:

t | x
--|---
0 | 0
1 | 1
2 | 0
3 | 1
4 | 0
5 | 1
6 | 0
7 | 1
8 | 0
9 | 1
… | …

Computing the first differences we get:

t | x | dx
--|---|----
0 | 0 | 1
1 | 1 | 1
2 | 0 | 1
3 | 1 | 1
4 | 0 | 1
5 | 1 | 1
6 | 0 | 1
7 | 1 | 1
8 | 0 | 1
9 | 1 | 1
… | … | …

Computing the second differences we get:

t | x |dx |d²x| …
--|---|---|-------
0 | 0 | 1 | 0 | …
1 | 1 | 1 | 0 | …
2 | 0 | 1 | 0 | …
3 | 1 | 1 | 0 | …
4 | 0 | 1 | 0 | …
5 | 1 | 1 | 0 | …
6 | 0 | 1 | 0 | …
7 | 1 | 1 | 0 | …
8 | 0 | 1 | 0 | …
9 | 1 | 1 | 0 | …
… | … | … | … | …

This leads to thinking of the system X as having an extended state (x, dx, d²x, …, d^k x), 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
http://intersci.ss.uci.edu/wiki/index.php/Differential_Analytic_Turing_Automata

An extended treatment of these topics can be found in the following paper:

• Differential Logic and Dynamic Systems
http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0

Regards,

Jon

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