## All Process, No Paradox • 8

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

Re: Laws of FormSeth • James Bowery (1) (2) (3)Lyle Anderson

Dear Seth, James, Lyle,

Nothing about calling time an abstraction makes it a nullity.  I’m too much a realist about mathematical objects to ever think that.  As a rule, on the other hand, I try to avoid letting abstractions leave us so absent-minded 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 …”

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.

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.

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 $\mathbb{B}$ for a generic 2-point set, usually $\{ 0, 1 \}$ and typically but not always interpreted for logic so that $0 = \mathrm{false}$ and $1 = \mathrm{true}.$  I use “teletype” parentheses $\texttt{(} \ldots \texttt{)}$ for negation, so that $\texttt{(} x \texttt{)} = \lnot x$ for $x ~\text{in}~ \mathbb{B}.$  Later on I’ll be using teletype format lists $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$ for minimal negation operators.

As long as we’re reading $x$ as a boolean variable $(x \in \mathbb{B})$ the equation $x = \texttt{(} x \texttt{)}$ is not paradoxical but simply false.  As an algebraic structure $\mathbb{B}$ 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 $x := \texttt{(} x \texttt{)}$ 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 $x' = \texttt{(} x \texttt{)}$ and the same change is expressed even more succinctly by defining $\mathrm{d}x = x' - x$ and writing $\mathrm{d}x = 1.$

Now suppose we are observing the time evolution of a system $X$ with a boolean state variable $x : X \to \mathbb{B}$ and what we observe is the following time series. Computing the first differences we get: Computing the second differences we get: This leads to thinking of the system $X$ as having an extended state $(x, \mathrm{d}x, \mathrm{d}^2 x, \ldots, \mathrm{d}^k x),$ and this additional language gives us the facility of describing state transitions in terms of the various orders of differences.  For example, the rule $x' = \texttt{(} x \texttt{)}$ can now be expressed by the rule $\mathrm{d}x = 1.$

The following article has a few more examples along these lines.

### 1 Response to All Process, No Paradox • 8

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