CYBERNETICS & HUMAN KNOWING

A Journal of Second Order Cybernetics & Cyber-Semiotics

Volume 5, No.1 1998

Virtual Logic — The Calculus of Indication

By Louis H. Kauffman

This is the fourth column in this series on “Virtual Logic”. This time I will discuss the calculus of indications of G. Spencer-Brown as it is constructed in his book “Laws of Form” [1]. For an excellent (albeit self-referential - the author, James Keys, of “Only Two Can Play This Game” is Spencer-Brown himself) review of Laws of Form, see [2]. Spencer-Brown’s book is based on the notion of a distinction, and the consequences of what there would be if there could be a distinction.

It is possible to proceed very simply from the notion of a single distinction and to reconstruct in a most uncanny fashion the foundations of mathematics and the patterns that lie at the bottom of any communication at all. It is, however, very difficult to proceed simply. In order to do so it is necessary to put aside all sorts of notions, fantasies and stories that we tell. In the process of the telling we hypnotize ourselves into believing as real the very worlds that are our own constructions.

In order to proceed in the most economical way, I will tell you a story. But you must be forewarned that after I have told this story and we have all understood it, then it is necessary to throw away the story and live in a simpler world.

This column is divided into two parts beyond this introduction: Story and Dialogue. Dialogue is intended to complement Story and to place it in context - a number of contexts. Story is itself very condensed and may require rereading to be appreciated. Our intent is to explore a number of themes that are related to simplicity and vanishing. As things nearly vanish, we reach regions where apparently distinct domains touch, join and become one. As things come into being, apparently distinct domains appear from an undifferentiated ground. These new domains grow in great profusion and prolixity, sometimes obscuring the simple origins. We are interested in creative growth. It is by returning to the origin that the source of such newness is found. The calculus of indications is a gem retrieved by a descent into nothingness.

I. Story

Imagine a distinction.

Let this distinction be denoted by the curly brackets shown below.

{ }

By our already agreed upon conventions

(we cannot proceed without them)

(where did we get them?)

we understand that the distinction produced by the curly brackets consists of an inside enclosed by the brackets and an outside that lies beyond the brackets. Let us denote the inside of the brackets by I and the outside by O. Let us place the letters I and O on the inside and the outside of the brackets as shown below.

{ I } O

Our original curly distinction is now decorated with names, and we can use these names as points of reference to speak of journeys from one side to the other.

Let there be an operator called sharp bracket denoted by

< >

that indicates the passage from one side of the distinction to the other side. Thus <I> = O is an equation in the sharp operator that says “If you cross from the inside then you are outside.” And <O> = I says that if you cross from the outside, then you are inside. Now we have all sorts of things - names for inside and outside, operators that mediate the crossings from inside to outside, notations for the operator, depiction’s of the original distinction.

You could begin to do mathematics with all of this. The operator itself looks like the negation operator NOT in logic: NOT inside is outside. NOT outside is inside. NOT NOT outside is outside.

NOT NOT inside is inside.

<<O>> = <I> = O.

<<I>> = <O> = I.

NOT X <——-> <X>.

NOT NOT X = X <——-> <<X>>= X.

Yes indeed. This is a familiar world. And in this world, we are inclined to wonder where in the “actual” world is this miraculous operator NOT that can undo itself in a blink? Ludwig Wittgenstein in his “Tractatus Logico Philosophicus” [3] wondered (Tractatus - Statement 5.511) “How can logic - all embracing logic, which mirrors the world, use such peculiar crochets and contrivances? Only because they are all connected with one another in an infinitely fine network, the great mirror.”

The blindness has already begun to set in. A world of description and a world of actuality are beginning to distinguish themselves. Yet here, we can see that the world of description is actually more complex than the simple distinction that it describes! (Perhaps you would not swear to that in your entrancement.)

We shall simplify.

We shall simplify by releasing terms and indicators until the language and the world it describes are coincident. Parallel lines may not have to wait until infinity before they meet.

Let go of I.

Identify the inside of the distinction by the absence of a name. In speaking about it we shall say that the inside is unmarked. The inside is the unmarked state of the distinction. The outside is the marked state of the distinction. Since markedness is distinct from unmarkedness, the two sides of the distinction are distinguished from one another by dint of their respective properties. One side is marked. The other side is not marked.

The equations of operation simplify from

<I> = O

<O>= I

< > =O

<O>= .

In this second form, the first equation reads that the crossing from the (literally!) unmarked state yields the outside (marked) state. The second equation says that the passage from the marked (outside) state yields the unmarked state.

The distinction itself is now decorated only with an O on its outside and nothing on its inside.

{ } O

Mirabile dictu, the new equations suggest a further simplification! For the equation < > = O suggests that we can do away with the name of the outside, replacing it with the empty operator symbol: < >. After all, < > denotes the state obtained by crossing from the unmarked state and this is indeed the outside, the marked state. Making this second descent, we find new equations.

< > = < >

<< >> = .

The first equation reads “The value obtained upon crossing from the unmarked state (the inside of the sharp bracket is unmarked) is the marked state (indicated directly by the sharp bracket).” The second equation reads “The value obtained upon crossing from the marked state is the unmarked state.”

The distinction is now decorated only with sharp brackets, indicating the marked state, on its outside.

{ } <>

These two descents have brought us into a remarkable coincidence of name and operation. The sign < > is the name of the marked state. The sign < > is result of the act of crossing from the unmarked state. The symbol <X> denotes the state obtained by crossing from the state indicated on its inside. Name and act have condensed. The name is nothing but the act of crossing from the absence of name.

And yet we have not yet reached the bottom of the singularity. The symbol sharp bracket is itself a distinction and indeed a distinction between inside and outside, as we have been speaking it for the last two descents. We no longer need the curly brackets! Let < > itself be the first distinction.

Replace { } by < >.

Then < > is decorated on its outside by a copy of itself.

{ } < > <——-> < > < >

This is redundant. One copy of the distinction is sufficient for identification. (Professor Wittgenstein is himself with or without his name tag.)

Thus we can write < > < > = < >, simplifying the distinction plus name to the distinction itself. The name is the thing named.

Finally, we arrive at the two fundamental equations of the calculus of indications.

1. < > < > = < >

The value of a call made again is the value of the call.

2. << >> =

The value of a crossing made again is not the value of the crossing.

From here we see that there is no great mirror. The world and the description of the world are one. The observer is the mark. Our excursions into ever-spiraling description and discovery follow inexorably from what a distinction would be if it could be at all.

That a distinction cannot be is the last stage of the descent.

II. Dialogue

This section is in the form of a dialogue. The two voices in the dialogue are named Jeremy and Lou. They will alternate from paragraph to paragraph. Lou is the author of Story.

Jeremy - Wait one minute Lou. You just ended this story with the suggestion that there cannot be a distinction. Haven’t we just been looking at a primary distinction and its properties?

Lou - Really I mean that a distinction is a kind of fiction, or story that we tell. It has exactly the existential quality of any good fictional entity. Take Sherlock Holmes. Once he came into existence, he had always been and he always would be. On the other hand, if you try to make a fiction into an actuality there will be limitations and the absence of a good fit. Mathematical entities like numbers exist in the mathematical world and have very exact properties there. This mathematical world is a fictional world. The patterns of the mathematical world are, however, found in our experience.

Jeremy - It seems to me that you are being quite paradoxical here. In your Story you intimate that there is no difference among actuality, our experience and our models of our experience. Is our experience then only a fiction?

Lou - Precisely! Our experience is a fiction, and the mathematical part of that fiction is the most solid and long-lasting aspect of it. Mathematical entities change very little in time, and the more you look at them the more you learn about them. The exemplar of both our experience and our models is the experience of a distinction. A distinction is always an experience of distinction for some one. This brings us, as usual, to second order cybernetics where the only systems that really exist are systems that are observers. An aware system is a system that transcends boundaries.

Jeremy - My world is real.

Lou - I do not deny the reality of your world or mine. I only say that what you have of that world is your story of it. The story, being a good story, appears to have inevitability and immutability, but it is nevertheless your story. And I invite you to speak to the author!

Jeremy - So the great mirror does exist. It is the great book in which these stories are written. It is our conscious awareness. Oh, I see what you mean. I am writing a tale about the observer and his sensitive consciousness.

Lou - At best the great mirror is beautifully flawed. This part of the story goes back to Frege. Frege believed that there must be an extension to every concept. He believed that to every concept there would be the set of all those entities that satisfy the concept. For example, if I have the concept of honest men, then there is the set of all honest men. Here is a belief in the perfect mirroring of concept and realization of concept.

Jeremy - This sounds like a good story.

Lou - Bertrand Russell came along and produced the concept whose extension is R, the set of all sets that are not members of themselves. For example the set of all philosophers is not a philosopher, hence is not a member of itself. The set of all concepts is itself a concept and hence is a member of itself. Is R a member of itself? If R is a member of itself, then R must not be a member of itself. If R is not a member of itself then R must be a member of R. Either choice leads to a contradiction. The great mirror has a beautiful crack. Attempts to tell a story about an actuality and its perfect description are doomed to incompleteness. This is just a special case of the more general result that distinctions are fictions and that it is actually not possible to make a distinction that does not spring a leak in time or space. In fact, as you see from Russell, paradox generates time! Each attempt to construct R is incomplete. We keep including the old R in the new R in hopes of a completion, and this is time.

Jeremy - Well I am a practical mathematical person. I think that my mathematics is made of solid distinctions that have survived the tests of time and of consistency. Numbers exist and I can use them the model the world.

Lou - Yes. And what are your numbers? They are generalizations of single marks. The mark in our story, denoted by sharp brackets, could be the number zero, or perhaps the number one. (It is after all one mark.) And how do you know it? You know it by your ability to make a distinction. By now you will admit that ability has its limitations, not just the Russellian ones, but simpler ones such as the need to be awake in order to perform the duty of discriminating between the presence and the absence of the brackets. Thus numbers and all your other mathematical models are based upon your ability to make/imagine a distinction. You make this up. You play hide and seek with it.

Jeremy - How far do you go in this line of thought?

Lou - I go all the way. Appearance and reality are identical. You, in so far as you are distinct to yourself and to others, are a mark (of distinction). The world as reality and the world as map are identical. The map is the territory. However, in order for the world to perceive itself it must imagine that it is divided into a part that sees and a part that is seen. This condition is the fiction on which we base explanations and science.

Jeremy - What do you get from having the calculus of indications?

Lou - The calculus of indications is a gem. It is a rigorous mathematical system of extreme simplicity in which we can see the genesis of form and the dissolution of form. Furthermore, the calculus of indications lets us reverse the descent into an ascent and tell many new tales of mathematics, communication and cybernetics.

Jeremy - When will you tell those stories?

Lou - I will begin them in the next column.

References

1. G. Spencer-Brown. Laws of Form. George Allen and Unwin Ltd. , London, (1969).

2. James Keys. Only Two Can Play This Game. Julian Press Inc., NY. (1972).

3. Ludwig Wittgenstein. Tractatus Logico-Philosophicus. London (1922).

CYBERNETICS & HUMAN KNOWING

Contents:

Virtual Logic — The Calculus of Indication