Impulse topic for the Panel Discussion 1 of the *Laws of Form 50th Anniversary Conference* in Liverpool at August the 9th 2019

In my late years I am an enthusiastic newcomer and a lateral entrant in the field of Spencer-Brown and I am very happy to have the chance to speak here and certainly to learn a lot. My topic are the circuits developed by Spencer-Brown. In a way, this is a continuation of Bernie Lewin's contribution. Just as he developed the foundations of classical geometry from the basic rules of *Laws of Form*, I would like to show how the circuits introduced in Chapter 11 can be turned into mathematics of the 20th century. While he has largely completed his work, I am at the beginning and I want to present seven theses in this short impulse

**Modulator function (LoF, 67), naming of the switch elements and second output p added**

(1) What does this circuit mean? First of all, it is necessary to explain why Spencer-Brown indicates the input with *a* and the output with *f*. If chapter 11 is seen as a whole, *a* stands for argument and *f* for function in my opinion. His calculus knows for *a* only the two arguments cross ⃧ (marked state) and an empty space (unmarked state), which he represents in this diagram with a wave trough and a wave crest. The chapter 11 deals with how arguments *a*, whose properties and rules are introduced and studied in the preceding primary arithmetic and primary algebra, self-refer to functions *f*, whose arguments they are. To show this, Spencer-Brown introduces limit crossings (*Grenzübergänge*) that are formed like the successor relation of natural numbers. The transition from *a* to *f* is formally similar to that in traditional mathematics, where Cantor achieves transfinite numbers and expands the space of natural numbers. Similarly, Spencer-Brown creates a new type of object with the limit crossing from *a* to *f*, which can be included in the space of arguments *a* by re-entry and extends it.

If the re-entry of *f* occurs, a vibration (oscillation) passes through the set of arguments, whereby time arises within its logic and can be represented formally. If this happens and a logical concept of time is found, in a final step it can be examined which temporal processes occur in the transition from the arguments *a* to the functions *f*. They are the topic of my impulse lecture. For Spencer-Brown they are no longer linear processes, as they are known from the final chains and causal sequences of classical logic and natural science, but networks in the way he knew them from his time as a developer of circuits, and as they had been introduced by neurophysiology and cybernetics in the 1930s and 1940s. He examines how an input *a* is modulated by networks into an output *f*. He restricts himself to the simplest case, when a new wave of double wavelength is generated from an incoming wave. Everything else should develop from this.

(2) It is no coincidence that he considers networks with which waves are modulated. In the 19th century, Stokes and Weierstrass's theorem showed that any function can be interpreted as an approximation of simple waves. Hilbert constructed the Hilbert space from this, which became of fundamental importance for quantum mechanics and game theory. In my understanding Spencer-Brown therefore uses the modulation of a given wave into a double wavelength wave to design the nucleus from which the foundations of modern mathematics can gradually be developed.

(3) How does the incoming wave pass through the network? In order to understand his approach, it is important that the modulation of a wave in a network is not done for the wave as a whole, but in sections per wave crest and wave trough. Each section is seen as a single input, which Spencer-Brown calls *stroke*. Each stroke must find its own way through the network.

(4) What happens at the switching elements (*markers*) inside the network? Each switching element determines how incoming wave crests and troughs are converted or blocked. It is the engineer's art to carefully arrange the switching elements. As a result, the network is designed in such a way that two consecutive strokes will never follow the same path. Each stroke determines a state of the network with its individual path. The wave is split (dispersed) as it travels through the network and as a result it is combined to the resulting wave. The network therefore no longer simply describes a difference ›f | a‹ between the input *a* and the output *f*, but a difference field ›f ▒ a‹ (the use of the symbols | and ▒ I take over from Scheier, 18f).

(5) The system is set up in such a way that in the oscillation of its system states it can react to the oscillation of the input and, as a whole, produces a kind of dynamic equilibrium of which Bertalanffy spoke, and which can be described as Maturana and Varela as self-organization (autopoiesis).

(6) The developer designs the system in collaboration with the network. He does not know in advance what inputs will be made, nor can he fully plan which way they will find through the network. He can only vaguely predict the network's behavior and relies on the system's self-organization. The network can be understood as the medium (ground), its respective state as its form.

With the networks, the oneway procedure at the beginning of *Laws of Form* is left behind: It is no longer a single-directional acting subject that draws a form into a ground (»draw a distinction«), but the subject designs a network that finds its own form, which can then be improved by the subject. The relationship between medium and form, or in Spencer-Brown's language: the relationship between cross and unmarked space (ock) becomes dynamic. Spencer-Brown shows in detail how the network can alternatively be constructed according to the rules of the *Laws of Form* as a form that is only formed from crosses and feedbacks by re-entries:

**Modulator function (LoF, 66)**

But its specific property as a network is better shown only in its usual graphical form. Thus Spencer-Brown succeeds with the networks in a transition from the simple forms in the main text of *Laws of Form* to a new design of logic whose basic elements are networks. Thinking further, they result in the idea of a mathematics that is no longer based only on numbers and the elementary geometric figures (point, circle, straight line).

(7) The fields of application are diverse and basically unlimited. For me, the networks therefore stand for a logic of medial modernity. In this panel I would like to give the example of organizational theory.

Every organization is a network that processes certain resources (its input) in order to achieve a result (its output). Within the organization, each switching element can be described as a position occupied by the members of the organization. Each job creates value, and the network plan is a generalization of linear envisioned value chains. What is crucial is the insight not only to design a scheme of the organization, but to understand how this scheme affects the self-organization of the organization and brings it to life on a theoretical level. To this end, Spencer-Brown has created a formalism that could make it possible to develop the algorithms, which was not possible in the various fields of application, such as organizational theory in this example.

Note: See also the detailed commentary on Spencer-Brown Law of Form (German, English), and in particular the explanations on how the individual strokes run through the network.

Louis H. Kauffman 2006: Laws of Form – An Exploration in Mathematics and Foundations, Rough Draft

Bernie Lewin: Enthusiastic Mathematics, Melbourne 2018

Claus-Artur Scheier: Luhmanns Schatten, Hamburg 2016

George Spencer-Brown: Laws of Form, New York 1972 (Julian Press) [1969]; Link

George Spencer-Brown: An Introduction to Reductors, unveröffentlichtes Typoscript 1992

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