ARC25 SIGHTLINES | Anthology of Reflexive Contributors
I. Introduction
In his treatise, the Tractatus Paradoxico Philosophicus (Uribe, 1995), Ricardo Uribe points out how, by entering into paradox beyond the Boolean, one can achieve remarkable simplifications in the design of switching circuits. His specific example uses the Mobius band as a source of the design for an electrical switching circuit that has as many switches as you like, with each switch independently controlling whether one light bulb is on or off. We shall describe his solution and then discuss it.
You may at first wonder where is the paradox? In fact we will see the paradox appear in more than one way as the discussion proceeds.
Systemic design (Sevaldson, 2025) is an organic meshing of systems perspectives with design, where organic means that it is not based on theory and reasoning alone. In this article, we take as a case study for systemic design the very problem that was discussed by Uribe, indicated above. As we shall see, it was possible to face that problem (turning a light on or off with arbitrarily many local switches) without any theory of the type that we shall describe due to Claude Shannon. In that case, one can (and it seems that Uribe did) leap beyond reasoning, through imagination, to an elegant solution. The leap, in Uribe’s tale, is based on visualising paradox in the form of a Mobius band (two edges locally but one edge as a whole). The Mobius is locally two but globally one. It is two, and yet it is one. And the lightbulb of Uribe’s mind went on when he saw that it is the twist (usually a single twist) in the band that makes it one. With two twists, there are two edges. With three twists, there is only one edge. With an odd number of twists, there is one edge. With an even number of twists, there are two edges. Then another leap: What if a twist is a switch? Twist or no twist are two positions of the twist-switch or crossing-switch, as we call it below. Use the connectivity of the edge(s) of a band (Mobius or not) as the determinant for the current in the system. With odd twists, the bulb is lighted. With even twists, the bulb is dark. Each twist switch controls the bulb.
Here we have described in some detail a possible dialogue leading to the systemic design behind what Uribe did. We wish to ask further questions. What if one also knows a theory? How will that affect the process? Can new theory also evolve with the help of systemic design? What does it mean to creatively use theory and reasoning? Will not such use, if creative, go beyond standard logical reasoning into the use of dialogue and imagination? Will not creative use of theory lead both to new designs and to new theory? It is our contention that the answer to all these questions is a resounding yes.
Consider the first circuit shown in Figure 1. The light is off. Each line away from the light bulb and the battery ends in a loop, and these loops have nothing to do with one another. You can imagine the loops as the two edges of a band with no twist, a non-Mobius band.
Now consider the second circuit shown in Figure 1. The light is on. The two loops have become one loop due to the cross connection shown. You can imagine the loop configuration as the boundary of a Mobius band. The Mobius band has a single boundary, and it is this single nature of the boundary that ensures that the light will be on.
Figure 1. A two-loop circuit and a Mobius circuit.
Now consider the circuit shown in Figure 2. There are 3 crossing connections. The light is on. If we switch any of these connections back to parallel the light will be off.
There is a special switch S as illustrated in the figure that changes between two parallel lines and two crossed over lines. This switch can be used to make a circuit so that a light bulb can be controlled by any number of switches in a circuit whose complexity does not rise very much as the number of switches increases.
Figure 2. Many switch Mobius framework to control one light.
A Mobius band may seem to be paradoxical. It is locally two in the sense that a local observer in the band would see two bits of edge and would not know that these local bits are connected globally in the band. But an observer who walked along the edge of the band would find himself on the opposite edge in his local domain! This is a paradox that is resolved by the observer understanding the nature of the larger reality of the Mobius band as a whole.
It is the paradoxical state that lights the bulb. The Mobius is not two, nor is it one.
In the topology and constructive design of the band, there is no paradox. It is only the short-sighted local observer who finds the band a paradox. And yet we can use the possibility of a paradox, the capacity to look beyond given boundaries as a principle of reason that goes beyond the Boolean.
But now what is this about a short-sighted observer? Consider the general observer. Consider yourself. Wherein is our short sight? We look and look and look and make one construction after another, searching for the new, for control, for the structure of mathematics, physics, the natural world, the meaning of existence, the nature of the universe. Of course, some of this rhetoric is attenuated if you are a second-order cybernetics person. Perhaps you believe that it is not quite right to give independent existence to “the world.” Still, you have to admit, there is a multiplicity and a need to find unity. And yet you already know the answer. Looking for all things, finding all things, what is the source of things? The source is not in more things. The explanation cannot be in how things beget things. Behind things are your indications of them. Indications are indications of the possibility of something. But also, indications are denoting the possibility of a distinction meaningful to you. Indications as possibilities of distinction stand at the cusp of creation.
In creation, there are no things. Things come from creation, from being. When you turn to that nexus of creation, you enter a place where signifier, signified, and interpretant are one, where there are no signs and where signs arise. Where there is no thing and yet things appear. That which is, is not one nor is it many.
The Mobius can be seen as a metaphor for our own short sight of the whole.
I am many and I am one.
Any thing is identical with what it is not.
By going to the Mobius, as did Uribe, one obtains maximal control of the light.
By going to the Whole one obtains maximal understanding of the light.
In Section II we shall review a known theory of circuit design, using Boolean algebra, due to Claude Shannon. We shall see that this theory works well in principle but can produce very inefficient designs. Rather than a flight of imagination into paradox, a designer using this theory would encounter frustrating complexity. This leads to another form of dialogue where one may search, within the theoretical context, for a way out. Once again, one needs creativity and the possibility to see something not dictated by the theory. We show how this dialogue can also lead to the crossing switch, albeit without the help of the Mobius paradox.
In Section III we show the very interesting consequences of taking the the crossing switch to a new level by making a theory of the crossing switch. This is a new algebra, not Boolean, that I call the crossing algebra. The crossing algebra is very much like algebras that have been previously designed to handle paradox, but this algebra is very helpful in making circuit designs using the crossing switch. Here the exploration that led out into the Mobius and over to new switching circuits acquires a theory of its own. We will discuss how such processes may continue for the benefit of making and understanding.
II. Shannon and Boole
Claude Shannon (Shannon 1925) gave a symbolic analysis of switching circuits in terms of Boolean algebra. His work is a great and simple breakthrough in applied mathematics. Shannon associates letters a,b,c … to switches, and the simplest such switch, as illustrated in Figure 3, consists in an input wire followed by an output wire with either a gap in between (switch open) or a connection between the input and the output (switch closed). The switch is diagrammed with the letter “a” placed between the input line and the output line. One can then envisage a network of switches with such labels. If there are many switches labelled “a,” then it is understood that all the places with that label are simultaneously open or shut. Thus, there are actually infinitely many types of switches in the Shannon theory. In practice, we are familiar with “single-pole single-throw” switches as the same as the basic Shannon switch and others, such as double-pole, double-throw switches that can be obtained in a hardware store. With this concept of switch, we can describe Shannon’s Theory using the logical connectives:
1. a and b = a∧b = series connection of switches “a” and “b”.
2. a or b = a∨b = parallel connection of switches “a” and “b”.
3. not a = a = a switch that is open whenever “a” is closed and closed whenever “a” is open.
See Figure 3 for an illustration of this correspondence.
Figure 3. Shannon’s association of switching networks with Boolean logic.
Shannon’s method allows one to design a switching circuit with any given logical behaviour. For example, the network corresponding to
If we want a network that can control a light bulb with three switches, then the logical condition is
and we can use Shannon’s recipe to write the corresponding network.
Note that we obtain this logical condition by thinking about what we want the switches to do. If all three in closed position turn on the light, then flipping any two of them should also turn on the light. This is what is written in the logical condition. The resulting network is four rows in parallel of series connections with three elementary switches in each row. In Figure 4, we show this network, and we show how it can be “analysed” into a simpler design with two single-pole, double-throw switches at the ends, and a special switch S in the middle. Remarkably, the special switch S is a realisation of the switch we described in the introduction that leaves two lines parallel to one another, or cross-connects them. An intelligent designer would realise that this middle switch S is a good invention and that S is not something “known” to the logical algebra. It arises in the connection and translation that Shannon found between logical algebra and switching networks.
Figure 4. Three switches in Shannon mode and the birth of the cross line switch.
Note also that if one did not see the new switch S then the direct and literal application of Shannon’s correspondence for 4,5,6, … switches controlling a single light bulb creates networks with more and more complexity. For n switches, we will have 1 + C(n,2) = 1 + n(n-1)2 parallel lines with n switches in each line. Here, C(n,2) is the number of ways to choose 2 switches to flip from n switches. Thus, for 100 switches, we would need 1 + 50 x 99 = 4951 parallel lines with 100 elementary switches on each line. It is just too complicated. But we can use the S switch as in the introduction and make a perfectly good 100 switch control for a light with just 100 copies of the S switch and some extra wire.
This part of the discussion shows how taking a specific viewpoint (such as Shannon’s Theory) can be very significant, and yet holding slavishly to that viewpoint could obscure further breakthroughs. Shannon’s method provides a way to think about networks, and it can be used, in conjunction with a designer’s eye, to go farther than one could have gone without it.
III. A Symbolic Analysis of Switching Circuits beyond Shannon and Boole
We can consider bits of network that are like the switch S. Such bits have four attached lines as shown in Figure 5. I shall call them “tangles”. Thus the switch S in any of its states (parallel or crossed) is an elementary tangle. Given two tangles S and T we can form a new tangle ST by attaching the left ends of T to the right ends of S, as shown in Figure 5. Since left and right are determined by how we draw in the plane, I shall let T⎾ denote the result of turning T by ninety degrees. Note from the Figure 5, that T⎾ now has different right and left connecting lines. In the figure we show that
We can interpret the right-hand side of this glyph as the parallel lines state of S, which we shall call E. Then we have E⎾ = ⊃⊂ = V. We can think of V as a new state of S where the parallel lines are vertical rather than horizontal. Note that the crossed state of S is unchanged by this rotation:
X⎾ = X
We shall use the letter X to denote the crossed state of S.
Thus, we now have the beginning of a symbolic theory for putting together networks of switches like S.
Note that when we have two switches of S type attached together and both in the crossed state, then together they produce the parallel state: XX =E. This is the symbolic key to how the Mobius circuit works. When there are an even number of X’s on the circuit, then we have just E on the circuit. When you connect the top ends of E to each other and the bottom ends of E to each other, you get two disjoint circles and an open circuit. The light is off. But when you have an odd number of X’s on the circuit, then there is only one loop, and the light is on!
In our symbols, we have
X
XX=E
XXX=EX=X
XXXX=EE=E
XXXXX=EEX=EX=X
ad infinitum.
Note that we now have the following symbolic facts, supported by the geometry of Figure 5.
Figure 5. The Mobius Algebra.
EA=AE=A for any A.
This means that we really do not have to write E.
Thus, we have E⎾ = V, we can write, ⎾= V.
The figure shows that VV = V if we ignore the circle in the middle that is generated by the self-interaction of V. This can be ignored if we are only interested in the connectivity that will light the bulb. Thus we can write VV = ⎾⎾=⎾. Note also that VX = XV = V as is illustrated in the figure. V interacts with X and creates a loop, but this loop does not change the connectivity. Thus, we have ⎾X = ⎾ and = E = .
Figure 6. A network example.
We are now in possession of an algebra for dealing with networks composed of crossing switches, and we can calculate when changing the switch settings can make the lights go on and off. See Figure 6 for an illustration of these ideas. In that figure, you can see that the light is on by taking a walk through the circuit, but you can also calculate the algebraic reduction of the expression at the bottom of the diagram. If you try this, you will see that algebra can find connectivity. For example, you can see from the algebra (or by tracing the circuit) that flipping switches in sector 3 or in sector 6 will not affect the bulb. The reader is encouraged to experiment.
Let us call this new algebra the Crossing Algebra.
IV. From Laws of Form to the Crossing Algebra
If the reader is familiar with Laws of Form, they can see that we have evolved a structure that has the Spencer-Brown (Spencer-Brown, 1969) mark (and its calculus of indications) in combination with the state X with the special rules, along with the usual laws of crossing and calling (shown first in Figure 7).
From this point of view X, the crossed state of the switch S, is a paradoxical state. It satisfies an equation analogous to the Liar Paradox that states that X is identical with what it is not. X is its own anti-particle. When X interacts with itself it vanishes. XX = .
Figure 7. The usual Laws of Crossing and Calling and the new Laws of Crossing Algebra.
In Lukasiewicz three-valued logic and in the Calculus for Self-Reference initiated in the context of Laws of Form (Kauffman, 1987, 1991, 1994, 1996, 2002, 2005, 2009, 2012), we add an element @ to the calculus behaving like X above except that @@ = @. In multiple-valued logic contexts, it is not so usual to imagine an entity that annihilates itself, but a generalised law of calling is usually taken so that @@=@. The Mobius Algebra is an expression of the very different topological structure and paradox of the Mobius band. We will continue exploration of this algebra in a sequel to the present article.
The significance of the crossing algebra is that it is an extension of indication into the structure of the connectivity of form. In this way, the idea of distinction relates to the intricacies of networks and topology. The cybernetic implications of the Mobius algebra and its network interpretations are just beginning.
From the point of view of systemic design, we have come to a very general point. It is now possible to say what it may mean to go outside of logic, reason, the Boolean framework, and to work in creative dialectic beyond given systematic approaches. From our point of view, the key is to appreciate that each distinction that we make is prior to the systematic and reasoning processes. Such processes have been designed by us to handle the plethora of distinctions and discriminations that are available and that we find and create. By going into creative domains in the making of distinctions, we give ourselves up to possible new thoughts and new designs. When these new designs are found, they can become the basis for further theories in an unending process of creation.
References
- Kauffman, L. H. (1987). Self-reference and recursive forms. Journal of Social and Biological Structures, 10, 53–72.
- Kauffman, L. H. (1996). Virtual logic. Systems Research, 13(3), 293–310.
- Kauffman, L. H. (2012). Knots and physics (3rd ed.). World Scientific Publishing. (Original work published 1991)
- Kauffman, L. H. (1994). Knot logic. In L. H. Kauffman (Ed.), Knots and applications (pp. 1–110). World Scientific Publishing.
- Kauffman, L. H., & Lambropoulou, S. (2002). From tangle fractions to DNA. In M. Monastyrsky (Ed.), Topology in molecular biology: Proceedings of the International Workshop and Seminar on Topology in Condensed Matter Physics, Dresden, 16–23 June 2002 (pp. 69–108).
- Kauffman, L. H. (2005). Eigenform. Kybernetes, 34(1/2), 129–150.
- Kauffman, L. H. (2009). Reflexivity and eigenform: The shape of process. Kybernetes, 38(3), 442–450.
- Sevaldson, B. (2019). What is systemic design? Practices beyond modelling and analyses. Proceedings of Relating Systems Thinking and Design, RSD8. https://rsdsymposium.org/what-is-systemic-design/
- Spencer-Brown, G. (1969). Laws of form. George Allen and Unwin.
- Shannon, C. (1938). A symbolic analysis of relay and switching circuits. Transactions of the American Institute of Electrical Engineers, 57(12), 713–723. https://doi.org/10.1109/T-AIEE.1938.5057767
- Uribe, R. (1995). Tractatus paradoxico-philosophicus (Brown ed.). University of Illinois at Urbana-Champaign.
- von Foerster, H. (2003). Understanding understanding: Essays on cybernetics and cognition. Springer-Verlag.
Acknowledgement
It gives the author great pleasure to thank Thomas Fischer for helpful conversations and for a collaboration—in—process on the question of the relationship of Paradoxical Machines and Non-Trivial Machines, a comparison and inclusion of Uribe and von Foerster (H. von Foerster, 2003). This paper is written in appreciation and memory of Annetta Pedretti and the depths of her design understanding.
Sightlines is an annual anthology of reflexive contributors—RSD chairs, speakers, and other influential voices whose work is foundational or influential in systemic design. This article serves as a reference to the RSD14 workshop, Tangle Circuits and the Limits of Knowing, organised by Louis H. Kauffman and Thomas Fischer.
PROFILE
Louis H. Kauffman
Professor Emeritus, University of Illinois at Chicago
Louis Hirsch Kauffman is an American mathematician, mathematical physicist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. Kauffman's research interests are in the fields of cybernetics, topology, and mathematical physics. His work is primarily on the topics of knot theory and its connections with statistical mechanics, quantum theory, algebra, combinatorics, and foundations. In topology, he introduced and developed the bracket polynomial and Kauffman polynomial.
Kauffman received his B.S. at the Massachusetts Institute of Technology in 1966 and his PhD in mathematics from Princeton University in 1972, with thesis Cyclic Branched-Covers, O(n)-Actions and Hypersurface Singularities written under the supervision of William Browder. He has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institut des Hautes Études Scientifiques in Bures Sur Yevette, France, the Institut Henri Poincaré in Paris, France, the University of Bologna, Italy, the Federal University of Pernambuco in Recife, Brazil, and the Newton Institute in Cambridge, England.
He is the founding editor and one of the managing editors of the Journal of Knot Theory and Its Ramifications, and editor of the World Scientific Book Series On Knots and Everything. He writes a column entitled Virtual Logic for the journal Cybernetics and Human Knowing. From 2005 to 2008, he was president of the American Society for Cybernetics. He plays clarinet in the ChickenFat Klezmer Orchestra in Chicago.
Kauffman is the 1993 recipient of the Warren McCulloch Award of the American Society for Cybernetics (ASC) and the 2014 recipient of the Norbert Wiener Medal of the ASC.
Adapted from “Louis Kauffman,” Wikipedia, accessed July 16, 2025, https://en.wikipedia.org/wiki/Louis_Kauffman
Sightlines: ARC25
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