Keynote: Catjects - Dirk Baecker

Sociology's interest in the Laws of Form is at least threefold. First, the cross describes a communication generating memory out of oscillation. Second, the calculus of indications makes it possible to write networks reproducing as eigen-values of systems. And, third, such networks, or catjects, as I would propose to call them, help to design and reflect strategies of projects differentiated and reproduced as systems.

Dirk Baecker is Professor of Cultural Theory and Management at the University of Witten/Herdecke, Germany. His research work includes sociological theory, culture theory, economic sociology, organization research, and management education. In his current research project, the Catjects Project, Dirk Baecker explores catjects – categories underlying the differentiation and reproduction of specific types of communication – as a means of cultural analysis, relying on a culture theory which understands culture as the product and condition of a complex mutuality and polycontexturality of different observers (such as minds, bodies, machines, swarms, nets, encounters, or organizations) emerging from each other's second-order observation.

Keynote: Explorations in Laws of Form - Louis H Kauffman

This talk will discuss a number of mathematical and philosophical excursions I have had in relation to Laws of Form.
The remarks in the talk fall into the following categories.
0. The mark and the observer are, in the form, identical.
1. Mathematics is about what a distinction would be if there could be a distinction.
2. Arithmetic in the form.
3. Transfinite arithmetic in the form.
4. Idemposition, contiguity and the Four Color Theorem.
5. Imaginary Values, Iterants, Eigenforms, Complex Numbers and Clifford Algebras.
6. Modulators, stable states and transitions. Imaginary values in the transition of form.
7. Re-entry forms and Lambda Calculus.

The Design of Computation - William Bricken

Between 1993 and 2000 under the direction of Richard Shoup the Natural Computing Project at Paul Allen's Interval Research Corporation applied LoF to the creation of formal computation from first principles. The goal was to develop a thorough understanding of LoF while building a complete suite of verifiable and efficient software and hardware design tools. We built a LoF engine that achieved proof without propositional or predicate calculus and a software and silicon design optimizer. We discovered new computational methods and perspectives, including
— void-based reasoning that treats selected structure as meaningless,
— deep pattern-directed transformation that ignores function boundaries,
— imaginary query-based reduction that locates redundancy without symbolic manipulation,
— logic engines that can run both sequentially and asynchronously in distributed environments,
— several completely new hardware architectures, and
— postsymbolic mathematical models of computation.

These tools outperformed commercial silicon design tools on million transistor sequential and logic circuits, but our primary objective was provably error-free computation for multilevel sequential logic synthesis and optimization. When Interval closed, we were in the process of transferring the computational suite to reconfigurable hardware.

Dr. Bricken has spent over 40 years developing the tools and techniques of boundary mathematics. Ph.D. in Research Methods, Educational Psychology and Computer Science from Stanford University.

In Between Concept and Instance: On the Synthetic Role of the So-Called Example - Christina Weiss

“For example, in a plane space a circle draws a distinction” (Spencer-Brown, Laws of Form, p. 1).
Laws of Form seeks to explicate the laws of distinction by (re-)constructing distinctions. Epistemologically it hereby endeavours to translate the semantic knowledge of what distinction in general is into the pragmatic knowledge of how one is able to construct a concrete instance of distinction.
Spencer-Brown himself doesn't put larger effort on the reflection of the relationship between semantic and pragmatic aspects in his theorizing, probably because of a deep conviction of some sort of constructive idealism, in which semantic and pragmatic elements are identical by definition. However, if one reconstructs the procedure of Laws of Form with respect to its handling of the relationship between semantic and pragmatic elements, one can recognize a significant shift from the general conceptual account of form as a unity of complements, given in the introduction, and the spatialized variant of it as “perfect continence”, which renders possible the construction of a circle as a so-called example of distinction.
Making explicit the inherent logic of this shift, hereby showing that the so-called example does actually carry the epistemological function of a synthetic differentiation of the very concept of distinction itself, together with the related discussion of predicative and impredicative elements of Laws of Form is the goal of this paper.

Dr. phil. Christina Weiss, born in 1973, is a philosopher interested in foundational questions concerning the relationship between Phenomenology and Constructivism, in dialectical and dialogical logics, German Idealism and its relationship to constructive logics in particular, philosophical semantics in general. Throughout her work in the different fields of research she seeks to expatiate on the foundations of an epistemology of schematization. She currently works on her habilitation treatise and teaches as a lecturer at the Institute of Philosophy at the Technical University of Darmstadt, Germany.

Time and the Spiritual Laws of Form - R John Williams

Heinz von Foerster's review of George Spencer-Brown's “Laws of Form” in the Whole Earth Catalog took what might have seemed an altogether dry and mathematical treatise and propelled it into a world where “new age” forms of religiosity and psychotherapy were radically changing Western culture. This presentation illustrates this transformative moment in two correlative areas: first, Spencer-Brown's own spiritual explorations with psychedelics, Eastern religion, and existential psychology (examining never before seen letters between Spencer-Brown and the psychologist R.D. Laing); and second, Spencer-Brown's famous visit to Esalen in 1973, where he spoke with an enthusiastic group of countercultural icons. While transcripts of Spencer-Brown's visit to Esalen have circulated online for years, in this presentation I play excerpts from a newly discovered recording of the event. What emerges, I argue, is no simple innovation in mathematics, but a programmatic re-imagination of religious experience.

R. John Williams is Associate Professor of English and Media Studies at Yale University

An Arithmetic and its Geometry in the Higher Degrees - Bernie Lewin

Laws of Form is supposed to be presenting an arithmetic, and yet expected characteristics of arithmetic are absent. There are no numerals. There is no consistent way to perform standard operations like multiplication and division. And the author himself admitted that no mensural geometry has been developed. This paper finds a way to understanding Spencer Brown's project in the original idea of 'arithmetic' as practiced by the ancient Pythagoreans. It shows how their emanation of dimensional magnitudes guides us to an order in the higher degrees of formal arithmetic where a geometry is found to emerge. The investigations of this geometry in the plane are preliminary, nevertheless they suggest that there might be a way to classify geometric and algebraic numbers by degrees of infinity according to the degree of number in the formal arithmetic that are used to express them.

Bernie Lewin is an amateur historian and philosopher of science based in Melbourne, Australia. His historical interests range from ancient Pythagorean mathematics to the 'Foundation of Mathematics' controversy that was left unresolved a century ago. A recent interest in the corruption of post-World War II state-funded natural science led to his book, 'Searching for the Catastrophe Signal' (2016). Philosophically, Lewin is a Platonist. Platonism underlies his history investigations as well as his approach to George Spencer Brown's 'Laws of Form' (1969). For Lewin, 'Laws of Form' might well herald a revival of Platonic science by showing a better way -- a more Pythagorean way (i.e., self-referencing, non-analytical) -- to build a hierarchy of infinite numbers and, in doing so, forge a new relationship with geometry. This view is carefully developed through an historical narrative in his book, 'Enthusiastic Mathematics' (2018). Lewin is the founding director of the Platonic Academy of Melbourne.

Lessons from the Markable Mark - George Burnett-Stuart

Some reflections on Laws or Form arising from my work on the Markable Mark website, especially its primary algebra app. The form shines more brightly the more we rid ourselves of preconceptions. Our aim here will be to take one or two steps along this rocky path.

George got his D.Phil. at Oxford. He was in Roger Penrose's research group there. He researched black holes, gravitational waves and twistor theory. Later he switched to freelance mathematical consulting. He invented and marketed an astronomical (or is it astrological?) clock called Almagest. He has long been hooked on LoF and something made him create the LoF-based website The Markable Mark in 2012.

Panel Discussion I - Lars Clausen, Walter Tydecks

Lars Clausen
Impulse Topic: Competing Identities and Dirty Distinctions

When the systems theoretician and sociologist Niklas Luhmann first started referring to the Laws of Form in the beginning of the eighties, it was a mere addition to his already well-developed concept of distinctions and selections. His main theoretical oeuvre Soziale Systeme from 1984, did not take the implications of LoF seriously. Only from the late eighties did his selective reading of LoF emerge as a powerful theoretical device for reconstruction of his own systems theory, especially in connection with Fritz Heiders early medium theory.
Although his late masterpiece Theory of Society from 1997 did take LoF seriously, Luhmanns health had degraded too far and was aware, that the needed final conversion of his theory to adhere to the Laws of Form, would rest on the shoulders of future generations. Dirk Baecker, Steffen Roth and others have taken the challenge up. Their goal has been to describe and analyse the coherency of society and the theoretical structure. This presentation parts from their propositions and instead observes on the power of incoherent forms, invalid operations and illegal distinctions. According to Luhmannian theory, such operations were regarded as dangerous and productive paradoxes alike, and had to be subdued and invisibilized, as they otherwise would bring societal operations to a stalemate. This, I argue, is a reflex by Luhmann, stemming from the wish for an orderly society and universe in his theory; instead, the dirty distinctions of society are far more than paradoxes in need of invisibilization. Dirty distinctions are necessary and congruent forms of social operations, compatible with Luhmannian theory. What is more uncertain, is the coherence with the strict proctologic of Spencer-Brown in his Laws of Form.

Lars Clausen is doctoral candidate at the Europa - Universität Flensburg, Germany and consultant at the UCL University College, Denmark. His main interest is in the scholastic legacy present in Luhmannian systems theory and Laws of Form.

Walter Tydecks
Impulse Topic: The Self-organization of the Medium in Spencer-Brown's Circuits

The Laws of Form are based on a classical opposition of subject and object. Spencer-Brown addresses a subject 'to draw a distinction', to present it graphically according to given rules and to inscribe it in a medium. The medium remains in a motionless and uninvolved role. This changes with the circuits that not only remember, but also count and perhaps perform higher functions. When asked what happens inside these circuits, this can not be completely cleared up. It can be checked and confirmed that an input provides the anticipated output. If the input is broken down into a chain of single elements, it shows that the circuit not only processes the individual data, but sometimes changes its own state. Although it can be described for each individual step how the incoming sign passes through the circuit and which states the circuit takes, but it appears like an unexplained black box, as it comes to changing one state in the following. In my contribution, this topic will be illustrated by the example of the circuit for the modulator function shown in Laws of Form and later commenting by Spencer-Brown and Kauffman and interpreted as a hidden self-organization of the medium.

Tydecks, Walter, b. 1952, studied mathematics, political science and philosophy (Dipl.-Math.), Professional activity as a system developer, project manager and IT manager of medium-sized companies with a global orientation, philosophical work with a focus on philosophy and mathematics, recent developments in logic, Aristotle and the classical German philosophy

Structural Comparison of Laws of Form and Genesis 1,1-2,4a - Josef Freystetter

Laws of Form has a well-known precursor: Genesis 1,1-2,4a. The treatise is 2500 years old; it was written in Babylon and recorded in the Jewish Torah or Christian Pentateuch. The universe and the earth (ground) were created as eight opera in seven days (times). Each Theorem, Consequence, Initial, Rule, Canon, Axiom, the Oscillator- Memory- and Modulatorfunction of the Laws of Form is found in this passage and, without these elaborated laws, it would not be rediscovered as a calculus of indication. The origin of this calculus is its eternal self-explanation of signs – emptiness and formlessness (hb. tohu wabohu). Its idiosyncracy is the self-injunction of the user, indicated as power (hb. Elohim) and its beginning is not with space but with time. Thus all the following opera are entangled with time and lead to eight complex finite expressions. The latter indicated as pro-duce (hb. adam) is reentered in the created expressions in the sixth time. Another property is the descent in the seventh time, which leads each language-using system to the experience of its potentiality.
Gn 1,1-2,4a is an accomplished self-referencial calculus whith nine qualities of space and seven qualities of time without paradoxes! It provides a new cross-disciplinary worldview between science and religion, which discloses the unexplored potential of the Laws of Form. This short, universally known, and easily accessible "story" rehabilitates emotion for mathematical thoughts in creating the universe and the ground.

Consultant for higher educational institutes and teacher at several Austrian universities. Studied theology and philosophy at the University of Vienna and about to graduate his PhD at the German University Witten/Herdecke.

Two Men of Distinction - Parallels in the World Views of George Spencer-Brown & Douglas Harding - Tom Short

I summarise the work of the modern ‘mystic’ Douglas Harding and explain how his approach is essentially to draw our attention to how we make distinctions in our everyday way of seeing but that when we apply these distinctions phenomenologically to ourselves, we are confronted with emptiness - the experience, in his own words, of "having no head". I explain why Harding felt that these experiences were equivalent to those obtained by Zen Buddhist meditation. I map his ideas on distinction to the LoF. I invoke the familiar visual illusions of gestalt psychology and sensory-motor loop considerations to suggest that Harding’s two ways of seeing can be conceived both as gestalts and eigenstates of reality.

Further, I suggest ways in which these ideas might enrich traditional Buddhist approaches to mindfulness meditation.

Now mainly retired, I have spent most of my professional life working in universities. Initially lecturing in physics then, via an MSc in Computer Science, becoming a Principal Lecturer in IT. Eventually I drifted into university IT management as Director of Administrative IT for two universities. Late in my career I discovered systems thinking via an OU Diploma in Systems Practice and I realised that there were other people who thought like me!

In the interstices in my career I developed an interest in Buddhism and led a double life as a Visiting Buddhist Chaplain in UK prisons for 20+ years and then as a Visiting Buddhist Chaplain in a University for 16 years, which is ongoing. I have an MA in Buddhist Studies.

I’m still very puzzled by the universe and still looking for answers.

The BF Cyclic Calculus - A Purely Containment-Based Extension to Laws of Form - Arthur Collings

This paper describes the BF (Brown 4) Calculus, a four-valued extension of Laws of Form in which the Law of Crossing is reinterpreted as a cycle of period four. It serves as a companion to the paper entitledThe BF Calculus and the Square Root of Negation, which also describes BF but employs a notation based on ordered pairs. BF is described without reference to ordered pairs, as a system built upon a single symbol, the imaginary mark. Based on a new axiom set introduced by this author, demonstrations are given for a series of new consequences, none of which exist in the Primary Algebra. An entirely new normal form is derived, and BF is shown to be functionally and axiomatically complete. We conclude by demonstrating the construction of Kauffman and Varela's Waveform Algebra within the BF Calculus.

Arthur M. Collings is an independent researcher and lives in Red Hook, NY. He is currently Treasurer of the American Society for Cybernetics. He is a cartographer and conservationist by profession, and is Vice President of Dutchess Land Conservancy, a conservation land trust in the New York's Hudson Valley.

Performance: Flatland: A Romance of Many Distinctions - Divyamaan Sahoo

Light illuminates the space that we intend to cleave, allowing us to form the primary distinction. Chapter 1 of the ‘Laws of Form’ begins with the following: ‘we take as given the idea of distinction and the idea of indication’. ‘Idea’ has many interpretations, rooting back to ‘form, pattern, shape, space’ and ‘idein’ (Greek), which means ‘to see’. This suggests an a priori source of light, or life, that allows us to see distinctions in the first place.

During my explorations of Edwin Abbott Abbott's ‘Flatland’, I decided to work with a fixed source of light that I took as given, and various incident objects such as the puppet and the puppeteer. The process was embodied and physical, and involved notation using boundaries and networks. I decided to then connect the arrangements/ compositions using equivalences and indicative shifts, and build a score that allowed me to move from one arrangement to another.

Divyamaan Sahoo from Kolkata, India, is a mathematician, puppeteer, and sound artist, deeply influenced by anima, the soul or essence of living things, in the alchemy of puppets and instruments.