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This paper aims to show how a sociological description – a swarm analysis of the Nazi dictatorship – initially made with the means borrowed from George Spencer-Brown’s Calculus of Indications, can be transformed into a digital circuit and with which methods and tools of digital mathematics this digital circuit can be analyzed and described in its behavior. Thus, the paper also aims to contribute to a better understanding of Chapter 11 of “Laws of Form.”

The analysis uses methods of automata theory for finite, deterministic automata. Basic set operations of digital mathematics and special set operations of the Boolean Differential Calculus are used to calculate digital circuits. The software used is based on ternary logic, in which the binary Boolean logic of the elements {0, 1} is extended by the third element “Don’t care” to {0, 1, −}.

The paper confirms the method of transforming a form into a digital circuit derived from the comparative functional and structural analysis of the Modulator from Chapter 11 of “Laws of Form” and defines general rules for this transformation. It is shown how the indeterminacy of re-entrant forms can be resolved in the medium of time using the methods of automata theory. On this basis, a refined definition of the degree of a form is presented.

The paper shows the potential of interdisciplinary approaches between sociology and information technology and provides methods and tools of digital mathematics such as ternary logic, Boolean Differential Calculus and automata theory for application in sociology.

Blake is relatively well-known, but who was J M Robertson? What's his connection with George Spencer-Brown? And how exactly did J M Robertson influence George Spencer-Brown?

George Spencer-Brown (1923–2016) is the author (among other works) of the undeservedly little-known book, Laws of Form (1969/2011), which was a key inspiration for Niklas Luhmann (1927–1998). But what inspired George Spencer-Brown? This paper explores two key influences on George Spencer-Brown and his work: the English poet and artist, William Blake (1757–1827) and the Scottish rationalist, politician and author, J M Robertson (1856–1933).

The paper points to a broken link between George Spencer-Brown's work and Niklas Luhmann's.

These questions are explored from two perspectives: first, George Spencer-Brown's works and their debt to (1) Blake's work, from which he quotes in a number of instances and to (2) J M Robertson's (in particular, the latter's Letters on Reasoning (1905) and Rationalism (1912)); second, my personal connection to Spencer-Brown, who mentored me through Laws of Form and with whom I developed a close friendship involving regular weekly telephone conversations for the greater part of the last four years of Spencer-Brown's life. I share anecdotes and stories that connect George Spencer-Brown and J M Robertson that span George Spencer-Brown's lifetime – from his school days to his dying days. Both Blake's and Robertson's influences are relevant to Spencer-Brown's view of morality. The paper looks at specific connections between Blake's work and J M Robertson's on the one hand and George Spencer-Brown's on the other.

George Spencer Brown is best known for his book Laws of Form, which elaborates a primary algebra of distinctions and forms capable of dealing with self-referential equations reflective of paradoxes in logic. The book has received little attention in mathematics, but it has greatly influenced cybernetics, communications, and ecological theories. But Spencer Brown also published poetry and stories, often under different names, and he practiced as a psychotherapist. Our chapter elaborates the utility of Laws of Form relating to organizational paradox before considering Spencer Brown’s other works in relation to his mathematics. Invoking philosophy, psychoanalysis and art, we suggest that these indicate a further distinction that sets all forms against the “nothing”: a wholeness or unity from out of which all distinctions, all words, meaning and life – but also all silence, nonsense and death – emerge in paradoxical opposition. Reading Spencer Brown not through the prism of mathematics, but as an evocative invitation to engage with the fissures that animate art and human life, highlights the paradoxical interplay of organization and violence; and how tragedy, suffering, sympathy and love should be more prominent in organizational research.

Cross tables are omnipresent in management, academia and popular culture. The Matrix has us, despite all criticism, opposition and desire for a way out. This paper draws on the works of three agents of the matrix. The paper shows that Niklas Luhmann criticised Talcott Parsons’ traditional matrix model of society and proceeded to update systems theory, the latest version of which is coded in the formal language of George Spencer Brown. As Luhmann failed to install his updates to all components of his theory platform, however, regular reoccurrences of Parsonian crosstabs are observed, particularly in the Luhmannian differentiation theory, which results in compatibility issues and produces error messages requesting updates. This paper aims to code the missing update translating the basic matrix structure from Parsonian into Spencer Brownian formal language.

This paper draws on work by Boris Hennig and Louis Kauffman and a yet unpublished manuscript by George Spencer Brown, to demonstrate that the latter introduced his cross as a mark to indicate NOR gates in circuit diagrams. The paper also shows that this NOR gate marker has been taken out of and may be observed to contain the tetralemma, an ancient matrix structure already present in traditional Indian logic. It then proceeds to translate the basic structure of traditional contingency tables into a Spencer Brownian NOR equation and to demonstrate the difference this translation makes in the modelling of social systems.

The translation of cross tables from Parsonian into Spencer Brownian formal language results in the design of a both matrix-shaped and compatible test routine that works as a virtual window for the observation of the actually unobservable medium in which a form is drawn, and can be used for consistency checks of expressions coded in Spencer Brownian formal language.

This paper quotes from and discusses a so far unpublished manuscript finalised by Spencer Brown in April 1961. The basic matrix structure is translated from Parsonian into Spencer Brownian formal language. A Spencer Brownian NOR matrix is coded that may be used to detect errors in expressions coded in Spencer Brownian formal language.