George Spencer-Brown Research Articles

Where George Spencer-Brown went wrong – re-entry recalculated

PurposeThe paper uncovers a mathematical error in George Spencer-Brown's genesis of re-entry. It distinguishes between the two interpretations of re-entry presented in “Laws of Form”: recursive versus sequential. The Indeterminacy inferred by George Spencer-Brown from his recursive genesis of re-entry is refuted in three different ways. The calculation of the Modulator from “Laws of Form” demonstrates that only the sequential interpretation of re-entry is reasonable. This contributes to the demystification of re-entry and enables a deeper understanding. Finally, six differences between the concept of form from “Laws of Form” and Niklas Luhmann's sociological systems theory are presented.Design/methodology/approachMethodologically, the paper uses the ternary logic of discrete mathematics, which extends {0, 1} by “don't care” to {0, 1, -}. George Spencer-Brown's Indeterminicy is refuted by using three different methods: complete induction, Theorems 14 and 15 and the software XBOOLE. For the calculation of the Modulator, the only practical application of re-entry in “Laws of Form”, techniques from automata theory are used.FindingsThe paper reveals a mathematical error in George Spencer-Brown's genesis of the re-entry of “Laws of Form” and refutes the assumption of Indeterminicy. The analysis of the only practical application of re-entry presented by George Spencer-Brown shows that the functioning of this Modulator can only be described correctly with the sequential interpretation of re-entry.Originality/valueThe paper emphasizes the interdisciplinary potential of sociology and information technology and provides methods and tools of discrete mathematics for use in the analysis of the works of George Spencer-Brown and Niklas Luhmann.

Spencer-Brown, Luhmann and Klein on Symmetry

Abstract This article discusses the concept of symmetry as Felix Klein developed it in his works on geometry as a measure of invariance. The objective is to generalize from Klein’s model and principles to be able to apply the concept of symmetry to the study of social structures or social formations in sociology. To achieve this goal, the article goes on to consider how the concept of symmetry bears on the Laws of Form of George Spencer-Brown and on Niklas Luhmann’s theory of functional differentiation. Both of them are reinterpreted in terms of symmetry breaking.

Iconic Mathematics: Math Designed to Suit the Mind.

Mathematics is a struggle for many. To make it more accessible, behavioral and educational scientists are redesigning how it is taught. To a similar end, a few rogue mathematicians and computer scientists are doing something more radical: they are redesigning mathematics itself, improving its ergonomic features. Charles Peirce, an important contributor to ordinary symbolic logic, also introduced a rigorous but non-symbolic, graphical alternative to it that is easier to picture. In the spirit of this iconic logic, George Spencer-Brown founded iconic mathematics. Performing iconic arithmetic, algebra, and even trigonometry, resembles doing calculations on an abacus, which is still popular in education today, has aided humanity for millennia, helps even when it is merely imagined, and ameliorates severe disability in basic computation. Interestingly, whereas some intellectually disabled individuals excel in very complex numerical tasks, others of normal intelligence fail even in very simple ones. A comparison of their wider psychological profiles suggests that iconic mathematics ought to suit the very people traditional mathematics leaves behind.

Roots, shoots, fruits: William Blake and J M Robertson: two key influences on George Spencer-Brown's work and the latter's relationship to Niklas Luhmann's work

PurposeBlake 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?Design/methodology/approachGeorge 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).FindingsThe paper points to a broken link between George Spencer-Brown's work and Niklas Luhmann's.Originality/valueThese 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.

Digital analysis of a form

PurposeThis 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.”Design/methodology/approachThe 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, −}.FindingsThe 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.Originality/valueThe 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.

A Sociological Reading of George Spencer-Brown's Laws of Form

A sociological reading of George Spencer-Brown’s Laws of Form consists in reading it as a theory of the observer. The paper looks at the “cross” established by the calculus of indications as a universal operator of general, or reflective, negation, presents second-order observation as a means to introduce indeterminacy as a precondition to communication and reads Spencer-Brown’s primary arithmetic and primary algebra as steps towards an understanding of the (socio-)logical space comprehending any arrangement and re-arrangement of indications and distinctions. A short overview of the history of the notion of “form,” or “idea,” as developed by Plato, disclaimed by Kant and Hegel, and employed by Marx, Simmel, and Cassirer shows that this notion from the beginning hides, and passes on, problems of self-reference, even if disguised as transcendental subjectivity. A way to deal with these problems may be shown by Spencer-Brown’s introduction of imaginary states within equations of the second degree. Imaginary states, or values, allow time, society, nature, and technology to be introduced as references accounting for, exploring, and exploiting the indeterminacy created by them. v5

Niklas Luhmann's Task of Processing Complexity and George Spencer-Brown's Exploration of Non-mathematical Mathematics for Construction

Niklas Luhmann's Task of Processing Complexity and George Spencer-Brown's Exploration of Non-mathematical Mathematics for Construction

On a Generality of the Observing-Operation: The Calculus of Indication by George Spencer-Brown and its Epistemological Potentials

On a Generality of the Observing-Operation: The Calculus of Indication by George Spencer-Brown and its Epistemological Potentials

From Hardware to Hardcore

The state and relevance of Systems as a field of research and a specific form of scientific inquiry into complex real-world problem situations, can be enhanced significantly by developing and applying more formalized and coherent tools: a new ‘hardware' enabling to build a new ‘hardcore' for systems science. The basis of this new hardware stems from a line of thought emanating from George Spencer-Brown and the ‘Laws of Form', running through the work of Francisco Varela and his calculus for self-reference, being radicalized by Niklas Luhmann and his views on ‘Social Systems', and continued by Dirk Baecker with the application of form theory to management and organizations. In this contribution, the author develops an understanding and appreciation of the potentials of a form-theoretical approach to formalizing systems (real-world phenomena) as well as Systems (field of research). Central aspects will be the power of the form-theoretical hardware as regards systems storytelling, systems diagnostics and abductive reasoning.

Spencer-Brown, Peirce, Girard, and the Origin of Logic

Abstract This article will show how René Girard’s hypothesis of the scapegoat victim is the indirect basis of the discipline that unfolds the necessary consequences of the distinction between two possible states; that is, formal logic. To support this unusual view on the foundation of logic, two systems of formal logic will be analyzed: the calculus of indications created by George Spencer-Brown in the 1960s and the system of existential graphs created by Charles S. Peirce in the late nineteenth century. The first system shows that logic is based on elemental difference, and the second displays that logic is based on an original act of expulsion, the expulsion of the opposite of logic, which will be called the illogical. It will then be argued that Girard’s theory allows the full understanding of the expulsion of the illogical; that is, the emergence of the elemental difference upon which logic is based. It is not argued that logic, as a symbolic system of inferences, is directly based on Girard’s theory. It is rather argued that, as shown by the work of Spencer-Brown, formal logic is based on elemental distinction and negation from where they had their roots in the scapegoat mechanism. Finally, concerning a real and specific pattern of inference, a link is suggested between apagoge and the founding murder.

The Be-ing of Objects

The paper is a reading of Martin Heidegger's Contributions to Philosophy (Of the Even) by means of Ranulph Glanville's notions of black box, cybernetic control and objects as well as by George Spencer-Brown's notion of form and Fritz Heider's notion of medium. In fact, as Heidegger was among those who emphasized systems thinking as the epitome of modern thinking, did in his lecture on Schelling's Treatise on the Essence of Human Freedom a most thorough reading of this thinking, and considered cybernetics the very fulfilment of modern science it is interesting to know whether second-order cybernetics, as it was not known to Heidegger and as it delves into an understanding of inevitable complexity and foundational ignorance, falls within that verdict mere modernity or goes beyond it. If modern science in its rational understanding considers its subjects to be objects sitting still while being observed, then indeed second-order cybernetics is different. It looks into the observer's interactions with black boxes, radically uncertain of where to expect operations of a self, but certain that we cannot restrict it to human consciousness.

Dificultades y paradojas de la observación de segundo orden: Reflexiones en torno al cálculo de la forma

This paper focuses on the particularities of the second-order observation from the point of view of the calculus of form devised by George Spencer-Brown. In a first step, it highlights the potential opening up for the social sciences with the use of the form as observation strategy. In a second step, it provides eleven theses on the practical use of the second-order observation. Specifically, it redefines the theory of second-order observation from systems theory of Niklas Luhmann.

A Calculus for Autopoiesis

The paper looks once more at the understanding and definition of autopoiesis as developed by Humberto R. Maturana, Francisco J. Varela, and Ricardo Uribe. We will focus on the question whether George Spencer-Brown's Laws of Form presents us with a possibility of translating Maturana's definition into a kind of a calculus. We look at Maturana's emphasis on components, networks, and boundaries and try to figure out how this emphasis can translate into an understanding of form that knows about self-reference, paradox, and play.

Aristotle and George Spencer-Brown

The paper deals with Aristotelian logic as the special case of more general epistemology and sociology of both science and common sense. The Aristotelian principles of identity, of noncontradiction, and of excluded middle are to be supplemented by the second-order cybernetic, or cybernethic principles of paradox, of ambivalence, and of control. In this paper we collect some ideas on how to evaluate the scope of Aristotelian logic with respect to the laws of thought they tried to determine and to do so within the historical moment of the impact of the invention of writing possibly triggering this determination. We look at some modern doubts concerning these laws and discovering an understanding of complexity that is not to be resumed under any principle of identity. The invention of sociology, epistemology, and the mathematics of communication follow suit in focusing not only on the observer but more importantly on the distinction between observers to further contextualize any talk of identities and operationalize both talk and fact of contradiction, paradox, and ambivalence.

Spencer-Brown vs. Probability and Statistics: Entropy’s Testimony on Subjective and Objective Randomness

This article analyzes the role of entropy in Bayesian statistics, focusing on its use as a tool for detection, recognition and validation of eigen-solutions. “Objects as eigen-solutions” is a key metaphor of the cognitive constructivism epistemological framework developed by the philosopher Heinz von Foerster. Special attention is given to some objections to the concepts of probability, statistics and randomization posed by George Spencer-Brown, a figure of great influence in the field of radical constructivism.

Systems, Network, and Culture

Zusammenfassung Der Aufsatz vergleicht die Theorien sozialer Systeme und sozialer Netzwerke im Hinblick auf ihre jeweilige Problemstellung. Die Systemtheorie konzentriert sich auf Probleme der Differenz und Reproduktion, während sich die Netzwerktheorie mit Problemen der Identität und Kontrolle beschäftigt. Erstere hat es mit Fragen der Kommunikation, letztere mit Fragen der Handlung zu tun. Um diese unterschiedlichen Akzentsetzungen zu verstehen, mag es sinnvoll sein, sich daran zu erinnern, dass die Systemtheorie ein Zeitgenosse der Erfindung des Computers ist, während die Netzwerktheorie trotz älterer Wurzeln ihren Erfolg der Einführung des Internets und damit einhergehender Phänomene verdankt. Der Aufsatz vergleicht die beiden Ansätze im Hinblick auf Fragen der mathematischen Modellierung, der Kultur und der Selbstreferenz, die interessanterweise eng miteinander zusammenhängen. Der Beitrag schließt mit einer Erinnerung an Bronislaw Malinowskis »wissenschaftliche Theorie der Kultur« und macht einen Versuch, diese mithilfe einer Spencer-Brown-Gleichung mathematisch zu modellieren. Man erhält die Form der Unterscheidung von Kommunikation, Bewusstsein und Leben und damit das Netzwerk drei reproduktionsfähiger Systeme.