Berkowitz G. C., Greenberg D. R. & White C. A. (1988) An approach to a mathematics of phenomena: Canonical aspects of reentrant form eigenbehavior in the extended calculus of indications. Cybernetics and Systems: An International Journal 19(2): 123–167.

Self-reference and recursion characterize a vast range of dynamic phenomena, particularly biological automata. In this paper we investigate the dynamics of self-referent phenomena using the Extended Calculus of Indications (ECI) of Kauffman and Varela, who have applied the ECI to mathematics, physics, linguistics, perception, and cognition. Previous studies have focused on the algebraic structure of the ECI, and on form dynamics using only the arithmetic of Spencer-Brown. We here examine the temporal behavior of self-referent or reentrant forms using the full power of the ECI to represent tangled hierarchies and multiple enfolded dimensions of space-time. Further, we explore the temporal convolution of static and recursive states in coherent fluctuation, providing a foundation for going beyond the Turing model of computation in finite automata. Novel results are presented on the structure of reentrant forms and the canonical elements of form eigenbehavior, the characteristic self-determined dynamic inherent in reentrant forms.

Ene P. (2013) Descriptions as Distinctions. George Spencer Brown’s Calculus of Indications as a Basis for Mitterer’s Non-dualistic Descriptions. Constructivist Foundations 8(2): 202–208. https://cepa.info/862

Context: Non-dualistic thinking is an alternative to realism and constructivism. Problem: In the absence of a distinct definition of the term “description,” the question comes up of what exactly can be included in non-dualistic descriptions, and in how far the definition of this term affects the relation between theory and empirical practice. Furthermore, this paper is concerned with the question of whether non-dualism and dualism differ in their implications. Method: I provide a wider semantic framework for the term “description” by means of George Spencer Brown’s terminology in his calculus of indications as laid out in Laws of Form. The connection of descriptions and distinctions enables descriptions to comprise reflections and language as well as empirical observations. Results: Non-dualism can be thought of in different ways but still has essential elements in common with dualism. Implications: Non-dualism, as well as dualism, is an argumentation technique suitable for specific situations, but without significant differences in implications.

This is the sixth column in this series on ‘Virtual Logic’. In this column we shall give a short exposition of how symbolic logic is illuminated by the calculus of indications. Columns four and five began an introduction to the calculus of indications. Nevertheless, we shall be self-contained here and recall this construction in section 1.

Kauffman L. H. (1978) Network synthesis and Varela’s calculus. International Journal of General Systems 4: 179–187. https://cepa.info/1822

Network models are given for self-referential expressions in the calculus of indications (of G. Spencer Brown). A precise model is presented for the behavior of such expressions in time. The extension of Brown’s calculus by F. Varela is then shown to describe behavior invariant properties of these networks. Network design is discussed from this viewpoint.

Kauffman L. H. (1987) Imaginary values in mathematical logic. In: Proceedings of the Seventeenth International Conference on Multiple Valued Logic. IEEE: 282–289. https://cepa.info/1842

We discuss the relationship of G. Spencer-Brown’s Laws of Form with multiple-valued logic. The calculus of indications is presented as a diagrammatic formal system. This leads to new domains and values by allowing infinite and self-referential expressions that extend the system. We reformulate the Varela/Kauffman calculi for self-reference, and give a new completeness proof for the corresponding three-valued algebra (CSR).

This is the sixth column in this series on “Virtual Logic.” In this column we shall give a short exposition of how symbolic logic is illuminated by the calculus of indications. Columns four and five began an introduction to the calculus of indications. Nevertheless, we shall be self-contained here and recall this construction in section 1.

Martin R. J. (2017) Moving Toward a Paradigm Shift by Developing that Paradigm Shift. Constructivist Foundations 13(1): 25–27. https://cepa.info/4386

Open peer commentary on the article “Mathematical Work of Francisco Varela” by Louis H. Kauffman. Upshot: Kauffman’s target article explicates Spencer Brown’s Laws of Form and Varela’s Calculus of Indications as a way of thinking about the observer and the observed. This commentary points out that thinking about observer and observed in this way can also be a way of thinking about learning and meaning.

Schiltz M. (2009) Space is the place: The laws of form and social systems. In: Clarke B. & Hansen M. (eds.) Emergence and embodiment: New essays on second-order systems theory. Duke University Press, Durham: 157–178. https://cepa.info/4670

Excerpt: The single most striking characteristic of George Spencer Brown’s Laws of Form is the variety of misunderstandings concerning its reception. Its basic idea is actually quite easy: “form” or “something” is identical to the difference it makes (with anything else) and (thus) eventually different from itself. All “something” or “form” or “being” is explained as the residual of a more fundamental level of operations (namely, the construction of difference), including the “calculus of indications” explaining the very Laws of Form. Due to its constructivist nature, the calculus has enjoyed admiration from a variety of people, some of whom are regarded of major importance in their respective scientific disciplines. After a meeting with Spencer Brown in 1965, the philosopher and logician Bertrand Russell congratulated the young and unknown mathematician for the power and simplicity of this calculus with its extraordinary notation. In 1969, shortly after the publication of LoF ‘s first edition, the father of neocybernetics, Heinz von Foerster, enthusiastically described it as a book that “should be in the hands of all young people.” In the cybernetic tradition, by the way, LoF ‘s resonance is undiminished. The international journal Cybernetics and Human Knowing published a Charles Sanders Peirce and George Spencer Brown double issue in 2001; there exist two extensive Web sites with LoF material and new Spencer Brown mathematical work (see “Spencer Brown–related sources” in the notes below); and a revised English edition of LoF is forthcoming. One would conclude that LoF is very much alive indeed. But as noted above, appraisal for the calculus is certainly not univocal. There exist (some very advanced) criticisms of the calculus. Some authors regard it as misconstrued from its very beginning: for Cull and Frank, the Laws of Form is no more than the Flaws of Form. The greater bulk of disapproving comments is, however, less than a spelled-out, intricate argument. In general, it aims at the status of LoF within the mathematical tradition and rejects it as a mere variant of Boolean algebra, simply using a new notation. Nil novum sub sole, so to speak. Whatever be the case, LoF ‘s thinking, especially where it concerns its far-reaching constructivist implications, has clearly not yet been well established. Spencer Brown’s (promising) claims notwithstanding, the context of his work, its notation, and its exotic vocabulary need a great deal of clarification.

Schiltz M. & Verschraegen G. (2002) Spencer-Brown, Luhmann and Autology. Cybernetics & Human Knowing 9(3–4): 55–78. https://cepa.info/3210

In this article we try to show how a social theory can be constructed which takes the requirement of self-implication or autology serious. For this purpose, we turn to Niklas Luhmann’s theory of self-referential, autopoietic systems. For it is our thesis that the specific conatus of Luhmann’s enterprise is to construct an autological social theory i.e. a theory which is sufficiently complex to imply itself, to describe itself in the course of describing its objects of investigation. To demonstrate this claim, the line of argument has firmly been rooted in George Spencer-Brown’s Law of Forms, a work central to systems theory. The architecture of Luhmann’s systems theory is thus presented in accordance with Spencer-Brown’s calculus of indications. Special attention is being paid to systems theory’s Leitdifferenz system/ environment. For this distinction is literally maintained as the alpha and the omega of the theory: the Leitdifferenz carries theoretical observations, and at the same time permits the self-observation. By means of the figure of ‘re-entry’, drawn from Spencer-Brown’s calculus, we show how the starting distinction between system and environment, can ‘re-enter’ the construction founded upon it and makes it possible for the theory to observe itself as a system within an environment.

Varela F. J. (1975) A calculus for self-reference. International Journal of General Systems 2: 5–24. https://cepa.info/1840

An extension of the calculus of indications (of G. Spencer Brown) is presented to encompass all occurrences of self-referential situations. This is done through the introduction of a third state in the form of indication, a state seen to arise autonomously by self-indication. The new extended calculus is fully developed, and some of its consequences for systems, logic and epistemology are discussed.