Purpose: The paper discusses the concept of a reflexive domain, an arena where the apparent objects as entities of the domain are actually processes and transformations of the domain as a whole. Human actions in the world partake of the patterns of reflexivity, and the productions of human beings, including science and mathematics, can be seen in this light. Methodology: Simple mathematical models are used to make conceptual points. Context: The paper begins with a review of the author’s previous work on eigenforms - objects as tokens for eigenbehaviors, the study of recursions and fixed points of recursions. The paper also studies eigenforms in the Boolean reflexive models of Vladimir Lefebvre. Findings: The paper gives a mathematical definition of a reflexive domain and proves that every transformation of such a domain has a fixed point. (This point of view has been taken by William Lawvere in the context of logic and category theory.) Thus eigenforms exist in reflexive domains. We discuss a related concept called a “magma.” A magma is composed entirely of its own structure-preserving transformations. Thus a magma can be regarded as a model of reflexivity and we call a magma “reflexive” if it encompasses all of its structure-preserving transformations (plus a side condition explained in the paper). We prove a fixed point theorem for reflexive magmas. We then show how magmas are related to knot theory and to an extension of set theory using knot diagrammatic topology. This work brings formalisms for self-reference into a wider arena of process algebra, combinatorics, non-standard set theory and topology. The paper then discusses how these findings are related to lambda calculus, set theory and models for self-reference. The last section of the paper is an account of a computer experiment with a variant of the Life cellular automaton of John H. Conway. In this variant, 7-Life, the recursions lead to self-sustaining processes with very long evolutionary patterns. We show how examples of novel phenomena arise in these patterns over the course of large time scales. Value: The paper provides a wider context and mathematical conceptual tools for the cybernetic study of reflexivity and circularity in systems.

Key words: reflexive, eigenform, cybernetics, Boolean algebra, knots, magma, Russell paradox, cellular automata

G. Spencer-Brown’s Laws of Form is summarized and the philosophical implications examined. Laws of Form is a mathematical system which deals with the emergence of anything out of the void. It traces how a single distinction in a void leads to the creation of space, where space is considered at its most primitive, without dimension. This in turn leads to two seemingly self-evident “laws.” With those laws taken as axioms, first an arithmetic is developed, then an algebra based on the arithmetic. The algebra is formally equivalent to Boolean algebra, though it satisfies all 2-valued systems. By following the implications of the algebra to its logical conclusions, self-reference emerges within the system in the guise of re-entry into the system. Spencer-Brown interprets this re-entry as creating time in much the same way in which distinction created space. Finally the paper considers the question of self-reference as seen in Francisco Varela’s Principles of Biological Autonomy, which extended Spencer-Brown’s Laws of Form to a 3-valued system.

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.