Volume 4 · Number 3 · Pages 121–137
Reflexivity and Eigenform: The Shape of Process

Louis H. Kauffman

Log in to download the full text for free

> Citation > Similar > References > Add Comment

Abstract

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

Citation

Kauffman L. H. (2009) Reflexivity and eigenform: The shape of process. Constructivist Foundations 4(3): 121–137. http://constructivist.info/4/3/121

Export article citation data: Plain Text · BibTex · EndNote · Reference Manager (RIS)

Similar articles

Kauffman L. H. (2016) Cybernetics, Reflexivity and Second-Order Science
Kauffman L. H. (2017) Eigenform and Reflexivity
Kauffman L. H. (2017) Mathematical Work of Francisco Varela
Müller K. H. & Riegler A. (2016) Mapping the Varieties of Second-Order Cybernetics
Kauffman L. H. (2012) The Russell Operator

References

Aczel P. (1988) Non-wellfounded sets. CSLI Lecture Notes Number 14. Center for the Study of Language and Information, Stanford CA. ▸︎ Google︎ Scholar
Barendregt H. P. (1984) The lambda calculus. Its syntax and semantics. North Holland, Amsterdam. ▸︎ Google︎ Scholar
Barwise J. & Moss L. (1996) Vicious circles. On the mathematics of non-wellfounded phenomena. CSLI Lecture Notes Number 60. Center for the Study of Language and Information, Stanford CA. ▸︎ Google︎ Scholar
Bohm D. (1980) Wholeness and the implicate order. Routledge & Kegan Paul, London. ▸︎ Google︎ Scholar
Bortoft H. (1971) The whole: Counterfeit and authentic. Systematics 9(2): 43–73. ▸︎ Google︎ Scholar
DeHornoy P. (2000) Braids and self-distributivity. Birkhäuser, Basel. ▸︎ Google︎ Scholar
Foerster H. von (1981) Notes on an epistemology for living things. In: Foerster H. von, Observing systems. Intersystems, Salinas CA: 258–271. Originally published as: Foerster H. von (1972) Notes on an epistemology for living things. Biological Computer Laboratory Report 9.3, BCL Fiche No. 104/1. University of Illinois, Urbana. ▸︎ Google︎ Scholar
Foerster H. von (1981) Objects: Tokens for (eigen-)behaviors. In: Foerster H. von, Observing systems. Intersystems, Salinas CA: 274–285. Originally published as: Foerster H. von (1976) Objects: Tokens for (eigen-)behaviors. ASC Cybernetics Forum 8(3/4): 91–96. ▸︎ Google︎ Scholar
Foerster H. von (1981) On constructing a reality. In: Foerster H. von, Observing systems. Intersystems, Salinas CA: 288–309. Originally published as: Foerster H. von (1973) On constructing a reality. In: Preiser F. E. (ed.) Environmental design research. Volume 2. Dowden, Hutchinson & Ross, Stroudberg: 35–46. ▸︎ Google︎ Scholar
Gardner M. (1970) Mathmatical Games –The fantastic combinations of John Conway’s new solitaire game “Life.” Scientific American 223: 120–123. ▸︎ Google︎ Scholar
Hintikka J. (1962) Knowledge and belief. An introduction to the logic of the two notions. Cornell University Press, Ithaca. ▸︎ Google︎ Scholar
Kauffman L. H. (1987) Self-reference and recursive forms. Journal of Social and Biological Structures 10: 53–72. ▸︎ Google︎ Scholar
Kauffman L. H. (1995) Knot logic. In: Kauffman L. H. (ed.) Knots and applications. World Scientific, Singapore: 1–110. ▸︎ Google︎ Scholar
Kauffman L. H. (2001) The mathematics of Charles Sanders Peirce. Cybernetics and Human Knowing 8(1/2): 79–110. ▸︎ Google︎ Scholar
Kauffman L. H. (2003) Eigenforms: Objects as tokens for eigenbehaviors. Cybernetics and Human Knowing 10(3/4): 73–90. ▸︎ Google︎ Scholar
Kauffman L. H. (2005) Eigenform. Kybernetes 34(1/2): 129–150. ▸︎ Google︎ Scholar
Lawvere F. W. (1972) Introduction to toposes, algebraic geometry and logic. Springer Lecture Notes on Mathematics Volume 274: 1–12. ▸︎ Google︎ Scholar
Lefebvre V. A. (1982) Algebra of conscience. Reidel, Dordrecht. ▸︎ Google︎ Scholar
Mandelbrot B. B. (1982) The fractal geometry of nature. W. H. Freeman, San Francisco. ▸︎ Google︎ Scholar
Piechocinska B. (2005) Physics from wholeness. PhD. Thesis. Uppsala Universitet, Sweden. ▸︎ Google︎ Scholar
Scott D. S. (1980) Relating theories of the lambda calculus.In: Seldin P. & Hindley R. (eds.) To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism. Academic Press, New York: 403–450. ▸︎ Google︎ Scholar
Spencer-Brown G. (1969) Laws of form. George Allen and Unwin, London. ▸︎ Google︎ Scholar
Webster’s New Collegiate Dictionary (1956) G. C. Merriam Publishers, Springfield MA. ▸︎ Google︎ Scholar
Wolfram S. (2002) A new kind of science. Wolfram Media, Champaign IL. ▸︎ Google︎ Scholar

Comments: 0

To stay informed about comments to this publication and post comments yourself, please log in first.