# Key word "reflexive domain"

Kauffman L. H. (2009) Reflexivity and Eigenform: The Shape of Process. Constructivist Foundations 4(3): 121–137. Fulltext at https://cepa.info/133

Kauffman L. H.
(

2009)

Reflexivity and Eigenform: The Shape of Process.
Constructivist Foundations 4(3): 121–137.
Fulltext at https://cepa.info/133
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.

Kauffman L. H. (2016) Cybernetics, Reflexivity and Second-Order Science. Constructivist Foundations 11(3): 489–497. Fulltext at https://cepa.info/2856

Kauffman L. H.
(

2016)

Cybernetics, Reflexivity and Second-Order Science.
Constructivist Foundations 11(3): 489–497.
Fulltext at https://cepa.info/2856
Context: Second-order cybernetics and its implications have been understood within the cybernetics community for some time. These implications are important for understanding the structure of scientific endeavor, and for researchers in other fields to see the reflexive nature of scientific research. This article is about the role of context in the creation and exploration of our experience. Problem: The purpose of this article is to point out the fundamental nature of the circularity in cybernetics and in scientific work in general. I give a point of view on the nature of objective knowledge by placing it in the context of reflexivity and eigenform. Method: The approach to the topic is based on logical analysis of the nature of circularity. Mathematics and cybernetics are both fundamentally concerned with the structure of distinctions, but there can be no definition of a distinction without circularity, since such a definition would itself be a distinction. The article proceeds by explicating the structure of reflexive domains D where the transformations of the domain are in one-to-one correspondence with the domain itself. Results: I show that every element of a reflexive domain has a fixed point. This means that eigenforms arise naturally in reflexive domains. Furthermore, a reflexive domain is itself an eigenform (at a higher level. This supports the context of a second-order science that would study domains of science as part of a larger cybernetic landscape. Implications: The value of the article is in its concise reformulation of the scientific endeavor as a search for eigenforms in reflexive domains. This new view of science is promising in that it includes the former worlds of apparent objectivity and it embraces those newer worlds of science where the theories and theorists become active participants in the ongoing process of creating knowledge. Constructivist content: I argue that the perspective of reflexive domains constitutes a new way to think about the practice of science, with observers deeply imbedded, and objectivity understood as the mutual search for eigenforms.

Kauffman L. H. (2017) Eigenform and Reflexivity. Constructivist Foundations 12(3): 246–252. Fulltext at https://cepa.info/4162

Kauffman L. H.
(

2017)

Eigenform and Reflexivity.
Constructivist Foundations 12(3): 246–252.
Fulltext at https://cepa.info/4162
Purpose: I introduce the concept of eigenform in the context of second-order cybernetics and discuss eigenform and eigenbehavior in the context of reflexivity. The point of eigenform is that it is a concept arising along with the observer at the point where the observer and the observed are apparently the same and yet apparently different. It is this nexus of the observer and the observed that is central to second-order cybernetics. Method: The article is designed as a formal introduction with excursions into the applications and meanings of these constructions. Results: I show how objects in our immediate experience can be seen to be eigenforms and that in this context such objects are a construct of our interactions, linguistic and otherwise experiential. In this way we can investigate scientifically without the need for an assumption of objectivity. Implications: The implications of this research are important for the performance and exploration of science. We can explore our role in that creation and find that what we create is independent of significant subsets of our actions. The practical implications of this study are strongest for the logical understanding of our constructions and actions. The social implications are in accord with the practical implications. We each produce eigenform models of the others and of ourselves.

Kauffman L. H. (2017) Mathematical Work of Francisco Varela. Constructivist Foundations 13(1): 11–17. Fulltext at https://cepa.info/4382

Kauffman L. H.
(

2017)

Mathematical Work of Francisco Varela.
Constructivist Foundations 13(1): 11–17.
Fulltext at https://cepa.info/4382
Purpose: This target article explicates mathematical themes in the work of Varela that remain of current interest in present-day second-order cybernetics. Problem: Varela’s approach extended biological autonomy to mathematical models of autonomy using reflexivity, category theory and eigenform. I will show specific ways that this mathematical modeling can contribute further to both biology and cybernetics. Method: The method of this article is to use elementary mathematics based in distinctions (and some excursions into category theory and other constructions that are also based in distinctions) to consistently make all constructions and thereby show how the observer is involved in the models that are so produced. Results: By following the line of mathematics constructed through the imagination of distinctions, we find direct access and construction for the autonomy postulated by Varela in his book Principles of Biological Autonomy. We do not need to impose autonomy at the base of the structure, but rather can construct it in the context of a reflexive domain. This sheds new light on the original approach to autonomy by Varela, who also constructed autonomous states but took them as axiomatic in his calculus for self-reference. Implications: The subject of the relationship of mathematical models, eigenforms and reflexivity should be reexamined in relation to biology, biology of cognition and cybernetics. The approach of Maturana to use only linguistic and philosophical approaches should now be reexamined and combined with Varela’s more mathematical approach and its present-day extensions.

Key words: Autonomy,

autopoiesis,

eigenform,

reflexivity,

reflexive domain,

observer,

self-reference,

category,

functor,

adjoint functor,

distinction.
Kauffman L. K. (2017) A concise approach to eigenform and reflexivity. Kybernetes 46(1): 1542–1554. Fulltext at https://cepa.info/4567

Kauffman L. K.
(

2017)

A concise approach to eigenform and reflexivity.
Kybernetes 46(1): 1542–1554.
Fulltext at https://cepa.info/4567
Purpose This paper aims to introduce the concept and praxis for eigenform in the context of second-order cybernetics. Design/methodology/approach The paper is designed as a formal (partly mathematical) introduction with excursions into the applications and meanings of these constructions. Mathematics studies what a distinction would be if there could be a distinction. Mathematics is a special form of fictional design. This study raises the question of “What it would mean to go beyond mathematics or for mathematics to go beyond itself?.” Findings This study shows how objects in the author’s experience can be seen to be eigenforms and that in this context such objects are a construct of their interactions, linguistic and otherwise experiential. In this way, the author can investigate scientifically without the need for an assumption of objectivity. The author cocreates the universe through the discovery of distinctions and eigenforms in their dialogue with what can be. Research limitations/implications The implications of this research are profound for the performance and exploration of science. The author can explore their role in that creation and find that what they create is independent of significant subsets of their actions. Practical implications The practical implications of this study are strongest for the logical understanding of the author’s constructions and actions. They have used eigenform and reflexivity to maintain a clear view of their participation in their own worlds. Social implications The social implications are in accordance with the practical implications. The author can now admit that they each produce eigenform models of the others and for themselves. These models have in-depth usage, in that it is understood that one is not identical with their models. Originality/value This paper presents a highly original and very simple way to incorporate second-order cybernetics into all thought and action.

Soto-Andrade J. & Varela F. J. (1984) Self-reference and fixed points. Acta Applicandae Mathematicae 2(1): 1–19.

Soto-Andrade J. & Varela F. J.
(

1984)

Self-reference and fixed points.
Acta Applicandae Mathematicae 2(1): 1–19.
We consider an extension of Lawvere’s Theorem showing that all classical results on limitations (i.e. Cantor, Russel, Gödel, etc.) stem from the same underlying connection between self-referentiality and fixed points. We first prove an even stronger version of this result. Secondly, we investigate the Theorem’s converse, and we are led to the conjecture that any structure with the fixed point property is a retract of a higher reflexive domain, from which this property is inherited. This is proved here for the category of chain complete posets with continuous morphisms. The relevance of these results for computer science and biology is briefly considered.

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