This article examines in some technical detail the application of Maturana and Varela’s biology of cognition to a simple concrete model: a glider in the game of Life cellular automaton. By adopting an autopoietic perspective on a glider, the set of possible perturbations to it can be divided into destructive and nondestructive subsets. From a glider’s reaction to each nondestructive perturbation, its cognitive domain is then mapped. In addition, the structure of a glider’s possible knowledge of its immediate environment, and the way in which that knowledge is grounded in its constitution, are fully described. The notion of structural coupling is then explored by characterizing the paths of mutual perturbation that a glider and its environment can undergo. Finally, a simple example of a communicative interaction between two gliders is given. The article concludes with a discussion of the potential implications of this analysis for the enactive approach to cognition.

Beer R. D. (2018) On the origin of gliders. In: Ikegami T., Virgo N., Witkowski O., Oka M., Suzuk R. & Iizuka H. (eds.) Proceedings of the 2018 Conference on Artificial Life. MIT Press, Cambridge MA: 67–74. https://cepa.info/6304

Using a glider in the Game of Life cellular automaton as a toy model, we explore how questions of origins might be approached from the perspective of autopoiesis. Specifically, we examine how the density of gliders evolves over time from random initial conditions and then develop a statistical mechanics of gliders that explains this time evolution in terms of the processes of glider creation, persistence and destruction that underlie it.

Beer R. D. (2020) Bittorio revisited: Structural coupling in the Game of Life. Adaptive Behavior 28(4): 197–212. https://cepa.info/7089

The notion of structural coupling plays a central role in Maturana and Varela’s biology of cognition framework and strongly influenced Varela’s subsequent enactive elaboration of this framework. Building upon previous work using a glider in the Game of Life (GoL) cellular automaton as a toy model of a minimal autopoietic system with which to concretely explore these theoretical frameworks, this article presents an analysis of structural coupling between a glider and its environment. Specifically, for sufficiently small GoL universes, we completely characterize the nonautonomous dynamics of both a glider and its environment in terms of interaction graphs, derive the set of possible glider lives determined by the mutual constraints between these interaction graphs, and show how such lives are embedded in the state transition graph of the entire GoL universe.

Dewhurst J. & Villalobos M. (2017) The enactive automaton as a computing mechanism. Thought: A Journal of Philosophy 6(3): 185–192. https://cepa.info/7513

Varela, Thompson, and Rosch illustrated their original presentation of the enactive theory of cognition with the example of a simple cellular automaton. Their theory was paradigmatically anti-computational, and yet automata similar to the one that they describe have typically been used to illustrate theories of computation, and are usually treated as abstract computational systems. Their use of this example is therefore puzzling, especially as they do not seem to acknowledge the discrepancy. The solution to this tension lies in recognizing a hidden background assumption, shared by both Varela, Thompson, and Rosch and the computational theories of mind which they were responding to. This assumption is that computation requires representation, and that computational states must bear representational content. For Varela, Thompson, and Rosch, representational content is incompatible with cognition, and so from their perspective the automaton that they describe cannot, despite appearances, be computational. However, there now exist several accounts of computation that do not make this assumption, and do not characterize computation in terms of representational content. In light of these recent developments, we will argue that it is quite straightforward to characterize the enactive automaton as a non-representational computing mechanism, one that we do not think they should have any objections to.

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

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. (2015) Self-reference, biologic and the structure of reproduction. Progress in Biophysics and Molecular Biology 10(3): 382–409. https://cepa.info/2844

Rodríguez Gómez S. (2022) Conjectural artworks: Seeing at and beyond Maturana and Varela’s visual thinking on life and cognition. AI & Society 37: 1307–1318. https://cepa.info/8115

This article delineates the notion of conjectural artworks – that is, ways of thinking and explaining formal and relational phenomena by visual means – and presents an appraisal and review of the use of such visual ways in the work of Chilean biologists and philosophers Humberto Maturana and Francisco Varela. Particularly, the article focuses on their recurrent uses of Cellular Automaton, that is, discrete, locally interacting, rule-based mathematical models, as conjectural artworks for understanding the concepts of autopoiesis, structural coupling, cognition and enaction: (i.e. Protobio and Bittorio). Additionally, the article proposes a new model of conjectural artwork based on an extension of cellular automaton: random Boolean networks namely, binary systems with variable local connections. Such model, as it is argued, is useful to connect the theoretical frameworks by Maturana and Varela, especially structural coupling and enaction, with other relevant fields such as biosemiotics’ Umwelt-research and cognitive landscapes in neurodynamics, and to advance and explore the concepts of structurally coupled categorization and generalization.