Autopoietic theory is more than a mere characterization of the living, as it can be applied to a wider class of systems and involves both organizational and epistemological aspects. In this paper we assert the necessity of considering the relation between autopoiesis and emergence, focusing on the crucial importance of the observer’s activity and demonstrating that autopoietic systems can be considered intrinsically emergent processes. From the attempts to conceptualize emergence, especially Rosen’s, autopoiesis stands out for its attention to the unitary character of systems and to emergent levels, both inseparable from the observer’s operations. These aspects are the basis of Varela’s approach to multiple level relationships, considered as descriptive complementarities.

Key words: Autopoiesis, emergence, observer, descriptive complementarity, Robert Rosen, Francisco Varela

In this paper we criticize the “Ashbyan interpretation” (Froese & Stewart, 2010) of autopoietic theory by showing that Ashby’s framework and the autopoietic one are based on distinct, often incompatible, assumptions and that they aim at addressing different issues. We also suggest that in order to better understand autopoiesis and its implications, a different and wider set of theoretical contributions, developed previously or at the time autopoiesis was formulated, needs to be taken into consideration: among the others, the works of Rosen, Weiss and Piaget. By analyzing the concepts of organization and closure, the idea of components, and the role of materiality in the theory proposed by Maturana and Varela, we advocate the view that autopoiesis necessarily entails selfproduction and intrinsic instability and can be realized only in domains characterized by the same transformative and processual properties exhibited by the molecular domain. From this theoretical standpoint it can be demonstrated that autopoietic theory neither commits to a sharp dualism between organization and structure nor to a reflexive view of downward causation, thus avoiding the respective strong criticisms.

An approach which has the purpose to catch what characterizes the specificity of a living system, pointing out what makes it different with respect to physical and artificial systems, needs to find a new point of view – new descriptive modalities. In particular it needs to be able to describe not only the single processes which can be observed in an organism, but what integrates them in a unitary system. In order to do so, it is necessary to consider a higher level of description which takes into consideration the relations between these processes, that is the organization rather than the structure of the system. Once on this level of analysis we can focus on an abstract relational order that does not belong to the individual components and does not show itself as a pattern, but is realized and maintained in the continuous flux of processes of transformation of the constituents. Using Tibor Ganti’s words we call it “Order in the Nothing”. In order to explain this approach we analyse the historical path that generated the distinction between organization and structure and produced its most mature theoretical expression in the autopoietic biology of Humberto Maturana and Francisco Varela. We then briefly analyse Robert Rosen’s (M, R)-Systems, a formal model conceptually built with the aim to catch the organization of living beings, and which can be considered coherent with the autopoietic theory. In conclusion we will propose some remarks on these relational descriptions, pointing out their limits and their possible developments with respect to the structural thermodynamical description.

Key words: Autonomy, Autopoiesis, (M, R)-Systems, Organization, Relational Biology

The major insight in Robert Rosen’s view of a living organism as an (M, R)-system was the realization that an organism must be “closed to efficient causation”, which means that the catalysts needed for its operation must be generated internally. This aspect is not controversial, but there has been confusion and misunderstanding about the logic Rosen used to achieve this closure. In addition, his corollary that an organism is not a mechanism and cannot have simulable models has led to much argument, most of it mathematical in nature and difficult to appreciate. Here we examine some of the mathematical arguments and clarify the conditions for closure.

Key words: (M, R) systems, Metabolic circularity, Robert Rosen, Definitions of life, Autopoiesis, Autocatalytic sets, Chemoton, Impredicativity

This paper has two primary aims. The first is to provide an introductory discussion of hyperset theory and its usefulness for modeling complex systems. The second aim is to provide a hyperset analysis of several perspectives on autonomy: Robert Rosen’s metabolism-repair systems and his claim that living things are closed to efficient cause, Maturana and Varela’s autopoietic systems, and Kauffman’s cataytically closed systems. Consequences of the hyperset models for Rosen’s claim that autonomous systems have non-computable models are discussed.

Key words: Autonomy, Autopoiesis, Catalytic closure, Francisco Varela, Hyperset, Robert Rose

First paragraph: The interest in Jakob von Uexküll in semiotics is obvious – he was a starter and pioneer of the semiotic approach in biology in thetwentieth century. The extension of semiotics from humanities, wheresemiotics has been centered during the period of its most intensive develo-pment, to the field of biology, has taken place gradually, with the mostdecisive steps taken onlyvery recently. The conclusion of this development is most fundamental – sign systems embrace all living systems, and the roots of semiotics lie in biology. On the other hand, Uexküll’s main ideawas to build up a biology which can handle the vital processes, and whichcan include the subject, the living self; or life itself, in the sense of Robert Rosen…

From the many attempts to produce a conceptual framework for the organization of living systems, the notions of (M, R) systems and Autopoiesis stand out for their rigor, their presupposition of the circularity of metabolism, and the new epistemologies that they imply. From their inceptions, these two notions have been essentially disconnected because each has defined its own language and tools. Here we demonstrate the existence of a deep conceptual link between (M, R) systems and Autopoietic systems. This relationship permits us to posit that Autopoietic systems, which have been advanced as capturing the central aspects of living systems, are a subset of (M, R) systems. This result, in conjunction with previous theorems proved by Rosen, can be used to outline a demonstration that the operation of Autopoietic systems cannot be simulated by Turing machines. This powerful result shows the potential of linking these two models. Finally, we suggest that the formalism of (M, R) systems could be used to model the circularity of metabolism.

Key words: Autopoiesis, (M, R) systems, Circularity, Metabolism, Closure, Turing–Church hypothesis

This article analyses the work of Robert Rosen on an interpretation of metabolic networks that he called (M, R) systems. His main contribution was an attempt to prove that metabolic closure (or metabolic circularity) could be explained in purely formal terms, but his work remains very obscure and we try to clarify his line of thought. In particular, we clarify the algebraic formulation of (M, R) systems in terms of mappings and sets of mappings, which is grounded in the metaphor of metabolism as a mathematical mapping. We define Rosen’s central result as the mathematical expression in which metabolism appears as a mapping f that is the solution to a fixed-point functional equation. Crucially, our analysis reveals the nature of the mapping, and shows that to have a solution the set of admissible functions representing a metabolism must be drastically smaller than Rosen’s own analysis suggested that it needed to be. For the first time, we provide a mathematical example of an (M, R) system with organizational invariance, and we analyse a minimal (three-step) autocatalytic set in the context of (M, R) systems. In addition, by extending Rosen’s construction, we show how one might generate self-referential objects f with the remarkable property f(f)=f, where f acts in turn as function, argument and result. We conclude that Rosen’s insight, although not yet in an easily workable form, represents a valuable tool for understanding metabolic networks.

Key words: (M, R) systems, Metabolic network, Metabolic closure, Infinite regress, Systems biology

A similarity between the concepts of reproduction and explanation is observed which implies a similarity between the less well understood concepts of complete self-reproduction and complete self-explanation. These latter concepts are shown to be independent from ordinary loglcal-mathematical-biological reasoning, and a special form of complete selfreproduction is shown to be axiomatizable. Involved is the question whether there exists a function that belongs to its own domain or range. Previously, Wittgenstein has argued, on intuitive grounds, that no function can be its own argument. Similarly, Rosen has argued that a paradox is implied by the notion of a function which is a member of its own range. Our result shows that such functions indeed are independent from ordinary logical-mathematical reasoning, but that they need not imply any inconsistencies, Instead such functions can be axiomatized, and in this sense they really do exist. Finally, the introduced notion of complete self-reproduction is compared with “self-reproduction” of ordinary biological language. It is pointed out that complete self-reproduction is primarily of interest in connection with formal theories of evolution.

Starting from the modeling relation, as introduced by the late Robert Rosen, we propose a relational method for studying impredicative systems, which are natural systems that have models containing hierarchical cycles. The general theory of impredicative systems vastly generalizes autopoietic systems, and has implications in the biological, psychological, and social realms, from which we offer many exemplifications. The method is formulated in category theory in terms of alternate descriptions and their functorial connections.