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

Our work aims at developing means to facilitating the access to stochastic thinking, especially for non-mathematically oriented learners. To this end, we draw on metaphoric and enactive approaches to the teaching and learning of randomness. More precisely, we report on a challenging didactical situation implemented in various classrooms, with students and prospective and practicing teachers, concerning problem posing and solving in the context of randomness that is approached through enactive metaphoring. The findings suggest that this sort of approach allows non-mathematically oriented learners to make sense of and abduct otherwise inaccessible mathematical notions and facts.

Key words: enaction, metaphor, randomness, embodiment, waiting time.

We are interested in exploring the role of enactive metaphoring in mathematical thinking, especially in the context of problem posing and solving, not only as a means to foster and enhance the learner’s ability to think mathematically but also as a mean to alleviate the cognitive abuse that the teaching of mathematics has turned out to be for most children and adolescents in the world. We present some illustrative examples to this end besides describing our theoretical framework.

Key words: metaphors, enaction, representation, visualisation, cognitive bullying.

Abstract: We offer a response to three themes arising from the commentators’ inquiries and critiques: (a) The epistemological compatibility of enactivism and conceptual metaphor theory; (b) the way enactive metaphorization works in the teaching and learning of mathematics, particularly in problem-posing and problem-solving activities; and (c) the nature of mathematical abstraction and its relation with enactive metaphorizing.

Context: How can an enactive approach to the teaching and learning of mathematics be implemented, which fosters mathematical thinking, making intensive use of metaphorizing and taking into account the learner’s experience? Method: Using in-person and remote ethnographic participant observation, we observe students engaged in mathematical activities suggested by our theoretical approach. We focus on their idiosyncratic metaphorizing and affective reactions while tackling mathematical problems, which we interpret from our theoretical perspective. We use these observations to illustrate our theoretical approach. Results: Our didactic examples show that alternative pathways are possible to access mathematical thinking, which bifurcate from the metaphors prevailing in most of our classrooms, like teaching as “transmission of knowledge” and learning as “climbing a staircase.” Our participant observations suggest that enacting and metaphorizing may indeed afford a new and more meaningful kind of experience for mathematics learners. Implications: Our observations highlight the relevance of leaving the learners room to ask questions, co-construct their problems, explore, and so on, instead of just learning in a prescriptive way the method to solve each type of problem. Consequently, one kind of solution to the current grim situation regarding mathematics teaching and learning would be to aim at relaxing the prevailing didactic contract that thwarts natural sense-making mechanisms of our species. Our conclusions suggest a possible re-shaping of traditional teaching practice, although we refrain from trying to implement this in a prescriptive way. A limitation of our didactic experience might be that it exhibits just a couple of illustrative examples of the application of our theoretical perspective, which show that some non-traditional learning pathways are possible. A full fledged ethnomethodological and micro-phenomenological study would be commendable. Constructivist content: We adhere to the enactive approach to cognition initiated by Francisco Varela, and to the embodied perspective as developed by Shaun Gallagher. We emphasize the cognitive role of metaphorization as a key neural mechanism evolved in humans, deeply intertwined with enaction and most relevant in our “hallucinatory construction of reality,” in the sense of Anil Seth.

Key words: Metaphor, enaction, embodiment, mathematical experience, participant observation, emotional anesthesia, negative emotions, Francisco Varela, Shaun Gallagher

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

We argue that an approach to the learning of mathematics based on enactive (bodily acted out) metaphorising may significantly help in alleviating the cognitive abuse millions of children worldwide suffer when exposed to mathematics. We present illustrative examples of enactive metaphoric approaches in the context of problem posing and solving in mathematics education, involving geometry and randomness, two critical subjects in school mathematics. Our examples show to what extent the way a mathematical situation is metaphorised and enacted by the learners shapes their emerging ideas and insights and how this may help to bridge the gap between the ‘mathematically gifted’ and those apparently not so gifted or mathematically inclined. Our experimental background includes a broad spectrum of prospective secondary math teachers, in-service primary teachers and their pupils, first-year university students majoring in social sciences and humanities and university students majoring in mathematics.

Key words: metaphor, enacting, enactivism, learning, mathematics

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.

Carlton (1988) has proposed an attractive hypothesis to link perceived visual images to brain electrical patterns via a linear representation of the Euclidean group onto an appropriate functional space. We show that the construction she proposes is (1) biologically restrictive, and (2) cannot be completed in the desired way. We conclude by presenting other possible means to pursue Carlton’s approach.

We are interested in exploring and developing an enactivist approach to problem posing and problem solving. We use here the term “enactivist approach” to refer to Varela’s radically nonrepresentationalist and pioneering “enactive approach to cognition” (Varela et al., The embodied mind: Cognitive science and human experience. Cambridge, MA: The MIT Press, 1991), to avoid confusion with the enactive mode of representation of Bruner, which is still compatible with a representationalist view of cognition. In this approach, problems are not standing “out there” waiting to be solved, by a solver equipped with a suitable toolbox of strategies. They are instead co-constructed through the interaction of a cognitive agent and a milieu, in a circular process well described by the metaphor of the Ouroboros (the snake eating its own tail). Also, cognition as enaction is metaphorized by Varela as “lying down a path in walking.” In this vein, we present here some paradigmatic examples of enactivist, and metaphorical, approaches to problem solving and problem posing, involving geometry, algebra, and probability, drawn from our didactical experimenting with a broad spectrum of learners, which includes humanities-inclined university students as well as prospective and in-service maths teachers. Our examples may be metaphorized as cognitive random walks in the classroom, stemming and unfolding from a situational seed.

Key words: enaction, metaphor, problem posing, problem solving