Context: During the 1980s, Ernst von Glasersfeld’s reflections nourished various studies conducted by a community of mathematics education researchers at CIRADE, Quebec, Canada. Problem: What are his influence on and contributions to the center’s rich climate of development? We discuss the fecundity of von Glasersfeld’s ideas for the CIRADE researchers’ community, specifically in didactique des mathématiques. Furthermore, we take a prospective view and address some challenges that new, post-CIRADE mathematics education researchers are confronted with that are related to interpretations of and reactions to constructivism by the surrounding community. Results: Von Glasersfeld’s contribution still continues today, with a new generation of researchers in mathematics education that have inherited views and ideas related to constructivism. For the post-CIRADE research community, the concepts and epistemology that von Glasersfeld put forward still need to be developed further, in particular concepts such as subjectivity, viability, the circular interpretative effect, representations, the nature of knowing, errors, and reality. Implications: Radical constructivism’s offspring resides within the concepts and epistemology put forth, and that continue to be put forth, through a large number of new and different generations of theories, thereby perpetuating von Glasersfeld’s legacy.

Key words: radical constructivism, mathematics education, viability, reification, representation

Biology gives us numerous examples of self-assertional systems whose essence does not precede their existence but is rather revealed through it. Immune system is one of them. The fact of behaving in order not only to satisfy external constraints as a pre-fixed set of possible environments and objectives, but also to satisfy internal “viability” constraints justifies a sharper focus. Adaptability, creativity and memory are certainly interesting “side-effects” of such a tendency for self-consistency. However in this paper, we adopted a largely pragmatic attitude attempting to find the best hybridizing between the biological lessons and the engineering needs. The great difficulty, also shared by neural net and GA users, remains the precise localisation of the frontier where the biological reality must give way to a directed design.

Context: Meeting Ernst von Glasersfeld for the first time in 1985, when about 70% of his work had still to be conceived, written and published, was a great stroke of fortune for me; it was based on my collaboration with Silvio Ceccato that had started in 1981 and it profoundly influenced my contributions to radical constructivism in the following 25 years of our friendship. Problem: Presenting the details of how it all began can shed a light on the development of constructivist ideas. Method: Anecdotes from 1979 to 1985 about how I came to meet Silvio Ceccato in Milan in 1981 and the influence of these events on preparing the 1985 meeting with Ernst von Glasersfeld, also in Milan. Results: The article describes the timeline of 50 years of publications by von Glasersfeld, an anecdote about a connection between Ceccato and the University of Zurich in the 60s, the attempt to present Ceccato’s ideas as compatible and complementary with the neuroscience discourse in 1985, von Glasersfeld’s opinion about this attempt, and this attempt’s potential influence on the emergence of a new concept in neuroscience, “EEG microstates.” Implications: The events and facts reported in the article help us to understand some aspects of an early phase in the development of radical constructivism, especially the relationship between Ceccato, von Glasersfeld and other members of the Italian Operational School such as Bruna Zonta, Felice Accame, and the author himself.

Key words: history of science, operational methodology, Silvio Ceccato, neuroscience, single activity state, EEG microstate, viability

Open peer commentary on the article “Applying Radical Constructivism to Machine Learning: A Pilot Study in Assistive Robotics” by Markus Nowak, Claudio Castellini & Carlo Massironi. Upshot: Applying radical constructivism to machine learning is a challenge that requires us to dive very deeply into its theory of knowing and learning. We need to clarify its fundamental concepts, if possible, in operational terms. This commentary aims at outlining how this kind of clarification could look in the case of 3 such concepts: (a) the construction of experiential reality; (b) learning as a constructive activity; (c) the viability of conceptual structures.

Living systems employ several mechanisms and behaviors to achieve robustness and maintain themselves under changing internal and external conditions. Regulation stands out from them as a specific form of higher-order control, exerted over the basic regime responsible for the production and maintenance of the organism, and provides the system with the capacity to act on its own constitutive dynamics. It consists in the capability to selectively shift between different available regimes of self-production and self-maintenance in response to specific signals and perturbations, due to the action of a dedicated subsystem which is operationally distinct from the regulated ones. The role of regulation, however, is not exhausted by its contribution to maintain a living system’s viability. While enhancing robustness, regulatory mechanisms play a fundamental role in the realization of an autonomous biological organization. Specifically, they are at the basis of the remarkable integration of biological systems, insofar as they coordinate and modulate the activity of distinct functional subsystems. Moreover, by implementing complex and hierarchically organized control architectures, they allow for an increase in structural and organizational complexity while minimizing fragility. Finally, they endow living systems, from their most basic unicellular instances, with the capability to control their own internal dynamics to adaptively respond to specific features of their interaction with the environment, thus providing the basis for the emergence of minimal forms of cognition.

Key words: regulation, control, functional integration, organization, autonomy, cognition.

This article revisits the concept of autopoiesis and examines its relation to cognition and life. We present a mathematical model of a 3D tesselation automaton, considered as a minimal example of autopoiesis. This leads us to a thesis T1: “An autopoietic system can be described as a random dynamical system, which is defined only within its organized autopoietic domain.” We propose a modified definition of autopoiesis: “An autopoietic system is a network of processes that produces the components that reproduce the network, and that also regulates the boundary conditions necessary for its ongoing existence as a network.” We also propose a definition of cognition: “A system is cognitive if and only if sensory inputs serve to trigger actions in a specific way, so as to satisfy a viability constraint.” It follows from these definitions that the concepts of autopoiesis and cognition, although deeply related in their connection with the regulation of the boundary conditions of the system, are not immediately identical: a system can be autopoietic without being cognitive, and cognitive without being autopoietic. Finally, we propose a thesis T2: “A system that is both autopoietic and cognitive is a living system.”

Key words: Autopoiesis, cognition, life, tessela- tion automaton, random dynamical system, boundary conditions

This Introduction is our attempt to clarify further the cluster of key notions: autonomy, viability, abduction and adaptation. These notions form the conceptual scaffolding within which the individual contribution contained in this volume can be placed. Hopefully, these global concepts represent fundamental signposts for future research that can spare us a mere flurry of modelling and simulations into which this new field could fall.

Key words: cognitivism, connectionism, enaction, autonomy, viability, abduction, adaptation

Constructivism rejects the metaphysical position that “truth,” and thus knowledge in science, can represent an “objective” reality, independent of the knower. It modifies the role of knowledge from “true” representation to functional viability. In this interview, Ernst von Glasersfeld, the leading proponent of Radical Constructivism underlines the inaccessibility of reality, and proposes his view that the function of cognition is adaptive, in the biological sense: the adaptation is the result of the elimination of all that is not adapted. There is no rational way of knowing anything outside the domain of our experience and we construct our world of experiences. In addition to these philosophical claims, the interviewee provides some personal insights; he also gives some suggestions about better teaching and problem solving. These are the aspects of constructivism that have had a major impact on instruction and have modified the manner many of us teach. The process of teaching as linguistic communication, he says, needs to change in a way to involve actively the students in the construction of their knowledge. Because knowledge is not a transferable commodity, learning is mainly identified with the activity of the construction of personal meaning. This interview also provides glimpses on von Glasersfeld’s life.

Key words: radical constructivism, education

This paper outlines an original interactivist–constructivist (I-C) approach to modelling intelligence and learning as a dynamical embodied form of adaptiveness and explores some applications of I-C to understanding the way cognitive learning is realized in the brain. Two key ideas for conceptualizing intelligence within this framework are developed. These are: (1) intelligence is centrally concerned with the capacity for coherent, context-sensitive, self-directed management of interaction; and (2) the primary model for cognitive learning is anticipative skill construction. Self-directedness is a capacity for integrative process modulation which allows a system to “steer” itself through its world by anticipatively matching its own viability requirements to interaction with its environment. Because the adaptive interaction processes required of intelligent systems are too complex for effective action to be prespecified (e.g. genetically) learning is an important component of intelligence. A model of self-directed anticipative learning (SDAL) is formulated based on interactive skill construction, and argued to constitute a central constructivist process involved in cognitive development. SDAL illuminates the capacity of intelligent learners to start with the vague, poorly defined problems typically posed in realistic learning situations and progressively refine them, transforming them into problems with sufficient structure to guide the construction of a solution. Finally, some of the implications of I-C for modelling of the neuronal basis of intelligence and learning are explored; in particular, Quartz and Sejnowski’s recent neural constructivism paradigm, enriched by Montague and Sejnowski’s dopaminergic model of anticipative–predictive neural learning, is assessed as a promising, but incomplete, contribution to this approach. The paper concludes with a fourfold reflection on the divergence in cognitive modelling philosophy between the I-C and the traditional computational information processing approaches.

Excerpt: In this chapter, I have argued that radical constructivist interpretations of constructivism differ from other interpretations in that radical constructivism is an epistemological theory based in viability. It is suggested that viability commits one to the expectation of and support for diversity in the classroom. Moreover, it obliges the radical constructivist to also reinterpret the mathematical meaning of concepts in light of the students’ inventions. To do this effectively, the radical constructivist must learn techniques of close listening and follow these by the articulation of student voice and the examination of the changes in his or her own perspective. It is the “voice-perspective” relationship which makes radical constructivism capable of deep reform in mathematics instruction.