Inspired by Enactivist philosophy yet in dialog with it, we ask what theory of embodied cognition might best serve in articulating implications of Enactivism for mathematics education. We offer a blend of Dynamical Systems Theory and Sociocultural Theory as an analytic lens on micro-processes of action-to-concept evolution. We also illustrate the methodological utility of design-research as an approach to such theory development. Building on constructs from ecological psychology, cultural anthropology, studies of motor-skill acquisition, and somatic awareness practices, we develop the notion of an “instrumented field of promoted action”. Children operating in this field first develop environmentally coupled motor-action coordinations. Next, we introduce into the field new artifacts. The children adopt the artifacts as frames of action and reference, yet in so doing they shift into disciplinary semiotic systems. We exemplify our thesis with two selected excerpts from our videography of Grade 4–6 volunteers participating in task-based clinical interviews centered on the Mathematical Imagery Trainer for Proportion. In particular, we present and analyze cases of either smooth or abrupt transformation in learners’ operatory schemes. We situate our design framework vis-à-vis seminal contributions to mathematics education research.

Key words: mathematics, education, motor learning, conceptual learning, motor problem, conceptual metaphor

The paradox of scientific expertise is that the growth of science leads to a fragmentation of scientific expertise. To resolve this paradox, this paper probes three hypotheses: 1) All scientific knowledge is perspectival. 2) The perspectival structure of science leads to specific forms of knowledge asymmetries. 3) Such perspectival knowledge asymmetries must be handled through second order perspectives. We substantiate these hypotheses on the basis of a perspectivist philosophy of science grounded in Peircean semiotics and autopoietic systems theory. Perspectivism is an important elaboration of constructivist approaches to help overcome problems in cross-disciplinary collaboration and use of science, and thereby make society better able to solve complex, real-world problems.

Context: The problems that are most in need of interdisciplinary collaboration are “wicked problems,” such as food crises, climate change mitigation, and sustainable development, with many relevant aspects, disagreement on what the problem is, and contradicting solutions. Such complex problems both require and challenge interdisciplinarity. Problem: The conventional methods of interdisciplinary research fall short in the case of wicked problems because they remain first-order science. Our aim is to present workable methods and research designs for doing second-order science in domains where there are many different scientific knowledges on any complex problem. Method: We synthesize and elaborate a framework for second-order science in interdisciplinary research based on a number of earlier publications, experiences from large interdisciplinary research projects, and a perspectivist theory of science. Results: The second-order polyocular framework for interdisciplinary research is characterized by five principles. Second-order science of interdisciplinary research must: 1. draw on the observations of first-order perspectives, 2. address a shared dynamical object, 3. establish a shared problem, 4. rely on first-order perspectives to see themselves as perspectives, and 5. be based on other rules than first-order research. Implications: The perspectivist insights of second-order science provide a new way of understanding interdisciplinary research that leads to new polyocular methods and research designs. It also points to more reflexive ways of dealing with scientific expertise in democratic processes. The main challenge is that this is a paradigmatic shift, which demands that the involved disciplines, at least to some degree, subscribe to a perspectivist view. Constructivist content: Our perspectivist approach to science is based on the second-order cybernetics and systems theories of von Foerster, Maruyama, Maturana & Varela, and Luhmann, coupled with embodied theories of cognition and semiotics as a general theory of meaning from von Uexküll and Peirce.

Key words: Perspectivism, semiotics, complex phenomena, social systems theory, differentiation of science, perspectival knowledge asymmetries.

The article is an introduction to a special section in TCS on the work of Niklas Luhmann. The first part of the article provides a general introduction to Luhmann’s work with an emphasis on the basic elements of Luhmann’s general systems theory, in particular Luhmann’s notions of autopoiesis and meaning, and the traditions on which it is based. The second part of the text is a presentation of the articles in the special section.

Key words: autopoiesis, communication, functionalism, meaning, phenomenology, systems theory

This article presents a unifying theory of the embodied, situated human brain called the Hierarchically Mechanistic Mind (HMM). The HMM describes the brain as a complex adaptive system that actively minimises the decay of our sensory and physical states by producing self-fulfilling action-perception cycles via dynamical interactions between hierarchically organised neurocognitive mechanisms. This theory synthesises the free-energy principle (FEP) in neuroscience with an evolutionary systems theory of psychology that explains our brains, minds, and behaviour by appealing to Tinbergen’s four questions: adaptation, phylogeny, ontogeny, and mechanism. After leveraging the FEP to formally define the HMM across different spatiotemporal scales, we conclude by exploring its implications for theorising and research in the sciences of the mind and behaviour.

Key words: active inference, evolutionary systems theory, hierarchically mechanistic mind, free-energy principle, neuroscience, psychology

With reference to three seminal books on cybernetics, communication theory and the calculus of distinctions, this article discusses some main threads in Niklas Luhmann’s sociological systems theoretical thinking. It argues that the systems theory, despite its still lively reputation in some quarters of the humanities, is not technocracy’s last attempt to cope with the complexity of modern society. Rather, it is an inquiry into the improbability of communication and into its translation into social structure, or better, into social form.

Key words: causality, communication, cybernetics, distinction, form

Compared with traditional theories, systems theory presents a deviation. It replaces causal explanation by functional explanation. This paper shows what scandalon is inherent in this substitution and elucidates some models (self-organization, dance, non-triviality, structural coupling) which put the explanatory principle to work. The paper concludes by showing how systems theory aims at a general concept of communication that not only means a passing on of knowledge but above all the tracing of ignorance. Overall, systems theory is presented as a joker dealing with the paradox that the system is never identical to itself as soon as it is considered as a function of itself and its environment. The system has to withdraw into the function it is a function of in order to enfold this paradox.

The paper looks at a combination of systems theory, cybernetics, and sociological theory in search of a tool for inquiring into contemporary social forms. The idea of observing networks, drawing on Heinz von Foerster’s and Niklas Luhmann’s notion of observing systems and Harrison C. White’s network calculus of identity and control, is outlined to enable basic sociological intuitions about social forms to be integrated with an understanding of both complexity and recursivity organizing our perspective on the human condition in a precarious world. Social forms are shown to gain robustness not from substantial identity but from relational ambiguity. Observing networks, or so the hypothesis goes, combine bodies, minds, society, and – soon perhaps – intelligent machines. The paper looks at how an understanding of complexity, recursivity, system, form, and network may help flesh out the calculus of our human condition.

Key words: complexity, form, network, recursivity, society, system

The paper compares social systems theory and social network theory in terms of what it is they respectively seek to elucidate. Whereas systems theory focuses on problems of difference and reproduction, network theory deals with problems of identity and control, the former privileging communication and the latter action. To understand their different foci, it may help to keep in mind that systems theory is a child of computing’s formative years, whereas the more recent success of network theory, despite its roots in a far older tradition, accompanies the advent of the Internet. The paper goes on to compare the two theories with respect to questions of mathematical modeling, culture, and self-reference, which interestingly are closely related. It proposes a mathematical modeling of culture, which uses Spencer-Brown’s notion of form to combine variables of communication, consciousness, and life into one network relying on three systems capable of reproducing themselves. The paper is relevant for constructivist approaches because it shows how systems are constructed relying on networks within their own interpretation as culture.

Upshot: Are narratives systems on their own, or rather structures supporting and, if need be, subverting the reproduction of systems? Bruce Clarke inquires into the ability of social systems theory to help understand narratives - and comes across some “mysteries of cognition” concerning the questions of how systems emerge and which of them might be considered self-referential and autopoietic.