The interactivist model has explored a number of consequences of process metaphysics. These include reversals of some fundamental metaphysical assumptions dominant since the ancient Greeks, and multiple further consequences throughout the metaphysics of the world, minds, and persons. This article surveys some of these consequences, ranging from issues regarding entities and supervenience to the emergence of normative phenomena such as representation, rationality, persons, and ethics.

Campbell S. R. (2003) Reconnecting mind and world: Enacting a (new) way of life. In: Lamon S. J., Parker W. A. & Houston K. (eds.) Mathematical modelling. Woodhead Publishing, Sawston: 245–253.

A common assumption in teaching mathematical modelling and applications is that mind and world are ontologically distinct. This dualist view give rise to an explanatory gap as to how these two realms connect. An alternate view where mind and world are ontologically identical is explored here. This alternate view, grounded in a monist ontology of embodied cognition, undermines and attempts to fill this explanatory gap. Embodied cognition presents challenges of its own, but it also presents new pedagogical opportunities.

Cariani P. (2012) Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics. Constructivist Foundations 7(2): 116–125. https://cepa.info/254

Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism.

Deffuant G., Fuchs T., Monneret E., Bourgine P. & Varela F. J. (1995) Semi-algebraic networks: An attempt to design geometric autopoietic models. Artificial Life 2(2): 157–177. https://cepa.info/2076

This article focuses on an artificial life approach to some important problems in machine learning such as statistical discrimination, curve approximation, and pattern recognition. We describe a family of models, collectively referred to as semi-algebraic networks (SAN). These models are strongly inspired by two complementary lines of thought: the biological concept of autopoiesis and morphodynamical notions in mathematics. Mathematically defined as semi-algebraic sets, SANs involve geometric components that are submitted to two coupled processes: (a) the adjustment of the components (under the action of the learning examples), and (b) the regeneration of new components. Several examples of SANs are described, using different types of components. The geometric nature of SANs gives new possibilities for solving the bias/variance dilemma in discrimination or curve approximation problems. The question of building multilevel semi-algebraic networks is also addressed, as they are related to cognitive problems such as memory and morphological categorization. We describe an example of such multilevel models.

Díaz-Rojas D., Soto-Andrade J. & Videla-Reyes R. (2021) Authors’ Response: Fathoming the Enactive Metaphorizing Elephant in the Dark…. Constructivist Foundations 16(3): 289–294. https://cepa.info/7162

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.

Díaz-Rojas D., Soto-Andrade J. & Videla-Reyes R. (2021) Enactive Metaphorizing in the Mathematical Experience. Constructivist Foundations 16(3): 265–274. https://cepa.info/7155

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.

Ernest P. (1993) Constructivism, the psychology of learning, and the nature of mathematics: Some critical issues. Science & Education 2: 87–93. https://cepa.info/2948

Constructivism is one of the central philosophies of research in the psychology of mathematics education. However, there is a danger in the ambiguous and at times uncritical references to it. This paper critically reviews the constructivism of Piaget and Glasersfeld, and attempts to distinguish some of the psychological, educational and epistemological consequences of their theories, including their implications for the philosophy of mathematics. Finally, the notion of ‘cognizing subject’ and its relation to the social context is examined critically.

Ernest P. (1996) Varieties of constructivism: A framework for comparison. In: Steffe L. P., Nesher P., Cobb P., Goldin G. A. & Greer B. (eds.) Theories of mathematical learning. Lawrence Erlbaum, Hillsdale: 315–350. https://cepa.info/5282

Excerpt: Following the seminal influence of Jean Piaget, constructivism is emerging as perhaps the major research paradigm in mathematics education. This is particularly the case for psychological research in mathematics education, However, rather than solving all of the problems for our field, this raises a number of new ones. Elsewhere 1 have explored the differences between the constructivism of Piaget and that of von Glasersfeld (Ernest, 1991b) and have suggested how social constructivism can be developed, and how it differs in its assumptions from radical constructivism (Ernest, 1990, 1991a). Here I wish to begin to consider further questions, including the following: What is constructivism, and what different varieties are there? In addition to the explicit principles on which its varieties are based, what underlying metaphors and epistemologies do they assume? What are the strengths and weaknesses of the different varieties? What do they offer as tools for researching the teaching and learning of mathematics? In particular, what does radical constructivism offer that is unique? And last but not least: What are the implications for the teaching of mathematics?

Foerster H. von (1969) Laws of Form (Book Review of Laws of Form, G. Spencer Brown). In: Whole Earth Catalog. Portola Institute, Palo Alto CA: 14. https://cepa.info/1634

Frans J. (2012) The Game of Fictional Mathematics. Review of “Mathematics and Reality” by Mary Leng. Constructivist Foundations 8(1): 126-128. https://cepa.info/821

Upshot: Leng attacks the indispensability argument for the existence of mathematical objects. She offers an account that treats the role of mathematics in science as an indispensable and useful part of theories, but retains nonetheless a fictionalist position towards mathematics. The result is an account of mathematics that is interesting for constructivists. Her view towards the nominalistic part of science is, however, more in conflict with radical constructivism.