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.

Cariani P. (2012) Infinity and the Observer: Radical Constructivism and the Foundations of Mathematics. Constructivist Foundations 7(2): 116–125. Fulltext at http://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. Fulltext at http://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.

Ernest P. (1993) Constructivism, the psychology of learning, and the nature of mathematics: Some critical issues. Science & Education 2: 87–93. Fulltext at http://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. Fulltext at http://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. Fulltext at http://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. Fulltext at http://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.

Friend M. (2017) Remarks of a philosopher of mathematics and science. In: Riegler A., Müller K. H. & Umpleby S. A. (eds.) New horizons for second-order cybernetics. World Scientific, Singapore: 327–332. Fulltext at http://cepa.info/4103

Fuchs P. & Hoegl F. (2015) Die Schrift der Form: George Spencer-Browns Laws of Form. In: Pörksen B. (ed.) Schlüsselwerke des Konstruktivismus. Second edition. Springer, Wiesbaden: 165–196.

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