In this paper our main objective is to interpret the major concepts in Wittgenstein’s philosophy of mathematics, in particular, language games and forms of life, from a social constructivist point of view in an attempt to show that this philosophy is still very relevant in the way mathematics is being taught and practiced today. We start out with a brief discussion of radical constructivism followed by a rudimentary analysis of the basic tenets of social constructivism, the final blow against absolutism – the soulless landmark of mathematics as often construed by the uninitiated. We observe that, the social constructivist epistemology of mathematics reinstates mathematics, and rightfully so, as “…a branch of knowledge which is indissolubly connected with other knowledge, through the web of language” (Ernest 1999), and portrays mathematical knowledge as a process that should be considered in conjunction with its historical origins and within a social context. Consequently, like any other form of knowledge based on human opinion or judgment, mathematical knowledge has the possibility of losing its truth or necessity, as well. In the third section we discuss the main points expounded in Wittgenstein’s two books, Tractatus Logico-Philosophicus and Philosophical Investigations, as well as in his “middle period” that is characterized by such works as Philosophical Remarks, Philosophical Grammar, and Remarks on the Foundations of Mathematics. We then briefly introduce the two main concepts in Wittgenstein’s philosophy that will be used in this paper: forms of life and language games. In the fifth and final section, we emphasize the connections between social constructivism and Wittgenstein’s philosophy of mathematics. Indeed, we argue that the apparent certainty and objectivity of mathematical knowledge, to paraphrase Ernest (Ernest 1998), rest on natural language. Moreover, mathematical symbolism is a refinement and extension of written language: the rules of logic which permeate the use of natural language afford the foundation upon which the objectivity of mathematics rests. Mathematical truths arise from the definitional truths of natural language, and are acquired by social interaction. Mathematical certainty rests on socially accepted rules of discourse embedded in our forms of life, a concept introduced by Wittgenstein (Wittgenstein, 1956). We argue that the social constructivist epistemology draws on Wittgenstein’s (1956) account of mathematical certainty as based on linguistic rules of use and forms of life, and Lakatos’ (1976) account of the social negotiation of mathematical concepts, results, and theories.

Key words: radical constructivism, social constructivism, forms of life, language games

For most archaeologists the meaning of prehistoric art appears to be grounded upon, if not synonymous with, the notion of representation and symbolism. This paper explores the possibility that the depictions we see already 30,000 years before present, for instance, at the caves of Chauvet and Lascaux, before and beyond representing the world, they first bring forth a new process of acting within this world and at the same time of thinking about it. It is argued that the unique ability of those early depictions to disrupt or question the ways the world is experienced under normal conditions makes possible for the visual apparatus to interrogate itself and thus acquire a sense of perceptual awareness not previously available.

Constructivism is at the heart of a pedagogical philosophy going back to Vico, whose view of the interrelationship of the arts and sciences sought to reconstitute the classical paideia. The Vichian idea that human beings can only know the truth of what they themselves have made has theoretical and practical consequences for Vico’s pedagogy and view of the university. Vico’s ideas on education are extended in the modern period by such thinkers as Cassirer, Piaget and Bateson. At the basis of Cassirer’s pedagogical philosophy is his theory of the symbol, the symbol being a universal and transcendent modality in culture. The result of this unifying theory is that symbolism, which is pervasive across the disciplines, provides a moral and ethical means for integrating communication about teaching. Cassirer’s thought is compatible with Piaget’s, which emphasizes the pluralism of experience and the role of dynamic learning in the construction of meaningful order. Piaget’s constructivism assumes that an operational bridge exists to link together the hard sciences, the human sciences, and the historical disciplines. This systems view of epistemological matters is similar in many respects to the one advanced by Gregory Bateson, which is explored in the paper’s final section.

Key words: Constructivism, Vico, Piaget, Cassirer, Bateson

Constructivism is at the heart of a pedagogical philosophy going back to Vico, whose view of the interrelationship of the arts and sciences sought to reconstitute the classical paideia. The Vichian idea that human beings can only know the truth of what they themselves have made has theoretical and practical consequences for Vico’s pedagogy and view of the university. Vico’s ideas on education are extended in the modern period by such thinkers as Cassirer, Piaget and Bateson. At the basis of Cassirer’s pedagogical philosophy is his theory of the symbol, the symbol being a universal and transcendent modality in culture. The result of this unifying theory is that symbolism, which is pervasive across the disciplines, provides a moral and ethical means for integrating communication about teaching. Cassirer’s thought is compatible with Piaget’s, which emphasizes the pluralism of experience and the role of dynamic learning in the construction of meaningful order. Piaget’s constructivism assumes that an operational bridge exists to link together the hard sciences, the human sciences, and the historical disciplines. This systems view of epistemological matters is similar in many respects to the one advanced by Gregory Bateson, which is explored in the paper’s final section.

Purpose: In order to develop transdisciplinary working across the disciplines, clear epistemological foundations are required. The purpose of this paper is to show that sociocybernetics to provides the required unifying metadisciplinary epistemological foundations and transdisciplinary frameworks. Design/methodology/approach – The authors note that second-order cybernetics provides a metadisciplinary framework for discerning the causes and cures for the schisms within the natural and cognitive sciences. The particular contributions of sociocybernetics are to extend the second-order understandings to unify the social sciences and, by incorporating extant sociological theory back into the transdisciplinary pursuits of cybernetics and systems theory, to enlighten and enrich those pursuits. Findings: In order to highlight the power and fruitfulness of these contributions from sociocybernetics, the authors problematise, deconstruct and reconstruct key concepts concerned with human communication. To do this, they take as central the question, What is a symbol? and present a sociocybernetic, transdisciplinary solution. In doing so they make clear the epistemological poverty of approaches in cognitive science that are based on the thesis that brains and computers are both physical-symbol systems. Originality/value – The paper contributes to the metadisciplinary and transdisciplinary aims of cybernetics and, in particular, uses a sociocybernetic analysis to enlighten foundational issues in cognitive science.

Key words: Sociocybernetics, Symbolism, Epistemology, Cognition

This chapter examines the ways in which French philosopher Gilles Deleuze offers conceptual resources for an enactive account of language, in particular his extensive consideration of language in The Logic of Sense. Specifically Deleuze’s distinction between the nonsense of Lewis Carroll’s portmanteau creations and that of Antonin Artaud’s “translation” of Carroll’s Jabberwocky highlights the need for an enactive, rather than merely embodied, approach to sense-making, particularly with regard to the general category of what Jakobson and Halle (1956) call “sound symbolism”.

Piaget’s logical model of formal operations is problematic since it is unclear how Piaget’s logic should be understood. In a recent review, Braine and Rumain conclude that neither of the two available interpretations is adequate. Under one interpretation, formulae which should be compatible are incompatible in Piaget’s logic. Under the other interpretation, formulae which should be incompatible are compatible in Piaget’s logic. A constructivist interpretation is outlined so as to overcome these weaknesses. It is proposed that the attribution of formal operational thinking depends upon an individual’s generalization of classificatory abilities which are present during concrete operations. Each of the formal operations has a unique specification in terms of the 16 patterns inherent in such classifications. Access to formal operations does not require conscious awareness of propositional symbolism. The interpretation is shown to be minimally adequate in avoiding the objections which invalidate existing interpretations. The interpretation is taken to exemplify two features of Piaget’s constructivism, namely the differentiation and integration of understanding which has a modal character.

This paper describes the semiotic approach to organism in two proto-biosemiotic thinkers, Susanne K. Langer and Hans Jonas. Both authors develop ideas that have become central terms of biosemiotics: the organism as subject, the realisation of the living as a closed circular self, the value concept, and, in the case of Langer, the concept of symbol. Langer tries to develop a theory of cultural symbolism based on a theory of organism as a self-realising entity creating meaning and value. This paper deals mainly with what both authors independently call “feeling.” Both authors describe “feeling” as a value-based perspective, established as a result of the active self interest manifested by an organic system. The findings of Jonas and Langer show the generation of a subject pole, or biosemiotic agent, under a more precise accent, as e.g. Uexküll does. Their ideas can also be affiliated to the interpretation of autopoiesis given by the late Francisco Varela (embodied cognition or “enactivism”). A synthesis of these positions might lead to insights how symbolic expression arises from biological conditions of living.