Two dichotomies in the philosophy of mathematics are discussed: the prescriptive – descriptive distinction, and the process – product distinction. By focusing on prescriptive matters, and on mathematics as a product, standard philosophy of mathematics has overlooked legitimate and pedagogically rewarding questions that highlight mathematics as a process of knowing which has social dimensions. In contrast the social-constructivist view introduced here can affect the aims, content, teaching approaches, implicit values, and assessment of the mathematics curriculum, and above all else, the beliefs and practices of the mathematics teacher.

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.

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: The value of Cifarelli & Sevim’s target article lies in the analysis of how reflective abstraction contributes to the description of mathematical learning through problem solving. The additional value of the article lies in its emphasis of some aspects of the learning process that goes beyond radical constructivist learning theory. I will look for common ground between the humanist philosophy of mathematics and radical constructivism. By doing so, I want to stress two converging elements: (i) the move away from traditionalist ontological positions and (ii) the central role of the students’ activity in the learning process.

A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget’s postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.

Key words: philosophy of mathematics, cognition, development

Review of Paul Ernest: The Philosophy of Mathematics Education. London/New York/Philadelphia: The Falmers Press, 1991. XIV + 329 p. – ISBN 1–85000–667–9 (pbk).

Key words: philosophy, mathematics

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

Excerpt: Dennis Lomas in his essay on “Paul Ernest’s Application of Social Constructivism to Mathematics and Mathematics Education” correctly indentifies Ernest as a major proponent of social constructivism in mathematics education. Lomas’s essay is quite circumscribed in its goals: he leaves aside whether Ernest has adequately, or otherwise, interpreted the arguments of I. Lakatos, Ludwig Wittgenstein, and L. S. Vygotsky that he appeals to develop his philosophy of mathematics; and Lomas declines to reflect on the more general relevance of social constructivism to “mathematics, mathematics education, or education in general.” Lomas wishes to focus upon Ernest’s account of mathematical objects, and to begin a critique of the “social, political, and ethical consequences that [Ernest] draws from his position” for the “great issues of freedom, justice, trust and fellowship.” I propose in this commentary to first take a broader view of Ernest’s work, locating his social constructivism on the larger canvas of constructivism in science and mathematics education, and then follow Lomas’s more narrowed concerns.

Context: As one of the major approaches within the philosophy of mathematics, constructivism is to be contrasted with realist approaches such as Platonism in that it takes human mental activity as the basis of mathematical content. Problem: Mathematical constructivism is mostly identified as one of the so-called foundationalist accounts internal to mathematics. Other perspectives are possible, however. Results: The notion of “meaning finitism” is exploited to tie together internal and external directions within mathematical constructivism. The various contributions to this issue support our case in different ways. Constructivist content: Further contributions from a multitude of constructivist directions are needed for the puzzle of an integrative, overarching theory of mathematical practice to be solved.

Key words: mathematical constructivism, conceptualism, internalism vs externalism, meaning finitism, mathematical practice