A Constructivist Approach to Experiential Foundations of Mathematical Concepts Revisited

This is a revised version of a paper which, as an anonymous reviewer guessed, was written some time ago; in fact I wrote it in 1990/ 91 for the 2nd International Conference on the History and Philosophy of Science in Science Teaching . Queen’s University, Kingston, Ontario, 1992. Much has happened since then in the philosophy of mathematics, especially regarding the “naturalization” of the highly abstract concepts that were at the core of the debate on the reality of mathematical objects. As I understand it, naturalization in that discipline is the attempt to illuminate the foundations of mathematics by mathematical rather than philosophical thinking and it leads to the dismantling of the Platonist notion that mathematical objects “exist” in an absolute sense. I fervently agree with this dismantling, but my approach is on a much lower level of abstraction and focuses on how the most elementary concepts, such as unit, plurality, number, point, line, and plane could be derived from ordinary experience. I have added a postscript with references to and brief comments on publications by Brian Rotman, Penelope Maddy, and George Lakoff as samples of recent voices. During the 1980s radical constructivism gained a certain currency in the fields of science and mathematics education. Although cognitive constructivists have occasionally referred to the intuitionist approach to the foundational problems in mathematics, no effort has so far been made to outline what a constructivist’s own approach might be. This paper attempts a start in that direction. Whitehead’s (1956, p. 393) description of three processes involved in criticizing mathematical thinking is used to show discrepancies between a traditional epistemological stance and the constructivist approach to knowing and communication. The bulk of the paper then suggests tentative itineraries for the progression from elementary experiential situations to the abstraction of the concepts of unit, plurality, number, point, line, and plane, whose relation to sensory–motor experience is usually ignored or distorted in mathematics instruction. There follows a discussion of the question of certainty in logical deduction and arithmetic. Introduction