During the last decade, radical constructivism has 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 description of three processes involved in criticising mathematical thinking (1925) 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 ele-mentary 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.