The 2008 lecture used the methods of cognitive psychology to analyze the mental representation of one of the simplest yet most fundamental mathematical objects: the concept of the natural integer.
The nature and origin of mathematical objects have been debated since antiquity. Many mathematicians adhere, explicitly or implicitly, to the Platonic hypothesis that mathematics is simply the exploration of a world apart, governed by its own constraints and pre-existing the human brain. To quote Alain Connes in his debate with Jean-Pierre Changeux: "As he moves through the geography of mathematics, the mathematician gradually perceives the incredibly rich contours and structure of the mathematical world. He gradually develops a sensitivity to the notion of simplicity that gives him access to new regions of the mathematical landscape" (Changeux & Connes, 1989).
The developmental psychologist, however, cannot help but be struck by the difficulty with which the child gradually builds up mathematical competence. They can easily conclude that mathematical objects are purely mental constructs. For Piaget, logic is the foundation ("The whole number can thus be conceived as a synthesis of the class and the asymmetrical relation"). For others, language plays an essential role (cf. Vygotsky: "Thought is not only expressed in words: it comes into the world through them").
The position I have defended in this lecture, which could be described as intuitionist, belongs to neither camp. It postulates that the cognitive foundations of mathematics must be sought in a series of fundamental intuitions about space, time and number, shared by many animal species, and that we inherit from a distant past when these intuitions played an essential role in survival. Mathematics is built by formalizing and consciously linking these different intuitions. This position is linked to, but not identical with, the mathematical intuitionism of Brouwer and Poincaré. As early as the 13th century, Roger Bacon noted that "mathematical knowledge is almost innate in us... it is the easiest of the sciences, evidently, in that no brain rejects it: even common men and illiterates can count and calculate". For mathematicians Philip David and Reuben Hersh, "within ideas, within mental objects, ideas with reproducible properties are called mathematical objects, and the study of mental objects with reproducible properties is called mathematics. Intuition is the faculty that enables us to think about and examine these internal mental objects" (Davis, Hersh, & Marchisotto, 1995, page 399).