Complementarity, sets and numbers. (English)

Educ. Stud. Math. 53, No. 3, 203-228 (2003).

Niels Bohr’s term ‘complementarity’ has been used by several authors to capture the essential aspects of the cognitive and epistemological development of scientific and mathematical concepts. In this paper we will conceive of complementarity in terms of the dual notions of extension and intension of mathematical terms. A complementarist approach is induced by the impossibility to define mathematical reality independently from cognitive activity itself. R. Thom, in his lecture to the Exeter International Congress on Mathematics Education in 1972, stated “the real problem which confronts mathematics teaching is not that of rigor, but the problem of the development of ‘meaning’, of the ‘existence’ of mathematical objects”. Student’s insistence on absolute ‘meaning questions’, however, becomes highly counter-productive in some cases and leads to the drying up of all creativity. Mathematics is, first of all, an activity, which, since Cantor and Hilbert, has increasingly liberated itself from metaphysical and ontological agendas. Perhaps more than any other practice, mathematical practice requires a complementarist approach, if its dynamics and meaning are to be properly understood. The paper has four parts. In the first two parts we present some illustrations of the cognitive implications of complementarity. In the third part, drawing on Boutroux’ profound analysis, we try to provide an historical explanation of complementarity in mathematics. In the final part we show how this phenomenon interferes with the endeavor to explain the notion of number. (Author’s abstract)