id: 02354986
dt: j
an: 2003f.04809
au: Otte, M.
ti: Complementarity, sets and numbers.
so: Educ. Stud. Math. 53, No. 3, 203-228 (2003).
py: 2003
pu: Springer Netherlands, Dordrecht
la: EN
cc: E20
ut: hypostatic abstraction; set-theoretical explanation of numbers; attributive
and referential use of terms
ci:
li: doi:10.1023/A:1026001332585
ab: 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)
rv: