Complementarity, sets and numbers |
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| Authors: | M Otte |
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| Institution: | 1. Universit?t Bielefeld, Postfach 100131, D-33615, Bielefeld
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| Abstract: | 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 acomplementarist 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.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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| Keywords: | attributive and referential use of terms complementarity history and epistemology of mathematics hypostatic abstraction set-theoretical explanation of number |
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