Towards an evolutionary account of conceptual change in mathematics: Proofs and refutations and the axiomatic variation of concepts. (English) Zbl 1032.00006

Mormann, Thomas

Kampis, George (ed.) et al., Appraising Lakatos. Mathematics, methodology, and the man. Dordrecht: Kluwer Academic Publishers. Vienna Circ. Inst. Libr. 1, 139-156 (2002).

I.Lakatos’ notions of concept-formation and concept-stretching as presented in his “Proofs and refutations” [Cambridge, Cambridge UP (1976; Zbl 0334.00022)] are elaborated for justifying an evolutionary account for mathematical change based on axiomatic concept variation. Whereas Lakatos restricted the notion of concept-stretching to informal, non-axiomatic mathematics, the author stresses its creative role for inventing new mathematical concepts using the axiomatic method. W. Hamilton’s invention of the quaternions is used as an example for the axiomatic variation of concepts. This method led to concepts having rather different features than those Hamilton was originally looking for.
In conceiving the history of mathematics as being related to the evolution of mathematical concepts the author gives a re-interpretation of basic concepts of the Darwinian or Lamarckian theory of evolution for mathematics, in particular he discusses the principles of variation, of the struggle for existence, of variation of fitness, and of inheritance. These concepts are tested by applying them to the evolution of the concept of the integral, especially dealing with the Cauchy and Lebesgue variants.
The author concludes that the ongoing evolution of mathematical knowledge shows that “the ‘essence’ of mathematics is not to be found in some hidden nature of mathematical objects, but rather is revealed in the specifics of the ongoing evolutionary process of mathematical knowledge …” (p.152). The ubiquity of variation “renders modern mathematical knowledge a definitely un-Platonic epistemic endeavor, making variation (on which level whatsoever) the very essence of mathematics” (ibid.).
For the entire collection see [Zbl 0990.00016].

Reviewer: Volker Peckhaus (Erlangen)

MSC:

00A30

Philosophy of mathematics

Keywords:

axiomatic variation; concept-stretching; concept-formation; invention; creation; theory of evolution; the concept of the integral; quaternions

Citations:

Zbl 0334.00022