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On Paradigm and Method

In: From Topology to Computation: Proceedings of the Smalefest

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  • Philip Holmes

Abstract

This is not a technical paper. Rather than turning in a brief summary of a recent research enthusiasm, I have taken the occasion of this celebration of Stephen Smale’s many contributions to mathematics to reflect a little on the role of mathematics in general and dynamical systems theory in particular, the conduct of mathematicians, and the relations of all these to turbulence studies in engineering physics. The technical preoccupations which lie behind and have prompted these somewhat loose musings may be found in the works of Aubry [1990], Aubry et. al. [1988, 1990], Aubry and Sanghi [1989, 1990], Armbruster et al. [1988, 1989], Berkooz et al. [1991a, 1991b], Stone and Holmes [1989, 1990, 1991], and Holmes [1990a]. The last of these contains an introductory survey of the uses (and some misuses) of dynamical systems theory and other tricks in turbulence studies. Other papers in that same volume describe yet more tricks. These tricks are important: It seems to me that no one style or speciality of mathematics is adequate when faced with problems of this complexity. This is one of the main themes of the present essay.

Suggested Citation

  • Philip Holmes, 1999. "On Paradigm and Method," Springer Books, in: Morris W. Hirsch & Jerrold E. Marsden & Michael Shub (ed.), From Topology to Computation: Proceedings of the Smalefest, chapter 48, pages 534-544, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2740-3_48
    DOI: 10.1007/978-1-4612-2740-3_48
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