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Weyl and Intuitionistic Infinitesimals

  • Mark van Atten
Chapter
Part of the Studies in History and Philosophy of Science book series (AUST, volume 49)

Abstract

As Weyl was interested in infinitesimal analysis and for some years embraced Brouwer’s intuitionism, which he continued to see as an ideal even after he had convinced himself that it is a practical necessity for science to go beyond intuitionistic mathematics, this note presents some remarks on infinitesimals from a Brouwerian perspective. After an introduction and a look at Robinson’s and Nelson’s approaches to classical nonstandard analysis, three desiderata for an intuitionistic construction of infinitesimals are extracted from Brouwer’s writings. These cannot be met, but in explicitly Brouwerian settings what might in different ways be called approximations to infinitesimals have been developed by early Brouwer, Vesley, and Reeb. I conclude that perhaps Reeb’s approach, with its Brouwerian motivation for accepting Nelson’s classical formalism, would have suited Weyl best.

Keywords

L.E.J. Brouwer Constructivism Infinitesimals Intuitionism Nonstandard analysis Physics G. Reeb A. Robinson R. Vesley H. Weyl 

Notes

Acknowledgements

Earlier versions of this paper were presented at the conference ‘Weyl and the Problem of Space: From Science to Philosophy’, University of Konstanz, May 2015, and at the workshop ‘Workshop on the Continuum in the Foundations of Mathematics and Physics’, University of Amsterdam, April 2017. I am grateful to the organisers for their invitations, and to the audiences for their questions and comments. I have also benefited from exchanges with Julien Bernard (who also shared his instructive, unpublished manuscript ‘New insights on Weyl’s Problem of Space, from the correspondence with Becker’ with me), Dirk van Dalen, Bruno Dinis, Mikhail Katz, Carlos Lobo, David Rabouin, Jean-Michel Salanskis, Sam Sanders, Wim Veldman, and Freek Wiedijk. Gödel’s shorthand notes on the non-Archimedean number system in Brouwer’s dissertation, mentioned in footnote 1.5, were kindly transcribed by Eva-Maria Engelen. These notes are owned by the Institute for Advanced Study and kept in the Department of Rare Books and Special Collections at the Firestone Library, Princeton University.55

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Authors and Affiliations

  • Mark van Atten
    • 1
  1. 1.Archives Husserl (ENS/CNRS)ParisFrance

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