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Brouwer’s Weak Counterexamples and the Creative Subject: A Critical Survey

Abstract

I survey Brouwer’s weak counterexamples to classical theorems, with a view to discovering (i) what useful mathematical work is done by weak counterexamples; (ii) whether they are rigorous mathematical proofs or just plausibility arguments; (iii) the role of Brouwer’s notion of the creative subject in them, and whether the creative subject is really necessary for them; (iv) what axioms for the creative subject are needed; (v) what relation there is between these arguments and Brouwer’s theory of choice sequences. I refute one of Brouwer’s claims with a weak counterexample of my own. I also examine Brouwer’s 1927 proof of the negative continuity theorem, which appears to be a weak counterexample reliant on both the creative subject and the concept of choice sequence; I argue that it provides a good justification for the weak continuity principle, but it is not a weak counterexample and it does not depend essentially on the creative subject.

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Acknowledgements

Thanks to James Appleby and Mark van Atten for helpful discussions, particularly in relation to §9. The conclusions drawn in this paper are my own responsibility.

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Fletcher, P. Brouwer’s Weak Counterexamples and the Creative Subject: A Critical Survey. J Philos Logic 49, 1111–1157 (2020). https://doi.org/10.1007/s10992-020-09551-y

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Keywords

  • Brouwer
  • Intuitionistic analysis
  • Intuitionistic logic
  • Weak counterexamples
  • Creative subject
  • Choice sequences