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

Fletcher

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.

• Acceptance Date: Feb 11, 2020
• Publication Date: May 9, 2020
• Journal: Journal of Philosophical Logic
• Print ISSN: 0022-3611
• Publisher: Springer Verlag
• Pages: 1111-1157
• DOI: https://doi.org/10.1007/s10992-020-09551-y
• Keywords: Brouwer, Intuitionistic analysis, Intuitionistic logic, Weak counterexamples,; Creative subject, Choice sequences
• Publisher URL: https://link.springer.com/article/10.1007%2Fs10992-020-09551-y