Brouwerian intuitionism

Mind 99 (396):501-534 (1990)

Authors
Michael Detlefsen
University of Notre Dame
Abstract
The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. The second is that the intuitionist's chief objection to the classical mathematician's use of logic does not center on the use of particular logical principles (in particular, the law of excluded middle and its ilk). Rather on the role the classical mathematician assigns (or at least extends) generally (i.e. regardless of the particular principles used) to the use of logic in the production mathematical proofs. Thus, the intuitionist critique of logic that we shall be presenting is far more radical than that which has commonly been presented. Concerning the second, more general, theme, my claim is this: some restriction of the role of logical inference in mathematical proof such as that mentioned above is necessary if one is to account for the seeming difference in the epistemic conditions of provers whose reasoning is based on genuine insight into the subject-matter being investigated, and would-be provers whose reasoning is based not on such insight, but rather on principles of inference which hold of every subject-matter indifferently.
Keywords intuitionism  Brouwer  Poincaré  excluded middle  the place of logical reasoning in mathematical proof  Brouwerian Intuitionism and logical reasoning  First Act of Intuitionism  Dummett
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DOI 10.1093/mind/XCIX.396.501
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Poincaré Against the Logicians.Michael Detlefsen - 1992 - Synthese 90 (3):349 - 378.
Against Against Intuitionism.Dirk Schlimm - 2005 - Synthese 147 (1):171-188.
Brouwer's Conception of Language, Mind and Mathematics'.Hiroshi Kaneko - 2002 - Annals of the Japan Association for Philosophy of Science 11 (1):35-49.
Undecidability, Epistemology and Anti-Realist Intuitionism.Sanford Shieh - 1997 - Nordic Journal of Philosophical Logic 2:55-67.

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