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