Whither "intuitionistic"?

The terms >intuitionism< and -- derived from the latter -- >intuitionistic<
are due to Luitzen Egbertus Jan Brouwer, a dutch mathematician who in 1912
(in his inaugural lecture at Amsterdam) introduced the term for a position
opposed to what he called formalism: the axiomatic approach to the
foundation of mathematics favoured by Hilbert, Zermelo et al. Brouwer did
not initiate intuitionism -- before him, others like Poincar\'e and
Kronecker had voiced similar objections against formalism -- but he coined
the term and became, during the 20s and 30s the most fervent proponent of
the related views on the foundation of mathematics.

Brouwer suggested to base Mathematics and all science on simple intuitions:
unity, difference and counting. He was strictly opposed to the the
employment of the law of the excluded middle in the reasoning about
infinite sets. A proof of existence should be given by constructive means,
not ex negativo by the proof of the impossibility of nonexistence.

In the context of his work on natural deduction, Gentzen considered --
besides, what, when opposed to >intuitionistic<, has come to be called
>classical< logic, including the law of the excluded middle -- the
formalization of intuitionistic reasoning. Thereby he draw on earlier work
done by Arend Heyting.

A good source to enhance the knowledge about intuitionism ist still this one:

Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie.
Berlin: Springer, 1934 (Reprint 1974).

Rainer Fischbach


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