FOM: Intuitionism, Goedel

[Charles Parsons]

Vaughan Pratt asks for a specific page number of the Collected Works where
Goedel speaks negatively of intuitionism. Such is to be found in *1938a
(see my posting of 10 March), but there the specific context is
constructivism and the prospects for an extended Hilbert program.

Although the point of view of the post-war writings is certainly
incompatible with intuitionism, very little is said in them about
intuitionism one way or the other; where there is polemic the target is
either the views of the Vienna Circle or the nominalistic tendencies of the
later Russell. The affirmations of a realistic point of view and the
defense of set theory were no doubt also aimed at intuitionism, but that
wasn't what he was concerned to emphasize.

The battles over intuitionism in earlier times concerned first of all the
foundations of analysis, which are not the center of attention in Goedel's
post-war writings. Also, against the Vienna Circle one thing that Goedel
was concerned to argue is that mathematics has a "real content" that can't
be reduced to conventions about the use of symbols. In that point,
connected with his defense of mathematical intuition, he would have agreed
with intuionism, even though the conception of mathematical intuition he
defended differed fundamentally from Brouwer's.

(I discuss these matters in my paper in the Bulletin of Symbolic Logic,
March 1995.)

Charles Parsons