Dummett on

Intuitionism

The two most influential early proponents of intuitionism were Brouwer and his student Heyting. Brouwer's justification for the intuitionistic natural numbers is based on a "fundamental intuition of one-twoicity." His basis for intuitionism, while it has many Kantian components, is generally agreed to be mystical, and often mystical in a pejorative sense. There is a whole literature of Brouwer reinterpretation, since no one but Brouwer was ever happy with the original version.

Brouwer took very seriously the idea that mathematics, (that is, intuitionistic mathematics) is about mental constructions, prior to any form of language. He therefore thought that formalization was inappropriate and misleading to the spirit of mathematics. The chief contribution of Heyting, his most influential student, was a formalization of many of Brouwer's mathematical ideas. He more or less ignored the philosophical issues.

Dummett presents what he takes to be the best argument for adopting intuitionistic logic as the fundamental basis of mathematical proof. His approach is very far from that of Brouwer, indeed is largely Wittgensteinian, though one can see how an attempt to reconstruct Brouwer is to some extent lurking in the background. He does not, in our reading, pursue aspects of intuitionistic mathematics that go beyond the basic logic.

He is not interested in defending intuitionistic mathematics as one form of interesting mathematics among many. He is rather interested in defending Brouwer's claim that classical logic is wrong and intuitionistic logic is the only correct way to reason.

The claim aroused, and continues to arouse, lots of strong feelings:

Brouwer: One day the belief in the law of the excluded middle will be regarded as a quaint superstition like the belief that the earth is flat.

Hilbert: Depriving a mathematician of the law of the excluded middle is like depriving a boxer of the use of his fists.

When Bishop, in the 1960s (?), submitted his book on constructive mathematics to a publisher, one of the reviewers said that it should not be published on the grounds that it was "subversive."

Let me show you that something is "wrong" with classical logic: The Generalized Continuum Hypothesis (GCH) is a statement about set theory that is independent of all known accepted mathematical principles. Nonetheless, there is a computer program that prints out "true" if GCH is true and "false" if GCH is false.

Proof: There is a computer program that prints out "true" and there is a computer program that prints out "false." One of them is the desired program.

Dummett bases his argument for intuitionistic logic on three theses:

1. The use thesis: the meaning of any piece of language is fully determined by its use.
2. The molecularist thesis: The units of meaning in a language are no larger than sentences, and the meanings of sentences are determined by the fixed contributions of their components.
3. It follows from the use thesis that if there is a fundamental component to meaning, that component is verification conditions.

What are the verification conditions for statements in mathematics? Computations and proofs. We will regard computations as a special case of proof. Thus, given Dummett's desire to have a shot at showing that our logical practices are coherent, the logical operators must have their meaning fully specified by the way in which they are used in proofs. -- ProfessorShaughanLavine - 26 Jan 2005