The course is an introduction to intuitionistic mathematics. This school of mathematical thinking was founded by L.E.J. Brouwer (1881-1966). He attacked the principle of the excluded middle in 1908 and developed the main ideas of his intuitionistic mathematics in the period 1910-1930. Giving some examples, we first explain the intuitionistic view that the meaning of a theorem has to be found in its proof and that different proofs of the same theorem very often, if one looks carefully, prove different things. We then show why many theorems in usual, “classical” mathematics are misleading and in what ways they might perhaps be repaired. We then go into the constructive interpretation of the logical constants that underlies Brouwer’s counterexamples. We explain the new axioms Brouwer introduces, the Continuity Principle, the Fan Theorem and the Principle of Bar Induction. We treat a number of mathematical applications of these axioms. We compare intuitionistic mathematics to other varieties of constructive mathematics.

Prerequisites

The student should have followed a course in mathematical analysis. No very specialized knowledge is required.