17 March 2004, General Mathematics Colloquium, John Kuiper

In the beginning of the twentieth century a new movement was added to the existing two that attempted to lay a solid foundation for the mathematical building. After Frege, Russell and Couturat, who viewed logic as the ultimate basis for mathematics, and Hilbert's formalist approach in which mathematics is just a manipulation with meaningless signs and symbols, Brouwer worked out earlier ideas by Poincaré and Borel: mathematics has an extra-logical content too.

For Brouwer, the ultimate basis for all mathematics is the ur-intuition of `the move of time', that is, the experience of the fact that two not-coinciding mental events are connected by a time continuum. Departing from this ur-intuition, the whole of mathematics, hence including set theory and geometry, can be constructed. In is early years as an active mathematician (in his own terms: his `first intuitionistic period', between 1907 and, say, 1914; note that most of his time during those years was spent on topology) his constructivistic requirements were very strict: only that what is constructed by the individual mind (mathematics is essentially languageless) counts as a mathematical object. In this lecture we will work this out for the logical figure of the hypothetical judgement in a mathematical context, and we will see that, in hindsight, Brouwer went too far in his constructivism.

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