Intellectica 2009/1, n° 51

The Mathematical Continuum. New Conceptions, New Challenges

Dirk van Dalen

The Return of the Flowing Continuum

 Abstract: After the arithmetization of the continuum in the 19th century, L.E.J. Brouwer, in his 1907 PhD thesis, brought back the intuitive continuum on the basis of his intuitionistic philosophy. The so-called ur-intuition (or intuition of time) yielded at the same time the natural numbers, and their derivatives, and the continuum. We show that intuitionistic mathematics, being a constructive mental activity of the subject, had to break with the traditional laws of logic, in particular with the principle of the excluded middle. We go into the properties of the continuum and the natural numbers, as spelled out by Brouwer. In particular we argue that the ur-intuition (embodied in the move of time) encompasses induction and recursion. The second installment of the intuitionistic revolution, beginning in 1918, resulted in an autonomous Brouwerian universe of mathematics, based on choice sequences and continuity.

 Key words: continuum, ur-intuition, choice sequence, induction, natural numbers, principle of the excluded middle, continuity principle, implication.