The irreflexivity of Brouwer ’ s philosophy ∗

I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for the kind of knowledge that such an account of itself would amount to. Brouwer tried to go ‘from philosophy to mathematics’ and grounded his intuitionistic mathematics in a more general philosophy. This background philosophy can be characterized as a transcendental one. That is, it purports to explain how a non-mundane subject builds up its world in consciousness. It is a radical transcendental philosophy in that this ‘world’ does not contain just physical objects but everything, including abstract objects and the mundane subject (the subject as part of the world). From the empirical point of view, such a non-mundane subject is an idealized one. Like fellow transcendentalists Kant and Husserl, Brouwer sought to account for mathematics by referring to structural features of acts of this idealized subject [18]. Because of its solipsistic tendencies (but see [17]) and its mystical characteristics, this part of Brouwer’s thought is generally dismissed. In fact, more sense can be made of it than is commonly assumed (see Van Dalen’s overview mentioned in footnote 1); however the claim I want to defend here is that, even if we grant Brouwer these features, his background philosophy could still not function as a basis for intuitionist mathematics. This philosophy, taken at face value, is not able to reflect on itself. It cannot thematize itself and, a fortiori, it can neither account for itself, nor be self-critical. Therefore it cannot do as the grounding for mathematics that Brouwer wanted it to be. For what is its ∗I wish to thank Dirk van Dalen, Charles Parsons, Richard Tieszen, and Robert Tragesser for comments on the draft version. I am also thankful to an anonymous reader of the draft for bringing up several further objections to my approach. This paper was written during a stay at Harvard in 1998, partially made possible by a grant from the Netherlands Organisation for Scientific Research (NWO), which is gratefully acknowledged. †Department of Philosophy, Utrecht University, Heidelberglaan 8, NL-3584 CS Utrecht, The Netherlands. Email: [email protected]. 1 Recently, Van Dalen gave an overview [14], and Van Stigt presented a translation of Brouwer’s seminal paper ‘Leven, Kunst en Mystiek’ [13, 20].