Did Brouwer Really Believe That?

This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses Brouwer (who, incidentally, was normally known not as “Jan” but as “Bertus”, a shortening of his second name, Egbertus),1 he says: ...later in his career, he [Brouwer] became the most forceful proponent of the so-called intuitionist philosophy of mathematics, which not only forbids the use of the Axiom of Choice but also rejects the axiom that a proposition is either true or false (thereby disallowing the method of proof by contradiction). The consequences of taking this position are dire. For instance, an intuitionist would not accept the existence of an irrational number! In fact, in his later years, Brouwer did not view the Brouwer Fixed Point Theorem as a theorem. These sentences contain a number of outdated but still common misconceptions about Brouwer and intuitionism,2 misconceptions which tend to propagate the myths that intuitionism is somehow based on arbitrary rejections of classical principles and that, as a result, it is not possible for the intuitionist to prove theorems as deep as those provable by standard methods. First, consider Ok’s words that intuitionism “forbids the use of the Axiom of Choice but also rejects the axiom that a proposition is either true or false”. The exclusion of choice and of truth values by the intuitionist ultimately stem from the fundamental premiss of intuitionism: that the objects of mathematics are mental creations, and hence that they can be said to exist if and only if those creations have actually been carried out. In other words, for the intuitionist, “existence” 1For more about Brouwer in person, see the masterly two-volume biography by van Dalen [8]. 2Although, following Ok, I use “intuitionism” and “intuitionist” throughout this note, it might have been better to use the words “constructivism” and “constructive mathematician” instead, since my comments apply to (Bishop-style) mathematics with intuitionistic logic [2], without the additional continuity principles and fan theorem that single out intuitionistic mathematics.