Brouwer’s ϵ-fixed point and Sperner’s lemma

It is by now common knowledge that in 1911 Brouwer gave mathematics a miraculous tool, the fixed point theorem, and that later in life, he disavowed it. It usually came as a shock when he replied to the question “is the fixed point theorem correct?” with a point blank “no”. This rhetoric exchange deserves some elucidation. At the time that Brouwer did his revolutionary topological work, he had suspended his constructive convictions for the time being. He was well aware that he was using the principle of the excluded middle, indeed in Brouwer (1919) [1], p. 950, he remarked that “In my philosophy-free mathematical papers I have regularly used the old methods, while at the same time attempting to deduce only those results, of which I could hope that they would find a place and be of value, if necessary in a modified form, in the new doctrine after the carrying out of a systematic construction of intuitionistic set theory”. And in the case of the fixed point theorem we are presented with exactly such a result. From the intuitionistic point of view the theorem is not correct because the fixed point that is promised can in general not be found, that is to say, approximated.