Brouwer and Euclid

Abstract: We explore the relationship between Brouwer's intuitionistic mathematics and Euclidean geometry. Brouwer wrote a paper in 1949 called "The contradictority of elementary geometry". In that paper, he showed that a certain classical consequence of the parallel postulate implies Markov's principle, which he found intuitionistically unacceptable. But Euclid's geometry, having served as a beacon of clear and correct reasoning for two millenia, is not so easily discarded. Brouwer started from a "theorem" that is not in Euclid, and requires Markov's principle for its proof. That means that Brouwer's paper did not address the question whether Euclid's "Elements" really requires Markov's principle. In this paper we show that there is a coherent theory of "non-Markovian Euclidean geometry." We show in some detail that our theory is an adequate formal rendering of (at least) Euclid's Book~I, and suffices to define geometric arithmetic, thus refining the author's previous investigations (which include Markov's principle as an axiom). Philosophically, Brouwer's proof that his version of the parallel postulate implies Markov's principle could be read just as well as geometric evidence for the truth of Markov's principle, if one thinks the geometrical "intersection theorem" with which Brouwer started is geometrically evident.