MATHIEU MARION WITTGENSTEIN AND BROUWER ⋆ ABSTRACT. In this paper, I present a summary of the philosophical relationship between Wittgenstein and Brouwer, taking as my point of departure Brouwer’s lecture on March 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not done serious philosoph- ical work for years, if one is to understand the impact of that lecture on him, it is better to compare its content with the remarks on logics and mathematics in the Tractactus. I thus show that Wittgenstein’s position, in the Tractactus, was already quite close to Brouwer’s and that the points of divergence are the basis to Wittgenstein’s later criticisms of intu- itionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic. During the 1920s, L. E. J. Brouwer promoted his foundational standpoint through a series of lectures at meetings of the German Mathematical Soci- ety; his Berlin lectures in 1927 were very well received. On March 10, 1928, he gave a lecture in Vienna on ‘Mathematics, Science and Lan- guage’, which was followed, four days later, by another one on ‘The Structure of the Continuum’. 1 It is easy to imagine the atmosphere of intellectual excitement that must have surrounded his visit. Brouwer held a radical stance about mathematics that entailed, as he would himself put it in his first lecture, the “collapse” of “considerable portions of the pre- vious mathematical edifice” (1996a, 1185). 2 Brouwer’s intuitionism was perceived at the time by many as revolutionary and dangerous: in Cam- bridge Frank Ramsey had written about a “Bolshevic menace” (1978, 207), while David Hilbert, the doyen of German mathematics, spoke of a “coup” (1998, 200) and waged throughout the 1920s a war against it; this Grundla- genstreit was to turn very bitter later on that year, with the somewhat illegal expulsion of Brouwer, at Hilbert’s request, from the editorial committee of Mathematische Annalen. 3 The sense of excitement provoked by Brouwer’s presence was not only due to his polemical stance about foundations: Brouwer was one if not the founder of modern topology; not only did he introduce fundamental notions of twentieth-century topology such as that of a simplicial approx- ⋆ To the memory of Michael Wrigley. Synthese 137: 103–127, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.