Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but his (...) commitment to twodifferent types of intuition, which explains his rather unusual and tormented philosophy of the mathematical continuum. I would like to thank Geoff Gorham, David McCarty, and Rosamond Rodman for reading an earlier draft of this work. I should also thank those who provided helpful comments on several distant ancestors of this paper: Emily Carson, Ulrich Majer, Erhard Scholz, John Schuerman, Stewart Shapiro, and Richard Tieszen. I am indebted to two anonymous referees for pointing out some problems and for pointing me to work on Weyl I did not previously know about. In particular, the recent articles in [Feist, 2004a] turned out to be (somewhat uncomfortably) relevant to the focus of this paper. In this last revision I have tried to show where I agree and disagree with the authors of those papers; I apologize for whatever repetition still exists, but it was tere before I read those papers. This paper has a long history, and comes out of several talks I gave some years ago. Audiences at the Center for Philosophy of Science, University of Pittsburgh (colloquium 1995), St Andrews University Philosophy of Mathematics Workshop (1996), the British Society for the History of Mathematics meeting (1996), the University of Mainz Mathematics Department (colloquium 1996), the Canadian Philosophical Association (1997 and 1999), and the University of British Columbia (colloquium 1998) should be thanked for their helpful comments. I also thank Neil Tennant for encouraging me to resurrect this work. CiteULike Connotea Del.icio.us What's this? (shrink)

A new form of skepticism is described, which holds that objectivity and understanding are incompossible ideals of modern science. This is attributed to Weyl, hence its name: Weylean skepticism. Two general defeat strategies are then proposed, one of which is rejected.