Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘ mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence (...) may possess a common explainer, they have no common explanation ; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no mathematical coincidence. (shrink)

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)