Maddy and Mathematics: Naturalism or Not

Abstract

Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's account, on the other hand, appears to be unable to similarly explain the reliability of mathematical practice without violating one of its central tenets. 1 Introduction 2 Mathematical naturalism 3 Desiderata and the attraction of naturalism 4 Assessment: Naturalism and names 4.1 Taking 'naturalism' seriously 4.2 Second philosophy (or what's in a name) 5 A way out? 6 Or out of the way?

On Naturalizing the Epistemology of Mathematics

“…See, e.g., Quine, 1948Quine, , 1951Quine, , 1954Quine, , 1969aQuine, , 1969bQuine, , 1969c See, e.g., Kitcher, 1983Kitcher, , 1988and Maddy, 1995and Maddy, , 1996and Maddy, , 1997and Maddy, , 1998aand Maddy, , 1998band Maddy, , 2003 I'll often not specify the epistemology of mathematics, it being understood that by naturalizing mathematics I intend naturalizing the epistemology of mathematics. 6 See, e.g., Dieterle, 1999;Paseau, 2005;Roland, 2007Roland, , 2008Rosen, 1999;and Weir, 2005. 7 Though this notion of falling within our scientific worldview is obviously imprecise, it should be precise enough for present purposes.…”

“…11 See, e.g., Quine, 1975. 12 Specific to Maddy, see Dieterle, 1999;Roland 2007;and Rosen, 1999. 13 For more on disciplinary holism, see Roland 2007.…”

“…I have argued elsewhere that the attempts of both fail, and they fail for essentially the same reasons as the views discussed in §3. (See Roland, 2007 and 2008.) Thus I opted not to address the views of Maddy or Kitcher in the present article.…”

Kitcher and the Obsessive Unifier*

“…Neither have I offered a conclusive argument in favor of causal realism. My purpose has been the more restricted one of pointing out that Kitcher's position is subject to a particular challenge, that some obvious responses to that challenge don't work, and that the challenge can be met by supplementing Kitcher's position in a way that Kitcher himself 23 For more on this, see Boyd (1980Boyd ( , 1989Boyd ( , 1991, Kornblith (1993), and Roland (2007). 24 This should not be taken to suggest that merely adopting a robust causal realism yields a map of the causal order of the world, or that mapping that order thereby becomes an easy matter, or even that there is a unique map of the causal order.…”