Maddy On The Multiverse

In Deniz Sarikaya, Deborah Kant & Stefania Centrone (eds.), Reflections on the Foundations of Mathematics. Berlin: Springer Verlag. pp. 43-78 (2019)
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Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'.
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