In the Univalent Foundations of mathematics spatial notions like “point” and “path” are primitive, rather than derived, and all of mathematics is encoded in terms of them. A Homotopy Type Theory is any formal system which realizes this idea. In this paper I will focus on the question of whether a Homotopy Type Theory can be justified intuitively as a theory of shapes in the same way that ZFC can be justified intuitively as a theory of collections. I first clarify (...) what such an “intuitive justification” should be by distinguishing between formal and pre-formal “meaning explanations” in the vein of Martin-Löf. I then go on to develop a pre-formal meaning explanation for HoTT in terms of primitive spatial notions like “shape”, “path” etc. (shrink)

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'. (shrink)