Hume’s discussion of space in the Treatise addresses two main topics: divisibility and vacuum. It is widely recognized that his discussion of divisibility contains an answer to Bayle, whose Dictionary article “Zeno of Elea” presents arguments about divisibility as support for fideism. It is not so widely recognized that, elsewhere in the same article, Bayle presents arguments about vacuum as further support for fideism. This paper aims to show that Hume’s discussion of vacuum contains an answer to these vacuum-based fideistic (...) arguments. Key to this answer is a distinction between two ways in which vacuum was conceived in the early modern period: i) as a genuine thing that has spatial properties, and yet is immobile, indivisible, and penetrable ; ii) as a mere absence of spatial things. This paper also aims to provide a novel defense of Hume against the long-standing objection that he is inconsistent in denying that we can conceive of a vacuum, while allowing that we can conceive of “invisible and intangible distance.” As I interpret him, Hume consistently denies that we can conceive of a positive vacuum, while allowing that we can conceive of two or more objects’ being arranged so as to have privative vacuum between them. (shrink)