This dissertation focuses on David Lewis’s Patchwork Principle. That principle says, very roughly, that scenarios that are “patched” together from other possible scenarios are themselves possible. Thus, there are no metaphysically necessary connections between distinct things. More broadly, my dissertation is concerned with the many components of Lewis’s metaphysical system which are inspired by Hume’s claim that the world’s contents are “entirely loose and separate.” I first show that if one makes certain natural assumptions, the Patchwork Principle implies the rest of those Hume-inspired components. It particular, it appears to imply that causal, nomic, and temporal-direction relations are all extrinsic (and always extrinsic to their bearers), and that the causal and nomic relations globally supervene on the non-causal non-nomic properties and relations. Thus, Lewis’s set of metaphysical views is systematic not only insofar as many of its members are inspired by Hume’s denial of necessary connections, but in the more profound way that its least contentious member appears to imply many of its more contentious members: it seems to present us with a stark dilemma, a dilemma about whether to accept or reject Lewis’s entire Humean system. Such a dilemma would be difficult indeed. On the one hand, I go on to offer an argument for the Patchwork Principle, an argument which seems as compelling as arguments get in this area. On the other hand, I also argue that the Patchwork Principle’s apparent implications are untenable, because they are inconsistent with the existence (or even the possibility) of conscious beings like ourselves. The dilemma is thus a conundrum.The crux of my solution to this conundrum (and dilemma) is that the apparent implications of the Patchwork Principle are merely apparent, since one of the “very natural assumptions” I made in arguing for those implications – that there can be concrete objects that do not share any parts – is false. Indeed, my solution assumes that of necessity, any two concrete objects overlap. Thus, while the Patchwork Principle is maintained, it is emptied of its content. And Hume’s claim that the world’s contents are “entirely loose and separate,” along with several components of Lewis’s metaphysical system which are inspired by it, are shown to be far wide of the mark.