Metaphysics is largely an a priori business, albeit a business that is sensitive to the findings of the physical sciences. But sometimes what the physical sciences tell us about our own world underdetermines what we should think about the metaphysics of how things actually are, and even how they could be.
This chapter has two aims. The first is to defend a particular conception of the methodology of a priori metaphysics by, in part, exemplifying that methodology and revealing its results. The second is to present a new account of holes. These two aims dovetail nicely. We are independently interested in providing a better analysis of the concept <hole> that yields a more plausible metaphysical story about holes. But focusing on holes is also a good way to explore the methodology we endorse: for this is an area of metaphysics that is sufficiently self-contained and narrow in focus that it provides a manageable case study, while at the same time raising interesting and deep issues about the nature of space. Ultimately we defend a new, functionalist, analysis of holes, which, unlike its rivals, neither misidentifies nor renders us implausibly eliminativist about holes under various different metaphysical suppositions about the nature of space. In the process, we set out the complex relations between the intension of “hole,” and its extension at various worlds under different suppositions about the nature of space. In explicating these relations our account exemplifies what we take to be the core methodology in a priori metaphysics.