As developed in Chap.  5, the attempts to fcapture the ontology of space and spacetime using the conceptual apparatus of substantivalism and relationism—that space is, respectively, either an independently existing entity or a relation among material substances—have proved to be quite problematic since the sophisticated versions of both substantivalism and relationism seem to be identical in the context of general relativity (GR), our best theory on the large scale structure of space. But, whether space is a substance or a relation is just one of the many conceptual distinctions that have entered the modern debate on the nature of space. In addition to substantivalism and relationism, one also finds references to realism versus anti-realism, platonism versus nominalism, and background independence versus background dependence. In this chapter, we will begin the examination of these new dichotomies, their impact on the traditional substantival/relational distinction, and their relationship to one of the least discussed, but quite central, components of the spatial ontology debate, namely, how the philosophy of mathematics factors into the philosophy of space and time.