 Nonetheless, many eminent physicists seem convinced that general relativity stands in need of such a construction, and have exerted extraordinary efforts in trying to devise one. This fact raises several philosophical problems. Though physicists sometimes offer as strong motivation the possibility of gaining the ability to analyze singular phenomena locally in a mathematically well-defined manner, they more often speak in terms that strongly suggest they suffer a metaphysical itch that can be scratched only by the sharp point of a localizable, spatiotemporal entity serving as the locus of their theorizing. Even were such a construction forthcoming, however, what sort of physical and theoretical status could accrue to these missing points? They would not be idealizations of a physical system in any ordinary sense of the term, since they would not represent a simplified model of a system formed by ignoring various of its physical features, as, for example, one may idealize the modeling of a fluid by ignoring its viscosity. Neither would they seem necessarily to be only convenient mathematical fictions, as, for example, are the physically impossible dynamical evolutions of a system one integrates over in the variational derivation of the Euler-Lagrange equations. To the contrary, as we have remarked, many physicists and philosophers seem eager to find such a construction for the purpose of bestowing substantive and clear ontic status on singular structure. What sorts of theoretical entities, then, could they be, and how could they serve in physical theory? 