Recently Carolyn Brighouse and Jeremy Butterfield have argued that David Lewis's counterpart theory makes it possible both to believe in the reality of spacetime points and to consider general relativity to be a deterministic theory, thus avoiding the ‘hole argument’ of John Earman and John Norton. Butterfield's argument relies on Lewis's own counterpart-theoretic analysis of determinism. In this paper, I argue that this analysis is inadequate. This leaves a gap in the Butterfield–Brighouse defence against the hole argument.

In this paper non-Hausdorff manifolds as potential basic objects of General Relativity are investigated. One can distinguish four stages of identifying an appropriate mathematical structure to describe physical systems: kinematic, dynamical, physical reasonability, and empirical. The thesis of this paper is that in the context of General Relativity, non-Hausdorff manifolds pass the first two stages, as they enable one to define the basic notions of differential geometry needed to pose the problem of the evolution-distribution of matter and are not in (...) conflict with the Einstein equations. With regard to the third stage, various potential conflicts with physical reasonability conditions are considered with a tentative conclusion that non-Hausdorff manifolds are more likely to pass this stage than is typically assumed. When dealing with some of these problems, the modal interpretation of non-Hausdorff manifolds is invoked, according to which they represent bundles of alternative possible spacetimes rather than single spacetimes. (shrink)

In this paper I claim that Earman and Norton 's hole argument against substantivalist interpretations of General Relativity assumes that the substantivalist must adopt a conception of determinism which I argue is unsatisfactory. Butterfield and others have responded to the hole argument by finding a conception of determinism open to the substantivalist that is not prone to the hole argument. But, unfortunately for the substantivalist, I argue this conception also turns out to be unsatisfactory. Accordingly, I search for a conception (...) of determinism that is both independently plausible and capable of blocking the hole argument. (shrink)