Online research in philosophy

I reject Norton and Earman's hole argument that spacetime substantivalism is incompatible with determinism. I reconcile these both technically and philosophically. There is a technical definition of determinism that is not violated by pairs of models of the kind used in the hole argument. And technicalities aside, the basic idea of determinism is not violated if we claim that at most one of the two models represents a possible world. This claim can be justified either by metrical essentialism (advocated by Maudlin), or by denying transworld identity for points: I prefer the latter.

This paper offers an alternative view of spacetime different from both substantivalism and relationism. Using basic ideas underlying the fiber bundle formulation of field theories, it illustrates the function of spacetime in individuating local fields. As the system of numerical identities for entities that we can individually refer to, spacetime is an intrinsic, indispensable, and inalienable structure of the physical world with distinct entities.

The debate between substantivalists and relationists about spacetime was given a new lease of life approximately twenty years ago, when John Earman and John Norton published an argument for the conclusion that, in the light of general relativity, substantivalism is untenable. Responses to Earman and Norton’s argument generated a proliferation of ‘substantivalisms’, and a debate between them that was, to the ears of at least some, distinctively metaphysical in character.

In this paper Modern Essentialism is used to solve a problem of individuation of spacetime points in General Relativity that has been raised by a New Leibnizian Argument against spacetime substantivalism, elaborated by Earman and Norton. An earlier essentialistic solution, proposed by Maudlin, is criticized as being against both the spirit of metrical essentialism and the fundamental principles of General Relativity. I argue for a modified essentialistic account of spacetime points that avoids those obstacles.

Einstein algebras have been suggested (Earman 1989) and rejected (Rynasiewicz 1992) as a way to avoid the hole argument against spacetime substantivalism. In this article, I debate their merits and faults. In particular, I suggest that a gauge-invariant interpretation of Einstein algebras that avoids the hole argument can be associated with one approach to quantizing gravity, and, for this reason, is at least as well motivated as sophisticated substantivalist and relationalist interpretations of the standard tensor formalism.

Einstein algebras have been suggested (Earman 1989) and rejected (Rynasiewicz 1992) as a way to avoid the hole argument against spacetime substantivalism. In this article, I debate their merits and faults. In particular, I suggest that a gauge‐invariant interpretation of Einstein algebras that avoids the hole argument can be associated with one approach to quantizing gravity, and, for this reason, is at least as well motivated as sophisticated substantivalist and relationalist interpretations of the standard tensor formalism.

In their modern classic ``What Price Substantivalism? The Hole Story'' Earman and Norton argued that substantivalism about spacetime points implies that general relativity is indeterministic and, for that reason, must be rejected as a candidate ontology for the theory. More recently, Earman has cottoned on to a related argument (in fact, related to a \emph{response} to the hole argument) that arises in the context of canonical general relativity, according to which the enforcing of determinism along standard lines---using the machinery of gauge theory---leads to a `frozen universe' picture (grounded in an absence of changes in values of general relativity's observables). \emph{Prima facie} this would seem to land the anti-substantivalist in waters at least as deep as those that Earman and Norton argued troubled substantivalism. In this paper I introduce the argument in what I think are clearer terms than Earman's, and assess his treatment of the problem. For the most part I agree with Earman about the nature of the problem, but I find aspects of his discussion wanting, especially as regards his proposed ontology. I argue that ontological sense can be made of the changelessness if a structuralist stance is adopted with respect to a natural class of observables.

Algebraic substantivalism, as an interpretation of general relativity formulated in the Einstein algebra formalism, avoids the hole argument against manifold substantivalism. In this essay, I argue that this claim is well-founded. I first identify the hole argument as an argument against a specific form of semantic realism with respect to spacetime. I then consider algebraic substantivalism as an alternative form of semantic realism. In between, I justify this alternative form by reviewing the Einstein algebra formalism and indicating the extent to which it is expressively equivalent to the standard formalism of tensor analysis on differential manifolds.

Spacetime substantivalism leads to a radical form of indeterminism within a very broad class of spacetime theories which include our best spacetime theory, general relativity. Extending an argument from Einstein, we show that spacetime substantivalists are committed to very many more distinct physical states than these theories' equations can determine, even with the most extensive boundary conditions.

I give an informal outline of the hole argument which shows that spacetime substantivalism leads to an undesirable indeterminism in a broad class of spacetime theories. This form of the argument depends on the selection of differentiable manifolds within a spacetime theory as representing spacetime. I consider the conditions under which the argument can be extended to address versions of spacetime substantivalism which select these differentiable manifolds plus some further structure to represent spacetime. Finally, I respond to the criticisms of Tim Maudlin and Jeremy Butterfield.