"On the Viability of Galilean Relationalism," British Journal for the Philosophy of Science, 68, pp. 1183–1204.
I explore the viability of a Galilean relational theory of spacetime---a theory that includes among its stock of basic relations a three-place collinearity relation. Two formal results are established. First, I prove the existence of a class of dynamically possible models of Newtonian mechanics in which collinearity is uninstantiated. Second, I prove that the dynamical properties of Newtonian systems fail to supervene on their Galilean relations. On the basis of these two results, I argue that Galilean relational spacetime is too weak of a structure to support a relational interpretation of classical mechanics.
I present an argument against a relational theory of spacetime that regards spacetime as a "structural quality of the field." The argument takes the form of a trilemma. To make the argument, I consider a general relativistic world in which there exist just two fields, an electromagnetic field and a gravitational field. Then there are three options: either spacetime is a structural quality of each field separately, both fields together, or one field but not the other. I argue that the first option founders on a problem of geometric coordination and that the second and third options collapse into substantivalism. In particular, on the third option it becomes clear that the relationalist's path to Leibniz equivalence is no simpler or more straightforward than the substantivalist's.
"Bundles, Metrics, and Indiscernibles"
I argue for the incompatibility of three claims. First, that objects are bundles of universals. Second, that there exist indiscernible objects. And third, that the world is a geometrically structured object, having, in particular, metric spatial structure. Holding any two fixed, one can derive the negation of the third. This furnishes us with a novel argument against the bundle theory of substance: 2 and 3 are true; therefore, 1 is false.
I introduce a new relational theory of spacetime which I call quantitative Galilean relationalism. The theory assumes, as a primitive, a three-place quantitative relation \alpha(p,q,r) which measures, intuitively, the angle subtended by points p, q, and r along the worldline of a point-sized particle. Among other things, I prove that \alpha(p,q,r) can be used to define the curvature of a worldline. The result is interesting because it entails the existence of a relationally kosher notion of absolute acceleration, which suggests the possibility of a relational foundation for Newton's laws. However, the victory is short-lived: I go on to prove that even though Newton's laws are supported, they lack a well-posed initial value problem. Consequences for the substantival-relational debate are discussed.
All physical theories, from classical Newtonian mechanics to relativistic quantum field theory, entail propositions concerning the geometric structure of spacetime. In my dissertation, I take the structural commitments of our theories seriously and ask: how is such structure instantiated in the physical world? For example, consider the property 'being curved'. Such a property is perfectly well-defined in a mathematical sense—there is no question of what it means for a mathematical space to be curved. But what could it mean to say that the physical world is curved? Call this the problem of physical geometry.
The problem of physical geometry is a plea for foundations—a request for fundamental truth conditions for physical-geometric propositions. My chief claim is that only a substantival theory of spacetime—a theory according to which spacetime is an entity in its own right—can supply the necessary truth conditions.
The dissertation is broken up into three parts, with each part examining an alternative, anti-substantivalist account of the structure of spacetime. Part one examines relational theories of spacetime. Part two examines accounts that invoke natural laws in a central way. And part three examines accounts that invoke the notion of a field. In each case, I show that the accounts considered face serious difficulties and argue that only a substantival theory of spacetime can offer a solution.