Against Pointillisme about Geometry

Abstract

This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts. More specifically, this paper argues against pointillisme as regards the structure of space and-or spacetime itself. To keep matters simple, I will conduct the argument almost entirely in the context of “Newtonian” ideas about space and time, and in particular in the context of the classical mechanics of point-particles, i.e. extensionless particles moving in a void. This paper argues that the geometrical structure of space, and-or the chrono-geometrical structure of spacetime, involves extrinsic properties, typically spatially extrinsic properties. Agreed, philosophers attracted by pointillisme typically concede this point. But how exactly to state the qualification remains unresolved. Here, the main question is whether properties represented by vectors, tensors, connections etc. can be intrinsic to a point. I address this question in three stages. In the main stage, I focus on Bricker's (1993) discussion, which is an unusually thorough attempt to relate vectors and tensors in modern geometry to the metaphysics of properties. But Bricker exemplifies a tendency I reject: the tendency to reconcile pointillisme with the fact that vectorial etc. properties seem extrinsic to points and point-sized objects, by proposing some heterodox construal of the properties in question. Thus Bricker proposes that we should re-found geometry in terms of non-standard analysis. I reply that once the spell of pointillisme is broken, such proposed heterodox foundations of geometry are unmotivated. Mechanics provides another striking illustration of the lure of pointillisme, and the tendency to reconcile it to vectorial properties by reconstruing physical quantities. Thus Tooley and others argue that for the sake of securing that a particle's instantaneous velocity is intrinsic to it at a time, we should not construe velocity in the orthodox way as a limit of average velocities - but instead re-construe it along lines they propose. Again, my view is that there is no need for such heterodoxy: instead, we can and should reject pointillisme.