“The Relations between Things”
versus “The Things between Relations”:
The Deeper Meaning of the
I first discussed the “hole argument” in 1980, when presenting my interpretation of the reason for Einstein's rejection, from mid-1913 to mid1915, of generally-covariant gravitational field equations.2 Yet I have only recently come to recognize the full import of the argument, thanks to stimulation provided by the work of Rynasiewiez (1994, 1996) and Liu (1997). In effect, if not intent, their work leads to an extension of the subject of the hole argument from the spatiotemporal relations between the points of a differentiable manifold to arbitrary relations between the elements of a set, and I am grateful to both for inspiring this extension.3 This paper considers several generalized and abstracted versions4 of the original hole argument.
The next section briefly recalls the original hole argument, which concerns a general-relativistic space-time, and its generalization to fibered manifolds. Section 3 discusses the process of abstraction by deletion from the differentiability and continuity properties of such manifolds, and the resulting possibility of a set-theoretical version of the hole argument within the context of G-spaces.5 Section 4 discusses the special case of a set with structure, with which the rest of the paper is concerned.6 After a digression in section 5 on Marx's use of reflexive definitions, section 6 returns to sets with structure and the problem of individuation of the elements of the set; section 7 then develops the analogue of the hole argument for the case in which the elements of the set are individuated by the relational structure.
Most treatments of the usual bole argument involve explicit reference to coordinate systems on the manifold, and this has been the source of much confusion and misunderstanding.7 Coordinates are a way of introducing names for the points of a manifold,8 so abstraction from