Spacetime, Ontology, and Structural Realism

Abstract

This essay explores the possibility of constructing a structural realist interpretation of spacetime theories that can resolve the ontological debate between substantivalists and relationists. Drawing on various structuralist approaches in the philosophy of mathematics, as well as on the theoretical complexities of general relativity, our investigation will reveal that a structuralist approach can be beneficial to the spacetime theorist as a means of deflating some of the ontological disputes regarding similarly structured spacetimes.

Notes

[1] An objection to grouping the similarly structured substantival and relationist theories might draw on Maudlin (1993), whose classificational scheme attempts to formulate both relationist and substantivalist versions of specific spacetime structures, e.g. Newtonian substantivalism, Newtonian relationism, etc. For many (R2) relationist theories, however, both the structures advocated, and the physical implications of those structures (such as a lone rotating body), seem identical to those espoused by the substantivalists. A case in point is Teller’s (1991) ‘liberalized’ relationism, which not only employs the same inertial structure as the substantivalist (381), but also may only require the existence of one object or event (396) to ground these structures. The vacuum solutions to GTR need not be seen as violating this ‘one object/event’ rule, moreover. Following Earman and Norton (1987, 519), one can identify the gravity waves in these models as material objects/events (since they can in principle be converted to other forms of energy); and, for the solutions without gravity waves, one could either accept Harré’s suggestion that they have ‘no reasonable physical interpretations’ (1986, 131), or simply side with Einstein’s hypothesis that the g field is a physical field.

[2] In the succeeding sections (Sections 3.2 and 4), the reason for not advocating the stronger claim, ‘all that exists is structure’, will become apparent.

[3] This is a fairly informal presentation of the identity of structuralist theories. Formulating an identity criterion for SR is a work in progress, as noted in Da Costa and French (2003, 122), but one could utilize their method of (partial) isomorphisms among the sub‐structures of models (48–52). Shapiro (1997, 91–93) also describes several means of capturing ‘sameness of (mathematical) structure’ across different systems through the use of a full, or partial, isomorphism of the objects and relations of the compared structures.

[4] Some of the different mathematical formulations of spacetime theories can be regarded in this manner, i.e. they may employ the same structures, but simply disagree on which structure is more fundamental. For instance, in the twistor formalism of Penrose, conformal structure is basic, with other structures as derivative, whereas the standard tensor formalism would consider conformal structure as derivative of the manifold and metric structures. The necessity of the mathematical structure to the function of the theory is the key point, here, regardless of its primary or derived status. This point is reminiscent of Quine’s famous critique of Carnap. Quine argued that the conventional element in Carnap’s treatment of geometric truth arose, not in determining the truths of geometry (since ‘the truths were there’), but in selecting from that interrelated set of pre‐existing truths which ones would serve as the fundamental Euclidean axioms, and which ones would serve as the derived results (Quine 1966, 108–109). In section 2.3, we will return to this topic, with the competing mathematical formulations of a spacetime theory (such as GTR) subsumed under the more general problem of underdeterminism, such that there are many different theoretical combinations of ‘geometry coordinated to physical processes’ (T = G + P) that save the phenomena.

[5] One of Hoefer’s reasons for singling out the g‐field is that a metric field without a global topology is possible ‘for at least small patches of space‐time’, although a manifold M without a metric cannot supply even a portion of spacetime (Hoefer 1998, 24). But, once again, the overall structure of GTR (i.e. for more than just small patches) mandates both M and g (in the standard formalism).

[6] The view of structure advocated in Brading and Landry (forthcoming), called ‘minimal structuralism’, would seem to be in accord with the ‘minimal nominalist’ structuralist position advocated in this essay. ‘What we call minimal structuralism is committed only to the claim that the kinds of objects that a theory talks about are presented through the shared structure of its theoretical models and that the theory applies to the phenomena just in case the theoretical models and the data models share the same kind of structure. No ontological commitment—nothing about the nature, individuality or modality of particular objects—is entailed’ (Brading and Landry, forthcoming, 20). If one interprets ‘objects’ as including substantival space, bodies, fields, etc., then this characterization of structure could apply to the minimal nominalist approach to spacetime structure described above.

[7] It should be noted that the problem of incongruent counterparts (Kant’s ‘handed’ objects) could stand as a counter‐argument to the claims of the causally inert status of spacetime. On the other hand, one could always invoke a maneuver similar to that advocated in Sklar (1985, 234–48), which accepts an ‘intrinsic’ property of handedness (i.e. a primitive property defined via continuous rigid motions) to avoid the commitment to substantival space.

[8] See Teller (1991), Sklar (1990), Bricker (1990), and Azzouni (2004, 196–212), for similar arguments about the causal irrelevance of spacetime structures for explaining accelerated motions and effects. An important early critique is Einstein (1923, 112–13), which labels absolute space as ‘a fictitious cause’ since rotation with respect to absolute space is not ‘an observable fact of experience’. Furthermore, with respect to the interrelationship between the metric and matter fields in GTR, (i) this relationship is not normally presented as ‘causal’, (ii) nor does it explain the non‐inertial forces associated with accelerated motion (rather, the interrelationship is relevant to explaining the metric curvature).

[9] See Barbour (1982). The attempt to explain non‐inertial force effects as due to the relative motion of matter through the Higgs field (in contemporary quantum field theories) would thus constitute an exception to this line of reasoning, for a direct causal explanation is intended. However, we are dealing with the classic GTR models in this essay.

[10] Earman’s text quoted above erroneously contains an extra 1/dt ² term in the left‐hand side of the equation for the covariant derivative in coordinate form, as verified through personal correspondence with Earman.

[11] Since the minimal nominalists reject Platonism, they do not claim, of course, that mathematical entities and their relationship enter into spacetime theories. But, spacetime structures are not just the physical relationships between the physical objects, either, since (as argued above) causation is not a spacetime relationship (whereas causation is most definitely a physical relationship among physical objects). Spacetime structures, like all other mathematical structures, are unique in this regard: systems can exemplify the structures, but the structures are not identical to the systems, nor (for the nominalists) can they exist in the absence of systems.

[12] A viewpoint remarkably similar to the SR thesis developed above can be found in the work of Stein (1977) and DiSalle (1995). Not only do they reject the standard substantival versus relational classification, but they place special emphasis on denying a causal role to spacetime: ‘A spacetime theory does not causally explain phenomena of motion, but uses them to construct physical definitions of basic geometrical structures by coordinating them with dynamical laws’ (DiSalle 1995, 317). The underdetermination problem poses a threat to this conception, nevertheless, since DiSalle leaves open the potential for a viable coordination of the physical and geometrical aspects of theories alternative to general relativity: e.g. the Brans–Dicke theory, but only if a physical process can be located for the needed coordination with the long‐range scalar field (geometrical component), which provides a (relational) Machian conception of mass (DiSalle 1995, 334–35).

[13] I would like to thank Oliver Pooley for his invaluable help in the research for this essay, as well as two anonymous referees for International Studies in the Philosophy of Science for their instructive comments on an earlier draft.