An obvious initial dilemma that faces the prospective Structural Realist (SR) theorist is the ontological status of the spacetime structures themselves. Do they exist as a sort of Platonic universal, independent of all physical objects or events in spacetime, or are they dependent on matter/events for their very existence or instantiation? This problem arises for the mathematician in an analogous fashion, since they also need to explicate the origins of mathematical structures (e.g., set theory, arithmetic). Consequently, a critic of the SR spacetime project might seem justified in regarding this dispute in the philosophy of mathematics as a re-emergence of the traditional substantivalist versus relationist problem, for the foundation of all mathematical structures, including the geometric spacetime structures, is once again either independent of, or dependent on, the physical.
Nevertheless, a survey of the various positions in the mathematical ontology dispute may work to the advantage of the SR spacetime theory, especially when the relevant mathematical and spacetime options are paired together according to their analogous role within the wider ontology debate. First, mathematical structuralism can be classified according to whether the structures are regarded as independent or dependent on their instantiation in systems (ante rem and in re structuralism, respectively), where a “system” is loosely defined as a collection of “objects” and their interrelationships. Ante rem structuralism, as favored by Resnik and Shapiro, is thus closely akin to the traditional “absolute” conception of spacetime, for a structure is held to “exist independent of any systems that exemplify it” (Shapiro). Yet, since “system” (and “object”) must be given a broad reading, without any ontological assumptions associated with the basis of the proposed structure, it would seem that substantivalism would not fit ante rem structuralism, as well. The structure of substantivalism is a structure in a substance, namely, a substance called “spacetime”, such that this unique substance “exemplifies” the structure (whereas ante rem structure exists in the Platonic sense as apart from any and all systems that exemplify it). The substantivalist might try to avoid this implication by declaring that their spacetime structures are actually closer in spirit to a pure absolutism, without need of any underlying entity (substance) to house the structures (hence, “substantivalism” is simply an unfortunate label). While this tactic may be more plausible for interpreting Newtonian spacetimes, it is not very convincing in the context of GTR, especially for the sophisticated substantivalist theories. Given the reciprocal relationship between the metric and matter fields, it becomes quite mysterious how an non-substantival, “absolute” structure, g, can be effected by, and effect in turn, the matter field, T. For the ante rem structuralist, mathematical structures do not enter into these sorts of quasi-causal interrelationships with physical things; rather, things “exemplify” structures (see also endnote 8). Accordingly, one of the initial advantages of examining spacetime structures from within a mathematical ontology context is that it drives a much needed wedge between an absolutism about quantitative structure and the metaphysics of substantivalism, although the two are typically, and mistakenly, treated as identical.
In fact, as judged against the backdrop of the ontology debate in the philosophy of mathematics, the mathematical structures contained in all spacetime theories would seem to fall within a nominalist classification. If, as the nominalists insist, mathematical structures are grounded on the prior existence of some sort of “entity”, then both the substantivalists and relationists would appear to sanction mathematical nominalism (with in re structuralism included among nominalist theories, as argued below): whether that entity is conceived as a unique non-material substance (substantivalism), physical field (metric-field relationism), or actual physical objects/events (relationism, of either the modal (R2) or strict (R1) type), a nominalist reading of mathematical structure is upheld. This outcome may seem surprising, but given the fact that traditional substantivalist and relationist theories have always based spatiotemporal structure on a pre-existing or co-existing ontology – either on a substance (substantivalism) or physical bodies (relationism) – a nominalist reading of spacetime structure has been implicitly sanctioned by both theories. Consequently, if both substantivalism and relationism fall under the same nominalist category in the philosophy of mathematics, then the deeper mathematical Platonist/nominalist issue does not give rise to a corresponding lower-level substantival/relational dichotomy as regards the basis of those spacetime structures (e.g., with substantivalism favoring a Platonic realism about mathematical structures, and relationism siding with a nominalist anti-realism). This verdict could change, of course, if a non-substantival “absolute” conception of spacetime becomes popular in GTR; but this seems unlikely, as argued above.
As there are a number nominalist reconstructions of mathematics, a closer examination of their content reveals that the different versions can be paired to different substantival and relational theories. For instance, a reductive (R1) spacetime relationism can be linked to some strict nominalist reconstructions of mathematics, as in Field’s attempt to treat mathematical objects and structures as entirely dispensable, or “fictional”. Field posits a continuum of spacetime points, conceived physically in the manner of a manifold substantivalist, in his effort to rewrite Newtonian gravitation theory along mathematically anti-realist lines. Modal (R2) relationists, like Teller, would not constitute the spacetime analogue of Field’s program, accordingly, since this form of relationism sanctions modal spacetime structures that can transcend the structures exhibited by the actually existing physical objects: e.g., the affine structure ∇ instantiated by a lone rotating body. Whereas Field requires an infinity of physical spacetime points (isomorphic to ℜ4 in order to capture the full content of the mathematician’s real numbers, the (R2) relationist can allow modal structures to serve this function, thus releasing the ontology of such extravagant demands. More importantly, if Field’s nominalist program is committed to manifold substantivalism, M then it is susceptible to the hole argument. All spacetime theories that utilize the metric in the identity of spacetime points, such as sophisticated substantivalism and metric-field relationism, would thereby incorporate a divergent set of structural scenarios (since they are not susceptible to the hole argument). Therefore, Field’s nominalist mathematics entails a spacetime structure that comprises a different SR theory than sophisticated substantivalism and metric-field relationism.
The mathematical equivalent of both sophisticated substantivalism and (R2) relationism is, rather, any of the less stringent nominalist theories that rejects Field’s strict nominalism, as in, e.g., Chihara in re structuralism. Much like the modal (R2) relationist theories surveyed above, the “minimal nominalists” do not allow structures to exist independently of the systems they exemplify, yet they do not believe that these structures can be dismissed as mere fictions, either. Contra Field, the minimal mathematical nominalists deny a purely instrumentalist construal of mathematical structures (while simultaneously rejecting a Platonic absolutism): they all insist, for instance, that mathematical structures cannot be excised from scientific theories without loss of valuable physical content. In re structuralism, moreover, employs “possible structures” as a means of avoiding a commitment to an infinite background ontology, a feature that helps to explain its common nominalist classification. The minimal nominalist theories often differ on how the mathematical structures are constructed from their basic ontology, as well as how to construe the truth-values of mathematical statements about mathematical structures. But these differences, such as in the kind of modality sanctioned, etc., can be quite subtle, and do not lend themselves to any drastic distinctions in type or basic intent. Hence, any attempt to rekindle the substantival/relational distinction among the competing minimal nominalist theories would seem implausible. A firm reliance on some form of modality and a non-instrumental construal of mathematical structures is the common, and crucial, similarity among these theories; and it is these aspects that are most important for the spacetime structuralist, whether of the sophisticated substantivalist or relationist variety.
These last observations are not meant to downplay the importance of the ongoing research in the philosophy of mathematics on the origin of structures, for it is always possible that substantial problems will arise for some of the minimal nominalist theories, thus eliminating them from contention. From the SR standpoint, in fact, the philosophy of mathematics would likely be considered a more proper arena for assessing the structures employed by spacetime theories, at least as opposed to the apparently unverifiable metaphysics of “substance versus body”. Not only has the traditional spacetime dichotomy failed to explain how these mathematical structures arise from their basic ontology, but, as we have seen, the underlying structures advocated by the sophisticated versions of both substantivalism and relationism are identical when judged within the wider philosophy of mathematics framework. Whether that foundational entity is called a substance or a physical object is irrelevant, and probably a conventional stipulation, since the real work, as judged from the mathematical perspective, concerns how the structures are constructed from the underlying entity—and the competing claims of substance or physical existent do not effect this mathematical construction. In essence, the only apparent difference between the sophisticated substantivalists and (R2) relationists are where those mathematical structures are located: either internal to the substance or field (for the substantivalists and metric-field relationists, respectively), or external to bodies/events (for non-field formulations of (R2) relationism, such as Teller’s). Needless to say, this internal/external distinction does not provide any information on how the mathematical structures are built-up; rather, it reveals the pervasive influence of the age-old substance/property dichotomy within the philosophy of science community, an unfortunate legacy that the SR theorist regards as hindering the advancement of the debate on spacetime theories.
Finally, since the competing minimal nominalist constructions are not being judged solely from a mathematical perspective, but from a scientific and empirical standpoint as well, a few words are in order on the relevance of empirical evidence in assessing spacetime structure. This issue will be addressed further in the remaining sections, but, in brief, it is unlikely that any physical evidence could provide a strong confirmation of any one of the competing nominalist theories. In effect, these nominalist constructions are only being utilized to explain the origins of the spacetime structures, such as M or g, that do appear in our best physical theories, with the important qualification that these nominalist constructions do not commit the physical theory to any problematic or meaningless physical outcomes (e.g., Field’s nominalism and the hole argument, as noted above). As for the sophisticated brands of both substantivalism and relationism, all of the minimal nominalist construction are apparently identical as regards their implications for possible spacetime scenarios and meaningful physical states. Unless other reasons are brought forward, the choice among the competing minimal nominalist constructions could thus be viewed as conventional, since it is difficult to conceive how empirical evidence could reach the deep mathematical levels where the differences in nominalist constructions of spacetime structures come into play.