Abstract
This essay examines the underdetermination problem that plagues structuralist approaches to spacetime theories, with special emphasis placed on the epistemic brands of structuralism, whether of the scientific realist variety or not. Recent non-realist structuralist accounts, by Friedman and van Fraassen, have touted the fact that different structures can accommodate the same evidence as a virtue vis-à-vis their realist counterparts; but, as will be argued, these claims gain little traction against a properly constructed liberal version of epistemic structural realism. Overall, a broad construal of spacetime theories along epistemic structural realist lines will be defended which draws upon both Friedman’s earlier work and the convergence of approximate structure over theory change, but which also challenges various claims of the ontic structural realists.
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Notes
Saatsi (2008) has argued that a scientific realism requires more than mere approximate continuity, but should also explain the success of the earlier theory from the succeeding theory’s vantage point. Yet, while quite correct as a final goal or heuristic of scientific realism, explanations of this sort would seem to require access to the ontology underlying these theories (or, at least, the succeeding theory’s ontology), and the liberal form of ESR endorsed in this essay denies that this requirement is necessary to establish the success of ESR relative to its non-realist rivals—the reason being that the non-realist alternatives have no grounds for claiming a future directed convergence of approximate continuity over the course of science (see, sections 3 and 4).
This is different from “metaphysical underdetermination” as mentioned in Ladyman (1998), which is generated by alternative realist interpretations of a single theory. Rather, entity underdetermination is the thesis that different theories, with different ontologies, will likely replace our currently successful theories. In essence, entity underdetermination is the ontological consequence of the pessimistic meta-induction.
Pooley (2006, 88) raises the formalism underdetermination issue, along with a number of metaphysical underdetermination objections (see, footnote 3). However, if the OSR theorist falls back upon their world-structure ontology, it is not clear that these metaphysical underdetermination cases can gain much traction against OSR. For instance, Pooley raises the specter that different interpretations of the measurement problem (e.g., de Broglie-Bohm versus other interpretations) lead to the underdetermination of the exact nature of the realist ontology, despite the use of the same mathematical formalism by these different interpretations (of the collapse of the wave function). Yet, since the same mathematical formalism is utilized, both collapse interpretations uphold the same world-structure ontology, with the difference lying in the mere details of how that ontology functions in the collapse case—and this is a much less troubling underdetermination than the entity or formalism underdetermination problems raised above, which do call into question the world-structure ontology itself.
Bain (2009) strives to address the formalism underdetermination in Pooley (2006) (dubbed, “Jones underdetermination” by both authors). Bain’s very promising analysis employs a category-theoretic approach in order to model a theory’s dynamical structure (say, solutions of the field equations in GR) via the symplectic manifold that encodes the phase space of dynamically possible states. This conception may blunt the worries associated with formalism underdetermination, but it depends on a category theory framework, which many find problematic (Bain 2009, 17). Also, the many different constructions of GR occupy different symplectic manifolds (e.g., twistor models are different from tensor models with/without boundary conditions; 19), and so neither the “entity” nor the “metaphysical” underdetermination problems, as we have called them, have been diminished (see footnote 3). The OSR theorist can, of course, always claim that future empirical data might decide between these competing models, given their different dynamical structure—but the liberal brand of ESR theorist will interpret this same argument as support for their more cautious brand of SR (since the evidence may forever fail to distinguish these alternatives).
This understanding of ESR, in the context of spacetime theories, should be contrasted with the interpretation offered by Esfeld and Lam. Since they interpret ESR as a commitment to intrinsic properties, they conclude: "applied to the framework of the standard tensor representation of space-time, epistemic structural realism implies that the identity of the space-time points is constituted by their fundamental intrinsic properties, independently of the space-time structure—that is, independently of the metric" (35). In other words, ESR becomes Hartry Field’s (1980) mathematical structure-reducing version of manifold substantivalism. But this does not follow at all, since ESR (on our liberal variety) remains agnostic as regards the underlying ontology, and thus the manifold points may have an ontological status that is either intrinsic, or, like OSR, is either derived from, or "on a par" with, the metric. In short, Esfeld and Lam's reading of ESR foists a commitment to a particular mathematical ontology (of points) when its only commitment is to some underlying ontology.
There is, of course, a great deal of subjective license involved in attempting to translate early twentieth century neo-Kantian theories into the modern OSR/ESR dichotomy, but, I would argue, their approach clearly falls on the epistemological side of that dichotomy, and not the ontological. Given the predominant emphasis on conceptual categories and their like in shaping our experience of the world, drawing purely ontological lessons, apart from these epistemic components, seems quite problematic. Friedman’s discussion of the goals of Cassirer’s Substance and Function makes this point clear: “[I]n accordance with the ‘critical’ theory of knowledge, . . . convergence, on this view, does not take place towards a mind- or theory-independent ‘reality’ of ultimate substantial ‘things’. . . . ‘Reality’, on this view, is simply the purely ideal limit or endpoint towards which the sequence of [theoretical] structures is mathematically converging—or, to put it another way, it is simply the series itself, taken as a whole” (Friedman 2005, 75).
A number of objections can be raised at this point. First, the neo-Kantian might claim that a mathematical invariant can always be found that links the totality of empirical data, thus rejecting the possibility just noted (although, as a counter-reply, this would be difficult to prove). Second, the anti-realist may insist that empirical adequacy dictates that all empirical data must be subsumed by more general theories (if not, empirical adequacy would be sacrificed). However, as argued previously, this demand would render anti-realism practically indistinguishable from ESR, since it basically admits that scientific theories must converge.
Can Friedman’s neo-Kantian insist, as does the realist, that the “world” provides the needed constraints to rule out the egalitarian option? While the “world” obviously plays a major role in scientific theorizing for Friedman, to claim that it explains convergence would clash with his insistence that his view is consistent with anti-realism. As explained above, van Fraassen’s anti-realism holds that convergence is contingent; so, if Friedman really accepts that his view is consistent with anti-realism, then he would need to add that convergence may also be a mere accident. I believe that Friedman’s past history of supporting realism, along with his demand for convergence (explored above), makes it unlikely that he would regard convergence as merely a contingent accident.
A famous counter-argument against structural realism, originally introduced by M. Newman, should be briefly addressed at this point, although we will only explore a version of the argument adapted for the semantic view of theories: if a mathematical structure represents the world by standing in an isomorphic relationship with the world’s structure, then one can undermine the uniqueness of this representation by introducing another world domain of the same cardinality, and carving out a structure that is also isomorphic to the mathematical structure in this new domain. As argued by French and Saatsi (2004), however, this problem can be overcome by including interpretations of the mathematical structure’s theoretical variables, so that they refer to a particular group of properties and relations. This reply to the Newman problem thus parallels Poincaré’s insight, i.e., that structures are always linked to the world via coordinating principles—and hence our G + P approach to structure naturally includes French’s and Saatsi’s defense. Friedman has himself contributed to the Newman argument, see, Demopolous and Friedman (1985).
That is, Friedman (1983) appears to endorse a realist commitment to spacetime structures, such as \( \langle M,D\rangle \), but he (apparently) never sanctions the existence of a distinct physical entity or substance that corresponds to \( \langle M,D\rangle \). For the advocates of ESR—and possibly Friedman (1983)—the substantival/relational dispute in ontology (i.e., whether spacetime is a substance or relation among substances) is a separate issue from the reality (=objectivity) of the spacetime structures themselves (in conjunction with the coordinating principles). In other words, both the substantivalist and relationist hypotheses are compatible with the minimalist ontological implications of our liberal brand of ESR (i.e., the convergence of spacetime theories and the manifest constraints/nonuniqueness of spacetime constructions), although both will obviously differ on the deeper ontological foundations.
Friedman (1983, 297–299). That is, the force \( {F^i} \) now explains why the particles that comprise an expanding or shrinking body, due to \( h\prime \), deviate from straight-line trajectories as the body moves, apparently inertially, throughout various regions of space. Here, the \( \bar{\Gamma }_{{jk}}^i \) are the flat components of \( \bar{D} \), and the \( \Gamma_{{jk}}^i \) are the non-flat components of \( D \), such that, \( {F^i} = (\Gamma_{{jk}}^i - \bar{\Gamma }_{{jk}}^i)({{{d{x_j}}} \left/ {{dt}} \right.})({{{d{x_k}}} \left/ {{dt}} \right.}) \).
An anti-realist or OSR realist may object, at this point, that the multiple routes form of ESR is tantamount to an equivalence of structure at the empirical level only, which is simply insufficient to qualify as a realist theory. This objection, however, fails to take into account the fact that our version of ESR is, indeed, a very liberal brand of realism (see footnote 1 and sections 2 and 3): given the two requirements of the type of ESR theory advocated in this essay—nonuniqueness, and that each nonunique G + P combination must be an invariant feature across all of the other nonunique combinations, G′ + P′—the “realism” in this theory is manifest in two important ways, (1) via predictions on future theoretical constructions (which runs counter to anti-realism, or is at least quite difficult for the anti-realist to explain), namely, that theories will converge; and (2) that there will exist constraints on the theoretical constructions (thus eliminating the egalitarian option along with its relativist implications for epistemology and ontology). Finally, it should be noted that a famous species of underdetermination concerning spacetime theories is consistent with this Poincaré-inspired exegesis, as well as with the brand of ESR defended in this essay: namely, the topological underdetermination manifest by observationally indistinguishable spacetime theories, as explored by Malament (1977) and Glymour (1977).
An earlier version of this paper was presented at the Philosophy of Science Association 2006 conference, Vancouver, B.C., Canada. I would like to thank the participants for their helpful discussions, especially Oliver Pooley, Carl Hoefer, Steven Weinstein, Vincent Lam, Nick Huggett, and Bas van Fraassen, and two anonymous referees from the European Journal for Philosophy of Science.
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Slowik, E. On structuralism’s multiple paths through spacetime theories. Euro Jnl Phil Sci 2, 45–66 (2012). https://doi.org/10.1007/s13194-011-0020-z
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DOI: https://doi.org/10.1007/s13194-011-0020-z