The Conditional Redundancy Thesis is modest: it claims that temporal and spatiotemporal ontology is unnecessary if change and relational structure discharge the empirical and explanatory roles attributed to time or spacetime. This section considers metric-field realism, sophisticated substantivalism, ontic structural realism, temporal realism, and explanatory necessity. The replies apply ONC without assuming that these realist positions fail.
5.1 Objection from Metric-Field Realism
A metric-field realist can argue that the metric is not merely a convenient representation of relations among material events. In general relativity, gravitational dynamics include degrees of freedom not fixed by the local distribution of matter. Vacuum solutions need not be geometrically trivial: non-zero Weyl curvature and propagating gravitational waves provide a serious candidate for physical content carried by the gravitational field itself (Wald, 1984). A realist may therefore contend that the metric contributes real dynamical structure even where no material clock or matter source is present. If that structure is irreducible, the paper’s appeal to change and relations among material events has not recovered the relevant role.
This is a substantive challenge, and the empirical audit does not answer it merely by observing that clocks register differential proper time. Nor does the fact that geometry can be represented relationally show that gravitational degrees of freedom are reducible to matter relations. The candidate residue is the independent dynamical content of the gravitational field, including vacuum and radiative degrees of freedom. A realist who establishes that such content is indispensable to prediction or explanation has met the kind of challenge ONC is designed to recognize.
The remaining issue is what the success establishes ontologically. It may support realism about the metric or gravitational field; it does not automatically show that a further temporal substance, or spacetime understood as an entity over and above the metric-field structure, must also be posited. To support that stronger conclusion, the realist would need to identify an additional contribution made by that further posit. Conversely, a recovery account that derives the relevant gravitational degrees of freedom and their dynamics from a more basic structure would strengthen the redundancy claim, but CST or relational language alone does not yet provide such a completed recovery. ONC therefore leaves metric-field realism open and makes the unresolved recovery question explicit.
5.2 Objection from Sophisticated Substantivalism
The hole argument targets a form of substantivalism that assigns primitive identities to manifold points independently of the physical fields, making diffeomorphically related models appear to represent distinct possibilities (Earman & Norton, 1987). Sophisticated substantivalists reject that haecceitistic commitment. They treat such models as physically equivalent, or they individuate spacetime points through the metric and other fields rather than through primitive identities of bare manifold points (Hoefer, 1996; Pooley, 2013). Spacetime is then a structured entity, not a featureless container.
This response avoids the standard hole-argument worry, but it leaves the present question open. ONC is compatible with sophisticated substantivalism if the structured spacetime so conceived contributes irreducible empirical or explanatory work. The proponent must specify that contribution beyond the metric relations and dynamics already represented. If the only work identified is the work already done by those fields and their lawful dependencies, the posit may remain representationally indispensable without being shown to be an additional ontological constituent in the sense ONC requires.
5.3 Objection from Ontic Structural Realism
Ontic structural realists argue that relational structure is ontologically basic and that objects are derivative or non-fundamental (Ladyman, 1998). They may therefore claim that temporal or spatiotemporal structure is fundamental because it belongs to the relational structure represented by our best physics.
OSR is compatible with ONC, but it sharpens the question of what counts as a residue. ONC does not deny that relational structure may be fundamental, and it does not infer that relations are merely representational because they are not objects. Rather, it asks whether the temporal or spatiotemporal structure posited by a particular OSR account is itself part of the irreducible structure, or whether the relevant ordering and geometric roles can be recovered from a more basic non-temporal structure. That is a substantive metaphysical and physical question, not a consequence of the label “structural realism.”
If OSR treats temporal relations as fundamental, its proponent must explain why that structure cannot be recovered from a more basic organisation. In CST, for example, causal order and cardinality are proposed as basic while continuum temporal and metric notions are recovered only in appropriate regimes; whether that recovery succeeds in the physically relevant cases remains open (Surya, 2019). OSR therefore faces the same evidential constraint as substantivalism: identify irreducible work or treat temporal vocabulary as representational.
5.4 Objection from Temporal Realism
A temporal realist may accept the semantic distinction between representation and ontology while maintaining that temporal structure represented in physics corresponds to a real physical structure. The realist might appeal to temporal ordering, temporal asymmetry, or the directionality of physical processes as evidence that time is not merely representational.
ONC does not reject this position at the outset. The relevant question is whether the realist can identify a residual empirical or explanatory contribution made by temporal structure as such. The existence of temporal asymmetries, for example, is not by itself enough: statistical structure, boundary conditions, coarse-graining, and system-environment relations can enter the physical explanation of asymmetry without requiring an additional temporal entity.
The same applies to temporal ordering. If ordering relations are indispensable to the physical explanation, the realist must still show why those relations require an additional temporal constituent rather than being features of the underlying physical structure and its lawful dependencies. Conversely, if a theory shows that a temporal degree of freedom has an irreducible explanatory role, ONC permits the realist inference. The criterion therefore does not settle temporal realism in advance; it identifies what would be needed to support it.
5.5 Objection from Explanatory Necessity
One might argue that even if temporal or spatiotemporal entities do not perform irreducible empirical work, they perform irreducible explanatory work. For example, temporal structure may be said to explain persistence, causation, or dynamical evolution.
ONC distinguishes empirical indispensability from explanatory indispensability.
Explanatory indispensability can justify ontology only if the explanation cannot be reformulated in terms of change and relational structure. Many explanations that appear to require temporal entities – persistence, causation, evolution – can be reformulated in terms of dependence relations among physical states. If an explanation genuinely requires a temporal entity, ONC permits the posit. If not, ONC treats the entity as explanatorily redundant.
5.6 Summary
These objections locate the relevant burden rather than settle it. Vacuum gravitational degrees of freedom are a serious candidate for irreducible metric content; sophisticated substantivalism and OSR offer developed accounts of structured ontology; temporal and explanatory realists may identify further residues. ONC permits these conclusions where the claimed contribution survives an adequate specification of physical states, relations, and dynamics. The audits establish no general elimination result.
