Against Radical Relationalism: in Defense of the Ordinal Structure of Time

\'Alvaro Mozota Frauca

REVIEW 3 major objections 4 minor 5 references

Against Radical Relationalism: in Defense of the Ordinal Structure of Time

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that the ordinal structure of time is an ineliminable part of Newtonian mechanics and general relativity, against "radical relationalist" attempts to reduce physics to unordered correlations.

desk verdict A clear, useful challenge to eliminativist readings of time in classical GR, but the central 'ineliminable' claim overreaches: the clock-variable reply is waved off rather than refuted, and the loop example misfires. read the letter →

arxiv 2505.08307 v1 pith:3SFZN6OP submitted 2025-05-13 physics.hist-ph gr-qc

Álvaro Mozota Frauca This is my paper · ORCID

classification physics.hist-phgr-qc

keywords philosophyoftimerelationalismgeneralrelativitychrono-ordinalstructurequantumgravityreparametrizationinvariancecausalphysics

topics Quantum Gravity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the ordinal structure of time—the fact that events form a before/after/in-between sequence—cannot be eliminated from classical physics. It targets the "radical relationalist" view, common in parts of the quantum gravity literature, that a model's physical content is just an unordered set of correlations between measurable quantities. Against this, the author claims that in Newtonian mechanics and general relativity the order of events is invariant under reparametrizations and diffeomorphisms, that it is needed to define relative durations, velocities, causation, and explanation, and that it belongs to the empirical content of the theories. If correct, the conceptual gap between timeless quantum gravity proposals and classical physics is larger than their defenders claim.

What carries the argument

The central object is the chrono-ordinal structure, defined as the in-betweeness relation between instants or events (not necessarily directed). The argument's load-bearing mechanism is the comparison of two formalisms: the Newtonian action with absolute time and the Jacobi action with an arbitrary parameter $\tau$, whose solutions form an equivalence class under reparametrization. The author shows that all members of the class agree on the ordinal structure but not on metric durations, so the moderate relationalist can retain order; the radical relationalist's further step to an unordered set of pairs discards a shared, invariant feature. In general relativity the same mechanism is the diffeomorphism-invariance of the past/future relation defined by the metric and causal structure.

What would settle it

Find two diffeomorphism-related solutions of general relativity that reproduce the same correlations between partial observables yet disagree about whether one event lies in the future of another; no such pair exists if the paper's invariance claim is right.

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Extended reading notes

Core claim

The central claim is that chrono-ordinal structure—the one-dimensional "in-between" ordering of instants in Newtonian mechanics and the partial order of past/future in general relativity—is a genuine, ineliminable part of the physical content of our classical theories. The author argues that even after quotienting by arbitrary temporal reparametrizations (the Jacobi action) or by spacetime diffeomorphisms (general relativity), the equivalence classes still share a definite order structure: if an event lies between two others in one representative, it does so in all. Therefore the radical relationalist's proposal to discard the order and keep only correlations throws away predictive and explanatory content, such as the fact that one planet-year lasts a definite number of planet-days or that two black-hole mergers are an ordered sequence ending in detection.

Load-bearing premise

The load-bearing premise is that radical relationalism is genuinely committed to treating a model's physical content as an unordered set of measured pairs, with no physical meaning to the sequence in which they occur; the paper sets aside the rival reading in which order is recovered from correlations with an external clock as "a bit strange" rather than proving it impossible.

Editorial extensions

If this is right

Radical relationalism cannot account for relative durations, since counting spins "between" the endpoints of a year presupposes an in-betweenness relation.
Velocities and rates of change are either abandoned or redefined in a way that smuggles in temporal derivatives, so the position is not a clean eliminativism.
The gauge-type arguments from reparametrization and diffeomorphism invariance do not support radical relationalism, because the order structure is common to all representatives of an equivalence class.
If the paper is right, timeless quantum gravity approaches face a larger explanatory burden: they must recover or approximate chrono-ordinal structure rather than deny its physical significance.
Spacetime functionalism, if it is to be viable, should aim to explain how ordinal structure could be recovered from a more fundamental theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the argument is right, a full radical-relationalist model of the universe would have to encode temporal order through correlations with clock variables, and the paper's "a bit strange" reply leaves that route open rather than closed.
A testable extension is to try to reformulate a simple Newtonian system—say, the orbiting and spinning planet—without any in-betweenness relation and check whether all order-sensitive predictions survive.
The argument suggests that approaches to quantum gravity that retain a discrete or continuous causal order are closer to classical physics than fully timeless proposals.
The recovery criterion implied by the paper can be sharpened: a timeless theory should be required to reproduce not just correlations but specific ordinal facts, such as "this event lies between those two," before being counted empirically adequate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me cut to the chase. This paper does something useful and long overdue: it names radical relationalism—the view that the physical content of classical GR is just correlations among partial observables, with no temporal order—and pushes back against it. The core observation, that chrono-ordinal structure is invariant under reparametrizations and diffeomorphisms and therefore cannot be dismissed as gauge, is correct and worth saying. On that basis the paper makes a fair case that the burden of proof sits with the eliminativist.

The new contribution is a careful conceptual characterization of the target and two toy examples (the solar-system model and the self-intersecting trajectory). The paper also connects the debate to spacetime functionalism in a helpful way: if you want to be a functionalist about time, you need to say how ordinal structure gets recovered, not just how it could be dropped.

Where it softens is the central ineliminability thesis. The whole argument leans on the claim that radical relationalism treats a model as an unordered set of measured pairs. But that characterization is exactly what Rovelli-style relationalists will contest. They don't need to say there is no order; they can say order is relative to a clock, and any clock variable defines an order within the set of correlations. The paper anticipates this reply in Section 3 and sets it aside as 'a bit strange' without argument. That is not a refutation, and it is the load-bearing point. If order can be encoded through correlations with a clock, then ordinal structure is not ineliminable; it is derivative.

The loop example also doesn't do the work assigned to it. The two trajectories are supposed to have identical correlations but different order. But in the setup, the planet's spin behaves differently in the two cases, so the set of pairs (x, α) actually differs. A relationalist who keeps the full set of correlations can tell them apart without invoking a primitive betweenness relation. The example needs to be redesigned or dropped.

The claim that ordinal structure is 'empirical content' is also stronger than what the argument shows. What the paper convincingly shows is that the order is an invariant, indispensable feature of the mathematical models, and that dropping it would cripple explanation and prediction. Whether it is 'in the world' in the metaphysically loaded sense—that is a further step the toy examples don't reach.

None of this sinks the paper. The core dialectical point stands: if classical GR has an invariant partial order, then a timeless quantum gravity needs a story about how that order emerges, and radical relationalism's story—just correlations, no order—is incomplete. The author should be asked to address the clock-variable reply directly, to test whether any ordering can be reconstructed from partial-observable correlations, and to qualify the ineliminability claim accordingly.

Worth serious peer review? Yes. It is a genuine, clearly argued contribution to a live debate, and the soft spots are fixable.

Referee Report

3 major / 4 minor

Summary. The paper argues against 'radical relationalism,' an eliminativist view of time endorsed or suggested by some authors in the quantum gravity community, according to which the ordinal structure of time is dispensable even in classical theories. The author's central claim is that in Newtonian mechanics and general relativity the chrono-ordinal structure of time is an ineliminable part of the models, is invariant under the relevant reparametrizations and diffeomorphisms, is needed to define relative durations, velocities, causation, and explanation, and should be counted as part of the empirical content of those theories. The argument proceeds through a toy solar-system model in Section 3, a generalization to general relativity in Section 4, and a discussion of consequences for quantum gravity in Section 5. The paper concludes that timeless approaches to quantum gravity should aim to recover or approximate chrono-ordinal structure rather than deny its physical significance.

Significance. If the argument succeeded, the paper would make a useful contribution to the philosophy of time and to the foundations of quantum gravity, because it would block a popular strategy for minimizing the conceptual gap between classical general relativity and timeless approaches to quantum gravity. The paper is clearly written and engagingly structured, and it correctly identifies important invariant content in reparametrization-invariant theories: the causal/chrono-ordinal structure is common to diffeomorphism-related models. It also gives a fair presentation of the moderate relationalist position via the Jacobi action. The main weakness is that the central ineliminability claim is stronger than the examples establish, because the paper does not refute the radical relationalist's natural reply that order can be reconstructed from correlations with clock variables. For this reason the paper is best read as a challenge to radical relationalism rather than a proof of its impossibility; the central claim is defensible but requires additional work.

major comments (3)

[Section 3, paragraph beginning 'I see two ways in which the radical relationalist could respond'] The clock-variable reply is set aside as 'a bit strange' rather than refuted. The ineliminability thesis requires showing that no set of correlations among partial observables can reproduce the betweenness structure of Newtonian or general-relativistic models. The paper does not provide such a proof, and it does not engage with the possibility that correlations with a monotonic clock variable, such as the spin angle α1, induce a derived ordering. Without an argument ruling out this reconstruction, the conclusion that chrono-ordinal structure is ineliminable is stronger than what the examples establish.
[Section 3, Figure 2 and surrounding discussion] The loop example does not by itself distinguish the two trajectories if the spin variable α1 is part of the correlated data. In the anticlockwise case, the set of pairs (x1, α1) differs from the clockwise case, because the spin angle at a given spatial location along the loop differs between the two traversals. A radical relationalist who retains the full set of correlations can therefore distinguish the two possibilities without positing a primitive betweenness relation. The example needs to be revised, or the author needs to explain why the relevant correlations are unavailable in the radical relationalist's picture.
[Section 2, characterization of radical relationalism] The paper characterizes radical relationalism as treating a model's physical content as the unordered set of measured pairs (x, α). This characterization risks building the conclusion into the premise: if 'order' is excluded by definition from the relationalist's descriptive apparatus, then the alleged loss of order is a consequence of the definition rather than an independent argument. A more dialectically effective approach would be to grant the relationalist the full set of correlations, including correlations with clock variables, and then show that those correlations still fail to capture something that our theories treat as physically significant.

minor comments (4)

[Section 3, second paragraph] The name 'Leibnitz' should be spelled 'Leibniz' for consistency with the rest of the paper and with the references.
[Figure 1] The right-hand panel is described as a set of points with no ordinal structure, but as drawn it can be misread as a trajectory projected onto a two-dimensional plane; a caption note clarifying that no betweenness relation is intended would prevent confusion.
[Section 4, footnote 8] The parenthetical remark about the bound on the difference between ways of defining Venus-year durations is sensible, but the phrase 'bounded above by the time it takes for light to travel between Earth and Venus' should specify that this is a rough estimate, since the actual value depends on orbital configuration.
[References] Several references contain line breaks inserted into URLs (e.g., Rovelli 2002 and Thébault 2021); the final published version should ensure that all URLs are rendered as continuous strings.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the argument is a philosophical critique whose characterization of the target view does not by itself entail the conclusion, and no load-bearing self-citation or fitted-input reduction is present.

full rationale

This paper contains no derivation chain in the sense relevant to circularity analysis: there are no fitted parameters, no empirically fitted inputs relabeled as predictions, and no uniqueness theorem imported from the authors' own prior work. The central claim is that chrono-ordinal structure is an ineliminable part of classical physical theories because it is needed for betweenness, relative duration, velocity, causation, explanation, and the empirical content of Newtonian mechanics and general relativity. The target position, radical relationalism, is characterized as treating a model's physical predictions as an unordered set of pairs (x1, α1); this characterization is not derived from the conclusion but is presented as the position's own claim. The paper's admitted weakness is dialectical rather than circular: it identifies the reply that temporal order could be encoded via correlations with an external clock and sets it aside as 'a bit strange' rather than proving it impossible. That is a gap in the argument against a determined relationalist, but it is not the reduction of a prediction to its own input. Self-citations occur in the paper but only as background references or as pointers to complementary arguments, not as the load-bearing justification for the central thesis. The invariance of the causal/chrono-ordinal structure under diffeomorphisms is invoked as a property of general-relativistic models, not as a result generated by this paper's own prior work. Accordingly, no circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The argument has no fitted parameters and invents no new physical entities. It rests on how current classical theories are to be read, on the invariance principle for physical content, and on the assertion that order is not recoverable from an unordered correlation set. These are philosophical premises, not empirical inputs.

assumptions (5)

domain assumption The physical content of a theory is what is invariant under its representational symmetries.
Invoked in Sections 3 and 4 when considering reparametrization and diffeomorphism equivalence classes; the author accepts this principle but disputes what is invariant.
domain assumption Relative duration and in-betweenness facts are empirical facts, not mere conventions.
Central to the claim that correlations alone are insufficient; footnote 8 concedes some conventionality in defining cross-body durations but asserts the facts remain.
ad hoc to paper The ordinal structure of standard solutions is not recoverable from an unordered set of measured correlations.
Introduced in Section 3 to separate radical relationalism from moderate relationalism; the paper does not prove it, and the relationalist clock reply targets this premise.
domain assumption Our best physical theories are characterized by one-dimensional trajectories in configuration space, forcing a one-dimensional chrono-ordinal structure.
Used in the final paragraphs of Section 3 to argue that even radical relationalist models retain a one-dimensional order; it relies on the contingent form of current theories.
standard math All monotonic reparametrizations preserve the order of instants.
Used in Section 3 to show that the Jacobi action's equivalence class retains ordinal structure despite losing Newtonian time.

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Cite this review

Pith. "Pith review of Against Radical Relationalism: in Defense of the Ordinal Structure of Time." pith.science (2026). https://pith.science/paper/3SFZN6OP

@misc{pith2026250508307,
  author       = {Pith},
  title        = {Pith review of: Against Radical Relationalism: in Defense of the Ordinal Structure of Time},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SFZN6OP}},
  note         = {Machine review of arXiv:2505.08307}
}

read the original abstract

Some authors in the quantum gravity community endorse, explicitly or implicitly, a radical relationalist view of time which states that the ordinal structure of time is not needed even in our classical theories, especially in general relativity. In this article I analyze this position and the arguments supporting it, and I argue that there are some serious concerns with some of the radical relationalists' arguments which make it an unattractive position. In this sense, I conclude that the chrono-ordinal structures of our theories play important theoretical and explanatory roles and that they can be taken to be part of the empirical content of our theories.

Figures

Figures reproduced from arXiv: 2505.08307 by the authors.

**Figure 2.** Two possible ways of ordering a trajectory. While t [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

5 extracted references · 4 canonical work pages

[1]

Barbour, J. B. (1994). The timelessness of quantum gravity: I. T he evidence from the classical theory [Publisher: IOP Publishing]. Classical and Quantum Gravity , 11 (12),

work page 1994
[21]

https://doi.org/10.1007/s10714-023-03067- Mozota Frauca, A. (2024a). GPS observables in Newtonian spacet ime or why we do not need ‘physical’ coordinate systems. European Journal for Philosophy of Science, 14 (4),

doi:10.1007/s10714-023-03067- 2024
[51]

(2024b, March)

https://doi.org/10.1007/s13194-024-00611-7 Mozota Frauca, A. (2024b, March). The Problem of Time for Non-D eparametrizable Models and Quantum Gravity. In F. Bianchini, V. Fano, & P. Graziani ( Eds.), Current Topics in Logic and the Philosophy of Science. Paper s from SILFS 2022 postgraduate conference.(Vol. 4). College Publications. https://philsci-archiv...

work page arXiv 2025
[119]

Lam, V., & W¨ uthrich, C. (2018). Spacetime is as Spacetime Does. Studies in His- tory and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 64, 39–51. https://doi.org/10.1016/j.shpsb.2018.04.003 Lam, V., & W¨ uthrich, C. (2021). Spacetime functionalism from a rea list perspective [arXiv: 2003.10172 Publisher: Springer]. S...

work page Pith review arXiv 2018
[2853]

https://doi.org/10.1088/0264-9381/11/12/005 Barbour, J. B. (2011). Shape Dynamics. An Introduction [arXiv: 1 105.0183 Publisher: Springer Basel]. Quantum Field Theory and Gravity, 257–297. https://doi.org/10.48550/arxiv.1105.0183 Colosi, D., & Rovelli, C. (2003). Simple background-independent Hamilt onian quan- tum model [arXiv: gr-qc/0306059]. Physical R...

work page Pith review arXiv doi:10.48550/arxiv.1105.0183 2011

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Barbour, J. B. (1994). The timelessness of quantum gravity: I. T he evidence from the classical theory [Publisher: IOP Publishing]. Classical and Quantum Gravity , 11 (12),

work page 1994

[2] [21]

https://doi.org/10.1007/s10714-023-03067- Mozota Frauca, A. (2024a). GPS observables in Newtonian spacet ime or why we do not need ‘physical’ coordinate systems. European Journal for Philosophy of Science, 14 (4),

doi:10.1007/s10714-023-03067- 2024

[3] [51]

(2024b, March)

https://doi.org/10.1007/s13194-024-00611-7 Mozota Frauca, A. (2024b, March). The Problem of Time for Non-D eparametrizable Models and Quantum Gravity. In F. Bianchini, V. Fano, & P. Graziani ( Eds.), Current Topics in Logic and the Philosophy of Science. Paper s from SILFS 2022 postgraduate conference.(Vol. 4). College Publications. https://philsci-archiv...

work page arXiv 2025

[4] [119]

Lam, V., & W¨ uthrich, C. (2018). Spacetime is as Spacetime Does. Studies in His- tory and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 64, 39–51. https://doi.org/10.1016/j.shpsb.2018.04.003 Lam, V., & W¨ uthrich, C. (2021). Spacetime functionalism from a rea list perspective [arXiv: 2003.10172 Publisher: Springer]. S...

work page Pith review arXiv 2018

[5] [2853]

https://doi.org/10.1088/0264-9381/11/12/005 Barbour, J. B. (2011). Shape Dynamics. An Introduction [arXiv: 1 105.0183 Publisher: Springer Basel]. Quantum Field Theory and Gravity, 257–297. https://doi.org/10.48550/arxiv.1105.0183 Colosi, D., & Rovelli, C. (2003). Simple background-independent Hamilt onian quan- tum model [arXiv: gr-qc/0306059]. Physical R...

work page Pith review arXiv doi:10.48550/arxiv.1105.0183 2011