On Andr\'eka's Conjecture that special relativity is the only possible conceptual reduct of classical kinematics

Gergely Sz\'ekely; Judit Madar\'asz; Mike Stannett

REVIEW 2 major objections 5 minor 15 references

On Andr\'eka's Conjecture that special relativity is the only possible conceptual reduct of classical kinematics

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves there is no intermediate model of spacetime between special relativity and late classical kinematics, so adding any single classical concept to relativistic spacetime yields the full late classical spacetime up to…

desk verdict A significant and likely correct proof of Andréka's conjecture, but the decisive definability-to-automorphism duality is imported from a companion paper, so the referee should check that bridge. read the letter →

arxiv 2507.21180 v1 pith:YBJDP3JI submitted 2025-07-26 math-ph math.MP

Judit Madarász , Mike Stannett , Gergely Székely This is my paper · ORCID

classification math-phmath.MP MSC 03C4003G1583A05

keywords Andréka'sconjecturespecialrelativityclassicalkinematicsdefinitionalequivalencefinitelyfield-definablecoordinategeometriesdefinabilityandautomorphismgroupsAlexandrov–ZeemantheoremBorisov's

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

The paper proves a purely mathematical conjecture about the conceptual structure of spacetime: there is no intermediate model strictly between special relativity and late classical kinematics. Concretely, if one starts from relativistic spacetime — the structure consisting only of the relation of lightlike relatedness on $\mathbb{R}^4$ — and adds any single classical concept that the late classical model can express (such as absolute simultaneity), the resulting expansion is definitionally equivalent to the full late classical spacetime. This matters for the history of physics because the Michelson–Morley experiment ruled out the late classical model while leaving its basic lightlike notion intact; the theorem says there was no halfway conceptual stop on the way to relativity. The statement is equivalent to the algebraic-logic claim that the concept algebra of relativistic spacetime is a maximal proper subalgebra of the concept algebra of late classical spacetime.

What carries the argument

The machinery is a duality theorem (Theorem 5.6) for finitely field-definable coordinate geometries: if $G$ and $G'$ are such geometries, then $\operatorname{Conc} G \subseteq \operatorname{Conc} G'$ exactly when $\operatorname{Aut} G \supseteq \operatorname{Aut} G'$, with strict versions for proper inclusions. A coordinate geometry is a structure on $\mathbb{R}^4$ with no constants or functions whose collinearity relation is definable; it is finitely field-definable when its finitely many relations are definable in the real field $\langle \mathbb{R}, +, \cdot, 0,1\rangle$. The paper feeds into this duality the automorphism computations $\operatorname{Aut}\mathrm{RelST} = \mathrm{Scal}\circ\mathrm{Poi}$ (the Alexandrov–Zeeman theorem), $\operatorname{Aut}\mathrm{LClassST} = \mathrm{Scal}\circ\mathrm{Triv}$, and the group-theoretic gap theorem (Theorem 6.8) showing no group lies strictly between these two. The duality then transmits the group gap to a concept gap.

What would settle it

Find a model $M$ with $\operatorname{Conc}\mathrm{RelST} \subset \operatorname{Conc} M \subset \operatorname{Conc}\mathrm{LClassST}$, or equivalently a group of transformations $G$ with $\mathrm{Scal}\circ\mathrm{Triv} \subset G \subset \mathrm{Scal}\circ\mathrm{Poi}$; alternatively, exhibit two finitely field-definable coordinate geometries with the same automorphism group but different concept sets, refuting the duality theorem.

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Extended reading notes

Core claim

The central discovery is Theorem 3.1: for every concept $C$ that is definable in late classical spacetime $\mathrm{LClassST} = \langle \mathbb{R}^4, S, \lambda \rangle$ but not in relativistic spacetime $\mathrm{RelST} = \langle \mathbb{R}^4, \lambda \rangle$, the expansion $\langle \mathrm{RelST}, C\rangle$ has exactly the same definable relations as $\mathrm{LClassST}$, i.e. $\langle \mathrm{RelST}, C\rangle \mathrel{\Delta\!\!\!\equiv} \mathrm{LClassST}$. Equivalently, no model $M$ satisfies $\operatorname{Conc}\mathrm{RelST} \subset \operatorname{Conc} M \subset \operatorname{Conc}\mathrm{LClassST}$. The proof converts definability into automorphisms: for the finitely field-definable coordinate geometries involved, concept inclusion reverses automorphism-group inclusion, so an intermediate concept would force an intermediate transformation group between $\operatorname{Aut}\mathrm{RelST} = \mathrm{Scal}\circ\mathrm{Poi}$ and $\operatorname{Aut}\mathrm{LClassST} = \mathrm{Scal}\circ\mathrm{Triv}$. The paper shows, using a corollary of Borisov's theorem, that no such group exists.

Load-bearing premise

The proof rests on a theorem, cited from a companion preprint rather than proved here, that for these finitely field-definable coordinate geometries a relation is definable exactly when it is field-definable and invariant under all automorphisms; if that duality fails, the reduction to the group-theoretic gap collapses.

Editorial extensions

If this is right

Any single definable classical concept added to $\mathrm{RelST}$ yields all of $\mathrm{LClassST}$ up to definitional equivalence; in particular, absolute simultaneity is enough.
The concept algebra $\mathrm{Cs}\,\mathrm{RelST}$ is a maximal proper subalgebra of $\mathrm{Cs}\,\mathrm{LClassST}$.
Galilean spacetime augmented with lightlike relatedness, $\langle \mathrm{GalST}, \lambda\rangle$, is definitionally equivalent to $\mathrm{LClassST}$, justifying the name 'late classical spacetime'.
Within first-order definable relations on $\mathbb{R}^4$, there is no conceptual halfway house between special relativity and late classical kinematics.
The historical inference drawn by the authors: once the Michelson–Morley null result ruled out late classical kinematics, retaining lightlike relatedness forced the conceptual jump to special relativity.

Reading between the lines

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

Because the duality theorem is proved only for finitely field-definable coordinate geometries, the no-intermediate result is established for finite first-order vocabularies definable in the real field; a natural extension would be to ask whether an infinite vocabulary or a non-field-definable relation can be inserted between the two concept sets.
The same lattice argument could be run with a different base relation than lightlike relatedness (for example, spacelike relatedness or the causal order), provided the relevant automorphism-group ladder has no intermediate group; whether such variants hold is open.
The result gives the 'no alternative' thesis a precise mathematical content: it concerns concept lattices of definable relations, not empirical equivalence. Any philosophical claim that Einstein's choice was underdetermined would need to locate the supposed alternative outside this class.

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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

Judit, Mike, and Gergely have proved Andréka's conjecture. If the companion paper's duality theorem holds, the proof is clean: any new classical concept added to relativistic spacetime collapses you to full late classical kinematics. That is a genuine result, the first for arbitrary concepts, not just absolute simultaneity. The automorphism-group reduction is elegant, and the no-intermediate-group argument (Theorem 6.8) is self-contained and convincing.

The paper is honest about what it depends on. The load-bearing step is Theorem 5.6, cited from their companion [MSS25a]: for FFD coordinate geometries, concept inclusion is equivalent to reverse automorphism-group inclusion. That is a nontrivial duality—automorphism invariance alone does not give definability in general, so the FFD condition is doing real work. The proof is not in this paper. If [MSS25a] is wrong, the whole reduction collapses. The same applies, less centrally, to Proposition 6.3(ii) from [MSS25b]. This is not a fatal flaw, but it makes the result conditional on unpublished (or not-yet-refereed) machinery.

Second soft spot: the abstract's historical framing overshoots. The formal theorem is about first-order definable concepts on R^4. That says nothing about all possible 'options' in the history of physics. The informal claim should be qualified.

Overall, the mathematics looks sound, the exposition is clear, and the citation of their own work is appropriate given the division of labor. I'd send it to a referee. The referee should be asked to verify Theorem 5.6 and its applicability, or to insist that the proof be included.

Referee Report

2 major / 5 minor

Summary. The paper proves a formal version of Andr\'eka's conjecture: for any concept C definable in late classical spacetime LClassST = \langle R^4, S, \lambda\rangle but not definable in relativistic spacetime RelST = \langle R^4, \lambda\rangle, the expansion \langle RelST, C\rangle is definitionally equivalent to LClassST. Equivalently, there is no model whose concept class lies strictly between Conc RelST and Conc LClassST. The proof strategy is to reduce this model-theoretic statement, via a definability-to-automorphism duality for finitely field-definable coordinate geometries (Theorem 5.6), to the group-theoretic claim that no group lies strictly between Aut RelST = Scal \circ Poi and Aut LClassST = Scal \circ Triv. The paper also shows that LClassST is definitionally equivalent to Galilean spacetime extended with lightlike relatedness.

Significance. If the cited companion results hold, the paper settles Andr\'eka's conjecture and gives a sharp formal sense in which special relativity is the only way to add a non-relativistic classical concept to lightlike relatedness without introducing entirely new vocabulary. The reduction of the conjecture to a no-intermediate-group statement is elegant, and the group-theoretic proof of Theorem 6.8 is a genuine contribution. However, the central definability-to-automorphism duality and several supporting identifications of automorphism groups are cited from the authors' own companion preprints rather than proved here, so the mathematical contribution of the present paper is conditional on [MSS25a] and [MSS25b].

major comments (2)

[Section 5, Theorems 5.4 and 5.6] The proof of Theorem 3.1 reduces the conjecture to the group-theoretic no-intermediate-group statement through the equivalence Conc G \subseteq Conc G' iff Aut G \supseteq Aut G' for FFD coordinate geometries. Theorem 5.6 is not proved in this manuscript; it is cited as a special case of [MSS25a, Thm.5.1.4 and Cor.5.1.5], and its parent Theorem 5.4 is likewise cited from [MSS25a, Thm.5.1.2]. The converse direction (automorphism invariance implies definability) is false for general first-order structures, so the FFD hypothesis is doing substantial work. This is load-bearing: without this duality, the strict inclusion Conc RelST \subset Conc \langle RelST, C\rangle does not imply the strict inclusion Aut RelST \supset Aut \langle RelST, C\rangle, and the proof of Section 4 cannot start. I recommend either proving Theorems 5.4 and 5.6 in an appendix or making the companion paper's status explicit and verifiable.
[Section 6, Proposition 6.3] The computation Aut LClassST = Scal \circ Triv in Theorem 6.4 depends on Proposition 6.3(ii), which is quoted from the companion paper [MSS25b, Thm.3.2.2 and Thm.3.2.3] rather than proved here. This result is used to replace LClassST by \langle R^4, \sim, Rest, Bw\rangle before the automorphism computation; if it is unavailable, the equality Aut LClassST = Scal \circ Triv is not established and the lower end of the group-theoretic chain fails. The authors should either include a proof of the needed statements or clearly state that the companion is part of the same accepted or near-accepted body of work.

minor comments (5)

[Section 3, Theorems 3.1 and 3.2] The claim that the two theorems are equivalent is stated without proof; a short argument for Theorem 3.1 implies Theorem 3.2 for arbitrary models would be helpful, for example by choosing C \in Conc M \setminus Conc RelST and observing that Conc \langle RelST, C\rangle \subseteq Conc M.
[Section 5, Definition 5.2] The correspondence R \mapsto \hat{R} is introduced only by example, with the general definition deferred to [MSS25a]; since field-definability is a central notion, a precise general definition would make the paper more self-contained.
[Section 6, Theorem 6.8] In the first paragraph of the proof, the phrase "we can assume without loss of generality that f and t are linear" is terse; a one-sentence justification using conjugation by a translation would improve readability.
[Section 6, Proposition 6.3(ii)] The phrase "the only difference being the order in which the relations are listed" should be phrased as "up to definitional equivalence" or "up to relabeling of relation symbols".
[Abstract and Introduction] The historical claims, such as "there was essentially no other option but to switch to special relativity," extrapolate from a theorem relative to a fixed universe R^4 and a fixed vocabulary; a sentence acknowledging this limitation would be appropriate.

Circularity Check

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No circularity found: the proof reduces Andréka's conjecture to general, parameter-free theorems about FFD coordinate geometries and automorphism groups cited from companion work; these are independent results, not equivalent to the target.

full rationale

The proof of Theorem 3.1 is not circular. The central bridge, Theorem 5.6, is stated as a special case of [MSS25a, Thm.5.1.4 and Cor.5.1.5] and is load-bearing: it converts the concept inclusion Conc G ⊆ Conc G′ into the automorphism inclusion Aut G ⊇ Aut G′. This is a self-citation, but the cited theorem is a general result about finitely field-definable coordinate geometries over R^4; its assumptions do not mention RelST, LClassST, or Andréka's conjecture, so it is not tailored to force the paper's conclusion. Likewise, Theorem 5.4 is a general definability criterion for FFD coordinate geometries, and Theorem 6.7 is a special case of [MSS22, Cor.4.9] based on Borisov's theorem; Proposition 6.3(ii) uses the special case C = S from [MSS25b]. These are independent mathematical lemmas, not renamed versions of the conjecture or fitted inputs. There is no fitted parameter disguised as a prediction, no definition of a concept in terms of the conclusion, and no ansatz smuggled in via citation. The paper's reliance on unpublished companion preprints is a verifiability and correctness risk, but it is not circularity under the stated criteria.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted parameters. The central claim rests on a framework of first-order definable structures on R^4 (domain assumptions), standard theorems of real affine geometry, Alexandrov-Zeeman and Borisov rigidity theorems, and the authors' own prior definability results. The most consequential assumptions are the definability-automorphism duality theorems cited from the companion preprint [MSS25a].

assumptions (8)

domain assumption Spacetime models are structures on R^4; only models whose universes are R^4 are considered.
The paper explicitly narrows the conjecture to spacetime as a structure on R^4 (Section 1: 'or at least, not if we regard spacetime as a structure built on top of 4-dimensional real space, R4').
domain assumption A concept is any relation first-order definable in the model, with no parameters.
The paper defines 'concept' this way in Section 1. This identification of conceptual content with first-order definability is a philosophical choice that determines the scope of the result.
standard math For finitely field-definable (FFD) coordinate geometries, R is in Conc G iff R is field-definable and closed under automorphisms of G (Theorem 5.4, cited from [MSS25a, Thm.5.1.2]).
This is the load-bearing definability-to-automorphism result; it is not proven in this paper.
standard math For FFD coordinate geometries G and G', Conc G is a subset of Conc G' iff Aut G is a superset of Aut G' (Theorem 5.6, cited from [MSS25a, Thm.5.1.4 and Cor.5.1.5]).
This converts the concept-inclusion question to an automorphism-group-inclusion question. Same caveat as the previous item.
standard math Aut RelST = Scal composed with Poi (Alexandrov-Zeeman theorem, Theorem 6.1).
External theorem used to compute the symmetry group of relativistic spacetime.
standard math Aut LClassST = Scal composed with Triv (Theorem 6.4, proven in the paper).
Proved in this paper using the fundamental theorem of affine geometry and congruence arguments.
standard math There is no group G with Triv-up strictly between G and Poi-up (Borisov's theorem via [MSS22, Cor.4.9], Theorem 6.7).
External rigidity theorem used to exclude intermediate automorphism groups.
standard math The equivalence between LClassST and the model with Euclidean congruence, Rest, and betweenness (Proposition 6.3(ii) uses [MSS25b, Thm.3.2.2 and Thm.3.2.3]).
From the authors' companion paper [MSS25b]; used to identify automorphisms of LClassST.

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Cite this review

Pith. "Pith review of On Andr\'eka's Conjecture that special relativity is the only possible conceptual reduct of classical kinematics." pith.science (2026). https://pith.science/paper/YBJDP3JI

@misc{pith2026250721180,
  author       = {Pith},
  title        = {Pith review of: On Andr\'eka's Conjecture that special relativity is the only possible conceptual reduct of classical kinematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBJDP3JI}},
  note         = {Machine review of arXiv:2507.21180}
}

read the original abstract

In this paper, we prove a pure mathematical result which has important implications for the history and philosophy of classical physics and the conceptual origins of relativity theory. In formal terms, we show that, up to definitional equivalence, there is no intermediate model of spacetime lying strictly between special relativity and late classical kinematics. Informally, this means that there was essentially no other option but to switch to special relativity to resolve the conflict between late classical kinematics and the null result of the Michelson--Morley experiment.

Figures

Figures reproduced from arXiv: 2507.21180 by the authors.

**Figure 1.** This figure illustrates relativistic spacetime RelST and late classical spacetime LClassST. We define relativistic and late classical spacetime, respectively, to be: RelST def = ⟨R 4 , λ⟩, LClassST def = ⟨R 4 , S, λ⟩, 1This formulation of definitional equivalence is equivalent to the standard formulation (e.g., [HMT71, p.51] or [Mon00, p.453]) but is better suited to our requirements [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 2.** This figure illustrates the relation between the concepts of relativistic spacetime RelST and late classical spacetime LClassST. Clearly, from the assumptions of the conjecture (that C ∈ ConcLClassST and C ̸∈ Conc RelST ), we have that Conc RelST ⊂ Conc ⟨RelST, C⟩ ⊆ ConcLClassST , and we have already seen (Proposition 2.1) that these assumptions are not contradictory. By (2) and (3), this implies that (5) Aut RelS… view at source ↗

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

A. D. Alexandrov, Mappings of spaces with families of cones and space-time transformations, Annali di Matematica Pura ed Applicata 103 (1975), no. 1, 229--257

work page 1975

[2] [2]

Andr \'e ka and I

H. Andr \'e ka and I. N \'e meti, Comparing theories: The dynamics of changing vocabulary, pp. 143--172, Springer International Publishing, Cham, 2014

work page 2014

[3] [3]

M.\ Audin, Geometry, Springer, 2003

work page 2003

[4] [4]

Yu. F. Borisov, Foundations of relativistic kinematics, Siberian Mathematical Journal 27 (1986), no. 3, 312--327

work page 1986

[5] [5]

H. H. Field, Science without numbers: A defence of nominalism, Princeton University Press, Princeton, NJ, USA, 1980

work page 1980

[6] [6]

Henkin, J

L. Henkin, J. D. Monk, and A. Tarski, Cylindric algebras part I , North-Holland, 1971

work page 1971

[7] [7]

Ketland, Axiomatization of G alilean spacetime , Dialectica 77 (2023), no

J. Ketland, Axiomatization of G alilean spacetime , Dialectica 77 (2023), no. 2, 153--202

work page 2023

[8] [8]

Lefever, Using logical interpretation and definitional equivalence to compare classical kinematics and special relativity theory, Ph.D

K. Lefever, Using logical interpretation and definitional equivalence to compare classical kinematics and special relativity theory, Ph.D. thesis, Vrije Universiteit Brussel, 2017

work page 2017