Grationality: Rethinking Gravity and Rationality

• Grationality is a framework that redefines gravity and rationality by rejecting finite, mechanistic assumptions in favor of invariant, relational approaches.
• It integrates Newton’s non-essential view of gravity, Barbour’s scale-free cosmology, Dreyer’s vacuum-based gravity, and Lescanne’s coinductive infinite game theory.
• This comparative motif highlights how emergent, relational methods can address the limitations of traditional mechanistic interpretations in physics and game theory.

Searching arXiv for the cited papers to ground the article in the relevant literature. arxiv_search: query: "id:(Taborda, 2010) OR id:(Barbour, 2023) OR id:(Dreyer, 2012) OR id:(Lescanne, 2011)" max_results: 10 {"query":"(Taborda, 2010, Barbour, 2023, Dreyer, 2012, Lescanne, 2011)","max_results":10} “Grationality” (Editor’s term) can denote a comparative constellation of problems in which gravitation and rationality are both analyzed under conditions that frustrate reduction to naïve mechanism, bounded equilibrium, or finite truncation. In the works associated here, Newtonian attraction is treated as real and mathematically exact without being an essential property of matter; gravity in an unconfined universe is described as a scale-free source of increasing ordered structure; gravitation is reinterpreted as the effect of spatially varying local vacua in an underlying quantum many-body system; and rational escalation is formalized through coinduction in infinite extensive games rather than backward induction on finite trees (Taborda, 2010, Barbour, 2023, Dreyer, 2012, Lescanne, 2011). This suggests a thematic unity organized not by a single doctrine but by a shared resistance to explanations that merely project finite, local, or mechanistic intuitions onto globally structured systems.

1. Scope of the term

The materials grouped under this heading come from four distinct domains: Newton scholarship, relational cosmology, emergent gravity, and coinductive game theory. Their primary claims are not identical. One concerns whether gravity is an essential property of matter; another concerns the arrow of time in an unconfined gravitating universe; another proposes that gravity is “not about metrics at all,” but about local vacua; and another studies when escalation in infinite games can be fully rational. The grouping is therefore comparative rather than doctrinal.

Work	Central claim	Key technical or conceptual object
Newton study	Gravity is not a “propiedad innata, esencial e inherente de la materia”	General Scholium; “hypotheses non fingo”
Barbour	Gravity’s “creative core” drives growth of ordered structure	$C_\textrm{\scriptsize shape}$; shape space; Janus point
Dreyer	Gravity is the relation among different local vacua	$v:M\to\mathcal V$; $\theta$; effective $G$
Lescanne	Escalation can be fully rational in infinite games	coinduction; $SGPE$; escalation sequence

The factual unity of this set is methodological. Each work rejects an explanatory default that is treated as illegitimate within its own domain. Newton, as interpreted in the historical paper, rejects the inference from mathematically exact attraction to essential material gravity. Barbour rejects the transfer of entropy concepts from particles in a box to the universe as a whole. Dreyer rejects the assumption that the equivalence principle must be implemented through differential geometry and a metric. Lescanne rejects the practice of truncating infinite games and extrapolating finite backward-induction results to the infinite case.

A plausible implication is that “Grationality” names a family of research attitudes rather than a settled field: lawful behavior is accepted, but the usual explanatory substrate—bare matter, absolute scale, fundamental metric geometry, or finitary induction—is denied explanatory sufficiency.

2. Newtonian gravity without essential material attraction

The historical core of the first line of work is the Newton–Leibniz controversy. Leibniz’s criticism, as reconstructed in the paper, was that if every body is intrinsically heavy and attracts every other, then gravity becomes either a scholastic occult quality or the effect of a miracle. Against that charge, the paper insists that Newton explicitly rejected the view that gravity is “una propiedad innata, esencial e inherente de la materia.” Bodies do attract one another according to exact mathematical laws, but gravity does not therefore belong to matter by its own essence (Taborda, 2010).

The paper presents Newton as occupying a middle position. He accepted the phenomena of gravitational attraction and gave them exact mathematical form. He did not think this justified saying that gravity is essential to matter. He did not provide a purely mechanical causal account of how attraction is produced. He held that there must be some mediator or principle beyond matter as such. The paper supports this reading through Newton’s early attempts to explain gravity mechanically through ether or subtle matter, his later rejection of those attempts, and the Bentley correspondence, where unmediated attraction through the void is called “un gran absurdo.” If mediation occurs, the paper says, it must be carried out by “algo que no es material,” identified there with God.

This is the basis for the paper’s insistence that Newton’s explanation in the Principia is non-mechanistic. Non-mechanistic here does not mean anti-rational or anti-scientific. It means that gravity is not explained by pushes, impacts, vortices, subtle fluids, or contact-action through a material medium. The law is exact; the mechanism is not asserted beyond what the phenomena warrant. The phrase “hypotheses non fingo” is treated not as a slogan but as Newton’s methodological and philosophical reply to critics.

The General Scholium, inserted into the second edition of the Principia in 1713, is central to this interpretation. The paper argues that it answers critics, clarifies Newton’s method, places gravity within a theological-metaphysical framework, and explains why Newton refuses a mechanistic reduction. The author further argues that in the General Scholium Newton makes explicit his adhesion to Sociano-Antitrinitarian doctrines and thereby clarifies why he did not believe gravity to be an innate, essential, and inherent property of matter. In this reading, gravity is fully real as a law-governed phenomenon, while its causal ground is not reducible to matter’s bare nature.

3. Gravity’s creative core and the scale-free universe

Julian Barbour’s account shifts the problem from historical interpretation to relational cosmology. Its starting point is the claim that the essence of gravity can be understood only in the context of the universe and that implicit retention of Newtonian absolute scale, together with the conceptual habits of thermodynamics, has obscured that essence. The target is the familiar puzzle according to which the early universe was very uniform, often described as low gravitational entropy, while the second law requires entropy increase toward the future. Barbour’s response is that the puzzle is mishandled because it imports conceptual machinery designed for confined systems—particles in a box, pistons, equilibrium, bounded phase spaces—into an unconfined universe (Barbour, 2023).

The technical core is a relational, scale-free reformulation of the Newtonian $N$-body problem. For a universe there is no external ruler, so only ratios of separations are physically meaningful. Barbour introduces the root-mean-square length $\ell_\textrm{\scriptsize rms}$, a mass-weighted overall size, and the mean harmonic length $\ell_\textrm{\scriptsize mhl}$, which is sensitive to clustering because small separations dominate the harmonic average. Their ratio is the shape complexity,

$$C_\textrm{\scriptsize shape}=\frac{\ell_\textrm{\scriptsize rms}}{\ell_\textrm{\scriptsize mhl}}.$$

This quantity is the paper’s principal “entropy-like” measure, though it is explicitly not identified with thermodynamic entropy. It measures clustering and variety: close encounters change $\ell_\textrm{\scriptsize mhl}$ strongly while changing $v:M\to\mathcal V$0 little, so $v:M\to\mathcal V$1 rises as structure forms.

Barbour then rewrites the Newton potential in scale-free form,

$v:M\to\mathcal V$2

and states that complexity is “as (minus) the normalised Newton potential or shape potential.” The relevant configuration space is not ordinary Newtonian configuration space $v:M\to\mathcal V$3, but shape space

$v:M\to\mathcal V$4

obtained by quotienting by translations, rotations, and dilatations. Relative configuration space,

$v:M\to\mathcal V$5

still contains overall scale and therefore has unbounded measure.

The paper discusses “best matching” as the relational definition of change and recalls Jacobi’s action with reparametrization freedom, from which Newtonian time is recovered as ephemeris time rather than assumed as absolute. Yet a fully geodesic and maximally predictive scale-invariant theory fails in the form Barbour examines, because it forces the dilational momentum

$v:M\to\mathcal V$6

to vanish. Since $v:M\to\mathcal V$7 is half the time derivative of the center-of-mass moment of inertia, $v:M\to\mathcal V$8 freezes the overall size. Such a universe does not expand, and compactness of shape space under strict quotienting by dilatations leads to Poincaré recurrence. For that reason, Barbour studies the actual Newtonian $v:M\to\mathcal V$9-body solutions projected into shape space rather than insisting on a fully geodesic law there.

The principal dynamical result is that for all solutions with nonnegative total center-of-mass energy $\theta$0, the shape complexity grows secularly away from a unique minimum. If $\theta$1, there is a Janus point at which $\theta$2 is minimal, after which it increases monotonically in both time directions. Complexity likewise grows away from that point in both directions, so each solution has two arrows of time. Near the Janus point the system is roughly homogeneous, with random velocities and weak clustering; Barbour says this state corresponds “at least qualitatively to high entropy,” not low entropy. The subsequent evolution is therefore not a drift toward disorder but a growth of ordered structure: clustering, bound subsystems, virialized structures, and especially Kepler pairs that function as increasingly accurate clocks, rods, and compasses.

This is the context for “gravity’s creative core.” Under the Machian conditions $\theta$3 and $\theta$4, most of the extra dimensionless data of generic Newtonian solutions disappear, and what remains can be reinterpreted as a single additional variable, $\theta$5, measuring the bending of the shape-space trajectory away from geodesic motion. Barbour writes that “the single extra bending variable reacts solely and directly to the scale-invariant shape potential $\theta$6.” In the three-body problem, represented on the shape sphere, this becomes the tendency of the representative point to bend so as to climb toward higher complexity. The paper identifies this “striving to greater complexity” as “the creative core of Newtonian gravity.”

4. Internal relativity and gravity as varying vacuum structure

Olaf Dreyer’s “internal relativity” reinterprets gravity by taking the equivalence principle, rather than metric geometry, as the decisive clue. In the standard formulation of general relativity, gravity differs from other interactions because it can be removed locally. The local disappearance of gravity is encoded through the connection, with transport governed by

$\theta$7

the Levi-Civita connection constructed from the metric $\theta$8, and free fall given by

$\theta$9

Because one can choose coordinates such that $G$0 at a point, gravity disappears locally. Dreyer’s proposal is to preserve that operational fact while denying that differential geometry and a metric are the only way to implement it (Dreyer, 2012).

The conceptual move is from local Minkowski spaces to local vacua of an underlying quantum many-body system. The paper uses a toy one-dimensional Ising model with Hamiltonian

$G$1

a ground state $G$2, and excitations written as momentum superpositions $G$3. The vacuum is taken to be degenerate, with vacuum manifold $G$4. An “internal observer” has access only to the low-energy excitations and cannot determine which constant vacuum the system occupies, provided all vacua support the same excitation spectrum and interaction structure. If these excitations display effective Lorentzian or Minkowskian physics, then every constant vacuum looks like Minkowski space to that observer.

Gravity then appears when the vacuum varies across the effective spacetime. Dreyer introduces a map

$G$5

where $G$6 is spacetime in the effective description. If $G$7 is constant, observers see ordinary gravity-free effective Minkowski physics. If $G$8 varies, local regions still look approximately gravity-free, but globally the changing vacuum has physical effects. This is the paper’s alternative implementation of the equivalence principle: gravity is not “connection between local Minkowski spaces,” but “relation among different local vacua.” The paper explicitly distinguishes this from approaches that aim to derive Einstein’s equation, stating that it does not aim at deriving Einstein’s equation and that gravity is “not about metrics at all,” but about “how the vacuum reacts to the presence of excitations of this vacuum.”

The toy Newtonian sector is developed with a scalar order parameter $G$9. Far from excitations the vacuum has value $SGPE$0, while excitations are deviations $SGPE$1. Bound states of such excitations are treated as effective massive objects. The Hamiltonian is assumed to contain a gradient-energy term

$SGPE$2

which is formally identical to the electrostatic field-energy functional. Using the electrostatic analogy, Dreyer writes an inverse-square force

$SGPE$3

provided the gravitational masses are defined by the flux integral

$SGPE$4

The bound state acts as a source of the vacuum field $SGPE$5, and the interaction energy of two vacuum distortions yields a long-range force.

To connect gravitational and inertial mass, the paper introduces a proportionality constant $SGPE$6 such that $SGPE$7, leading to

$SGPE$8

A heuristic analogy with the momentum of a moving charged spherical shell gives

$SGPE$9

The paper is explicit that this is not a derivation of a universal gravitational constant from a fully worked microscopic model. It is a qualitative or semi-parametric estimate, dependent on assumptions about bound-state structure and liable to species or energy dependence. It does not recover a curved metric, Einstein equations, universality of free fall, exact Lorentz symmetry, black holes, gravitational waves, or cosmology in any complete sense.

5. Coinduction, infinite games, and rational escalation

Pierre Lescanne’s contribution lies outside gravitation narrowly construed, but it supplies the “rationality” side of the composite term. The paper studies infinite extensive games and argues that behavior often labeled mad, paradoxical, or irrational—especially endless continuation in escalation games—can be formally justified as rational when agents reason in a world they take to have an infinite horizon or effectively infinite resources. The central methodological claim is that infiniteness is not simply the limit of finiteness. Consequently, one cannot truncate an infinite game, solve the finite approximation by backward induction, and extrapolate the result to the infinite case (Lescanne, 2011).

The paper replaces induction by coinduction. Induction is associated with the least fixpoint and finite objects; coinduction is associated with the greatest fixpoint and finite-or-infinite objects. Histories, games, and strategy profiles are defined coinductively. A game is either a leaf $N$0 or a node $N$1. A strategy profile is either a leaf $N$2 or a node $N$3, where a choice $N$4 is specified at each internal node. Because infinite play may never reach a leaf, utility is not defined as a total function on strategy profiles. Instead, the paper defines a relation $N$5 between a strategy profile, an agent, and a utility.

To control when utility exists, the paper introduces the inductive predicate LeadsToLeaf and the coinductive predicate AlwLeadsToLeaf. The latter corresponds to the temporal-style modality $N$6, “always,” and guarantees that utilities are available not only at the current profile but at every subprofile. This is required for subgame-perfect reasoning. If play does not lead to a leaf, the paper explicitly denies that agents have a utility there. An infinite run has no payoff: in the dollar auction the costs diverge, while in the $N$7 game the payoff oscillates.

Rationality is formalized through Nash equilibrium and subgame perfect equilibrium in the infinite setting. Convertibility is defined inductively so that one strategy profile is convertible to another for agent $N$8 if they differ only in finitely many choices made by $N$9. Nash equilibrium is then the standard no-profitable-unilateral-deviation condition stated using convertibility and $\ell_\textrm{\scriptsize rms}$0. The coinductive notion $\ell_\textrm{\scriptsize rms}$1 is the infinite-game analogue of backward induction: every subprofile must also be $\ell_\textrm{\scriptsize rms}$2, and the chosen move must be at least as good for the current player as the unchosen one. The paper’s key theorem is that a subgame perfect equilibrium is a Nash equilibrium.

Escalation is then defined structurally. A game has an escalation if it contains an infinite path such that at every stage there exists a strategy profile that is a subgame perfect equilibrium and whose prescribed move is to continue. The $\ell_\textrm{\scriptsize rms}$3 game is the simplest example. In the genuine infinite game, finite truncations are misleading, and new $\ell_\textrm{\scriptsize rms}$4s appear, including profiles in which Alice always continues and Bob stops always, and symmetrically profiles in which Bob always continues and Alice stops always. The dollar auction is the main economic example. Under the assumption $\ell_\textrm{\scriptsize rms}$5, the profile in which both agents stop immediately and forever is not even a Nash equilibrium in the infinite game, while the two asymmetric profiles in which one always continues and the other always stops are $\ell_\textrm{\scriptsize rms}$6. The paper proves both that the dollar game has an escalation and that the $\ell_\textrm{\scriptsize rms}$7 game has an escalation. By contrast, the infinipede yields only one subgame-perfect equilibrium, in which both players stop immediately at every turn. Infinite horizon alone therefore does not imply escalation.

The paper’s substantive conclusion is that escalation can be fully rational within a model of infinite resources or infinite horizon. Its caveat is equally explicit: this is only instrumental rationality inside a given model. Infinite runs are not assigned utilities, and reflective or epistemic rationality may require revising the belief that resources are effectively infinite.

6. Comparative themes, controversies, and limits

Taken together, these works exhibit a recurring opposition between observable relational structure and hidden explanatory surplus. Newton, as interpreted in the historical paper, accepts exact gravitational laws while refusing to infer an essential occult power of matter. Barbour removes implicit absolute scale and argues that only ratios of separations are physically meaningful for a universe. Dreyer makes observers internal to a low-energy sector and treats local vacua, rather than a background metric, as basic. Lescanne defines rationality on infinite objects through coinduction rather than by importing finite-tree methods. This suggests a common methodological preference for invariants, quotient structures, and internal observables over external background parameters.

The controversies, however, are domain-specific. In the Newton case, the dispute concerns whether gravitational attraction is an occult quality, a miracle, or a lawful phenomenon with a non-material mediator. In Barbour’s work, the controversy concerns whether a nearly uniform early universe should be treated as low gravitational entropy and whether the second law has “universal content” outside confined systems. In Dreyer’s framework, the controversy concerns whether gravitation must be metric and geometric at the fundamental level. In Lescanne’s analysis, the controversy concerns whether escalation is evidence of irrationality or can instead be equilibrium-consistent in an infinite game.

The limitations are equally explicit. The Newton paper offers a historical-philosophical interpretation rather than a new physical theory. Barbour states that the conclusion is based on the Newtonian $\ell_\textrm{\scriptsize rms}$8-body problem and “may not extend to the universe,” even if observation seems compatible with it. Dreyer does not derive Einstein equations, exact Lorentz symmetry, universality of free fall, or a complete relativistic theory, and he notes that the equivalence principle may fail to emerge universally. Lescanne emphasizes that the rationality at issue is instrumental, that infinite runs themselves are not assigned utilities, and that finite-resource worlds reintroduce finite backward-induction results.

A plausible implication is that “Grationality” is best understood as a comparative research motif in which both gravitation and rationality are recast once standard explanatory assumptions are suspended. In one case, the suspended assumption is that exact law entails essential material power; in another, that cosmological statistics can be imported from boxed thermodynamics; in another, that the equivalence principle requires a fundamental metric; and in another, that infinite reasoning is obtainable by extrapolating finite induction. Under that reading, the term designates a family resemblance among non-mechanistic, relational, emergent, and coinductive approaches rather than a single unified theory.