Nonconservation of Energy and Loss of Determinism II: Colliding with an Open Set

David Atkinson; Porter Johnson

REVIEW 3 major objections 4 minor 14 references

Nonconservation of Energy and Loss of Determinism II: Colliding with an Open Set

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An actually infinite set of colliding balls is logically inconsistent.

desk verdict A clear, honest philosophy-of-physics paper; the new numerics are solid, but the claim that actual infinity is logically inconsistent rests on an asserted barrier rather than a fully derived contradiction. read the letter →

arxiv 1908.09865 v1 pith:5L5KKJ6M submitted 2019-08-26 physics.hist-ph physics.class-ph

David Atkinson , Porter Johnson This is my paper · ORCID

classification physics.hist-phphysics.class-ph

keywords supertasksZenoballsactualinfinitypotentialenergyconservationdeterminismNewtonianmechanicslogicalinconsistency

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 a scenario with an actually infinite line of stationary point masses at the positions 1, 1/2, 1/4, ..., plus an identical ball approaching from the left, contains a logical contradiction rather than a physical surprise. The approaching ball can collide with none of the stationary balls, yet it also cannot pass them, so the natural conditions describing the system are jointly inconsistent. The authors conclude that actual infinity must be excluded from the domain of mechanics, leaving only finite or potentially infinite collections. If correct, this blocks supertask-based arguments that mechanics fails to conserve energy or to be deterministic, and it removes the possibility of a ball appearing or vanishing at the accumulation point. Finite numerical simulations show the approaching ball either rebounds or comes to rest as the number of stationary balls grows, consistent with the potential-infinite limit.

What carries the argument

The load-bearing structure is the configuration of an infinite array of zero-size point masses with an accumulation point at the origin, together with an approaching point mass. The contradiction is generated by the collision rule that a moving point mass transfers all its momentum to an identical stationary point mass and stops, which implies the traveling ball can never reach a stationary ball because infinitely many others shield the leftmost one, yet it also cannot pass through the infinite array. In the finite approximation, the machinery is the elementary elastic collision law $v_{\mathrm{new}} = \frac{M-m}{M+m} v$, iterated to give $v_{AB} = \left(\frac{m_{AB}-1}{m_{AB}+1}\right)^{n+1}$ for $n$ identical unit-mass stationary balls, which determines the approaching ball's final velocity as a function of its mass and the number of balls.

What would settle it

The central claim would be settled by finding an explicit solution of the collision equations in which a point mass passes through an infinite accumulating set of stationary point masses without stopping, or by a finite-system simulation in which the approaching ball's final velocity fails to approach the value $\left(\frac{m_{AB}-1}{m_{AB}+1}\right)^{n+1}$ as $n$ increases.

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Extended reading notes

Core claim

The central claim is that a configuration of an actually infinite set of stationary point masses at the Zeno points $1, \frac12, \frac14, \ldots$ plus an identical point mass moving at constant speed toward the origin is logically inconsistent. The inconsistency is expressed by four incompatible conditions: the stationary point masses sit at the Zeno points; the traveling point mass moves uniformly and reaches the origin; a moving point mass stops only when it coincides with a stationary one and otherwise continues at constant speed; and the traveling point mass comes to rest before reaching the point $1$. Since these cannot all hold, any conclusion drawn from the scenario—such as the claim that the ball vanishes at the origin, or that energy and determinism fail—is a non sequitur. The paper therefore restricts mechanics to finite or potentially infinite systems and supports this with numerical simulations of finite systems whose behavior approaches a well-defined limit.

Load-bearing premise

The conclusion that the approaching ball cannot pass the origin assumes that an infinite set of zero-size point masses with an accumulation point forms an impenetrable barrier; that assumption is asserted rather than derived from collision mechanics, and if the ball could traverse the measure-zero gaps between point masses, the contradiction would not follow.

Editorial extensions

If this is right

If actual infinity is excluded from mechanics, the Zeno-ball supertask no longer demonstrates energy nonconservation or indeterminism.
For any finite number of stationary balls, energy and momentum are conserved; the potential-infinite limit inherits these conservation laws.
The inconsistency is present at all times, not just at the moment of arrival at the accumulation point, so no instantaneous disappearance or creation can be singled out.
The domain restriction would also block the time-reversed scenario in which a ball spontaneously appears at the accumulation point.
In finite systems of identical stationary balls, the approaching ball rebounds when lighter than a stationary ball and, when equal or heavier, its final velocity tends to zero as the number of balls grows.

Reading between the lines

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

The inconsistency may be an artifact of idealizing point masses as both zero-size and impenetrable; a model with finite-radius balls or with point masses that can pass one another would not produce the contradiction, suggesting the impenetrability assumption carries the argument.
The same prohibition on actual infinity would apply to any physical idealization that relies on an infinite collection, such as infinite lattices or thermodynamic limits, which are usually treated as potential limits; drawing that boundary is left open.
The paper's finite-system asymptotics yield a concrete check: simulate an incoming ball against a growing chain of stationary balls with decreasing masses and test whether the final velocity converges to the predicted limit, which would test the domain restriction without invoking actual infinity.

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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

Colleague,

Quick take: this is a short, readable paper arguing that Alper and Bridger's Zeno-ball supertask is logically inconsistent, so actual infinity should be kept out of mechanics. The genuinely new piece is the finite-n numerical study with geometrically decreasing masses; the inconsistency argument itself is explicitly credited to the authors' earlier joint work with Peijnenburg.

What it does well: the finite collision mechanics is correct, formula (3) for equal masses is right, and the appendix gives enough detail that the numerical results are reproducible in principle, even without shipped code. The paper is honest about the 'discontinuity at infinity' and about the Cesaro-mean smoothing being a heuristic rather than a theorem. That is good practice for this genre.

The soft spots are in the philosophical core. Section 2's claim that an infinite set of point masses 'should have blocked its way' is asserted, not derived. The formal four-condition inconsistency does real work only if you accept that the collision rule applies globally to an infinite, non-well-ordered sequence with no first collision. The stress-test note is right that alternative continuations—pass-through, stop at the origin, or Alper-Bridger disappearance—are not genuinely ruled out within the text. The paper's conclusion that actual infinity is logically inconsistent is therefore a bit stronger than what is shown; what is shown is that one particular formalization has no model. That is a legitimate position, but it is a philosophical choice, not a forced theorem.

Also, the restriction of mechanics to potentially infinite systems is stipulated rather than derived. That is fine as a proposal, but it should be labeled more carefully as a stipulation.

Who is this for? Philosophers of physics working on supertasks and determinism. Physicists will find the numerics trivially correct and the conclusion unsurprising. The paper deserves a serious referee: it engages the literature, is internally consistent, and the numerical extension is new. A referee should press on the barrier assumption and on whether the contradiction survives without condition 4.

My recommendation: send it to peer review, but with the expectation that the logical-inconsistency claim needs to be either sharpened or softened to 'the conditions are inconsistent given these background assumptions.'

Referee Report

3 major / 4 minor

Summary. The paper considers the Alper–Bridger variant of the Zeno-balls supertask, in which a ball approaches an actually infinite set of point masses located at Zeno points. It argues that the four stated conditions describing this system are logically inconsistent, so that the correct response to supertask arguments for energy non-conservation and indeterminism is to exclude actual infinities from the domain of Newtonian and relativistic mechanics, keeping only finite or potentially infinite systems. The paper derives the final velocity of the moving ball for finite systems of identical balls and presents numerical simulations for geometrically decreasing masses, using these to infer the behavior of the potentially infinite system.

Significance. If the inconsistency claim is correct, the paper would provide a principled resolution of a long-standing debate: supertask-based violations of energy conservation and determinism would be shown not to follow from mechanics, because the scenarios that generate them involve an actual infinity, which falls outside the domain of the laws. The finite-n formula (3) is transparent and useful, and the paper is commendably explicit about the 'discontinuity at infinity' and about the stipulative character of the proposed domain restriction. However, the central logical proof is compressed, and the numerical inference for the potential-infinite case is not rigorously grounded; both points need attention before the main conclusion is established.

major comments (3)

[§2, conditions (1)–(4) and Fig. 1] The central claim that the Alper–Bridger system is logically inconsistent is asserted too quickly. The sentence 'an infinite number of Zeno balls should have blocked its way' is not derived in the text; condition (3) only says that the moving ball comes to rest when it occupies the same position as a stationary ball, and it does not explicitly say that it remains at rest thereafter. Please expand the argument: after reaching the origin, the AB ball would coincide with a Zeno ball at each time 2^{-k}; by condition (3) and Newton's first law it would then remain at rest, so at time 1 it would have to be both at position 1/2 (from the rest at t=1/2) and at position 1 (from constant motion), a contradiction. If 'comes to rest' is intended as only an instantaneous event, an explicit axiom of permanent rest is needed; as printed, the inconsistency is conditional on an unstated reading.
[§3, Figs. 2–4 and the Cesàro-mean passage] The inference from the finite-n simulations to the potential-infinite behaviour relies on the Cesàro mean, but the authors explicitly note that the sequence v_AB does not converge as n→∞. A Cesàro mean is a summability method, not a physical law; different summation methods generally give different limits, so the 'asymptote' read off the smoothed curves is a numerical extrapolation rather than a consequence of Newtonian mechanics. Please either prove (or state as a definition) that the potential-infinite final velocity is the Cesàro or other specified mean of the finite-n results, or present the finite-n results only as numerical illustrations and refine the claim that 'we can easily answer the question as to what happens to the AB ball.'
[§4, 'forced by logic' and 'we stipulate'] There is a tension between saying that the restriction to potential infinity is 'forced by logic' and the later statement that 'we stipulate that systems involving an actual infinity of elements do not lie within the domain' of mechanics. If the restriction is stipulated, then the paper has not shown that Laraudogoitia-style conclusions are non sequiturs; it has chosen a convention that avoids them. Please state clearly which claim is intended: either the scenario is inconsistent under the stated physical rules (which requires the proof requested above), or the ban is a definitional choice, in which case 'forced by logic' should be withdrawn.

minor comments (4)

[Appendix] The collision-update equations in the Appendix appear to have the coefficients in the formula for vnew_{k-1} interchanged; for equal masses the printed formula leaves v_{k-1} unchanged, which contradicts the equal-mass collision behavior described in §3. Please verify and correct the formula.
[References] References [5] and [9] are the same Alper–Bridger paper and should be consolidated; references [3] and [7] are cited as 'to be published' and should be updated with publication data.
[Miscellaneous typos] The author affiliation line contains 'Netherland s' with a stray space, and §4 contains 'familiar' for 'familiar'; these typos should be corrected.
[Figure 6] Figure 6's vertical axis labels appear garbled ('10 10 10 10'); please relabel the logarithmic axis with proper powers of ten.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the §2 inconsistency is argued from the stated collision conditions, the finite-n simulations are independent checks, and the domain restriction is an explicit stipulation rather than a fitted prediction.

full rationale

The paper's central derivation is the inconsistency claim in Section 2. That claim is argued directly from the four listed conditions (stationary point masses at Zeno points; a moving AB ball; the equal-mass collision rule "when the moving mass point occupies the same position as a stationary mass point, it comes to rest"; and rest before reaching point 1), not from any fitted parameter or from prior papers. The finite-n calculations in Section 3 are independent numerical integrations of the elastic collision formula and are not used to fit the contradiction. The conclusion that restricting mechanics to finite or potentially infinite systems restores conservation and determinism is openly stipulated rather than derived: the paper states that the physicist "defines an infinite system to be the conceptual result of allowing a finite system to grow without bound" and later says "To remove the absurdity we stipulate that systems involving an actual infinity of elements do not lie within the domain of Newtonian, or of Einsteinian mechanics." These are explicit definitional choices, not hidden reductions. Self-citations to the authors' previous work I, [4], and [7] are background: they support the generalized relativistic version, time-reversal properties, and prior treatments of isomorphic systems, but the Section 2 contradiction does not reduce to those citations. The only genuinely contestable move is the sentence "But this is also impossible, since an infinite number of Zeno balls should have blocked its way," which is asserted rather than derived from the stated conditions; that is a missing justification or correctness risk, not a circular step, because the conclusion is not presupposed by the premises. Thus the derivation is self-contained in the sense relevant to circularity analysis.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests mainly on modeling assumptions about point particles and on the stipulation that a physical infinite system is a limit of finite ones. No new entities are introduced, and the scan variables mu and m_AB are not fitted to external data.

free parameters (2)

mu = 0.5, 0.7, 0.95 in figures
Mass ratio for the geometric Zeno masses mp = mu^p; chosen by hand to illustrate different regimes, not fitted to external data.
m_AB = varied from below 1 to 10000 (log scale)
Mass of the incoming AB ball; scanned to map collision outcomes, but the central claim does not depend on a single fitted value.

assumptions (4)

domain assumption Equal-mass elastic collision of point masses in one dimension exchanges velocities; general collision formula v_new = (M-m)/(M+m)v.
Used throughout Section 3 and the Appendix to advance the finite simulations; assumes instantaneous pairwise elastic collisions.
domain assumption The Zeno balls are point masses at exactly the Zeno points, with no ball at the accumulation point.
Section 2 setup; the accumulation point with no occupant is essential to the alleged contradiction.
domain assumption An infinite physical system is understood as the limit of finite systems, and physical quantities are those with finite limits.
Section 1 and Section 3; this stipulation is what makes energy conservation and determinism hold in the potential-infinite limit.
standard math Classical logic, including ex falso quodlibet, is used to reject the actual-infinite scenario once its conditions are inconsistent.
Section 2 uses logical inconsistency to disqualify the scenario, citing Peijnenburg and Atkinson [7].

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Cite this review

Pith. "Pith review of Nonconservation of Energy and Loss of Determinism II: Colliding with an Open Set." pith.science (2026). https://pith.science/paper/5L5KKJ6M

@misc{pith2026190809865,
  author       = {Pith},
  title        = {Pith review of: Nonconservation of Energy and Loss of Determinism II: Colliding with an Open Set},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5L5KKJ6M}},
  note         = {Machine review of arXiv:1908.09865}
}

read the original abstract

An actual infinity of colliding balls can be in a configuration in which the laws of mechanics lead to logical inconsistency. It is argued that one should therefore limit the domain of these laws to a finite, or only a potentially infinite number of elements. With this restriction indeterminism, energy non-conservation and (creatio ex nihilo) no longer occur. A numerical analysis of finite systems of colliding balls is given, and the asymptotic behavior that corresponds to the potentially infinite system is inferred.

Figures

Figures reproduced from arXiv: 1908.09865 by the authors.

**Figure 1.** AB ball and Zeno balls 0 1 There is no ball at 0, but the origin is a point of accumulation of the locations of the Zeno balls. If the AB ball were to collide with a Zeno ball, it would come to rest, thereby imparting all its energy to the Zeno ball, which would move off with the speed that the AB ball originally had. However, there is no Zeno ball with which it could collide. For suppose, per impossibile, that it d… view at source ↗

**Figure 2.** Final velocity of AB ball: mAB = 1 and µ = 0.5 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Final velocity of AB ball: mAB = 1 and µ = 0.7 In Figures 2 – 4 the final velocity of the AB ball, vAB, is shown for µ = 0.5, 0.7 and 0.95, as a function of the number of Zeno balls. As can be seen from these graphs, the qualitative behaviour of the AB ball is that it rebounds from the set of Zeno balls, but with reduced speed. This bears some resemblance to what would happen in an inelastic collision of the AB ball… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Final velocity of AB ball: mAB = 1 and µ = 0.95 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Final velocity of AB ball as a function of its mass: [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: As in the previous figure, but with a logarithmic mass-scale [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

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Mendell, H.: Aristotle and Mathematics. Stanford Encyclopedia of Phi- losophy, Chapter 12 (2004). http://plato.stanford.edu/entries/aristotle-mathematics/#12

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Atkinson, D.: Losing energy in classical, relativistic and quantum me - chanics. Stud. Hist. Phil. Mod. Phys. 38, 170-180 (2007). Atkinson, D.: A relativistic Zeno process. Synthese 160, 5-12 (2008)

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Alper, J.S., Bridger, M.: Newtonian Supertasks: A Critical Analysis . Synthese 114, 335-369 (1998)

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Angel, L.: Evens and Odds in Newtonian Collision Mechanics. British Journal for the Philosophy of Science 56, 179188 (2005)

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Peijnenburg, J., Atkinson, D.: Lamps, cubes, balls and walls, Zeno problems and solutions”. To be published in Philosophical Studies

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Earman, J., Norton, J.D.: Comments on Laraudogoitia’s ‘Classical P ar- ticle Dynamics, Indeterminism and a Supertask. British Journal for the Philosophy of Science 49, pp. 123-33 (1998)

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Stanford Encyclopedia of Phi- losophy, Chapter 12 (2004)

Mendell, H.: Aristotle and Mathematics. Stanford Encyclopedia of Phi- losophy, Chapter 12 (2004). http://plato.stanford.edu/entries/aristotle-mathematics/#12

work page 2004

[2] [2]

Mind 105, 81-83 (1996)

P´ erez Laraudogoitia, J.: A Beautiful Supertask. Mind 105, 81-83 (1996)

work page 1996

[3] [3]

Inﬁnitely many colliding balls

Atkinson, D., Johnson, P.: Nonconservation of Energy and Loss of De- terminism I. Inﬁnitely many colliding balls. to be published in Founda- tions of Physics

work page

[4] [4]

Atkinson, D.: Losing energy in classical, relativistic and quantum me - chanics. Stud. Hist. Phil. Mod. Phys. 38, 170-180 (2007). Atkinson, D.: A relativistic Zeno process. Synthese 160, 5-12 (2008)

work page 2007

[5] [5]

Synthese 114, 335-369 (1998)

Alper, J.S., Bridger, M.: Newtonian Supertasks: A Critical Analysis . Synthese 114, 335-369 (1998)

work page 1998

[6] [6]

British Journal for the Philosophy of Science 56, 179188 (2005)

Angel, L.: Evens and Odds in Newtonian Collision Mechanics. British Journal for the Philosophy of Science 56, 179188 (2005)

work page 2005

[7] [7]

To be published in Philosophical Studies

Peijnenburg, J., Atkinson, D.: Lamps, cubes, balls and walls, Zeno problems and solutions”. To be published in Philosophical Studies

work page

[8] [8]

British Journal for the Philosophy of Science 49, pp

Earman, J., Norton, J.D.: Comments on Laraudogoitia’s ‘Classical P ar- ticle Dynamics, Indeterminism and a Supertask. British Journal for the Philosophy of Science 49, pp. 123-33 (1998)

work page 1998