pith. sign in

arxiv: 1907.01598 · v2 · pith:JGGGP2KDnew · submitted 2019-07-02 · 🧮 math.HO · math.DS

Rod in a train: a mechanical problem of H.Whitney, or Much Ado About Nothing

Alexander Shen This is my paper

Pith reviewed 2026-05-25 10:12 UTC · model grok-4.3

classification 🧮 math.HO math.DS
keywords rod in traincontinuity argumentWhitney problemmathematical historyobjectionsCourant Robbinsconfiguration space
0
0 comments X

The pith

The objections to the continuity argument solution of the rod-in-train problem were not fully justified.

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

The paper traces the history of a 1941 mechanical puzzle about positioning a rod inside a moving train so it never touches the floor. A solution relying on a continuity argument was offered early on, yet drew repeated objections from mathematicians such as Littlewood, Broman, Poston, Stewart and Arnold. By reviewing the sequence of published comments, the paper concludes that those objections did not succeed in refuting the original continuity reasoning. The result is presented as a clarification that removes much of the accumulated confusion around the problem.

Core claim

The continuity argument that establishes the existence of a suitable initial rod position remains valid; the various published objections raised against it over subsequent decades did not demonstrate a genuine flaw in the argument.

What carries the argument

The continuity argument applied to the space of possible rod positions and orientations inside the moving train.

If this is right

  • The 1941 solution can be accepted as correct without additional topological machinery.
  • The recorded debate largely rests on misunderstandings rather than substantive mathematical disagreement.
  • Similar continuity arguments in other geometric problems require less additional justification than the objections suggested.
  • Historical reviews of such puzzles can reduce persistent confusion in the literature.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Retrospective examinations of other mid-century mathematical controversies may reveal comparable patterns of overstated objections.
  • The rod-in-train example could serve as a concise classroom illustration of how continuity arguments work in configuration spaces.
  • Language barriers between the original Russian discussion and later English comments may have amplified the apparent disagreements.

Load-bearing premise

The published comments and objections accurately capture the mathematical substance of the debate without significant miscommunication.

What would settle it

A concrete counter-example or rigorous proof showing that the continuity argument fails to guarantee a non-touching initial position for the rod would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.01598 by Alexander Shen.

Figure 294
Figure 294. Figure 294: Possible history of the moving rod for different initial conditions. [PITH_FULL_IMAGE:figures/full_fig_p011_294.png] view at source ↗
read the original abstract

In 1941 a mechanical problem about a rod in a moving train (there is a initial position such that rod does not touch the floor while train is moving) was published by R.Courant and H.Robbins in their popular book "What is mathematics?" and attributed to H.Whitney. Many mathematicians, including G.E.Littlewood, A.Broman, T.Poston, I.Stewart, V.Arnold, commented on this problem and its solution based on a continuity argument, and created a lot of confusion. In this paper we follow these developments and discuss at what extent the objections were justified. (In Russian)

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript is a historical review (in Russian) of the 'rod in a train' problem introduced in Courant and Robbins (1941) and attributed to Whitney. It traces the continuity-argument solution and the subsequent objections and comments by Littlewood, Broman, Poston, Stewart, and Arnold, concluding that those objections were not fully justified.

Significance. If the assessment of the objections holds, the paper clarifies a well-known episode in popular mathematics and the history of a continuity argument, documenting how interpretive confusion arose. As a literature review without new derivations or models, its primary contribution is historiographical rather than mathematical.

major comments (1)
  1. [Abstract and main discussion] The central claim that the objections were 'not fully justified' rests on a chronological survey of published comments. However, the manuscript does not reconstruct the continuity argument from first principles, quote and refute the precise mathematical steps in the cited objections (e.g., Littlewood on non-uniqueness or physical realizability, Arnold on topological issues), or exhibit a corrected model demonstrating that the argument survives those critiques. This leaves the justification assessment dependent on historical context alone.
minor comments (2)
  1. The manuscript is written in Russian with only an English abstract; an English translation or expanded English abstract would improve accessibility for an international readership.
  2. Ensure that all referenced works (including the original Courant-Robbins text and each objection) receive complete, consistent bibliographic citations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report and for recognizing the historiographical nature of the manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and main discussion] The central claim that the objections were 'not fully justified' rests on a chronological survey of published comments. However, the manuscript does not reconstruct the continuity argument from first principles, quote and refute the precise mathematical steps in the cited objections (e.g., Littlewood on non-uniqueness or physical realizability, Arnold on topological issues), or exhibit a corrected model demonstrating that the argument survives those critiques. This leaves the justification assessment dependent on historical context alone.

    Authors: The manuscript is a historical review whose stated aim (see abstract) is to follow the sequence of published comments on the Whitney rod-in-train problem and to assess the extent to which those objections were justified within the interpretive framework available at the time. Because the paper’s contribution is historiographical rather than mathematical, it deliberately limits itself to a chronological survey of the literature (Courant–Robbins, Littlewood, Broman, Poston, Stewart, Arnold) and does not attempt a first-principles reconstruction or a new corrected model. The assessment that the objections were “not fully justified” is therefore grounded in the documented sequence of publications and the interpretive confusion that arose, rather than in a technical refutation of each mathematical step. We maintain that this scope is appropriate for the paper’s genre and that adding a technical reconstruction would change its character. revision: no

Circularity Check

0 steps flagged

Historical review paper with no derivation chain or circular steps

full rationale

This is a historical discussion paper that chronologically follows published comments on the rod-in-train continuity argument and assesses the extent to which objections were justified. It contains no mathematical derivations, equations, predictions, or first-principles results. No steps match any of the enumerated circularity patterns (self-definitional, fitted-input-as-prediction, self-citation load-bearing, etc.). All references are to external prior works by other authors; the paper's conclusions are interpretive and do not reduce to any input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No mathematical derivations, free parameters, or new entities are introduced; the work is purely historical and interpretive.

pith-pipeline@v0.9.0 · 5631 in / 908 out tokens · 33178 ms · 2026-05-25T10:12:58.471132+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 2 internal anchors

  1. [1]

    М.: МЦНМО, 2002

    В.И.Арнольд, Что такое математика?. М.: МЦНМО, 2002. (2-е издание: 2008)

    work page 2002

  2. [2]

    М.: МЦНМО, 2009

    В.И.Арнольд, Математическое понимание природы. М.: МЦНМО, 2009. (2-е издание, исправленное: 2010, 3–5 издания, стереотипные: 2011, 2013, 2016.)

    work page 2009

  3. [3]

    С. В. Болотин, В. В. Козлов, Вариационное исчисление в целом, существование траекторий в области с границей и задача Уитни о перевернутом маятнике, Известия РАН, Сер. матем., 2015, том 79, выпуск 5, 39–46,http://www.mathnet. ru/links/517f47a75c35b951a03b01b699d9abef/im8413.pdf

    work page 2015

  4. [4]

    Reviewed by Brian E

    What Is Mathematics? An Elementary Approach to Ideas and Methods. Reviewed by Brian E. Blank,Notices of the AMS, December 2001, 1325–1329,https://www.ams. org/notices/200111/rev-blank.pdf

    work page 2001

  5. [5]

    Whitney,Nordisk Matematisk Tidskrift, 6(2) p

    Arne Broman, A mechanical problem by H. Whitney,Nordisk Matematisk Tidskrift, 6(2) p. 78–82 (1958). Published by: Mathematica Scandinavica,https://www.jstor. org/stable/24524634

    work page arXiv 1958

  6. [6]

    Review by: Leonard Gillman.The American Mathematical Monthly, 105(5), 485–488 (May 1998),https://www.jstor.org/stable/3109832

    Reviewed Work:What Is Mathematics?by Richard Courant, Herbert Robbins, Ian Stewart. Review by: Leonard Gillman.The American Mathematical Monthly, 105(5), 485–488 (May 1998),https://www.jstor.org/stable/3109832

    work page arXiv 1998

  7. [7]

    Oleg Zubelevich, Bounded Solutions to the System of 2-nd Order ODE and the Whitney pendulum, https://arxiv.org/abs/1502.04306. См. также журнальный вариант: Applicationes Mathematicae, 42, 159–165 (2015), DOI: 10.4064/am42-2-3

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4064/am42-2-3 2015

  8. [8]

    London, New York, Toronto: Oxford University press

    Richard Courant, Herbert Robbins,What is Mathematics? An elementary approach to ideas and methods. London, New York, Toronto: Oxford University press. 1941. Fourth printing: 1948. Ninth printing: 1958. Tenth printing: 1960. Eleventh printing: 1961. Русский перевод: Р.Курант, Г.Роббинс, Что такое математика? Элементарный очерк идей и методов. М.–Л.: Госуда...

    work page 1941

  9. [9]

    The Lever of Mahomet, by Richard Courant and Herbert Robbins, p. 2412 of the antology The World of Mathematics, A small library of the literature of mathematics from A’h-mos´ e the Scribe to Albert Einstein, presented with commentaries and notes by James R. Newman, London, George Allen and Unwin, Ltd, 1960

    work page 1960

  10. [10]

    (В книге указано: First published in 1941 by Oxford University Press

    Richard Courant and Herbert Robbins, revised by Ian Stewart,What is mathematics? An elementary approach to ideas and methods, 2nd edition, 1996, Oxford University Press. (В книге указано: First published in 1941 by Oxford University Press. First issued as Oxford University Press paperback, 1978. First published as a second edition, 1996.)

    work page 1996

  11. [11]

    J. E. Littlewood, A Mathematician’s Miscellany. London: Methuen & Co., Ltd.,

    work page

  12. [12]

    Переиздания: 2-е, стереотипное, 1965, 3-е, 1973, 4-е, стерео- типное, 1978, 5-е, исправленное, 1990

    Русский перевод: Дж.Литлвуд, Математическая смесь, перевод В.И.Левина, М.: Физматгиз,1962. Переиздания: 2-е, стереотипное, 1965, 3-е, 1973, 4-е, стерео- типное, 1978, 5-е, исправленное, 1990

    work page 1962

  13. [13]

    И. Ю. Полехин, Примеры использования топологических методов в задаче о пе- ревернутом маятнике на подвижном основании,Нелинейная динамика, 10(4), 465– 472 (2014), http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=nd& paperid=457

    work page 2014

  14. [14]

    Ivan Polekhin, An inverted pendulum with a moving pivot point: examples of topological approach, https://arxiv.org/pdf/1407.4787.pdf

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    Poston, Au courant with differential equations,Manifold, 18 (Spring 1976), 6–9

    T. Poston, Au courant with differential equations,Manifold, 18 (Spring 1976), 6–9. See also: Tim Poston, On deducing the presence of catastrophes Math´ ematiques et sciences humaines, tome 64 (1978), 71–99,http://www.numdam.org/item?id=MSH_ 1978__64__71_0

    work page 1976

  16. [16]

    Ian Stewart, Game, Set and Math: Enigmas and Conundrums, London: Basil Blackwell, Ltd., 1989

    work page 1989

  17. [17]

    Caroline Davis, Christopher Zeeman medal award lecture. Report. London mathematical society newsletter, No. 384 (September 2009),https://wwwf.imperial. ac.uk/~dgcrowdy/_news/LMS.pdf

    work page 2009

  18. [18]

    In:Handbook of topological fixed point theory, Springer, 2005,https: //doi.org/10.1007/1-4020-3222-6, pages 905–943,https://link.springer.com/ chapter/10.1007/1-4020-3222-6_23 17

    Roman Srzednicki, Klaudiusz W´ ojcik, Piotr Zgliczy´ ski, Fixed point results based on Wa˙ zewski method. In:Handbook of topological fixed point theory, Springer, 2005,https: //doi.org/10.1007/1-4020-3222-6, pages 905–943,https://link.springer.com/ chapter/10.1007/1-4020-3222-6_23 17

    work page doi:10.1007/1-4020-3222-6 2005

  19. [19]

    Roman Srzednicki, On periodic solutions in the Whitney’s inverted pendulum problem, https://arxiv.org/pdf/1709.08254.pdf. См. также журнальный ва- риант: Discrete and Continuous Dynamical Systems, Series S,12(7) (November 2019), 2127–2141, http://dx.doi.org/10.3934/dcdss.2019137 18

    work page doi:10.3934/dcdss.2019137 2019