Pith. sign in

REVIEW 3 major objections 4 minor 36 references

Poincar\'e on Gibbs and on Probability in Statistical Mechanics

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A long-overlooked 1906 Poincaré paper, read against Gibbs's 1902 treatise, introduced the uncertainty-of-initial-conditions rationale and the fine/coarse-grained entropy distinction to statistical mechanics.

desk verdict Careful historical recovery of Poincaré's 1906 paper with two plausible priority claims; the ontic-probability conclusion is asserted, not argued, but the history stands on its own. read the letter →

arxiv 2505.12168 v1 pith:46FMRS2Z submitted 2025-05-17 physics.hist-ph cond-mat.stat-mech

Bruce D. Popp This is my paper · ORCID
classification physics.hist-phcond-mat.stat-mech PACS 01.65.+g
keywords PoincaréGibbsstatisticalmechanicsfine-grainedentropycoarse-grainedhomoclinicpointsonticprobabilityfrequentist
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 Poincaré's 1906 reflections on kinetic theory introduced two ideas that now anchor statistical mechanics: probability enters because initial conditions are uncertain, and entropy has to be evaluated on finite cells rather than at the infinitesimal limit. It reads the 1906 paper as a companion to Gibbs's 1902 Elementary Principles and as an application of Poincaré's own 1890 discovery that nearby trajectories in Hamiltonian systems can diverge sharply. If the reading is right, Poincaré deserves credit for the uncertainty-of-initial-conditions rationale and for the terms fine-grained and coarse-grained entropy, and the need for ensembles follows from deterministic chaos rather than from practical convenience. The paper also concludes that both Poincaré and Gibbs wanted an account of irreversibility and did not reach one.

What carries the argument

The load-bearing object is the Hamiltonian phase space with coordinates $(\mathbf{q},\mathbf{p})$ and the entropy functional $S=-k\int P\log P\,d\tau$. Under deterministic Hamiltonian flow the fine-grained entropy built from the exact probability density $P$ and infinitesimal phase-space volume $d\tau$ is constant, whereas the coarse-grained entropy evaluated over finite cells $\delta$ changes with time. The mechanism that makes $P$ genuinely probabilistic is sensitive dependence on initial conditions near homoclinic points, established in Poincaré's 1890 study of the three-body problem: small differences in starting data can decide between stable and unstable trajectories, so no single trajectory can stand for the system.

What would settle it

A pre-1906 publication using the equivalent of fine-grained or coarse-grained entropy would falsify the priority claim, and a rigorous argument that sensitive dependence on initial conditions produces only epistemic uncertainty would falsify the ontic-frequentist conclusion.

Watch

Extended reading notes

Core claim

Poincaré's 1906 paper Réflexions sur la théorie cinétique des gaz presented statistical mechanics as a many-body problem in Hamiltonian mechanics and introduced a choice between two hypotheses: if initial conditions are fully known, the deterministic equations make probability unnecessary and the entropy sum diverges; if initial conditions are uncertain, probability is unavoidable and the entropy stays finite. Poincaré adopted the second hypothesis. His earlier 1890 work on the three-body problem had shown that near homoclinic points nearly identical initial conditions can separate into stable and unstable trajectories, so the uncertainty is structural rather than a removable inconvenience. From this the paper concludes that ensemble probability is ontic and frequentist, with no a priori value. On entropy, Poincaré distinguished fine-grained entropy, computed in the infinitesimal limit and constant under the dynamics, from coarse-grained entropy, computed over finite cells and capable of changing; he asserted without proof that the coarse entropy of physicists always increases.

Load-bearing premise

The load-bearing premise is that deterministic chaos makes probability a real property of the system rather than a statement of our ignorance, a step the paper asserts rather than defends; a second unproven premise is that Poincaré's 1906 paper really is the first to use the terms fine-grained and coarse-grained entropy.

Editorial extensions

If this is right

  • Poincaré, not later authors, would be credited with the uncertainty-of-initial-conditions argument and with the terms fine-grained and coarse-grained entropy.
  • The use of Gibbsian ensembles would be a consequence of deterministic chaos rather than a practical approximation, so phase-averaging is required even though the underlying equations are deterministic.
  • Coarse-grained entropy changes while fine-grained entropy is constant, which locates the source of macroscopic entropy change in the finite resolution of observation or in the finite size of the constituents.
  • The Hamiltonian foundation extends statistical mechanics beyond gases to asteroids, stars, oscillators and other many-body systems whose forces derive from a potential.
  • The long-standing goal of deriving macroscopic irreversibility from reversible mechanics remains open, since the paper records that neither Poincaré nor Gibbs reached it.

Reading between the lines

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

  • The paper does not settle whether deterministic chaos produces ontic probability or merely epistemic ignorance; a reader who wants the modern claim to stand would need that step defended.
  • The priority claim for fine-grained and coarse-grained entropy is qualified in the paper as appearing to be the first use; a systematic search of pre-1906 sources would test it directly.
  • If the ontic-frequentist reading is accepted, philosophical treatments of statistical mechanics that treat probability as a bookkeeping device would have to confront the chaos-based argument rather than dismiss ensembles as a convenience.
  • The unproved assertion that coarse entropy always increases suggests a concrete open problem: whether some mixing or decay condition on the flow would turn that assertion into a theorem.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript examines Poincaré's 1906 article "Réflexions sur la théorie cinétique des gaz" as a previously neglected contribution to the foundations of statistical mechanics. It argues that Poincaré introduced the uncertainty-of-initial-conditions rationale for using probability and the distinction between fine-grained and coarse-grained entropy, that the terms "fine-grained" and "coarse-grained" were first used by Poincaré, and that Poincaré's 1890 results on sensitivity to initial conditions justify Gibbsian ensembles and imply that probability is ontic and frequentist and has no a priori value. The paper also reconstructs the reception of Poincaré's paper by Kroo, the Ehrenfests, Zermelo, Burbury, and others, and concludes that neither Poincaré nor Gibbs succeeded in explaining irreversibility.

Significance. The paper has genuine historical value: it quotes primary sources with page numbers, distinguishes carefully between Gibbs's concepts and Poincaré's terminology, and is admirably explicit about open questions such as whether Gibbs had read Poincaré and whether irreversibility has been explained. If the priority claims are correct, the paper would be a useful correction to the standard attribution of fine- and coarse-grained entropy terms and of the uncertainty-of-initial-conditions rationale. The analysis of Burbury's alternative, physically motivated basis for coarse graining is also a valuable contribution. However, the paper's broader significance is weakened by an unsupported philosophical inference in Sections 4 and 5: the move from deterministic chaos to "ontic and frequentist" probability is asserted rather than argued, and it is in tension with the paper's own epistemic account of the continuous hypothesis in Section 3.3. The historical description of Poincaré's reasoning is defensible, but the modern-relevance claim built on the ontic reading is not established.

major comments (3)
  1. [§5 (also §4 and the abstract)] The conclusion that sensitivity to initial conditions "means that ensemble probability is ontic and frequentist and does not have an a priori value" is asserted without adequate support. The paper's own reconstruction in §3.3 is epistemic: the continuous hypothesis is that initial conditions "are not fully known" and that "we can only evaluate the probability." Deterministic Hamiltonian equations with uncertain initial data yield uncertainty about trajectories; chaos amplifies the consequences of ignorance but does not by itself make probability a property of the system. The paper neither cites a Poincaré passage in which he treats probability as ontic nor engages the substantial philosophical literature on chance in deterministic systems. This is load-bearing because the abstract and §5 frame the ontic claim as the main modern relevance of the historical account. The claim should be removed or supported by explicit textual evidence from Poincaré and by an argument that deterministic chaos establishes ontic rather than epistemic probability. As it stands, even the "frequentist" label is underdetermined: an ensemble frequency can be an epistemic tool.
  2. [§3.4 and §4] The terminological priority claim is stated more strongly than the evidence warrants. Section 3.4 says Poincaré's 1906 paper "appears to be the first use of the terms fine- and coarse-grained," but Section 4 states without the hedge that "The terms were however first used by Poincaré." The evidence cited—Poincaré's 1906 paper and the 1911 Ehrenfest review—does not rule out earlier or independent uses, and no systematic search of the relevant literature is reported. Since the paper's significance includes giving Poincaré credit for the terms, this claim needs either a documented search or a consistently provisional formulation.
  3. [§4 and §5] The statement that "there is no genuine a priori outcome for a system" overstates what Poincaré's 1890 sensitivity result implies. The 1890 work concerns nearby trajectories that separate near homoclinic points within a deterministic Hamiltonian framework; it does not show that the system lacks a determinate future evolution. At most it shows that outcomes are unpredictable in practice given coarse knowledge of initial conditions. This stronger reading is exactly what supports the paper's "ontic" conclusion, so the paper should distinguish practical unpredictability from indeterminism and from ontic probability.
minor comments (4)
  1. [§5] The sentence "the analytical mechanics is Gibbsian statistical mechanics" is ungrammatical as printed; it should read something like "the analytical-mechanics foundation of Gibbsian statistical mechanics."
  2. [§3.3] In the last sentence of §3.3, "and here shown that" should be "and here showed that" (or "and here shows that").
  3. [§2.1] The distinction between Poincaré 1906d, 1906e, and 1906f is relegated to a footnote; since the paper's entire analysis rests on 1906f, the distinction deserves one or two sentences in the main text.
  4. [§4 and §5] The two conclusions repeat nearly identical content, including the same opening phrase "In summary, we focused on two concepts." Combining them or clearly separating the historical summary from the philosophical discussion would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the historical-interpretive argument is anchored in primary texts and does not reduce by construction to its own inputs.

full rationale

No circularity of the kind defined here is present. The paper's central claims are historical-interpretive: that Poincaré's 1906 article introduced the uncertainty-of-initial-conditions rationale and the fine/coarse-grained-entropy terminology to statistical mechanics, and that his 1890 dynamical-systems results justify the use of ensembles. These claims are anchored in quoted primary texts (Poincaré 1906f; Gibbs 1902; Burbury 1903; Ehrenfest and Ehrenfest-Afanassjewa 1911) rather than in fitted parameters or definitions that presuppose the conclusion. The sole self-citation (Popp 2022) is a supporting reference for an application to globular clusters and is not load-bearing. The inference from sensitivity to initial conditions to 'ontic and frequentist' probability is a philosophical argument that may be contestable, but contestability is a correctness risk, not circularity; it does not reduce to an equation or to a self-citation. The priority claim about 'fine-grained' and 'coarse-grained' is explicitly hedged with 'appears to be the first use,' and the relevant historical evidence is external. The appended footnote expressing personal uncertainty about non-deterministic irreversibility is an honest limitation, not a circular step. Thus the derivation is self-contained and the appropriate score is 0.

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

No free parameters or invented entities appear, as the paper is an historical and conceptual analysis rather than a mathematical construction. The load-bearing assumptions are interpretive choices about which texts to privilege and which inferences to draw from them, especially the ontic-frequentist reading of chaos-induced probability.

assumptions (4)
  • domain assumption Hamiltonian dynamics with differentiable potentials is an adequate model for the systems of statistical mechanics (gases, oscillators, star clusters).
    The paper's whole reading of Poincare and Gibbs depends on their shared Hamiltonian framework, introduced in Section 2.3. It excludes collisional kinetic theory assumptions and is presented as the foundation for everything that follows.
  • domain assumption The continuous hypothesis (uncertain initial conditions) is the correct choice for describing physical systems, and the discontinuous hypothesis is rejected because it yields infinite entropy.
    Poincare's choice, described in Section 3.3, is adopted by the paper as the basis for probability. The rejection argument is taken from Poincare's text and not independently examined.
  • ad hoc to paper Sensitivity to initial conditions (chaos) entails that probability in statistical mechanics is ontic and frequentist, not merely epistemic.
    This inference, stated in Sections 4 and 5, is central to the author's interpretive claim but is asserted without engaging the philosophical literature on chance in deterministic systems.
  • standard math Standard results of Hamiltonian mechanics: phase-space volume invariance (Liouville), recurrence, and constancy of fine-grained entropy.
    Invoked in Sections 2.3 and 3.4 as background; accepted as proven in the cited primary literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Poincar\'e on Gibbs and on Probability in Statistical Mechanics." pith.science (2026). https://pith.science/paper/46FMRS2Z

@misc{pith2026250512168,
  author       = {Pith},
  title        = {Pith review of: Poincar\'e on Gibbs and on Probability in Statistical Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/46FMRS2Z}},
  note         = {Machine review of arXiv:2505.12168}
}
read the original abstract

This paper reviews a paper from 1906 by J. Henri Poincar\'e on statistical mechanics with a background in his earlier work and notable connections to J. Willard Gibbs. Poincar\'e's paper presents important ideas that are still relevant for understanding the need for probability in statistical mechanics. Poincar\'e understands the foundations of statistical mechanics as a many-body problem in analytical mechanics (reflecting his 1890 monograph on The Three-Body Problem and the Equations of Dynamics) and possibly influenced by Gibbs independent development published in chapters in his 1902 book, Elementary Principles in Statistical Mechanics. This dynamical systems approach of Poincar\'e and Gibbs provides great flexibility including applications to many systems besides gasses. This foundation benefits from close connections to Poincar\'e's earlier work. Notably, Poincar\'e had shown (e.g. in his study of non-linear oscillators) that Hamiltonian dynamical systems display sensitivity to initial conditions separating stable and unstable trajectories. In the first context it precludes proving the stability of orbits in the solar system, here it compels the use of ensembles of systems for which the probability is ontic and frequentist and does not have an a priori value. Poincar\'e's key concepts relating to uncertain initial conditions, and fine- and coarse-grained entropy are presented for the readers' consideration. Poincar\'e and Gibbs clearly both wanted to say something about irreversibility, but came up short.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 32 canonical work pages

  1. [1]

    : Vorlesungen über Gastheorie : Th

    bbook Boltzmann , L. : Vorlesungen über Gastheorie : Th . Theorie der Gase Mit Einatomigen Molekülen , Deren Dimensionen Gegen die Mittlere Weglänge verschwinden . Vorlesungen über Gastheorie . Johann Ambrosius Barth , Leipzig ( 1896 ) bbook

    work page

  2. [2]

    : Lectures on Gas Theory

    bbook Boltzmann , L. : Lectures on Gas Theory . Dover Publications , New York ( 1995 ) bbook

    work page 1995

  3. [3]

    : On the Variation of Entropy as Treated by J

    barticle Bumstead , H.A. : On the Variation of Entropy as Treated by J . Willard Gibbs . Philosophical Magazine and Journal of Science 7 ( XXXVII ), 8 -- 14 ( 1904 ) barticle

    work page 1904

  4. [4]

    : On the Variation of Entropy as Treated in Willard Gibbs ' Statistical Mechanics

    barticle Burbury , S.H. : On the Variation of Entropy as Treated in Willard Gibbs ' Statistical Mechanics . Philosophical Magazine and Journal of Science 6 ( 32 ), 251 -- 9 ( 1903 ) barticle

    work page 1903

  5. [5]

    : Le Potentiel Thermodynamique et Ses Applications à la Mécanique Chimique et à L'étude des Phénomènes électriques

    bbook Duhem , P.M.M. : Le Potentiel Thermodynamique et Ses Applications à la Mécanique Chimique et à L'étude des Phénomènes électriques . A. Hermann , Paris ( 1886 ). https://gallica.bnf.fr/ark:/12148/bpt6k62445r bbook

    work page

  6. [6]

    Statistischen Mechanik

    barticle Ehrenfest-Afanassjewa , T. , Ehrenfest , P. : Bemerkung zur Theorie der Entropiezunahme in der " Statistischen Mechanik " von W . Gibbs . Sitzungsberichte der Kaiserliche Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Klasse 115 , 89 -- 98 ( 1906 ) barticle

    work page 1906

  7. [7]

    , Ehrenfest-Afanassjewa , T

    bchapter Ehrenfest , P. , Ehrenfest-Afanassjewa , T. : Begriffliche Grundlagen der statistischen Auffassung in der Mechanik . In: Encyklopädie der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen vol. IV, Issue 6, Article 32 , pp. 3 -- 90 . B. G. Teubner , Leipzig ( 1911 ) bchapter

    work page 1911

  8. [8]

    , Ehrenfest-Afanassjewa , T

    bbook Ehrenfest , P. , Ehrenfest-Afanassjewa , T. : The Conceptual Foundations of the Statistical Approach in Mechanics . Dover books on physics and chemistry . Dover Publications , New York ( 1990 ) bbook

    work page 1990

Show all 36 references
  1. [9]

    : Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundations Of Thermodynamics

    bbook Gibbs , J.W. : Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundations Of Thermodynamics . C. Scribner's Sons , New York ( 1902 ) bbook

    1902

  2. [10]

    : The Scientific Papers of J

    bbook Gibbs , J.W. : The Scientific Papers of J . Willard Gibbs , Vol . II , Dynamics , Vector Analysis and Multiple Algebra , Electromagnetic Theory of Light , etc. vol. 2 . Longmans, Green, and Co. , London ( 1906 ). http://archive.org/details/in.ernet.dli.2015.90573 bbook

    1906

  3. [11]

    : Book Review : Elementary Principles in Statistical Mechanics , Developed with Especial Reference to the Rational Foundations of Thermodynamics

    barticle Hadamard , J. : Book Review : Elementary Principles in Statistical Mechanics , Developed with Especial Reference to the Rational Foundations of Thermodynamics . Bulletin of the American Mathematical Society 12 ( 4 ), 194 -- 211 ( 1906 ) 10.1090/S0002-9904-1906-01319-2...

    1906 doi

  4. [12]

    : Poincaré, celestial mechanics, dynamical-systems theory and “chaos”

    barticle Holmes , P. : Poincaré, celestial mechanics, dynamical-systems theory and “chaos” . Physics Reports 193 ( 3 ), 137 -- 163 ( 1990 ) 10.1016/0370-1573(90)90012-Q barticle

    1990 doi

  5. [13]

    : Gibbs vs Boltzmann Entropies

    barticle Jaynes , E.T. : Gibbs vs Boltzmann Entropies . American Journal of Physics 33 ( 5 ), 391 -- 398 ( 1965 ) 10.1119/1.1971557 barticle

    1965 doi

  6. [14]

    : The Physics of J

    bchapter Klein , M.J. : The Physics of J . Willard Gibbs in His Time . In: Proceedings of the Gibbs Symposium , Yale University , May 15–17, 1989 , pp. 1 -- 21 . American Mathematical Society , Providence, RI ( 1990 ) bchapter

    1989

  7. [15]

    : Über den Fundamentalsatz der statistischen Mechanik

    barticle Kroo , J. : Über den Fundamentalsatz der statistischen Mechanik . Annalen der Physik, ser. 4 34 ( 5 ), 907 -- 35 ( 1911 ) barticle

    1911

  8. [16]

    , Beale , P.D

    bbook Pathria , R.K. , Beale , P.D. : Statistical Mechanics , Fourth edition edn. Academic Press , London San Diego Cambridge, MA Oxford ( 2022 ) bbook

    2022

  9. [17]

    : Sur le problème des trois corps et les équations de la dynamique

    barticle Poincaré , H. : Sur le problème des trois corps et les équations de la dynamique . Acta Mathematica 13 , 1 -- 271 ( 1890 ) barticle

  10. [18]

    : Les Méthodes Nouvelles de la Mécanique Céleste vol

    bbook Poincaré , H. : Les Méthodes Nouvelles de la Mécanique Céleste vol. 1 . Gauthier-Villars , Paris ( 1892 ). https://archive.org/details/lesmthodesnouv001poin/page/n5/mode/2up bbook

  11. [19]

    : Le mécanisme et l’expérience

    barticle Poincaré , H. : Le mécanisme et l’expérience . Revue de métaphysique et de morale 1 , 534 -- 7 ( 1893 ) barticle

  12. [20]

    : Entropy

    barticle Poincaré , H. : Entropy . The Electrician 50 , 688 -- 689 ( 1903 ) barticle

    1903

  13. [21]

    : L’état actuel et l’avenir de la physique mathématique

    barticle Poincaré , H. : L’état actuel et l’avenir de la physique mathématique . Bulletin des sciences mathématiques 28 , 302 -- 324 ( 1904 ) barticle

    1904

  14. [22]

    : The Principles of Mathematical Physics

    barticle Poincaré , H. : The Principles of Mathematical Physics . The Monist 15 ( 1 ), 1 -- 24 ( 1905 ) barticle

    1905

  15. [23]

    : La Voie lactée et la théorie des gaz

    barticle Poincaré , H. : La Voie lactée et la théorie des gaz . Bulletin de la société astronomique de France 20 , 153 -- 165 ( 1906 ) barticle

    1906

  16. [24]

    : The Milky Way and the Theory of Gases

    barticle Poincaré , H. : The Milky Way and the Theory of Gases . Popular Astronomy 14 , 475 -- 88 ( 1906 ) barticle

    1906

  17. [25]

    : The Present and the Future of Mathematical Physics

    barticle Poincaré , H. : The Present and the Future of Mathematical Physics . Bulletin of the American Mathematical Society 12 , 240 -- 260 ( 1906 ) barticle

    1906

  18. [26]

    : Réflexions sur la théorie cinétique des gaz

    botherref Poincaré , H. : Réflexions sur la théorie cinétique des gaz. Bulletin des séances de la Société française de physique, 67--68 (1906) botherref

    1906

  19. [27]

    : Réflexions sur la théorie cinétique des gaz

    botherref Poincaré , H. : Réflexions sur la théorie cinétique des gaz. Bulletin des séances de la Société française de physique, 150--184 (1906) botherref

    1906

  20. [28]

    : Réflexions sur la théorie cinétique des gaz

    barticle Poincaré , H. : Réflexions sur la théorie cinétique des gaz . Journal de Physique Théorique et Appliquée 5 ( 1 ), 369 -- 403 ( 1906 ) 10.1051/jphystap:019060050036900 barticle

    1906 doi

  21. [29]

    : Sur la théorie des quanta

    barticle Poincaré , H. : Sur la théorie des quanta . Journal de Physique Théorique et Appliquée 2 ( 1 ), 5 -- 34 ( 1912 ) 10.1051/jphystap:0191200200500 barticle

    1912 doi

  22. [30]

    : The Three - Body Problem and the Equations Of Dynamics

    bbook Poincaré , H. : The Three - Body Problem and the Equations Of Dynamics . Astrophysics and Space Science Library , vol. 443 . Springer , Cham ( 2017 ). 10.1007/978-3-319-52899-1 bbook

    2017 doi

  23. [31]

    : Early Application of Kinetic Theory of Gases to Star Clusters

    barticle Popp , B.D. : Early Application of Kinetic Theory of Gases to Star Clusters . Journal for the History of Astronomy 53 ( 2 ), 141 -- 62 ( 2022 ) 10.1177/00218286221078338 barticle

    2022 doi

  24. [32]

    : The Principles of Statistical Mechanics

    bbook Tolman , R.C. : The Principles of Statistical Mechanics . Dover Publications , New York ( 1979 ) bbook

    1979

  25. [33]

    : Compendium of the Foundations of Classical Statistical Physics

    bchapter Uffink , J. : Compendium of the Foundations of Classical Statistical Physics . In: Butterfield , J. , Earman , J. (eds.) Philosophy of Physics . Handbook of the philosophy of science , pp. 923 -- 1074 . Elsevier , Amsterdam ( 2007 ) bchapter

    2007

  26. [34]

    : Notizen und Besprechungen

    barticle Zermelo , E. : Notizen und Besprechungen . ( Elementary Principles in Statistical Mechanics ) . Jahresbericht der Deutschen Mathematiker-Vereinigung 15 , 232 -- 42 ( 1906 ) barticle

    1906

  27. [35]

    : Ernst Zermelo : Collected Works / Gesammelte Werke II vol

    bbook Zermelo , E. : Ernst Zermelo : Collected Works / Gesammelte Werke II vol. II . Springer , Berlin, Heidelberg ( 2013 ). 10.1007/978-3-540-70856-8 bbook

    2013 doi

  28. [36]

    write newline

    " write newline "" before.all 'output.state := FUNCTION string.to.integer 't := t text.length 'k := #1 'char.num := t char.num #1 substring 's := s is.num s "." = or char.num k = not and char.num #1 + 'char.num := while char.num #1 - 'char.num := t #1 char.num substring FUNCTI...

Pith tools

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