The Logical Problem of the Definition of Irrational Numbers [1927e]

  • Robert S. Cohen
  • John J. Stachel
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 21)


A legend records “that the author of the theory of incommensurables was swallowed up in a shipwreck. Thus heaven punished the one who had ‘expressed the inexpressible, represented the unfigurable, unveiled that which should have remained forever hidden’”.1


Rational Number Logical Problem Irrational Number Continuous Series Arithmetic Theory 
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  1. 1.
    P. Boutroux., L’idéal scientifique des mathématiciens ( Alcan, Paris, 1920; P.U.F., Paris. 1955 ), p. 48.Google Scholar
  2. 2.
    From Heath, A History of Greek Mathematics. 2 vols. (Oxford 1921); see the index.Google Scholar
  3. 3.
    See Natucci. II Concetto di Numero e le sue Estensioni (Turin 1923) pp. 207 ff.Google Scholar
  4. 4.
    Ibid., pp. 357 ff.Google Scholar
  5. 5.
    C. Burali-Forti and R. Marcolongo., Analyse vectorielle générale, vols. I and II (Mattel, Pavia, 1912–13).Google Scholar
  6. 6.
    O. Perron, Irrationalzahlen (Leipzig 1921 ).Google Scholar
  7. 7.
    Revue des Sociétés Savantes (2). vol. IV. 1869: Leçons nouvelles sur l’analyse infintésimale. vol. I. ch. II. 1894.Google Scholar
  8. 8.
    Die Elemente der Funktionenlchre’. Crelle’s Journal, 1872.Google Scholar
  9. 9.
    For a comparison of the Cantorian and Dedekindian definitions of the continuum, see B. Russell, Introduction to Mathematical Philosophy, ch. X (George Allen and Unwin. London. 1919 ); B. Russell and A. N. Whitehead, Principia Mathematica, vol. III. *275 (Cambridge University Press, Cambridge, England. 1913 ).Google Scholar
  10. 10.
    Stetigkeit und irrationale Zahlen (Braunschweig 1872). I am quoting from a translation (unpublished) of the principal passages of the book that I made several years ago. (We have quoted the standard English translation: R. Dedekind. Essays on the Theory of Numbers (Open Court, Chicago. 1901), pp. 11–12, 15–Eds.]Google Scholar
  11. 11.
    Natucci. op. cit. (3), p. 259; cf. also Natucci, ‘Origine e Sviluppo del Concetto di Numbero irrazionale’, Scientia, 1925, pp. 293 ff.Google Scholar
  12. 12.
    Principia Mathematica, 3 vols., 1910–1911–1913. — Of course Russell and Whitehead found many much more important things than this.Google Scholar
  13. 13.
    Introduction to Mathematical Philosophy, op. cit., Note 9, p. 71.Google Scholar
  14. 14.
    Op cit., Note 11Google Scholar
  15. 15.
    Cf. Natucci., op. cit., Note 3. pp. 446–449: Poincaré. Science et méthode [English transi, by Ü B. Halsted in The Foundations of Science (Science Press, New York, 1913) — Ed.] and Dernières pensées (passim) (Flammarion. Paris. 1913). [English transl, by J. Bolduc: Mathematics and Science last Essays (New York. Dover reprint. 1963) — Ed.]Google Scholar
  16. 16.
    I presented this analysis at the Congress of the Association française pour l’Avancement des Sciences, held in Lyon in 1926.Google Scholar
  17. 17.
    For more precise definitions, see Russell and Whitehead, Principia Mathematica, vols. 2 and 3.Google Scholar
  18. 18.
    The numbers preceded by asterisks arc references to Principia Mathematica, vol. 3.Google Scholar
  19. 19.
    Op. cit., Note 6. p. 57.Google Scholar
  20. 20.
    Deruyts. Congress of the Association française pour l’Avancement des Sciences. Liège. 1924.Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1979

Authors and Affiliations

  • Robert S. Cohen
  • John J. Stachel

There are no affiliations available

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