Abstract
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
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Notes
P. Boutroux., L’idéal scientifique des mathématiciens ( Alcan, Paris, 1920; P.U.F., Paris. 1955 ), p. 48.
From Heath, A History of Greek Mathematics. 2 vols. (Oxford 1921); see the index.
See Natucci. II Concetto di Numero e le sue Estensioni (Turin 1923) pp. 207 ff.
Ibid., pp. 357 ff.
C. Burali-Forti and R. Marcolongo., Analyse vectorielle générale, vols. I and II (Mattel, Pavia, 1912–13).
O. Perron, Irrationalzahlen (Leipzig 1921 ).
Revue des Sociétés Savantes (2). vol. IV. 1869: Leçons nouvelles sur l’analyse infintésimale. vol. I. ch. II. 1894.
Die Elemente der Funktionenlchre’. Crelle’s Journal, 1872.
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 ).
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 I am quoting from a translation (unpublished) of the principal passages of the book that I made several years ago.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.]
Natucci. op. cit. (3), p. 259; cf. also Natucci, ‘Origine e Sviluppo del Concetto di Numbero irrazionale’, Scientia, 1925, pp. 293 ff.
Principia Mathematica, 3 vols., 1910–1911–1913. — Of course Russell and Whitehead found many much more important things than this.
Introduction to Mathematical Philosophy, op. cit., Note 9, p. 71.
Op cit., Note 11
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.]
I presented this analysis at the Congress of the Association française pour l’Avancement des Sciences, held in Lyon in 1926.
For more precise definitions, see Russell and Whitehead, Principia Mathematica, vols. 2 and 3.
The numbers preceded by asterisks arc references to Principia Mathematica, vol. 3.
Op. cit., Note 6. p. 57.
Deruyts. Congress of the Association française pour l’Avancement des Sciences. Liège. 1924.
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Cohen, R.S., Stachel, J.J. (1979). The Logical Problem of the Definition of Irrational Numbers [1927e]. In: Cohen, R.S., Stachel, J.J. (eds) Selected Papers of Léon Rosenfeld. Boston Studies in the Philosophy of Science, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9349-5_3
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