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Some Contributions of Peano to Analysis in the Light of the Work of Belgian Mathematicians

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Giuseppe Peano between Mathematics and Logic

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Abstract

The period of the main original contributions of Giuseppe Peano (1858–1932) to analysis goes from 1884 till 1900, and his work deals mostly with a critical analysis of the foundations of differential and integral calculus and with the fundamental theory of ordinary differential equations. During this period, the main analysts in Belgium were Louis-Philippe Gilbert (1832–1892) and his successor Charles-Jean de La Vallée Poussin (1866–1962), at the Université Catholique de Louvain, and Paul Mansion (1844–1919) at the Université de Gand. At the Université de Liège, Eugène Catalan (1814–1894) retired in 1884, and his successor was Joseph Neuberg (1840– 1926), an expert in the geometry of the triangle. Analysis at the Université Libre de Bruxelles was still waiting to be awakened by Théophile De Donder (1872–1957)1.

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References

  1. See e. g. J. Mawhin (2001), 99–115 and the references therein.

    Google Scholar 

  2. See e. g. J. Mawhin (1989), 385–396 and the references therein.

    Google Scholar 

  3. See e. g. J. Mawhin (2003), 196–215 and the references therein.

    Google Scholar 

  4. C. Jordan, Cours d’analyse, vol. 1, Paris, Gauthier-Villars, 1882.

    Google Scholar 

  5. T. Flett (1974), 69–70, 72; T. Flett (1980), 62–63.

    Google Scholar 

  6. P. Dugac (1979), 32–33.

    Google Scholar 

  7. Th. Guitard (1986), 20.

    Google Scholar 

  8. H.C. Kennedy (1980), 15–16.

    Google Scholar 

  9. H. Gispert (1982), 44–47; H. Gispert (1983), 55.

    Google Scholar 

  10. U. Bottazzini (1991), 45.

    Google Scholar 

  11. M.T. Borgato (1991), 68–76.

    Google Scholar 

  12. G. Michelacci (2005), 8–270.

    Google Scholar 

  13. Michelacci (2005), 168–170.

    Google Scholar 

  14. E. Luciano (2007), 226–227, 255–262.

    Google Scholar 

  15. Peano (1884a) 45–47.

    Google Scholar 

  16. Peano (1884a), 47.

    Google Scholar 

  17. C. Jordan, Cours d’analyse de l’École polytechnique, vol. 1, 2e éd., Paris, Gauthier-Villars, 1893.

    MATH  Google Scholar 

  18. P. Gilbert (1884a), 153–155.

    Google Scholar 

  19. Peano (1884b), 252–256.

    Google Scholar 

  20. P. Gilbert (1884b), 475–482.

    Google Scholar 

  21. T. Flett (1974), 69.

    Google Scholar 

  22. G. Peano (1882b), 439–446.

    Google Scholar 

  23. Peano (1884c), x–xi: “La dimostrazione di questo numero fu data da Cauchy, Analyse algébrique, Paris 1821, nota III. La dimostrazione geometrica (pure data dal Cauchy, id., pag. 44) in cui si ritiene che la linea di equazione y = f(x), che ha due punti giacenti da parte opposta dell’asse delle x, incontra questo asse in qualche punto, non è soddisfacente … sarebbe esatta qualora si definisse per funzione continua quella che non può passare da un valore ad un altro senza passare per tutti i valori intermedii. E questa definizione trovasi appunto in alcuni trattati, e fra i recenti citeró il Gilbert, Cours d’analyse infinitésimale, Louvain 1872; ma erroneamente, l’A. a pag. 55 cerca di dimostrare la sua equivalenza con quella di cui noi ci serviamo. Invero, se col tendere di x ad a, f(x) oscilla entro valori che comprendono f(a), senza tendere ad alcun limite, f(x) è discontinua per x = a, secondo la nostra definizione, ed è continua, secondo la definizione del Gilbert.”

    Google Scholar 

  24. P. Gilbert (1872), 53–55.

    Google Scholar 

  25. P. Gilbert, Cours d’analyse infinitésimale. Partie élémentaire, 3e éd., Paris, Gauthier-Villars, 1887, 57.

    MATH  Google Scholar 

  26. Luciano (2007), 252

    Google Scholar 

  27. A. Demoulin (1929), 1–71.

    Google Scholar 

  28. G. Peano (1889b), (1889c), 110–112; Peano (1889e), 182–183; Peano (1890d), 73–74; Peano (1890e), 153–154; Peano (1892s), 12–14.

    Google Scholar 

  29. U. Cassina (1952), 337–362.

    Google Scholar 

  30. Cassina (1952), 349.

    Google Scholar 

  31. G. Peano (1885a), 677–685.

    Google Scholar 

  32. Mansion (1891), p. 57

    Google Scholar 

  33. G. Peano (1890f), 182–228.

    Google Scholar 

  34. P. Mansion (1890), 222–224.

    Google Scholar 

  35. P. Mansion (1914), 168–174.

    Google Scholar 

  36. P. Mansion (1904), 254–257; P. Mansion (1907), 213–218.

    Google Scholar 

  37. See e. g. P. Butzer, J. Mawhin (2000), 3–9 and the references therein.

    Google Scholar 

  38. Ch.-J. de La Vallée Poussin (1893a), 1–82

    Google Scholar 

  39. P. Mansion (1892), 227–236.

    Google Scholar 

  40. Cette démonstration rédigée à l’aide des symboles de la logique algébrique, est d’une étude très pénible pour ceux qui ne sont pas familiers avec ces notations. Nous avons donné du même théorème une démonstration plus simple dans les Annales de la Société Scientifique de Bruxelles (t. XVII, 1ère partie, p. 8–12). Tout ce que nous disons ici de la démonstration de Peano s’applique aussi à celle-là, qui ne diffère pas essentiellement de celle de Peano.

    Google Scholar 

  41. Ch.-J. de La Vallée Poussin (1893a), 79.

    Google Scholar 

  42. Ch.-J. de La Vallée Poussin (1893b), 8–12.

    Google Scholar 

  43. G. Peano (1890f), 182–228.

    Google Scholar 

  44. G. Peano (1897c), 9–18.

    Google Scholar 

  45. G. Peano (1908a), 429.

    Google Scholar 

  46. G. Peano (1892a), 40–46.

    Google Scholar 

  47. Peano (1892a), 40–42.

    Google Scholar 

  48. Ch.-J. de La Vallée Poussin (1908), 193–254.

    Google Scholar 

  49. A. Denjoy (1935), 273–326.

    Google Scholar 

  50. J. Marcinkiewicz, A. Zygmund (1936), 1–43.

    Google Scholar 

  51. J. Marcinkiewicz (1937), 38–69.

    Google Scholar 

  52. E. Corominas Vigneaux (1946), 88–93.

    Google Scholar 

  53. E. Corominas Vigneaux (1947), 89–91.

    Google Scholar 

  54. E. Corominas Vigneaux (1953), 177–222.

    Google Scholar 

  55. H. Oliver (1954), 444–456.

    Google Scholar 

  56. A. Zygmund (1959), 59–60.

    Google Scholar 

  57. C. Kassimatis (1965), 1171–1172.

    Google Scholar 

  58. P.S. Bullen, S.N. Mukhopadhyay (1973), 127–140.

    Google Scholar 

  59. G. Peano (1889e), 182–183.

    Google Scholar 

  60. P.L. Butzer, R.J. Nessel (1993), 72–73; P.L. Butzer, R.J. Nessel (2004), 381.

    Google Scholar 

  61. See e. g. J. Mawhin (1992), 120.

    Google Scholar 

  62. J. Kurzweil (1957), 418–446.

    Google Scholar 

  63. R. Henstock (1961), 402–418.

    Google Scholar 

  64. For more details see e. g. J. Mawhin (1992) or A. Fonda (2001). For comments on its history, see e. g. J. Mawhin (2007), 47–63 and the references therein.

    Google Scholar 

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Authors and Affiliations

  1. Institut de recherche en mathématique et physique (IRMP), Université Catholique de Louvain-la-Neuve, Belgium

    Jean Mawhin

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  1. Jean Mawhin
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Editors and Affiliations

  1. Department of Mathematics “G. Peano”, University of Torino, Italy

    Fulvia Skof

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Mawhin, J. (2011). Some Contributions of Peano to Analysis in the Light of the Work of Belgian Mathematicians. In: Skof, F. (eds) Giuseppe Peano between Mathematics and Logic. Springer, Milano. https://doi.org/10.1007/978-88-470-1836-5_2

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