Late Nineteenth Century Background

  • Leo Corry
Part of the Archimedes book series (ARIM, volume 10)

Abstract

Hilbert’s first published, comprehensive presentation of an axiomatized mathematical discipline appeared in June of 1899, in the epoch-making Grundlagen der Geometrie (GdG).Based on a course taught in the winter semester of 1898–99, GdG was published as part of a Festschrift issued in Göttingen to celebrate the inauguration of a monument to honor two of its legendary scientists: Carl Friedrich Gauss (1777–1855) and Wilhelm Weber (1804–1891). Hilbert had been teaching courses on topics related to geometry and its foundations since 1891. Nevertheless, the conception and the results embodied in GdG signified a real innovation that was to make a deep impact on geometry and, indeed, on the whole of mathematics for decades to come. Like most of Hilbert’s early important works, this one had deep roots in central developments of the classical theories that thrived in the nineteenth century. In order to understand those roots and the actual historical significance of GdG, the present chapter is devoted to describing in some detail the relevant background related to those developments. It comprises four main themes, all of them spanning the late nineteenth century: Hilbert’s early career (§ 1.1), foundations of geometry (§ 1.2), foundations of physics (§ 1.3), and mathematics and physics in Göttingen at the time Hilbert arrived there (§ 1.4).

Keywords

Entropy Manifold Steam Expense Arena 

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References

  1. 1.
    Felix Klein to David Hilbert, December 6, 1894. (Frei (ed.) 1985, 115).Google Scholar
  2. 2.
    Cf. Rowe 2003a.Google Scholar
  3. 3.
    Cf. Corry 2003, §§ 1.2 & 2.2.4; Schappacher & Volkert 1991. DHN504 contains the notes taken by Hilbert at a rather high-level course on number thenry given by Weber in SS 1 RR2Google Scholar
  4. 4.
    Cf. Siebert 1966.Google Scholar
  5. 5.
    Filed as DHN506, 507–508, 511, 512, 513, 515, respectively.Google Scholar
  6. 6.
    Filed as DHN505, 509, 514, 516, 517, 518, respectively.Google Scholar
  7. 7.
    Wuliner 1870.Google Scholar
  8. 8.
    Cf. Olesko 1991, 278–279.Google Scholar
  9. 9.
    Cf. Jungnickel & McCormmach 1996, 212. However, as Kragh 1999, 10, justly stresses in relation with physics textbooks around 1900: “Textbooks of this period usually rested on a mechanical basis and did not reflect the change in worldview discussed in frontier theoretical physics. This is how textbooks are: They are by nature conservative and cautious in their attitude towards new ideas. ”Google Scholar
  10. 10.
  11. 11.
    For instance, in Hilbert 1905–06, 10.Google Scholar
  12. 12.
    Cf. Hilbert 1920a.Google Scholar
  13. 13.
    As documented in a letter by Weber to Dedekind, probably written at the end of 1879, and quoted in Strobl 1985, 144–145. In all likeliness, Minkowski may have attended the 1882 course of Weber on number theory, together with Hilbert. See above, note 4.Google Scholar
  14. 14.
    Cf. Schwermer 2003.Google Scholar
  15. 15.
    On Minkowski’s mathematical activities in Bonn, cf. Schwermer 1991, 79–93.Google Scholar
  16. 16.
    Hilbert 1910, 355.Google Scholar
  17. 17.
    Minkowski to Hilbert, December 20, 1890 (Rüdenberg & Zassenhaus (eds.) 1973, 39–40). Cf. Hilbert GAVol. 3, 355.Google Scholar
  18. 18.
    Cf. Schwermer 2003. Minkowski explicitly mentioned Voigt among the teachers that influenced him (Minkowski GAVol. L 1 59)Google Scholar
  19. 19.
    Notes are preserved in DHN 522 Google Scholar
  20. 20.
    Minkowski 1888.Google Scholar
  21. 21.
    Cf. Rüdenberg & Zassenhaus 1973, 110–117.Google Scholar
  22. 22.
    Cf. Crilly 1986; Parshall 1989, 158–162; Parshall & Rowe 1994, 67–68.Google Scholar
  23. 23.
    A classical historical account of the early development of invariant theory appears in Meyer 1890. See also Fisher 1966, 141–156; Kline 1972, 924–932; Parshall 1989, 170–176; 1990, 12–13. Detailed explanations of the basic concepts of invariant theory appear in Crilly 1986; Hilbert 1896; Weyl 1944, 618–624. A classic textbook on the issue is Study 1933. More recent expositions of the theory appear in Dieudonné & Carrell 1971; and Springer 1977.Google Scholar
  24. 24.
    For a detailed account of the differences between the English and the German schools of invariant theory, see Parshall 1989, 176–180.Google Scholar
  25. 25.
    Gordan 1868, 327.Google Scholar
  26. 26.
    Hilbert 1889; Mertens 1886.Google Scholar
  27. 27.
    Hilbert’s proof appeared first in Hilbert 1888–89, and in an improved version in Hilbert 1890. For a reconstruction of Hilbert’s proof in a more recent formulation, see Springer 1977, 15–42.Google Scholar
  28. 28.
    Cf. Ferreirós 1999. 282–285.Google Scholar
  29. 29.
    Cf. Blumenthal 1935, 194; Klein 1926–27, Vol. 1, 330. Gordan’s reaction to Hilbert’s proof is documented in the Klein-Hilbert correspondence, see Frei (ed.) 1985, 61–65. See also Rowe 1989, 196–198.Google Scholar
  30. 30.
    Gordan 1893.Google Scholar
  31. 31.
    Cf. Klein 1926–27, Vol. 1, 331.Google Scholar
  32. 32.
    Hilbert 1893, 287.Google Scholar
  33. 33.
    Hilbert 1893, 288. Hilbert returned to this list, with a minor change, in the lecture read before the 1893 International Congress of Mathematics in Chicago. Cf. Hilbert 1896, 377.Google Scholar
  34. 34.
    Hilbert 1896, 383.Google Scholar
  35. 35.
    Hilbert 1893, 344.Google Scholar
  36. 36.
    For various assessments of Hilbert’s work on invariants see Parshall 1990, 11 ff.; Weyl 1939, 27–29; 1944, 627.Google Scholar
  37. 37.
    Hilbert to Klein, September 14, 1892 (Frei (ed.) 1985, Doc. 71). According to an oft-quoted assertion of Blumenthal (1935, 395), Hilbert wrote a letter in the same spirit to Minkowski in 1892. Such a letter, however, has apparently not been preserved, and in particular it is not included in Rüdenberg & Zassenhaus (eds.) 1973. Hilbert published no further articles on invariants, but he maintained an occasional interest in the topic. DHN520 registers among Hilbert’s courses in Göttingen only two on invariants: one in 1897 and a second one, somewhat surprisingly, as late as WS 1929–30. From all the 68 dissertations he directed in Göttingen, only two dealt with invariants (cf. HGAVol. 3, 431). One was completed by Sophus Marxsen in 1900 and the second one was submitted in 1909 by Andreas Speiser (1885–1970) . Occasionally Hilbert also discussed recent research on invariants at the meetings of the GMG. One such presentation is recorded as late as December 3, 1913 (cf. JDMV22 (1913), 27).Google Scholar
  38. 38.
    Cf. Corry 2003, Ch. 2; Edwards 1975, 1977, 1980, 1987.Google Scholar
  39. 39.
    Cf. Blumenthal 1935, 397.Google Scholar
  40. 40.
    JDMV1 (1891), 12.Google Scholar
  41. 41.
    Noether & Brill 1892–3.Google Scholar
  42. 42.
    Schoenflies 1900.Google Scholar
  43. 43.
    Czuber 1899.Google Scholar
  44. 44.
    Minkowski’s letters to Hilbert during these years contain many references to Minkowski’s work on his planned section of the Zahlberichtand on his own book. Cf. Rüdenberg & Zassenhaus (eds.) 1973, esp. pp. 57 ff. Cf. also Zassenhaus 1973. Additional accounts appear in Blumenthal 1935, 396–399; Ellison & Ellison 1978, 191; Reid 1970, 42–45 & 51–53. At the annual meetings of the DMV, Minkowski’s report continued to be expected until at least 1897. Cf. JDMV,Vol. 6 (1898), 7.Google Scholar
  45. 45.
    Minkowski to Hilbert, March 31, 1896 (Rüdenberg & Zassenhaus eds. 1973, 79–80). English translation quoted from Rowe 2000, 61.Google Scholar
  46. 46.
    As late as 1944 Weyl wrote (p. 626) that “even today, after almost fifty years, a study of this book is indispensable for anybody who wishes to master the theory of algebraic numbers. ” Cf. also Hasse 1932, 529; Ellison & Ellison 1978, 191.Google Scholar
  47. 47.
    Hilbert 1897, 66–67. English translation quoted from Hilbert 1998, x.Google Scholar
  48. 48.
    Cf. Minkowski 1905, 162–163.Google Scholar
  49. 49.
    Of special significance and greatest influence were Hilbert 1898; idem, 1899a. See Hasse 1932.Google Scholar
  50. 50.
    English translation quoted from Rowe 1994, 190.Google Scholar
  51. 51.
    Blumenthal 1922, 72.Google Scholar
  52. 52.
    Toeplitz 1922.Google Scholar
  53. 53.
    Cf. Corry 2003, Ch. 3.Google Scholar
  54. 54.
    Riemann 1868.Google Scholar
  55. 55.
    Quoted from Ewald (ed.) 1996, 652–653.Google Scholar
  56. 56.
    Cf. Ferreirós 1999, 15–16.Google Scholar
  57. 57.
    Ewald (ed.) 1999, Vol. 2, 661.Google Scholar
  58. 58.
    Cf. Ferreirós 2000, xcii-cxvi;Laugwitz 1999; Torretti 1978, 82–109.Google Scholar
  59. 59.
    Cf. Scholz 1980, especially pp. 101–135.Google Scholar
  60. 60.
    Cf. Torretti 1978, 155–171.Google Scholar
  61. 61.
    Cf. Richards 1988.Google Scholar
  62. 62.
    Cf Reich 1994 29–34 & 59–65. Google Scholar
  63. 63.
    Cf. Hawkins 2000, 111–137; Torretti 1978, 176–186.Google Scholar
  64. 64.
    Poncelet 1822.Google Scholar
  65. 65.
    Cf. Freudenthal 1974.Google Scholar
  66. 66.
    Cf. Ziegler 1985.Google Scholar
  67. 67.
    Jordan 1870. Cf. Corry 2003, 28–30.Google Scholar
  68. 68.
    Cf. Hawkins 1989, 317, note 13.Google Scholar
  69. 69.
    Cf. Hawkins 1989, 283–284.Google Scholar
  70. 70.
    For details about Klein’s formative years, see Parshall and Rowe 1994, 154–166; Rowe 1989a.Google Scholar
  71. 71.
    For an account of Cayley’s contributions, see Klein 1926–7 Vol. 1, 147–151.Google Scholar
  72. 72.
    Klein 1871 and 1873. For comments on these contributions of Klein, see Rowe 1994, 194–195; Toepell 1986, 4–6; Torretti 1978, 110–152.Google Scholar
  73. 73.
    This is, of course, a very schematic presentation of the program, the ideas behind which also underwent important changes. In the original presentation, for instance, Klein did not include the obvious case of affine geometry. For additional details, see Hawkins 1984. For an account of Klein’s own original work in geometry immediately before the formulation of the program, see Rowe 1989a.Google Scholar
  74. 74.
    For an illuminating comparison between Klein’s and Killing’s differing perspectives on the scope of geometry, cf. Hawkins 2000, 130–137.Google Scholar
  75. 75.
    Quoted in Hawkins 2000, 137.Google Scholar
  76. 76.
    Klein 1898, 585.Google Scholar
  77. 77.
    Cf. Hawkins 1984.Google Scholar
  78. 78.
    Cf. Avellone, Brigagalia & Zapulla 2002, 385–399.Google Scholar
  79. 79.
    Cf. Hawkins 2000, 251–255, 291–292.Google Scholar
  80. 80.
    Klein 1874, 1880.Google Scholar
  81. 81.
    Cf. Ferreirós 1999, 119–124; Israel 1981.Google Scholar
  82. 82.
    Cf. Fisch 1999; Pycior 1981.Google Scholar
  83. 83.
    Cf. Rowe 1996a; Tobies 1996.Google Scholar
  84. 84.
    Grassmann 1995, 11.Google Scholar
  85. 85.
    Apelt to Grassmann. September 3, 1845. Quoted in Caneva 1978, 104.Google Scholar
  86. 86.
    Cf. Brigaglia 1996; Avellone, Brigagalia & Zapulla 2002, 374–380. Of course, the works of these two Turin mathematicians were also complementary in many other respects.Google Scholar
  87. 87.
    Cf Corry 2003, 67; Ferreirós 1999, 53–62.Google Scholar
  88. 88.
    Riemann, however, was not the only decisive influence on Dedekind’s career; Dirichlet also played a decisive role. Cf. Ferreirós 1999, 215–256.Google Scholar
  89. 89.
    Cf. Corry 2003, Ch. 2, see especially pp. 135–136.Google Scholar
  90. 90.
    Cf. Tazzioli 1994.Google Scholar
  91. 91.
    Dedekind Werke Vol. 3, 334. Google Scholar
  92. 92.
    This point has been illuminated in Ferreirós 1999, 246–248.Google Scholar
  93. 93.
    Quoted in Ewald 1996, 793–794.Google Scholar
  94. 94.
    Quoted in Ewald 1996, 809. Italics in the original. A very similar passage is found in §134 (p. 823).Google Scholar
  95. 95.
    As described in Ferreirós 1999, 238.Google Scholar
  96. 96.
    Pasch 1887, 129. See also Contro 1976; Nagel 1939, 193–199; Torretti 1978, 210–218.Google Scholar
  97. 97.
    Pasch & Dehn 1926, 19.Google Scholar
  98. 98.
    Pasch 1882, 105: “If the point c1 lies in the segment ab,and the segment ac1is extended over the congruent segment c1c2, this one over c2c3, and so forth, then a segment cnc nz+ 1is eventually reached, such that the point b lies within it. ”Google Scholar
  99. 99.
    Cf. Contro 1976, 287–290.Google Scholar
  100. 100.
    Klein GMAVol. 1, 397–398.Google Scholar
  101. 101.
    Wiener 1891, 46.Google Scholar
  102. 102.
    Wiener called the Pappus theorem, “Pascal’s theorem for two lines ”. Hilbert would also adopt this name later on. For a more or less contemporary formulation of the theorem, see Enriques 1903. Interestingly, Enriques explicitly remarked in the introduction to the German version of his book (p. vii)that he was following the classical approach introduced by von Staudt, and followed by Klein and others, rather than the more modern one developed recently by Pasch and Hilbert.Google Scholar
  103. 103.
    Wiener 1983, 72.Google Scholar
  104. 104.
    Schur 1898Google Scholar
  105. 105.
    Cf. Kennedy 1980; Segre 1994.Google Scholar
  106. 106.
    Cf. Ferreirós 1999, 251.Google Scholar
  107. 107.
    Cf. Torretti 1978, 221.Google Scholar
  108. 108.
    Cf. Kennedy 1981. 443: Borga. et al. 1985.Google Scholar
  109. 109.
    Quoted in Avellone, Brigagalia & Zapulla 2002, 413.Google Scholar
  110. 110.
    Cf. Marchisotto 1993, 1995.Google Scholar
  111. 111.
    Cf. Torretti 1978, 225–226.Google Scholar
  112. 112.
    In Veronese 1891. Cf. Avellone, Brigagalia & Zapulla 2002, 380–385.Google Scholar
  113. 113.
    On criticism directed at Veronese’s work by German mathematicians, see Toepell 1986, 56.Google Scholar
  114. 114.
    Cf. Boi 1990.Google Scholar
  115. 115.
    Cf. Gray 1999a, 65. It is likewise important to stress that Italian axiomatic works in the late nineteenth century, under Peano’s influence, were not limited to geometry. Of special importance are the early attempts by Cesare Burali-Forti (1861–9131) to formulate a system of axioms for set theory, before yet being aware of the existence of the set-theoretic paradoxes (Burali-Forti 1896, 1896a). Cf Moore 1982, 150–151Google Scholar
  116. 116.
    Cf. Contro 1985.Google Scholar
  117. 117.
    Cf Heilbron 1982, Jungnickel & McCormmach 1986, Vol. 2.Google Scholar
  118. 118.
    For an account of the processes leading to the creation of this new image of physics around 1850, see Caneva 1978; Harman 1982, 12–44; Jungnickel & McCormmach 1986, Vol 1.Google Scholar
  119. 119.
    Two classical, detailed accounts of the development of the kinetic theory of gases (particularly during the late nineteenth century) can be consulted: Brush 1976 and Klein 1970 (esp. 95–140). In the following naragranhs I have drawn heavily on them. See also CCercignani 1998, 71–131Google Scholar
  120. 120.
    Maxwell 1860.Google Scholar
  121. 121.
    Maxwell 1867.Google Scholar
  122. 122.
    See 5.2 below.Google Scholar
  123. 123.
    However, it must be stressed that, until Max Planck’s treatment of the issue in 1900, this connection was largely ignored by other physicists involved in the study of the macroscopic behavior of gases. Cf. Kuhn 1978, 20–21.Google Scholar
  124. 124.
    Boltzmann 1877. Cf. Brush 1976, 605–627.Google Scholar
  125. 125.
    The terms Umkehreinwandand Wiederkehreinwandwere introduced only in 1907 by Tatyana and Paul Ehrenfest. See Klein 1970, 115.Google Scholar
  126. 126.
    For the subtleties of Planck’s position on this issue see Heilbron 2000, 1–46; Hiebert 1971, 72–79; Kuhn 1978, 22–29.Google Scholar
  127. 127.
    Although several detailed studies of Zermelo’s contribution to set-theory and logic are available (e.g., Moore 1982, Peckhaus 1990, 76–122), his complete biography is yet to be written.Google Scholar
  128. 128.
    Cf. Kuhn 1978. 26–27: Jungnickel & McCormmach 1996.212–216.Google Scholar
  129. 129.
    This is different from Liouville’s theorem on analytic functions.Google Scholar
  130. 130.
    For a brief account of this trend and a comprehensive list of secondary literature on the development of the electromagnetic view of nature, see Jungnickel & McCormmach 1996, 231–242. A more recent, enlightened discussion appears in Darrigol 2000, Ch. 9, who makes appear this putative “worldview ” as much less monolithic than previous presentations did. In particular, Darrigol stresses different degress of commitment to the belief that inertial properties of matter could be fully reduced to electrodynamical forces. Thus whereas Wien, Abraham, and, later on, Gustav Mie advocated a strong view on this question, Lorentz, and even more so Einstein later on, favored a more relaxed apporach. See also 3.2 below.Google Scholar
  131. 131.
    Mach 1893, 495–496.Google Scholar
  132. 132.
    Cf. Hiebert 1968.Google Scholar
  133. 133.
    fßlackmore 1972 1 17–1 1 RGoogle Scholar
  134. 134.
    Cf. Blackmore 1972, 120.Google Scholar
  135. 135.
    Cf. DiSalle 1993. 345: Jungnickel& McCormmach 1986, Vol. 1, 1R1–1R5.Google Scholar
  136. 136.
    This trend is discussed in Barbour 1989, Ch. 12.Google Scholar
  137. 137.
    Neumann 1870, 3. Hereafter I refer to Neumann 1993.Google Scholar
  138. 138.
    Neumann 1993, 361. The reference is to Leibniz MS, Vol. 4, 135. Google Scholar
  139. 139.
    See Barbour 1989, 646–653; DiSalle 1993, 348–349.Google Scholar
  140. 140.
    Lange’s ideas are discussed in Barbour 1989, 655–662.Google Scholar
  141. 141.
    Neumann 1870, 22 (1993, 367).Google Scholar
  142. 142.
    Cf. Tobies & Rowe (eds.) 1990, 29.Google Scholar
  143. 143.
    Cf. Jungnickel & McCormmach 1986, Vol. 1, 184–185.Google Scholar
  144. 144.
    Cf. Maier 1998.Google Scholar
  145. 145.
    Cf. Fölsing 1997a, 474. Based on Hertz’s letters of reply to Klein, Fölsing assumes in his account that the large-scale project involved should be the Encyklopädie der Mathematischen Wissenschaften.However, as will be seen below (§ 1.4.1), this project was not initiated before 1893. There seems to be no known, additional documentation on the relationship between the two.Google Scholar
  146. 146.
    Cf. Fölsing 1997a, 475–476.Google Scholar
  147. 147.
    Cf. Blackmore 1972, 119.Google Scholar
  148. 148.
    For recent discussions, see Baird et al. (eds.) 1998.Google Scholar
  149. 149.
    Cf. Lützen 1998, 2004 (Forthcoming).Google Scholar
  150. 150.
    Hertz 1956, 145.Google Scholar
  151. 151.
    Hertz 1956, 146.Google Scholar
  152. 152.
    In all these passages, the term “permissible ” is intended, of course, in the sense discussed in Hertz’s introduction, namely, logically consistent.Google Scholar
  153. 153.
    Cf. Lützen 1995, 76–83.Google Scholar
  154. 154.
    Boltzmann 1899, 88.Google Scholar
  155. 155.
    Cf. Jungnickel & McCormmach 1986, Vol. 2, 144–148; Olesko 1991, 439–448; Ramser 1974.Google Scholar
  156. 156.
    Cf. Schwermer 2003.Google Scholar
  157. 157.
    DHN416, comprises fifteen letters of Volkmann to Hilbert, spanning the years 1886 to 1913, and the draft of one letter of Hilbert to Volkmann, dated January 15, 1897. Volkmann’s letters deal almost entirely with appointment questions involving, mainly, Arthur Schoenflies, Otto Hölder and Franz Meyer and they evince an absolute trust in Hilbert’s professional and personal judgment.Google Scholar
  158. 158.
    Volkmann 1892Google Scholar
  159. 159.
    Volkmann 1894.Google Scholar
  160. 160.
    For more details, see Volkmann 1900, 12–20. In pp. 78–79, he discusses in greater detail Newton’s laws of motion and the universal law of gravitation as examples of principles and laws of nature respectively.Google Scholar
  161. 161.
    Volkmann 1900, 363.Google Scholar
  162. 162.
    Volkmann to Hilbert, January 2, 1900 (DHN416, 12): “Der Standpunkt, von dem aus das Buch geschrieben ist, wird schwerlich den Anforderungen mathematischer Präzisions-Darstellungen entsprechen, er will und kann es auch gar nicht. In dem Sachregister finden Sie alle die schönen Schlagworte die sonst Gegensätze anzeigen, friedlich vereint. ”Google Scholar
  163. 163.
    See Boltzmann’s own account in Boltzmann 1899a, 109–111.Google Scholar
  164. 164.
    D’Agostino 2000, 201–222.Google Scholar
  165. 165.
    Boltzmann 1899a, 105–107.Google Scholar
  166. 166.
    Cf. JDMV8 (1900), 17–23.Google Scholar
  167. 167.
    Boltzmann 1899.Google Scholar
  168. 168.
    Cf. JDMV7 (1899), 4.Google Scholar
  169. 169.
    Boltzmann 1899, 90.Google Scholar
  170. 170.
    Reich 1985.Google Scholar
  171. 171.
    Toepell 1996. 235–238: Vollrath 1993.Google Scholar
  172. 172.
    Voss 1899.Google Scholar
  173. 173.
    Voss 1903 Google Scholar
  174. 174.
    Voss 1908 1913. 1914.Google Scholar
  175. 75.
    Voss to Hilbert, July 19. 1899 (DHN418. 1).Ouoted in Toenell 1996,410.Google Scholar
  176. 176.
    This term is actually used in the second edition, Voss 1913, 106–107, which slightly differs in this passage from the first one, Voss 1908, 86–87.Google Scholar
  177. 177.
    Voss 1901, 9 (note 2).Google Scholar
  178. 178.
    VOSS 1901, 11.Google Scholar
  179. 179.
    Voss 1901, 18–30.Google Scholar
  180. 180.
    Cf. note 151 above. Voss also referred the reader here to a Larmor 1900, 288. It is relevant here to quote a passage from that text since it illuminates the kind of ideas that he, and later on Hilbert, had in mind when speaking of mechanics as the most basic science, and the extent to which their mechanical reductionism was meant to be all-embracing. Thus Larmor wrote: “Moreover, mechanical science has to do with systems in being: it does not avail to trace the circumstances of growth or structural change even in inorganic material. What happens when two gaseous molecules unite to form a compound molecule is unknown except for the slight indirect indications of spectrum analysis. Now all initiations of organic activity seem to involve structural change, not merely mechanical disturbance, and are, in so far, outside the domain of mechanical laws. But the activities of an organism treated as a permanent system—such for example as propagation of nervous impulse—are likely enough, once they are started, to be of the nature of the interactions of matter in bulk, so that it is legitimate to seek for them a mechanical correlation.... There is room for complete mechanical coordination of all the functions of an organism, treated as an existing material system, without requiring any admission that similar principles are supreme in the more remote and infinitely complex phenomena concerned in growth and decay of structure. ”Google Scholar
  181. 181.
    Voss claimed that this was the approach strongly followed by Hertz. He was referring to the abovequoted discussion on the principle of inertia and how different kinds of bodies relate to it (Voss auoted Hertz 1894. nn. 53 & 157).Google Scholar
  182. 182.
    Cf. especially Ferreirós 1999, 62–64.Google Scholar
  183. 183.
    Voss 1901, 40. As an important reference Voss mentioned Wien 1900, on which see also § 3.2 below.Google Scholar
  184. 184.
    Cf. Darrigol 2000, 354–356; Greffe, et al. (eds.) 1996.Google Scholar
  185. 185.
    Rowe 2001, esp. 78–84.Google Scholar
  186. 186.
    Klein to Hilbert, December 6, 1894 (Frei (ed.) 1985, 115). English translation quoted from Rowe 2000, 62.Google Scholar
  187. 187.
    Schoenflies 1891. Cf. Scholz 1989. 137–148.Google Scholar
  188. 188.
    Cf. Klein GMAVol. III, App. I, 8–9.Google Scholar
  189. 189.
    Rowe 1989, 202.Google Scholar
  190. 190.
    Rowe 1989, 191–193.Google Scholar
  191. 191.
    Manegold 1970.Google Scholar
  192. 192.
    For a recent, detailed account of the Encyklopädieproject, cf. Hashagen 2003, 487–522.Google Scholar
  193. 193.
    Cf. Hashagen 2003, 501–503.Google Scholar
  194. 194.
    Respectively Schubert 1898, Netto 1898, Pringsheim 1898.Google Scholar
  195. 195.
    Cf Klein to Dyck. December 30. 1900. Ouoted in Hashagen 2003. 504.Google Scholar
  196. 196.
    Von Dvck 1904. xi.Cf also JDMV9 (19011. 69.Google Scholar
  197. 197.
    Von Dyck 1904, xv; JDMV9 (1901), 69.Google Scholar
  198. 198.
    Pareto 1911. On the importance of this article, cf Ingrao & Israel 1985.Google Scholar
  199. 199.
    As Warwick 2003, 253, points out, Klein visited the Cambridge mathematical physicists frequently and deeply appreciated their achievements in applied mathematics.Google Scholar
  200. 200.
    Von Dyck 1904, xv; Cf. Gispert 2001.Google Scholar
  201. 201.
    Klein to von Dyck. December 24, 1905. Quoted in Hashagen 2003, 520.Google Scholar
  202. 202.
    Cf. Rowe 1989. 209.Google Scholar
  203. 203.
    Cf. Perron 1952.Google Scholar
  204. 204.
    Cf. Gispert 2001, 94–96.Google Scholar
  205. 205.
    Cf e a P7 5(19040 470 ; P7R (19070 549–551Google Scholar
  206. 206.
    Cf. von Mises 1924, 88.Google Scholar
  207. 207.
    Respectively: Lorentz 1904 (see below § 3.2); Paul & Tatyana Ehrenfest 1912 (see below § 3.3.8); Boltzmann & Nabl 1907; Pauli 1921 (see below § 9.4).Google Scholar
  208. 208.
    Fano 1907. See Hawkins 2000. 290–316 for the influence of this article on Elie Cartan (1869–1951).Google Scholar
  209. 209.
    Hellinger & Toeplitz 1927 Cf Köthe 19R2575–5R4l Dieudonné 1981 112Google Scholar
  210. 210.
    Von Dyck 1904, xii. Google Scholar
  211. 211.
    Cf. Gispert 1999.Google Scholar
  212. 212.
    Hilbert 1900a.Google Scholar
  213. 213.
    Cf. Schubring 1989.Google Scholar
  214. 214.
    Klein & Riecke 1900.Google Scholar
  215. 215.
    Klein & Riecke 1904.Google Scholar
  216. 216.
    Jungnickel & McCormmach 1986, Vol. 2, 118–124.Google Scholar
  217. 217.
    Riecke 1896.Google Scholar
  218. 218.
    A list of existing sources on Voigt is summarized in Schirrmacher 2003a. 19. note 6.Google Scholar
  219. 219.
    Kuhn 1978, 135.Google Scholar
  220. 220.
    Cf Olesko 1991 400–402Google Scholar
  221. 221.
    Voigt 1889.Google Scholar
  222. 222.
    Voigt 1895–96.Google Scholar
  223. 223.
    Jungnickel & McCormmach 1986. Vol. 2. 268–273.Google Scholar
  224. 224.
    Cf. Olesko 1991, 387–388.Google Scholar
  225. 225.
    As expressed in Voigt 1915, esp. p. 416.Google Scholar
  226. 226.
    See Barkan 1999, 58–76.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2004

Authors and Affiliations

  • Leo Corry
    • 1
  1. 1.Cohn Institute for History and Philosophy of ScienceTel Aviv UniversityIsrael

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