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).
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References
Felix Klein to David Hilbert, December 6, 1894. (Frei (ed.) 1985, 115).
Cf. Rowe 2003a.
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 RR2
Cf. Siebert 1966.
Filed as DHN506, 507–508, 511, 512, 513, 515, respectively.
Filed as DHN505, 509, 514, 516, 517, 518, respectively.
Wuliner 1870.
Cf. Olesko 1991, 278–279.
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. ”
DHN 519.
For instance, in Hilbert 1905–06, 10.
Cf. Hilbert 1920a.
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.
Cf. Schwermer 2003.
On Minkowski’s mathematical activities in Bonn, cf. Schwermer 1991, 79–93.
Hilbert 1910, 355.
Minkowski to Hilbert, December 20, 1890 (Rüdenberg & Zassenhaus (eds.) 1973, 39–40). Cf. Hilbert GAVol. 3, 355.
Cf. Schwermer 2003. Minkowski explicitly mentioned Voigt among the teachers that influenced him (Minkowski GAVol. L 1 59)
Notes are preserved in DHN 522
Minkowski 1888.
Cf. Rüdenberg & Zassenhaus 1973, 110–117.
Cf. Crilly 1986; Parshall 1989, 158–162; Parshall & Rowe 1994, 67–68.
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.
For a detailed account of the differences between the English and the German schools of invariant theory, see Parshall 1989, 176–180.
Gordan 1868, 327.
Hilbert 1889; Mertens 1886.
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.
Cf. Ferreirós 1999. 282–285.
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.
Gordan 1893.
Cf. Klein 1926–27, Vol. 1, 331.
Hilbert 1893, 287.
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.
Hilbert 1896, 383.
Hilbert 1893, 344.
For various assessments of Hilbert’s work on invariants see Parshall 1990, 11 ff.; Weyl 1939, 27–29; 1944, 627.
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).
Cf. Corry 2003, Ch. 2; Edwards 1975, 1977, 1980, 1987.
Cf. Blumenthal 1935, 397.
JDMV1 (1891), 12.
Noether & Brill 1892–3.
Schoenflies 1900.
Czuber 1899.
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.
Minkowski to Hilbert, March 31, 1896 (Rüdenberg & Zassenhaus eds. 1973, 79–80). English translation quoted from Rowe 2000, 61.
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.
Hilbert 1897, 66–67. English translation quoted from Hilbert 1998, x.
Cf. Minkowski 1905, 162–163.
Of special significance and greatest influence were Hilbert 1898; idem, 1899a. See Hasse 1932.
English translation quoted from Rowe 1994, 190.
Blumenthal 1922, 72.
Toeplitz 1922.
Cf. Corry 2003, Ch. 3.
Riemann 1868.
Quoted from Ewald (ed.) 1996, 652–653.
Cf. Ferreirós 1999, 15–16.
Ewald (ed.) 1999, Vol. 2, 661.
Cf. Ferreirós 2000, xcii-cxvi;Laugwitz 1999; Torretti 1978, 82–109.
Cf. Scholz 1980, especially pp. 101–135.
Cf. Torretti 1978, 155–171.
Cf. Richards 1988.
Cf Reich 1994 29–34 & 59–65.
Cf. Hawkins 2000, 111–137; Torretti 1978, 176–186.
Poncelet 1822.
Cf. Freudenthal 1974.
Cf. Ziegler 1985.
Jordan 1870. Cf. Corry 2003, 28–30.
Cf. Hawkins 1989, 317, note 13.
Cf. Hawkins 1989, 283–284.
For details about Klein’s formative years, see Parshall and Rowe 1994, 154–166; Rowe 1989a.
For an account of Cayley’s contributions, see Klein 1926–7 Vol. 1, 147–151.
Klein 1871 and 1873. For comments on these contributions of Klein, see Rowe 1994, 194–195; Toepell 1986, 4–6; Torretti 1978, 110–152.
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.
For an illuminating comparison between Klein’s and Killing’s differing perspectives on the scope of geometry, cf. Hawkins 2000, 130–137.
Quoted in Hawkins 2000, 137.
Klein 1898, 585.
Cf. Hawkins 1984.
Cf. Avellone, Brigagalia & Zapulla 2002, 385–399.
Cf. Hawkins 2000, 251–255, 291–292.
Klein 1874, 1880.
Cf. Ferreirós 1999, 119–124; Israel 1981.
Cf. Fisch 1999; Pycior 1981.
Cf. Rowe 1996a; Tobies 1996.
Grassmann 1995, 11.
Apelt to Grassmann. September 3, 1845. Quoted in Caneva 1978, 104.
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.
Cf Corry 2003, 67; Ferreirós 1999, 53–62.
Riemann, however, was not the only decisive influence on Dedekind’s career; Dirichlet also played a decisive role. Cf. Ferreirós 1999, 215–256.
Cf. Corry 2003, Ch. 2, see especially pp. 135–136.
Cf. Tazzioli 1994.
Dedekind Werke Vol. 3, 334.
This point has been illuminated in Ferreirós 1999, 246–248.
Quoted in Ewald 1996, 793–794.
Quoted in Ewald 1996, 809. Italics in the original. A very similar passage is found in §134 (p. 823).
As described in Ferreirós 1999, 238.
Pasch 1887, 129. See also Contro 1976; Nagel 1939, 193–199; Torretti 1978, 210–218.
Pasch & Dehn 1926, 19.
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. ”
Cf. Contro 1976, 287–290.
Klein GMAVol. 1, 397–398.
Wiener 1891, 46.
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.
Wiener 1983, 72.
Schur 1898
Cf. Kennedy 1980; Segre 1994.
Cf. Ferreirós 1999, 251.
Cf. Torretti 1978, 221.
Cf. Kennedy 1981. 443: Borga. et al. 1985.
Quoted in Avellone, Brigagalia & Zapulla 2002, 413.
Cf. Marchisotto 1993, 1995.
Cf. Torretti 1978, 225–226.
In Veronese 1891. Cf. Avellone, Brigagalia & Zapulla 2002, 380–385.
On criticism directed at Veronese’s work by German mathematicians, see Toepell 1986, 56.
Cf. Boi 1990.
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, 150–151
Cf. Contro 1985.
Cf Heilbron 1982, Jungnickel & McCormmach 1986, Vol. 2.
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.
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, 71–131
Maxwell 1860.
Maxwell 1867.
See 5.2 below.
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.
Boltzmann 1877. Cf. Brush 1976, 605–627.
The terms Umkehreinwandand Wiederkehreinwandwere introduced only in 1907 by Tatyana and Paul Ehrenfest. See Klein 1970, 115.
For the subtleties of Planck’s position on this issue see Heilbron 2000, 1–46; Hiebert 1971, 72–79; Kuhn 1978, 22–29.
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.
Cf. Kuhn 1978. 26–27: Jungnickel & McCormmach 1996.212–216.
This is different from Liouville’s theorem on analytic functions.
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.
Mach 1893, 495–496.
Cf. Hiebert 1968.
fßlackmore 1972 1 17–1 1 R
Cf. Blackmore 1972, 120.
Cf. DiSalle 1993. 345: Jungnickel& McCormmach 1986, Vol. 1, 1R1–1R5.
This trend is discussed in Barbour 1989, Ch. 12.
Neumann 1870, 3. Hereafter I refer to Neumann 1993.
Neumann 1993, 361. The reference is to Leibniz MS, Vol. 4, 135.
See Barbour 1989, 646–653; DiSalle 1993, 348–349.
Lange’s ideas are discussed in Barbour 1989, 655–662.
Neumann 1870, 22 (1993, 367).
Cf. Tobies & Rowe (eds.) 1990, 29.
Cf. Jungnickel & McCormmach 1986, Vol. 1, 184–185.
Cf. Maier 1998.
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.
Cf. Fölsing 1997a, 475–476.
Cf. Blackmore 1972, 119.
For recent discussions, see Baird et al. (eds.) 1998.
Cf. Lützen 1998, 2004 (Forthcoming).
Hertz 1956, 145.
Hertz 1956, 146.
In all these passages, the term “permissible ” is intended, of course, in the sense discussed in Hertz’s introduction, namely, logically consistent.
Cf. Lützen 1995, 76–83.
Boltzmann 1899, 88.
Cf. Jungnickel & McCormmach 1986, Vol. 2, 144–148; Olesko 1991, 439–448; Ramser 1974.
Cf. Schwermer 2003.
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.
Volkmann 1892
Volkmann 1894.
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.
Volkmann 1900, 363.
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. ”
See Boltzmann’s own account in Boltzmann 1899a, 109–111.
D’Agostino 2000, 201–222.
Boltzmann 1899a, 105–107.
Cf. JDMV8 (1900), 17–23.
Boltzmann 1899.
Cf. JDMV7 (1899), 4.
Boltzmann 1899, 90.
Reich 1985.
Toepell 1996. 235–238: Vollrath 1993.
Voss 1899.
Voss 1903
Voss 1908 1913. 1914.
Voss to Hilbert, July 19. 1899 (DHN418. 1).Ouoted in Toenell 1996,410.
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.
Voss 1901, 9 (note 2).
VOSS 1901, 11.
Voss 1901, 18–30.
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. ”
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).
Cf. especially Ferreirós 1999, 62–64.
Voss 1901, 40. As an important reference Voss mentioned Wien 1900, on which see also § 3.2 below.
Cf. Darrigol 2000, 354–356; Greffe, et al. (eds.) 1996.
Rowe 2001, esp. 78–84.
Klein to Hilbert, December 6, 1894 (Frei (ed.) 1985, 115). English translation quoted from Rowe 2000, 62.
Schoenflies 1891. Cf. Scholz 1989. 137–148.
Cf. Klein GMAVol. III, App. I, 8–9.
Rowe 1989, 202.
Rowe 1989, 191–193.
Manegold 1970.
For a recent, detailed account of the Encyklopädieproject, cf. Hashagen 2003, 487–522.
Cf. Hashagen 2003, 501–503.
Respectively Schubert 1898, Netto 1898, Pringsheim 1898.
Cf Klein to Dyck. December 30. 1900. Ouoted in Hashagen 2003. 504.
Von Dvck 1904. xi.Cf also JDMV9 (19011. 69.
Von Dyck 1904, xv; JDMV9 (1901), 69.
Pareto 1911. On the importance of this article, cf Ingrao & Israel 1985.
As Warwick 2003, 253, points out, Klein visited the Cambridge mathematical physicists frequently and deeply appreciated their achievements in applied mathematics.
Von Dyck 1904, xv; Cf. Gispert 2001.
Klein to von Dyck. December 24, 1905. Quoted in Hashagen 2003, 520.
Cf. Rowe 1989. 209.
Cf. Perron 1952.
Cf. Gispert 2001, 94–96.
Cf e a P7 5(19040 470 ; P7R (19070 549–551
Cf. von Mises 1924, 88.
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).
Fano 1907. See Hawkins 2000. 290–316 for the influence of this article on Elie Cartan (1869–1951).
Hellinger & Toeplitz 1927 Cf Köthe 19R2575–5R4l Dieudonné 1981 112
Von Dyck 1904, xii.
Cf. Gispert 1999.
Hilbert 1900a.
Cf. Schubring 1989.
Klein & Riecke 1900.
Klein & Riecke 1904.
Jungnickel & McCormmach 1986, Vol. 2, 118–124.
Riecke 1896.
A list of existing sources on Voigt is summarized in Schirrmacher 2003a. 19. note 6.
Kuhn 1978, 135.
Cf Olesko 1991 400–402
Voigt 1889.
Voigt 1895–96.
Jungnickel & McCormmach 1986. Vol. 2. 268–273.
Cf. Olesko 1991, 387–388.
As expressed in Voigt 1915, esp. p. 416.
See Barkan 1999, 58–76.
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Corry, L. (2004). Late Nineteenth Century Background. In: David Hilbert and the Axiomatization of Physics (1898–1918). Archimedes, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2778-9_2
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