Abstract
A careful analysis is made, in the light of subsequent developments, of the background, content, aim, and influence of the ten German editions of Hilbert's “Grundlagen der Geometrie”. Especial attention is given to the sources of Hilbert's ideas, the critical reactions of contemporary geometers to the first edition, the latter's connections with Hilbert's famous problems, and its role as a model for later axiomatic approaches tho the foundations of mathematics and mathematical physics.
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References
[Arc] Raymond C., Archibald.,Remarks on Klein's Famous Problems of Elementary Geometry, Am. Math. Montly21 (1914), 247–59.
[Art] Emil Artin.,Geometric Algebra, Interscience, 1957.
[B-B] Birkhoff G., Bennett M.K.,Felix Klein and his Erlanger Programm, inHistory and Philosophy of Modern Mathematics, (W. Aspray and P. Kitcher, eds.) U. of Minn. Press, 1988.
[Ba] Rouse Ball W.W.,A Short Account of the History of Mathematics, 3rd ed, Macmillan, ca 1900.
[Be] Bell E.T.,The Development of Mathematics, McGraw-Hill, 1940.
[Ben] Bennett M.K.,Biatomic Lattices, in Algebra Universalis, 1988.
[Bo] Carl B Boyer.,History of Geometry, Scripta Mathematica, 1956.
[Br] Browder F.E.,Mathematical Developments Arising from Hilbert Problems, Proc. Symposia in Pure Math. 28, Providence, Am. Math. Soc., 1976.
[Co] Coolidge J.L.,A History of Geometrical Methods, Clarendon Press, Oxford, 1940.
[De] Réné Descartes.,La Géométrie (Paris ca. 1657). References are to the translation by David E. Smith and M. L. Latham, published under the title “The Geometry of Réné Descartes, Open Court Chicago, 1925.
[EB]Encyclopaedia Britannica, 1971 edition.
[EMW]Encyclopadie der Mathematische Wissenschaften, vol. III, Part 1 (first half). Teubner, 1907–10.
[En] Enriques F.,Prinzipien der Geometrie, pp. 1–129 of Enz. Math. Wiss. III. AB. (ca. 1907).
[Eu] Sir Thomas Heath.,The Thirteen Books of Euclid's Elements, 3 vols. Second ed., Cambridge Univ. Press, 1921; Dover reprint 1956.
[Fr1] Freudenthal H.,Zur Geschichte der Grundlagen der Geometrie, Niew Arch. voor. Wisk.5 (1957), 105–142.
[Fr2] Freudenthal H.,The Main Trends in the Foundations of Geometry in the 19th Century, inProc. International Congress on Logic Methodology and the Philosopy of Science, 1960 (E. Nagel, P. Suppes, A. Tarski, eds.).
[GdG] David Hilbert.,Grundlagen der Geometrie, First ed., Tuebner, 1899.
[H-B] David Hilbert., Bernays P.,Grundlagen der Mathematik, Springer, Berlin 1934, v. 2 1939.
[H1] David Hilbert.,Foundations of Geometry, Springer, Berlin 1934, v. 2 1939.
[H7] David Hilbert.,Grundlagen der Geometrie, 7te Aufl. Teubner, 1930.
[H10] David Hilbert.,Foundations of Geometry, English translation of the 10th German edition (P. Bernays, Ed.). Open Court, 1971.
[HGA] David Hilbert.,Gesammelte Abhandlungen, (3 vols.), Springer 1935.
[He] Hedrick E.R.,The English and French Translation of Hilbert's Grundlangen der Geometrie, Bull. Am. Math. Soc. 9 (1902), 158–65.
[JvH] J. van Heijenoort.,From Frege to Gödel, Harvard University Press, 1967.
[K1] Felix Klein.,Famous Problems of Elementary Geometry, Trans. by W.W. Beman and D.E. Smith New York Dover, 1956. Stechert. 1940; Dover, 1956. (German edition published in 1908).
[K2] Felix Klein.,Elementary Mathematics from an Advanced Standpoint, v. 2. Trans. by E.R. Hedrick and C.A. Nobel. Macmillan, 1939. (Original German edition published in 1908).
[Kl] Morris Kline.,Mathematical Thought from Ancient to Modern Times, Oxford Univ. Press. 1972.
[Lo] Gino Loria.,Pour une histoire de la géometrie analytique, Verh. 3rd Int. Math. Kongr., Heidelberg, 1904, 562–74.
[LT2] Garrett Birkhoff.,Lattice Theory, second edition. American Mathematical Society, 1948.
[Mo] E.H. Moore.,On the Projective Axioms of Geometry, Trans. Am. Math. Soc.9 (1903), 402–24.
[Mo2] E.H. Moore.,On the Foundations of Mathematics, Am. Math. Soc.9 (1903), 402–24.
[Mo3] Moore R.L.,Sets of Metrical Hypotheses for Geometry, Trans. TAMS9 (1908) 487–512.
[Po] Henri Poincaré.,Poincaré's Review of Hilbert's Foundations of Geometry, trans. by E.V. Huntington. Bull. Am. Math. Soc.9 (1903), 1–23.
[Sch] Schmidt A.,Zu Hilberts Grundlagen der Geometrie, in Hilbert, Ges. Werk vol. ii., pp. 400–410.
[To] Torretti R.,Philosophy of Geometry from Riemann to Poincaré, Reidel Boston 1978
[V] Vahlen K.Th.,Abstrakte Geometrie, Teuber, 1905.
[Ve] Veblen O.,A System of Axioms for Geometry, Trans. AMS5 (1904), 343–33.
[VY] Veblen O., Young J.W.,Projective Geometry, 2 vols. Ginn, Boston, 1910, 1918.
[We] Herman Weyl.,David Hilbert and His Mathematical Work, Bull. Am. Math. Soc.50 (1944), 612–54.
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Birkhoff, G., Bennett, M.K. Hilbert's “Grundlagen der Geometrie”. Rend. Circ. Mat. Palermo 36, 343–389 (1987). https://doi.org/10.1007/BF02844894
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DOI: https://doi.org/10.1007/BF02844894