Abstract
The observation that the geometry of Bolyai and Lobachevsky and related geometries can be derived from mere axioms about connection [7–10]1 and the actual development of the hyperbolic plane geometry in terms of concepts of alignment by the Notre Dame school of geometry (1937–46) [1, 3, 4, 5, 13] seem to have attracted little attention.2 Euclidean geometry cannot be developed from axioms about connection (the first of Hilbert’s five groups of axioms) and, in fact, requires postulates about congruence, which are usually supplemented by assumptions about alignment, parallelism , order, and perpendicularity. The aforementioned observation thus reveals an inherent elementary character of hyperbolic geometry which does not seem to bear out Poincaré’s claim that, compared to non-Euclidean geometries, the Euclidean geometry is distinguished by simplicity (it is ‘athe simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree’).
Numbers in brackets refer to the references at the end of the paper.
Note of the in many ways excellent treatises by borsuk and szmielew.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Rice Institute Pamphlets, 27. No. 1 (1940) 80–107.
K. Menger, ‘Statistical Metrics’, Proc. Nat. Acad. Sci. USA28 (1942), 535–537.
K. Menger, ’Probabilistic Geometry’, ibid. 37 (1951), 226–229.
K. Menger, ’Géométrie Générale’, Mem. Sci. Math. Paris, 124 (1954).
A. Wald, ’On a Statistical Generalization of Metric Spaces’, Proc. Nat. Acad. Sci. U.S.A., 29 (1943), 196–197.
B. Schweizer and A. Sklar, ’Espaces métriques aléatoires’, Comptes Rendus Acad. Sci. Paris, 247 (1958), 2092–2094.
B. Schweizer and A. Sklar, ’Statistical Metric Spaces’, Pacif. J. Math. 10 (1960), 313–334.
A. N. Šerstnev, ’On the Concept of a Stochastic Normalized Space’, Dokl. Akad. Nauk SSSR149 (1963), 380–283.
For a brief summary, see B. Schweizer, ’Probabilistic Metric Spaces - The First 25 Years’, The New York Statistician19 (1967), 3–6.
B. Schweizer and A. Sklar, ’Statistical Metric Spaces Arising from Sets of Random Variables in Euclidean n-Space’, Teoria Veroyatnostey7 (1962), 456–465.
E. Schrödinger, Ann. Phys. 63 (1920), 397–456.
K. Menger, ’Ensembles flous et fonctions aléatoires’, Comptes Rendus Acad. Sci. Paris232 (1951), 2001–2003.
R. Bellman, R. Kalaba, and L. Zadeh, ’Abstraction and Pattern Classification’, J. Math. Anal. Appl. 13 (1966), 1–7.
K. Menger, The Theory of Relativity and Geometry’, in Albert Einstein: Philosopher-Scientist, Evanston, I11., 1949, pp. 472–474.
E.g., in ’An Axiomatic Theory of Functions and Fluents’ in The Axiomatic Method (ed. by Henkin et al.), Amsterdam 1959, pp. 454–473.
’Mensuration and other Mathematical Connections of Observable Material’ in Measurement, Definitions and Theories (ed. by Churchman and Ratoosh), New York, 1959, pp. 97–128.
Cf. the papers quotes in 11 and ’Variables, Constants, Fluents’ in Current lssues in the Philosophy of Science (ed. by Feigl and Maxwell), New York, 1961 pp. 304–316; ’A Counterpart of Occam’s Razor in Pure and Applied Mathematics’, Synthese 12 (1960), 415–428, and 13 (1961), 331–349.
J. Mehlberg, ’A Classification of Mathematical Concepts’, Synthese14 (1962).
Cf. especially the second paper quoted in 11 and ’Rejoinder to Adams’ in Current lssues in the Philosophy of Science, quoted in 12, p. 318.
Houtappel, Van Dam, and Wigner, ’The Conceptual Basis and Use of the Geometrie Invariance Principles’, Rev. Mod. Phys. 37 (1965), especially pp. 598–600. Added in proof: For explicit references to procedures, see also H. Ekstein, ’Presymmetry’, Phys. Rev. 153 (1967), 1397–1402, and ’Presymmetry II’, Phys. Rev., Aug. 25,1969.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Wien
About this chapter
Cite this chapter
Menger, K. (2002). The New Foundation of Hyperbolic Geometry. In: Schweizer, B., et al. Selecta Mathematica. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6110-4_37
Download citation
DOI: https://doi.org/10.1007/978-3-7091-6110-4_37
Published:
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-7282-7
Online ISBN: 978-3-7091-6110-4
eBook Packages: Springer Book Archive