Abstract
This paper explores such questions as who actually discovered non-euclidean geometry, who actually believed in its consistency and why, and who can be said to have proved it to be free of contradiction. To this end I will analyze some views and results if ten or so philosophers and mathematicians from Kant to Hilbert. One main theme is that without some rudimentary idea of a model, the discovery and establishment of non-euclidean geometry would not have been possible. Another is that only the notion of a model enabled thinkers to conceive of properties of logical inference such as soundness and completeness of axioms and/or rules. These themes are surprisingly difficult to articulate clearly without compromising historical accuracy, but I believe that in most cases the attempt to do so leads to a better understanding of the writers involved.
Keywords
- Euclidean Geometry
- Projective Geometry
- Hyperbolic Geometry
- Spherical Triangle
- Constant Negative Curvature
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Beltrami, Eugenio: 1868, `Saggio di interpretazione della geometria non euclidea’, Giornale di matematiche 6, 289–312.
Brittan, Gordon, Jr.: 1978, Kant’s Theory of Science, Princeton University Press, Princeton. Dehn, Max: 1922, ‘Hilberts geometrische Werk’, Die Naturwissenschaften 27, Heft 4, 77–82.
Dehn, Max: 1926, `Die Grundlagen der Geometrie in historischer Entwicklung’, in Moritz Pasch (ed.), Vorlesungen über die neuer Geometrie, Zweite Auflage, Springer, Berlin, pp. 185–271.
Engel, Friedrich and Stäckel, Paul: 1913, Urkunden zur Geschide der nichteuklidischen Geometrie, I I Wolfgang and Johann Bolyai, B. G. Teubner, Leipzig und Berlin.
Frege, Gottlob: 1980, Philosophical and Mathematical Correspondence, University of Chicago Press, Chicago.
Freudenthal, Hans: 1960, `Die Grundlagen der Geometrie um die Wende des 19. Jahrhundrets’, Mathematisch-Physikalisch Semesterbericht 7, 2–25.
Freudenthal, Hans: 1964, `Zu Herrn Bottemas Kritik’, Mathematisch-Physikalisch Semesterbericht 10, 114–117.
Friedman, Michael: 1985, `Kant’s Theory of Geometry’, The Philosophical Review 94, 455–506.
Gómez, Ricardo: 1986, `Beltrami’s Kantian View of Non-euclidean Geometry’, Kantstudien 77, 102–107.
Gray, Jeremy: 1989, Ideas of Space, 2nd ed. Clarendon Press, Oxford.
Helmholtz, Hermann von: 1977, Epistemological Writings, ed. R. S. Cohen and Y. Elkana, D. Reidel, Dordrecht.
Hilbert, David: 1902a, `Mathematical Problems’, Bulletin of the American Mathematical Society 8, 437–479.
Hilbert, David: 1902b, The Foundations of Geometry, trans. E. J. Townsend, Open Court, Chicago.
Hilbert, David: 1971, Foundations of Geometry, trans. L. Unger, Open Court, La Salle.
Hintikka, Jaakko: 1988, `On the Development of the Model-theoretic Viewpoint in Logical Theory’, Synthese 77, 1–36.
Kant, Immanuel: 1965, Critique of Pure Reason, trans. Norman Kemp Smith, St. Martin’s Press, New York.
Kant, Immanuel: 1902–56, Kants gesammelte Schriften,Herausgegeben von der Preussischen Akademie der Wissenschaften zu Berlin, 23 vols., de Gruyter, Berlin. (Cited as `Akademie’ by volume number in the text).
Kästner, Abraham: 1790, Was heisst in Euclids Geometrie möglich?’, Philosophisches Magazin 2, 391–402.
Kästner, Abraham: 1790, `Über den mathematischen Begriff des Raums’, Philosophisches Magazin 2, 403–419.
Kästner, Abraham: 1790, `über die geometrischen Axiome’, Philosophisches Magazin 2, 420–430.
Klein, Felix: 1921, Gesammelte matematische Abhandlungen, hrsg. R. Fricke und A. Ostrowski, Springer, Berlin.
Lambert, Johann: 1895, `Theorie der Parallellinien’, in F. Engel and P. Stäckel (eds.), Theorie der Parallellinien von Euclid bis auf Gauss, B. G. Teubner, Leipzig, pp. 152–207.
Martin, Gottfried: 1967, `Das geradlinige Zweieck, ein offener Widerspruch in der Kritik der reinen Vernunft’, in W. Arnold und H. Zelmer (eds.), Tradition und Kritik, Friedrich Frommann Verlag, Stuttgart, pp. 229–235.
Russell, Bertrand: 1956, An Essay on the Foundations of Geometry, Dover Publications, Toronto.
Scanlon, Michael: 1989, `Beltrami’s Model and the Independence of the Parallel Postulate’, History and Philosophy of Logic 3, 13–34.
Sommer, Julius: 1900, ‘Hilbert’s Foundations of Geometry’, Bulletin of the American Mathematical Society 6, 287–299.
Stäckel, Paul: 1899, ‘Bemerkungen zu Lamberts Theorie der Parallellinien’, Bibliotheca mathematical 13, 107–110.
Toepell, Michael-Markus: 1986, Über die Entstehung von David Hilberts “Grundlagen der Geometrie”, Vandenhoeck und Ruprecht, Göttingen.
Torretti, Roberto: 1978, Philosophy of Geometry from Riemann to Poincare, D. Reidel, Dordrecht.
Webb, Judson: 1987, `Immanuel Kant and the Greater Glory of Geometry’, in A. Shimony and D. Nails (eds.), Naturalistic Epistemology, D. Reidel, Dordrecht, pp. 17–70.
Weyl, Hermann: 1970, `David Hilbert and his Mathematical Work’, in Constance Reid (ed.), Hilbert, Springer, Berlin, pp. 245–283.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Webb, J. (1995). Tracking Contradictions in Geometry: The Idea of a Model from Kant to Hilbert. In: Hintikka, J. (eds) From Dedekind to Gödel. Synthese Library, vol 251. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8478-4_1
Download citation
DOI: https://doi.org/10.1007/978-94-015-8478-4_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4554-6
Online ISBN: 978-94-015-8478-4
eBook Packages: Springer Book Archive