Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-22T20:51:27.046Z Has data issue: false hasContentIssue false
  • English
  • Français

HILBERT, DUALITY, AND THE GEOMETRICAL ROOTS OF MODEL THEORY

Published online by Cambridge University Press:  29 December 2017

GÜNTHER EDER  and
GEORG SCHIEMER
GÜNTHER EDER*
Affiliation:
Department of Philosophy, University of Salzburg
GEORG SCHIEMER*
Affiliation:
Department of Philosophy, University of Vienna
*
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF SALZBURG FRANZISKANERGASSE 1 A-5020 SALZBURG, AUSTRIA E-mail: [email protected]
†DEPARTMENT OF PHILOSOPHY UNIVERSITY OF VIENNA UNIVERSITÄTSSTRAßE 7 A-1010 VIENNA, AUSTRIA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’s Foundations of Geometry (1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’s Foundations.

Keywords

03A0500A30Hilberthistory of geometryearly metatheoryhistory of model theorymodern axiomatics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 11 , Issue 1 , March 2018 , pp. 48 - 86
Copyright
Copyright © Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Andersen, K. (2007). The Geometry of an Art – The History of the Mathematical Theory of Perspective from Alberti to Monge. Berlin Heidelberg: Springer.Google Scholar
Arana, A. & Mancosu, P. (2012). On the relationship between plane and solid geometry. The Review of Symbolic Logic, 5(2), 294–353.Google Scholar
Baldus, R. (1928). Zur Axiomatik der Geometrie. Über Hilberts Vollständigkeitsaxiom. Mathematische Annalen, 100, 321–333.Google Scholar
Blanchette, P. (2012). Frege’s Conception of Logic. Oxford: Oxford University Press.Google Scholar
Blanchette, P. (2014). The Frege-Hilbert controversy. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Spring 2014 Edition). Available at: http://plato.stanford.edu/archives/spr2014/entries/frege-hilbert/.Google Scholar
Coxeter, H. M. S. (1987). Projective Geometry (second edition). Berlin: Springer.Google Scholar
Eder, G. (2016). Frege’s ‘On the foundations of geometry’ and axiomatic metatheory. Mind, 125(497), 5–40.CrossRefGoogle Scholar
Field, J. V. & Gray, J. J. (1987). The Geometrical Work of Girard Desargues. New York: Springer.CrossRefGoogle Scholar
Frege, G. (1980). Philosophical and Mathematical Correspondence. Oxford: Blackwell Publishers.Google Scholar
Giovannini, E. (2016). Bridging the gap between analytic and synthetic geometry: Hilbert’s axiomatic approach. Synthese, 193, 31–70.CrossRefGoogle Scholar
Gray, J. (1989). Ideas of Space: Euclidean, NonEuclidean, and Relativistic. Oxford: Oxford University Press.Google Scholar
Gray, J. (2007). Worlds out of Nothing – A Course in the History of Geometry in the 19th Century. New York: Springer.Google Scholar
Gray, J. (2008). Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press.Google Scholar
Hallett, M. (1994). Hilbert’s axiomatic method and the laws of thought. In George, A., editor. Mathematics and Mind. Oxford: Oxford University Press, pp. 158–200.Google Scholar
Hallett, M. (2008). Reflections on the purity of method in Hilbert’s Grundlagen der Geometrie. In Mancosu, P., editor. The Philosophy of Mathematical Practice. Oxford: Oxford University Press, pp. 198–255.Google Scholar
Hallett, M. (2010). Frege and Hilbert. In Potter, M. and Ricketts, T., editors. The Cambridge Companion to Frege. Cambridge: Cambridge University Press, pp. 413–464.Google Scholar
Hilbert, D. (1968). Grundlagen der Geometrie (tenth edition). Leipzig: Teubner, An English translation is available as Foundations of Geometry, Unger, L. (ed.), La Salle: Open Court Press, 1971.Google Scholar
Hilbert, D. (1900a). Mathematische Probleme. Nachrichten von der königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, Vol. 3, 253–296.Google Scholar
Hilbert, D. (1900b). Über den Zahlbegriff. Jahresbericht der deutschen Mathematiker-Vereinigung, Vol. 8, 180–185.Google Scholar
Hilbert, D. (2004). David Hilbert’s Lectures on the Foundations of Geometry 1891–1902. Berlin Heidelberg: Springer-Verlag.Google Scholar
Hilbert, D. (2013). David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933. Berlin Heidelberg: Springer.Google Scholar
Hilbert, D. & Bernays, B. (1934). Grundlagen der Mathematik, Vol. 1. Berlin: Springer, An English translation is available as Foundations of Mathematics I, Wirth, C.-P. (ed.), London: College Publications, 2011.Google Scholar
Hintikka, J. (1988). On the development of the model-theoretic viewpoint in logical theory. Synthese, 77(1), 1–36.CrossRefGoogle Scholar
Hintikka, J. (2011). What is the axiomatic method? Synthese, 183(1), 69–85.Google Scholar
Hodges, W. (1993). Model Theory. Cambridge: Cambridge University Press.Google Scholar
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.Google Scholar
Lindström, P. (1997). Aspects of Incompleteness. Berlin: Springer.Google Scholar
Nagel, E. (1939). The formation of modern conceptions of formal logic in the development of geometry. Osiris, 7, 142–223.CrossRefGoogle Scholar
Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig: Teubner.Google Scholar
Poncelet, V. (1822). Traité des propriétés projectives des figures. Paris: Gauthier-Villars.Google Scholar
Schiemer, G. & Reck, E. (2013). Logic in the 1930s: Type theory and model theory. Bulletin of Symbolic Logic, 19(4), 433–472.CrossRefGoogle Scholar
Schlimm, D. (2010). Pasch’s philosophy of mathematics. Review of Symbolic Logic, 3(1), 93–118.Google Scholar
Sieg, W. (2014). The ways of Hilbert’s axiomatics: Structural and formal. Perspectives on Science, 22(1), 133–157.Google Scholar
Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica, 1, 261–405.Google Scholar
Toepell, M. (1986). Über die Entstehung von David Hilberts Grundlagen der Geometrie. Göttingen: Vandenhoeck & Ruprecht.Google Scholar
Torretti, R. (1978). Philosophy of Geometry from Riemann to Poincaré. Dordrecht/Boston/London: D. Reidel.Google Scholar
Veblen, O. & Young, J. W. (1938). Projective Geometry. New York: Ginn and Company.Google Scholar
Visser, A. (1997). An overview of interpretability logic. In Kracht, M., Rijke, M. d., Wansing, H., editors. Advances in Modal Logic ’96. Stanford, CA: CSLI Publications, pp. 307–359.Google Scholar