Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T04:17:10.594Z Has data issue: false hasContentIssue false
  • English
  • Français

PROJECTIVE DUALITY AND THE RISE OF MODERN LOGIC

Part of: History of mathematics and mathematicians Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  26 July 2021

GÜNTHER EDER
GÜNTHER EDER*
Affiliation:
DEPARTMENT OF PHILOSOPHY UNIVERSITY OF VIENNA UNIVERSITÄTSSTRAßE7 1010VIENNA, AUSTRIAE-mail: guenther.eder@univie.ac.at
Get access Rights & Permissions [Opens in a new window]

Abstract

The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we want to see in which ways geometers’ preoccupation with duality was shaped by developments that lead to modern logic towards the end of the nineteenth century, and how these developments in turn might have been influenced by reflections on duality.

Keywords

dualityreciprocityearly metatheoryprojective geometry

MSC classification

Primary: 01A55: 19th century
Secondary: 01A60: 20th century 03A05: Philosophical and critical
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 27 , Issue 4 , December 2021 , pp. 351 - 384
Check for updates
Copyright
The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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

Andersen, K., The Geometry of an Art—The History of the Mathematical Theory of Perspective from Alberti to Monge, Springer, Berlin–Heidelberg, 2007.Google Scholar
Apéry, F., Models of the Real Projective Plane, Friedrich Vieweg, Braunschweig–Wiesbaden, 1987.CrossRefGoogle Scholar
Arana, A. and Mancosu, P., On the relationship between plane and solid geometry. The Review of Symbolic Logic, vol. 5 (2012), no. 2, pp. 294–353.CrossRefGoogle Scholar
Avellone, M., Brigaglia, A., and Zappulla, C., The foundations of projective geometry in Italy from De Paolis to Pieri. Archive for History of Exact Sciences, vol. 56 (2002), no. 5, pp. 363–425.10.1007/s004070200052CrossRefGoogle Scholar
Awodey, S. and Reck, E., Completeness and categoricity, part 1: Nineteenth-century axiomatics to twentieth-century metalogic. History and Philosophy of Logic, vol. 23 (2002), pp. 1–30.CrossRefGoogle Scholar
Badesa, C., Mancosu, P., and Zach, R., The development of mathematical logic from Russell to Tarski, 1900–1935, The Development of Modern Logic (Haaparanta, L., editor), Oxford University Press, Oxford, 2009.Google Scholar
Beltrami, E., Saggio di interpretazione della geometria non-euclidea. Giornale di mathematiche, vol. 6 (1868), pp. 284–312.Google Scholar
Bertran-San Millán, J., Frege, Peano and the interplay between logic and mathematics, forthcoming.Google Scholar
Blanchette, P., Models in geometry and logic: 1870–1920, Logic, Methodology and Philosophy of Science—Proceedings of the 15th International Congress (S. S. Niniiluoto, editor), College Publications, London, 2017, pp. 41–61.Google Scholar
Cayley, A., An introductory memoir upon quantics. Philosophical Transactions of the Royal Society of London, vol. 144 (1854), pp. 245–258.Google Scholar
Cayley, A., A sixth memoir upon quantics. Philosophical Transactions of the Royal Society of London, vol. 149 (1859), pp. 61–90.Google Scholar
Chasles, M., Aperçu Historique sur l’Origine et le Développement des Méthodes en Géométrie, Hayez, Bruxelles, 1837.Google Scholar
Coffa, A., From geometry to tolerance: Sources of conventionalism in nineteenth-century geometry, From Quarks to Quasars: Philosophical Problems of Modern Physics (R. Colodny, G., editor), University of Pittsburgh Press, Pittsburgh, 1986, pp. 3–70.Google Scholar
Coolidge, J. L., The rise and fall of projective geometry. The American Mathematical Monthly, vol. 41 (1934), no. 4, pp. 217 – 228.CrossRefGoogle Scholar
Coxeter, H. S. M., Projective Geometry, second ed., Springer, Berlin, 1987.Google Scholar
Coxeter, H. S. M., The Real Projective Plane, Springer, New York, 1992.Google Scholar
Cremona, L., Elements of Projective Geometry, Clarendon Press, Oxford, 1885.Google Scholar
Dawson, J. W., Why Prove it Again? Alternative Proofs in Mathematical Practice, Springer, Heidelberg–New York–Dordrecht–London, 2015.CrossRefGoogle Scholar
Dedekind, R., Was sind und was sollen die Zahlen? fourth 1918 ed., Friedrich Vieweg, Braunschweig, 1888.Google Scholar
Demopoulos, W., Frege, Hilbert, and the conceptual structure of model theory. History and Philosophy of Logic, vol. 15 (1994), no. 2, pp. 211–225.CrossRefGoogle Scholar
Detlefsen, M., Duality, epistemic efficiency and consistency, Formalism & Beyond (Link, G., editor), De Gruyter, Berlin, 2014, pp. 1–24.Google Scholar
Ferreirós, J., Hilbert, logicism, and mathematical existence. Synthese, vol. 170 (2009), no. 1, pp. 33–70.CrossRefGoogle Scholar
Field, J. and Gray, J., The Geometrical Work of Girard Desargues, Springer, New York, 1987.CrossRefGoogle Scholar
Frege, G., Philosophical and Mathematical Correspondence, Blackwell, Oxford, 1980.Google Scholar
Frege, G., Collected Papers on Mathematics, Logic and Philosophy, Basil Blackwell, Oxford, 1984.Google Scholar
Freudenthal, H., The main trends in the foundations of geometry in the 19th century, Logic, Methodology, and Philosophy of Science (Nagel, E., Suppes, P., and Tarski, A., editors), Stanford University Press, Stanford, 1962, pp. 613–621.Google Scholar
Gergonne, J. D., Philosophie mathématique: Considérations philosophiques Sur les élémens de la science de l’étendue. Annales de Mathématiques Pures et Appliquées, vol. 16 (1825/1826), pp. 209–231.Google Scholar
Gergonne, J. D., Polémique mathématique: Réclamation de M. le capitaine Poncelet (extraite du bulletin universel des annonces et nouvelles scientifiques) avec des notes. Annales de Mathématiques Pures et Appliquées, vol. 18 (1827), pp. 125–142.Google Scholar
Giovannini, E. N. and Schiemer, G., What are implicit definitions? Erkenntnis , pp. 1–31, forthcoming.Google Scholar
Gray, J., Jean victor Poncelet, Traité des propriétés projectives des figures, Landmark Writings in Western Mathematics (Grattan-Guiness, I., editor), Elsevier, Amsterdam, 2005, pp. 366–376.CrossRefGoogle Scholar
Gray, J., Worlds out of Nothing—A Course in the History of Geometry in the 19th Century, Springer, New York, 2007.Google Scholar
Hallett, M., Reflections on the purity of method in Hilbert’s Grundlagen der Geometrie, The Philosophy of Mathematical Practice (Mancosu, P., editor), Oxford University Press, Oxford, 2008, pp. 198–255.CrossRefGoogle Scholar
Hallett, M. and Majer, U., David Hilbert’s Lectures on the Foundations of Geometry 1891–1902, Springer, Berlin–Heidelberg, 2004.CrossRefGoogle Scholar
Hawkins, T., Hesse’s principle of transfer and the representation of Lie algebras. Archive for History of Exact Sciences, vol. 39 (1988), no. 1, pp. 41–73.CrossRefGoogle Scholar
Hesse, O., Ein Uebertragungsprincip. Journal für die Reine und Angewandte Mathematik, vol. 66 (1866a), pp. 15–21.Google Scholar
Hesse, O., Vier Vorlesungen Aus der Analytischen Geometrie, B. G. Teubner, Leipzig, 1866b.Google Scholar
Heyting, A., Axiomatic Projective Geometry, North-Holland, Amsterdam, 1980.Google Scholar
Hilbert, D., Projektive Geometrie: Vorlesungsmanuskript (Vorlesung SS 1891), Niedersächsische Staats- und Universitätsbibliothek Göttingen, Handschriftenabteilung, 1891, Cod. Ms. D. Hilbert 535, partly published in Hallett & Majer 2004.Google Scholar
Hilbert, D., Grundlagen der Geometrie, 10th 1968 ed., Teubner, Leipzig, 1899.Google Scholar
Hintikka, J., On the development of the model-theoretic viewpoint in logical theory. Synthese, vol. 77 (1988), no. 1, pp. 1–36.CrossRefGoogle Scholar
Kennedy, H. C., The origins of modern axiomatics: Pasch to Peano. The American Mathematical Monthly, vol. 79 (1972), no. 2, pp. 133–136.CrossRefGoogle Scholar
Klein, F., Über sogenannte Nicht-Euklidische Geometrie. Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, vol. 17 (1871), pp. 419–433.Google Scholar
Klein, F., Vergleichende Betrachtungen über neuere geometrische Forschungen, Mathematische Annalen, vol. 43 (1893), no. 1, pp. 63–100.CrossRefGoogle Scholar
Klein, F., Über sogenannte Nicht-Euklidische Geometrie (2). Mathematische Annalen, vol. 6 (1873), pp. 311–343.CrossRefGoogle Scholar
Klein, F., Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (Teile 1 und 2), Springer, Berlin–Heidelberg–New York, 1926/1927.Google Scholar
Klein, F. and Blaschke, W., Vorlesungen über Höhere Geometrie, Springer, Berlin, 1926.Google Scholar
Krömer, R. and Corfield, D., The form and function of duality in modern mathematics. Philosophia Scientiae, vol. 18 (2014), no. 3, pp. 95–109.CrossRefGoogle Scholar
Lorenat, J., Polemics in public: Poncelet, Gergonne, Plücker, and the duality controversy. Science in Context, vol. 28 (2015), no. 4, pp. 545–585.CrossRefGoogle ScholarPubMed
MacLane, S., Duality for groups. Bulletin of the American Mathematical Society, vol. 56 (1950), no. 6, pp. 485–516.CrossRefGoogle Scholar
Marchisotto, E. A. and Smith, J. T., The Legacy of Mario Pieri in Geometry and Arithmetic, Birkhäuser, Boston, 2007.Google Scholar
Marquis, J.-P., From a Geometrical Point of View: A Study in the History and Philosophy of Category Theory, Springer, Berlin, 2009.Google Scholar
Möbius, A. F., Der Barycentrische Calcul. Ein neues Hülfsmittel zur Analytischen Behandlung der Geometrie, Johann Ambrosius Barth, Leipzig, 1827.Google Scholar
Nagel, E., The formation of modern conceptions of formal logic in the development of geometry. Osiris, vol. 7 (1939), pp. 142–223.CrossRefGoogle Scholar
Pasch, M., Vorlesungen über Neuere Geometrie, Teubner, Leipzig, 1882.Google Scholar
Pasch, M., Betrachtungen zur Begründung der Mathematik. Mathematische Zeitschrift (1924), pp. 231–240.CrossRefGoogle Scholar
Pasch, M., Begriffsbildung und Beweis in der Mathematik. Annalen der Philosophie und philosophischen Kritik, vol. 4 (1924/1925), no. 7, pp. 348–367.Google Scholar
Pieri, M., I principii della geomtria di posizione composti in sistema logico deduttivo. Memorie della Reale Accademia delle Scienze di Torino (Series 2), vol. 48 (1898), pp. 1–62.Google Scholar
Pieri, M., Della geometria elementare come sistema ipotetico deduttivo: Monografia del punto e del moto. Memorie della Reale Accademia delle Scienze di Torino (Series 2), vol. 49 (1900), pp. 173–222.Google Scholar
Plücker, J., Analytisch-Geometrische Entwicklungen, vol. 2, G. D. Baedeker, Essen, 1831.Google Scholar
Poncelet, J.-V., Traité des Propriétés Projectives des Figures, 1865/1866 ed., Gauthier-Villars, Paris, 1822.Google Scholar
Reck, E., Dedekind’s structuralism: An interpretation and partial Defense. Synthese, vol. 137 (2003), pp. 369–419.CrossRefGoogle Scholar
Reye, T., Die Geometrie der Lage, second 1877 ed., Carl Rümpler, Hannover, 1866.Google Scholar
Richter-Gebert, J., Perspectives on Projective Geometry, Springer, Berlin–Heidelberg, 2011.CrossRefGoogle Scholar
Schiemer, G., Transfer principles, Klein’s Erlangen program, and methodological structuralism, The Prehistory of Mathematical Structuralism (Reck, E. and Schiemer, G., editors), Oxford University Press, New York, 2020, pp. 106–141.CrossRefGoogle Scholar
Schiemer, G. and Eder, G., Hilbert, duality, and the geometrical roots of model theory. The Journal of Symbolic Logic, vol. 11 (2018), no. 1, pp. 48–86.Google Scholar
Schiemer, G. and Reck, E., Logic in the 1930s: Type theory and model theory, this Journal, vol. 19 (2013), no. 4, pp. 433–472.Google Scholar
Schiemer, G., Zach, R., and Reck, E., Carnap’s early metatheory: Scope and limits. Synthese, vol. 194 (2017), no. 1, pp. 33–65.CrossRefGoogle Scholar
Schlimm, D., Pasch’s philosophy of mathematics. The Review of Symbolic Logic, vol. 3 (2010), no. 1, pp. 93–118.CrossRefGoogle Scholar
Schlimm, D., The correspondence between Moritz Pasch and Felix Klein. Historia Mathematica, vol. 40 (2013), pp. 183–202.CrossRefGoogle Scholar
Schlimm, D., Pasch’s empiricism as a methodological structuralism, The Prehistory of Mathematical Structuralism (Reck, E. and Schiemer, G., editors), Oxford University Press, New York, 2020, pp. 88–105.CrossRefGoogle Scholar
Specker, E., Dualität. Dialectica, vol. 12 (1958), nos. 3–4, pp. 451–465.CrossRefGoogle Scholar
von Staudt, G. K. C., Geometrie der Lage, Bauer und Raspe, Nürnberg, 1847.Google Scholar
von Staudt, G. K. C., Beiträge zur Geometrie der Lage, Friedrich Korn’schen Buchhandlung, Nürnberg, 1856.Google Scholar
Steiner, J., Systematische Entwicklung der Abhängigkeit Geometrischer Gestalten von Einander, G. Fincke, Berlin, 1832.Google Scholar
Tappenden, J., Metatheory and mathematical practice in Frege, Gottlob Frege: Critical Assessment of Leading Philosophers, vol. 2 (Beaney, M. and Reck, E., editors), Routledge, New York, 2005, pp. 190–228.Google Scholar
Toepell, M., Über die Entstehung von David Hilberts Grundlagen der Geometrie, Vandenhoeck & Ruprecht, Göttingen, 1986.Google Scholar
Torretti, R., Philosophy of Geometry from Riemann to Poincaré, Reidel, Dordrecht–Boston-London, 1978.CrossRefGoogle Scholar
Veblen, O. and Young, J. W., Projective Geometry, vol. 1, Blaisdell, New York–Toronto–London, 1910.Google Scholar
Visser, A., An overview of interpretability logic, Advances in Modal Logic, Vol. 1 (Kracht, M., deRijke, M., Wansing, H. & Zakharyaschev, M., editors), CSLI Publications, Stanford, 1998, pp. 307–359.Google Scholar
Webb, J., Tracking contradictions in geometry: The idea of a model from Kant to Hilbert, From Dedekind to Gödel. Essays on the Development of the Foundations of Mathematics (Hintikka, J., editor), Springer, Dordrecht, 1995, pp. 1–20.Google Scholar
Wilson, M., Frege: The royal road from geometry. Noûs, vol. 26 (1992), no. 2, pp. 149–180.CrossRefGoogle Scholar