Cassirer and the Structural Turn in Modern Geometry
DOI:
https://doi.org/10.15173/jhap.v6i3.3439Abstract
The paper investigates Ernst Cassirer’s structuralist account of geometrical knowledge developed in his Substanzbegriff und Funktionsbegriff (1910). The aim here is twofold. First, to give a closer study of several developments in projective geometry that form the direct background for Cassirer’s philosophical remarks on geometrical concept formation. Specifically, the paper will survey different attempts to justify the principle of duality in projective geometry as well as Felix Klein’s generalization of the use of geometrical transformations in his Erlangen program. The second aim is to analyze the specific character of Cassirer’s geometrical structuralism formulated in 1910 as well as in subsequent writings. As will be argued, his account of modern geometry is best described as a “methodological structuralism”, that is, as a view mainly concerned with the role of structural methods in modern mathematical practice.References
Biagioli, Francesca, 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer.
Birkhoff, Garrett and M. K. Bennett, 1988. “Felix Klein and His ‘Erlanger Programm’.” In History and Philosophy of Modern Mathematics, edited by W. Aspray and P. Kitcher, pp. 145–76. Minneapolis: University of Minnesota Press.
Carnap, Rudolf, 1922. Der Raum: Ein Beitrag zur Wissenschaftslehre. Berlin: Reuther & Reichard.
———, 1929. Abriss der Logistik. Vienna: Springer.
———, 2000. Untersuchungen zur allgemeinen Axiomatik, edited by T. Bonk and J. Mosterin. Darmstadt: Wissenschaftliche Buchgesellschaft.
Cassirer, Ernst, 1907. “Kant und die moderne Mathematik.” Kant-Studien 12: 1–40.
———, 1910. Substanzbegriff und Funktionsbegriff. Untersuchungen über die Grundfrage der Erkenntniskritik. Berlin: Springer.
———, 1923. Substance and Function. Chicago: Open Court.
———, 1944. “The Concept of Group and the Theory of Perception.” Philosophy and Phenomenological Research 5: 1–36.
———, 2010. Vorlesungen und Vorträge zu philosophischen Problemen der Wissenschaften, edited by K. C. Köhnke, J. M. Krois and O. Schwemmer. Hamburg: Felix Meiner Verlag.
Chasles, Michel, 1837. Aperçu historique sur l’origine et le dévéloppement des méthodes en géométrie. Brussels: M. Hayez.
Coxeter, H. M. S., 1974/1987. Projective Geometry, 2nd ed. Berlin: Springer.
Dedekind, Richard, 1888. Was sind und was sollen die Zahlen? Brunswick: Vieweg and Son.
Desargues, Girard, 1639. Brouillon projet d’une atteinte aux événements des rencontres du cône avec un plan. Paris: M. Poudra.
Frege, Gottlob, 1980. Philosophical and Mathematical Correspondence, edited by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel and A. Veraart, translated by B. McGuinness and H. Kaal. Oxford: Blackwell.
Gergonne, Joseph Diez, 1825–1826. “Considérations philosophiques sur les éléments de la science de l’éntendue.” Annales de Mathématiques Pures et Appliquées 16: 209–31.
Gray, Jeremy, 2007. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. New York: Springer.
———, 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press.
Hallett, Michael, 2008. “Reflections on the Purity of Method in Hilbert’s Grundlagen der Geometrie.” In The Philosophy of Mathematical Practice, edited by P. Mancosu, pp. 198–255. Oxford: Oxford University Press.
Hawkins, Thomas, 2000. Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926. New York: Springer.
Heis, Jeremy, 2007. The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer. PhD Dissertation, University of Pittsburgh.
———, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.
———, 2014. “Ernst Cassirer’s Substanzbegriff und Funktionsbegriff .” Hopos: The Journal of the International Society for the History of Philosophy of Science 4: 241–70.
Hilbert, David, 1899. Grundlagen der Geometrie, 10th ed. Leipzig: Teubner, 1968.
Ihmig, Karl-Norbert, 1997. Cassirers Invariantentheorie der Erfahrung und seine Rezeption des Erlanger Programms. Hamburg: Felix Meiner Verlag.
———, 1999. “Ernst Cassirer and the Structural Conception of Objects in Modern Science: The Importance of the ‘Erlanger Programm’.” Science in Context 12: 513–29.
Klein, Felix, 1893. “Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen 1872.” Mathematische Annalen 43: 63–100.
———, 1926. Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Springer.
———, 1928. Vorlesungen über Nicht-Euklidische Geometrie. Berlin: Springer.
Linnebo, Øystein, 2008. “Structuralism and the Notion of Dependence.” Philosophical Quarterly 58: 59–79.
Marquis, Jean-Pierre, 2009. From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. New York: Springer.
Nagel, Ernest, 1939. “The Formation of Modern Conceptions of Formal Logic in the Development of Geometry.” Osiris 7: 142–223.
Natorp, Paul, 1910. Die logischen Grundlagen der exakten Wissenschaften. Leipzig: Tuebner.
Parsons, Charles, 1990. “The Structuralist View of Mathematical Objects.” Synthese 84: 303–46.
Pasch, Moritz, 1882. Vorlesungen über neuere Geometrie. Leipzig: Teubner.
Poncelet, Jean-Victor, 1822. Traité des propriétés projectives des figures. Paris: Gauthier-Villars.
Reck, Erich, 2003. “Dedekind’s Structuralism: An Interpretation and Partial Defense.” Synthese 137: 369–419.
Reck, Erich and Pierre Keller, forthcoming. “From Dedekind to Cassirer: Logicism and the Kantian Heritage.” In Kant’s Philosophy of Mathematics, vol. 2, edited by O. Rechter and C. Posy. Oxford: Oxford University Press.
Resnik, Michael D., 1997. Mathematics as a Science of Patterns. New York: Oxford University Press.
Rowe, David E., 1985. “Felix Klein’s Erlanger Antrittsrede.” Historia Mathematica 12: 123–41.
Schiemer, Georg and Günther Eder, 2018. “Hilbert, Duality, and the Geometrical Roots of Model Theory.” The Review of Symbolic Logic, forthcoming.
Schlimm, Dirk, 2010. “Pasch’s Philosophy of Mathematics.” Review of Symbolic Logic 3: 93–118.
———, 2013. “Axioms in Mathematical Practice.” Philosophia Mathematica 21: 37–92.
Shapiro, Stewart, 1997. Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press.
Sieg, Wilfried, 2014. “The Ways of Hilbert’s Axiomatics: Structural and Formal.” Perspectives on Science 22: 133–57.
Toepell, Michael-Markus, 1986. Über die Entstehung von David Hilberts Grundlagen der Geometrie. Göttingen: Vandenhoeck & Ruprecht.
Torretti, Roberto, 1978. Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel.
Weber, Heinrich and Josef Wellstein, 1903. Encyklopädie der Elementar-Mathematik: Ein Handbuch für Lehrer und Studierende. Leipzig: Teubner.
Wussing, Hans, 1984/2007. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory. New York: Dover.
Yap, Audrey, 2017. “Dedekind and Cassirer on Mathematical Concept Formation.” Philosophia Mathematica 25: 369–89.
Birkhoff, Garrett and M. K. Bennett, 1988. “Felix Klein and His ‘Erlanger Programm’.” In History and Philosophy of Modern Mathematics, edited by W. Aspray and P. Kitcher, pp. 145–76. Minneapolis: University of Minnesota Press.
Carnap, Rudolf, 1922. Der Raum: Ein Beitrag zur Wissenschaftslehre. Berlin: Reuther & Reichard.
———, 1929. Abriss der Logistik. Vienna: Springer.
———, 2000. Untersuchungen zur allgemeinen Axiomatik, edited by T. Bonk and J. Mosterin. Darmstadt: Wissenschaftliche Buchgesellschaft.
Cassirer, Ernst, 1907. “Kant und die moderne Mathematik.” Kant-Studien 12: 1–40.
———, 1910. Substanzbegriff und Funktionsbegriff. Untersuchungen über die Grundfrage der Erkenntniskritik. Berlin: Springer.
———, 1923. Substance and Function. Chicago: Open Court.
———, 1944. “The Concept of Group and the Theory of Perception.” Philosophy and Phenomenological Research 5: 1–36.
———, 2010. Vorlesungen und Vorträge zu philosophischen Problemen der Wissenschaften, edited by K. C. Köhnke, J. M. Krois and O. Schwemmer. Hamburg: Felix Meiner Verlag.
Chasles, Michel, 1837. Aperçu historique sur l’origine et le dévéloppement des méthodes en géométrie. Brussels: M. Hayez.
Coxeter, H. M. S., 1974/1987. Projective Geometry, 2nd ed. Berlin: Springer.
Dedekind, Richard, 1888. Was sind und was sollen die Zahlen? Brunswick: Vieweg and Son.
Desargues, Girard, 1639. Brouillon projet d’une atteinte aux événements des rencontres du cône avec un plan. Paris: M. Poudra.
Frege, Gottlob, 1980. Philosophical and Mathematical Correspondence, edited by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel and A. Veraart, translated by B. McGuinness and H. Kaal. Oxford: Blackwell.
Gergonne, Joseph Diez, 1825–1826. “Considérations philosophiques sur les éléments de la science de l’éntendue.” Annales de Mathématiques Pures et Appliquées 16: 209–31.
Gray, Jeremy, 2007. Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century. New York: Springer.
———, 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press.
Hallett, Michael, 2008. “Reflections on the Purity of Method in Hilbert’s Grundlagen der Geometrie.” In The Philosophy of Mathematical Practice, edited by P. Mancosu, pp. 198–255. Oxford: Oxford University Press.
Hawkins, Thomas, 2000. Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926. New York: Springer.
Heis, Jeremy, 2007. The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer. PhD Dissertation, University of Pittsburgh.
———, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.
———, 2014. “Ernst Cassirer’s Substanzbegriff und Funktionsbegriff .” Hopos: The Journal of the International Society for the History of Philosophy of Science 4: 241–70.
Hilbert, David, 1899. Grundlagen der Geometrie, 10th ed. Leipzig: Teubner, 1968.
Ihmig, Karl-Norbert, 1997. Cassirers Invariantentheorie der Erfahrung und seine Rezeption des Erlanger Programms. Hamburg: Felix Meiner Verlag.
———, 1999. “Ernst Cassirer and the Structural Conception of Objects in Modern Science: The Importance of the ‘Erlanger Programm’.” Science in Context 12: 513–29.
Klein, Felix, 1893. “Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen 1872.” Mathematische Annalen 43: 63–100.
———, 1926. Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Springer.
———, 1928. Vorlesungen über Nicht-Euklidische Geometrie. Berlin: Springer.
Linnebo, Øystein, 2008. “Structuralism and the Notion of Dependence.” Philosophical Quarterly 58: 59–79.
Marquis, Jean-Pierre, 2009. From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory. New York: Springer.
Nagel, Ernest, 1939. “The Formation of Modern Conceptions of Formal Logic in the Development of Geometry.” Osiris 7: 142–223.
Natorp, Paul, 1910. Die logischen Grundlagen der exakten Wissenschaften. Leipzig: Tuebner.
Parsons, Charles, 1990. “The Structuralist View of Mathematical Objects.” Synthese 84: 303–46.
Pasch, Moritz, 1882. Vorlesungen über neuere Geometrie. Leipzig: Teubner.
Poncelet, Jean-Victor, 1822. Traité des propriétés projectives des figures. Paris: Gauthier-Villars.
Reck, Erich, 2003. “Dedekind’s Structuralism: An Interpretation and Partial Defense.” Synthese 137: 369–419.
Reck, Erich and Pierre Keller, forthcoming. “From Dedekind to Cassirer: Logicism and the Kantian Heritage.” In Kant’s Philosophy of Mathematics, vol. 2, edited by O. Rechter and C. Posy. Oxford: Oxford University Press.
Resnik, Michael D., 1997. Mathematics as a Science of Patterns. New York: Oxford University Press.
Rowe, David E., 1985. “Felix Klein’s Erlanger Antrittsrede.” Historia Mathematica 12: 123–41.
Schiemer, Georg and Günther Eder, 2018. “Hilbert, Duality, and the Geometrical Roots of Model Theory.” The Review of Symbolic Logic, forthcoming.
Schlimm, Dirk, 2010. “Pasch’s Philosophy of Mathematics.” Review of Symbolic Logic 3: 93–118.
———, 2013. “Axioms in Mathematical Practice.” Philosophia Mathematica 21: 37–92.
Shapiro, Stewart, 1997. Philosophy of Mathematics: Structure and Ontology. New York: Oxford University Press.
Sieg, Wilfried, 2014. “The Ways of Hilbert’s Axiomatics: Structural and Formal.” Perspectives on Science 22: 133–57.
Toepell, Michael-Markus, 1986. Über die Entstehung von David Hilberts Grundlagen der Geometrie. Göttingen: Vandenhoeck & Ruprecht.
Torretti, Roberto, 1978. Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel.
Weber, Heinrich and Josef Wellstein, 1903. Encyklopädie der Elementar-Mathematik: Ein Handbuch für Lehrer und Studierende. Leipzig: Teubner.
Wussing, Hans, 1984/2007. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory. New York: Dover.
Yap, Audrey, 2017. “Dedekind and Cassirer on Mathematical Concept Formation.” Philosophia Mathematica 25: 369–89.
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