Articulating Space in Terms of Transformation Groups: Helmholtz and Cassirer
DOI:
https://doi.org/10.15173/jhap.v6i3.3436Abstract
Hermann von Helmholtz’s geometrical papers (1868–1878) have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial perception from those describable in terms of axiomatic geometry. This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences.References
Beltrami, Eugenio, 1868. “Saggio di interpretazione della geometria non-euclidea.” Giornale di Matematiche 6: 284–322.
Biagioli, Francesca, 2014. “Hermann Cohen and Alois Riehl on Geometrical Empiricism.” Hopos 4: 83–105.
———, 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer.
Cassirer, Ernst, 1910. Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik. Berlin: B. Cassirer. English translation in Cassirer (1923), pp. 1–346.
———, 1921. Zur Einstein’schen Relativitätstheorie: Erkenntnistheoretische Betrachtungen. Berlin: B. Cassirer. English translation in Cassirer (1923), pp. 347–465.
———, 1923. Substance and Function and Einstein’s Theory of Relativity, translated by M. C. Swabey and W. C. Swabey. Chicago: Open Court.
———, 1950. The Problem of Knowledge: Philosophy, Science, and History since Hegel, translated by W. H. Woglam and C. W. Hendel. New Haven: Yale University Press.
———, 2010. Vorlesungen und Vorträge zu philosophischen Problemen der Wissenschaften 1907–1945, edited by J. Fingerhut, G. Hartung, and R. Kramme. Hamburg: Meiner.
DiSalle, Robert, 2006. “Kant, Helmholtz, and the Meaning of Empiricism.” In Friedman and Nordmann (2006), pp. 123–39.
Friedman, Michael, 1997. “Helmholtz’s Zeichentheorie and Schlick’s Allgemeine Erkenntnislehre.” Philosophical Topics 25: 19–50.
Friedman, Michael and Alfred Nordmann, eds., 2006. The Kantian Legacy in Nineteenth-Century Science. Cambridge, MA: MIT Press.
Hawkins, Thomas, 1984. “The Erlanger Programm of Felix Klein: Reflections on Its Place in the History of Mathematics.” Historia Mathematica 11: 442–70.
Heis, Jeremy, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.
Helmholtz, Hermann von, 1867. Handbuch der physiologischen Optik. Leipzig: Voss.
———, 1868. “Über die Tatsachen, die der Geometrie zugrunde liegen.” Reprinted in Helmholtz (1921), pp. 38–55.
———, 1870. “Über den Ursprung und die Bedeutung der geometrischen Axiome.” Reprinted in Helmholtz (1921), pp. 1–24.
———, 1878. “Die Tatsachen in der Wahrnehmung.” Reprinted in Helmholtz (1921), 109–52.
———, 1887. “Zählen und Messen, erkenntnistheoretisch betrachtet.” Reprinted in Helmholtz (1921), 70–97.
———, 1921. Schriften zur Erkenntnistheorie, edited by P. Hertz and M. Schlick. Berlin: Springer. Translated into English as Helmholtz (1977).
———, 1977. Epistemological Writings, translated by M. Lowe, edited by R. Cohen and Y. Elkana. Dordrecht: Reidel.
Hyder, David, 2009. The Determinate World: Kant and Helmholtz on the Physical Meaning of Geometry. Berlin: De Gruyter.
Ihmig, Karl-Norbert, 1997. Cassirers Invariantentheorie der Erfahrung und seine Rezeption des “Erlanger Programms”. Hamburg: Meiner.
Kant, Immanuel, 1787. Critik der reinen Vernunft, 2nd ed. Riga: Hartknoch.
Klein, Felix, 1871. “Über die sogenannte Nicht-Euklidische Geometrie.” Mathematische Annalen 4: 573–625.
———, 1872. “Vergleichende Betrachtungen über neuere geometrische Forschungen.” Erlangen: Deichert. Translated into English as “A Comparative Review of Recent Researches in Geometry,” by M. W. Haskell. Bulletin of the New York Mathematical Society 2 (1892–1893): 215–49.
———, 1890. “Zur Nicht-Euklidischen Geometrie.” Mathematische Annalen 37: 544–72.
———, 1898. “Gutachten, betreffend den dritten Band der Theorie der Transformationsgruppen von S. Lie anlässlich der ersten Vertheilung des Lobatschewsky-Preises.” Mathematische Annalen 50: 583–600. (Based on a lecture given in 1897.)
———, 1925. Elementarmathematik vom höheren Standpunkte aus, vol. 2, Geometrie. Berlin: Springer.
———, 1928. Vorlesungen über nicht-euklidische Geometrie, edited by W. Rosemann, 2nd ed. Berlin: Springer.
Lenoir, Timothy, 2006. “Operationalizing Kant: Manifolds, Models, and Mathematics in Helmholtz’s Theories of Perception.” In Friedman and Nordmann (2006), pp. 141–210.
Lie, Sophus, 1893. Theorie der Transformationsgruppen, vol. 3. Leipzig: Teubner. Translated into English as Theory of Transformation Groups: General Properties of Continuous Transformation Groups: A Contemporary Approach and Translation, by J. Merker. Berlin: Springer, 2015.
Neuber, Matthias, 2012. “Helmholtz’s Theory of Space and its Significance for Schlick.” British Journal for the History of Philosophy 20: 163–80.
Patton, Lydia, 2009. “Signs, Toy Models, and the A Priori: From Helmholtz to Wittgenstein.” Studies in History and Philosophy of Science 40: 281–89.
———, 2014. “Hermann von Helmholtz.” Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/hermann-helmholtz/, accessed 3 February 2017.
Poincaré, Henri, 1898. “On the Foundations of Geometry.” The Monist 9: 1–43.
———, 1902. La Science et L’Hypothèse. Paris: Flammarion.
Pulte, Helmut, 2006. “The Space between Helmholtz and Einstein: Moritz Schlick on Spatial Intuition and the Foundations of Geometry.” In Interactions: Mathematics, Physics and Philosophy, 1860–1930, edited by V. F. Hendricks, K. F. Jørgensen, J. Lützen, and S. A. Pedersen, pp. 185–206. Dordrecht: Springer.
Rowe, David, 1992. “Klein, Lie, and the Erlanger Programm.” In 1830–1930: A Century of Geometry, Epistemology, History and Mathematics, edited by L. Boi, D. Flament, and J.-M. Salanskis, pp. 45–54. Berlin: Springer.
Russell, Bertrand, 1897. An Essay on the Foundations of Geometry. Cambridge: Cambridge University Press.
Ryckman, Thomas, 2005. The Reign of Relativity: Philosophy in Physics, 1915–1925. New York: Oxford University Press.
Schlick, Moritz, 1916. “Idealität des Raumes: Introjektion und psychophysisches Problem.” Vierteljahrsschrift für wissenschaftliche Philosophie und Soziologie 40: 230–54.
Torretti, Roberto, 1978. Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel.
Wussing, Hans, 2007. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, translated by A. Shenitzer. Mineola: Dover.
Yaglom, Isaak M., 1988. Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century, translated by S. Sossinsky. Boston: Birkhäuser.
Biagioli, Francesca, 2014. “Hermann Cohen and Alois Riehl on Geometrical Empiricism.” Hopos 4: 83–105.
———, 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer.
Cassirer, Ernst, 1910. Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik. Berlin: B. Cassirer. English translation in Cassirer (1923), pp. 1–346.
———, 1921. Zur Einstein’schen Relativitätstheorie: Erkenntnistheoretische Betrachtungen. Berlin: B. Cassirer. English translation in Cassirer (1923), pp. 347–465.
———, 1923. Substance and Function and Einstein’s Theory of Relativity, translated by M. C. Swabey and W. C. Swabey. Chicago: Open Court.
———, 1950. The Problem of Knowledge: Philosophy, Science, and History since Hegel, translated by W. H. Woglam and C. W. Hendel. New Haven: Yale University Press.
———, 2010. Vorlesungen und Vorträge zu philosophischen Problemen der Wissenschaften 1907–1945, edited by J. Fingerhut, G. Hartung, and R. Kramme. Hamburg: Meiner.
DiSalle, Robert, 2006. “Kant, Helmholtz, and the Meaning of Empiricism.” In Friedman and Nordmann (2006), pp. 123–39.
Friedman, Michael, 1997. “Helmholtz’s Zeichentheorie and Schlick’s Allgemeine Erkenntnislehre.” Philosophical Topics 25: 19–50.
Friedman, Michael and Alfred Nordmann, eds., 2006. The Kantian Legacy in Nineteenth-Century Science. Cambridge, MA: MIT Press.
Hawkins, Thomas, 1984. “The Erlanger Programm of Felix Klein: Reflections on Its Place in the History of Mathematics.” Historia Mathematica 11: 442–70.
Heis, Jeremy, 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geometry.” British Journal for the History of Philosophy 19: 759–94.
Helmholtz, Hermann von, 1867. Handbuch der physiologischen Optik. Leipzig: Voss.
———, 1868. “Über die Tatsachen, die der Geometrie zugrunde liegen.” Reprinted in Helmholtz (1921), pp. 38–55.
———, 1870. “Über den Ursprung und die Bedeutung der geometrischen Axiome.” Reprinted in Helmholtz (1921), pp. 1–24.
———, 1878. “Die Tatsachen in der Wahrnehmung.” Reprinted in Helmholtz (1921), 109–52.
———, 1887. “Zählen und Messen, erkenntnistheoretisch betrachtet.” Reprinted in Helmholtz (1921), 70–97.
———, 1921. Schriften zur Erkenntnistheorie, edited by P. Hertz and M. Schlick. Berlin: Springer. Translated into English as Helmholtz (1977).
———, 1977. Epistemological Writings, translated by M. Lowe, edited by R. Cohen and Y. Elkana. Dordrecht: Reidel.
Hyder, David, 2009. The Determinate World: Kant and Helmholtz on the Physical Meaning of Geometry. Berlin: De Gruyter.
Ihmig, Karl-Norbert, 1997. Cassirers Invariantentheorie der Erfahrung und seine Rezeption des “Erlanger Programms”. Hamburg: Meiner.
Kant, Immanuel, 1787. Critik der reinen Vernunft, 2nd ed. Riga: Hartknoch.
Klein, Felix, 1871. “Über die sogenannte Nicht-Euklidische Geometrie.” Mathematische Annalen 4: 573–625.
———, 1872. “Vergleichende Betrachtungen über neuere geometrische Forschungen.” Erlangen: Deichert. Translated into English as “A Comparative Review of Recent Researches in Geometry,” by M. W. Haskell. Bulletin of the New York Mathematical Society 2 (1892–1893): 215–49.
———, 1890. “Zur Nicht-Euklidischen Geometrie.” Mathematische Annalen 37: 544–72.
———, 1898. “Gutachten, betreffend den dritten Band der Theorie der Transformationsgruppen von S. Lie anlässlich der ersten Vertheilung des Lobatschewsky-Preises.” Mathematische Annalen 50: 583–600. (Based on a lecture given in 1897.)
———, 1925. Elementarmathematik vom höheren Standpunkte aus, vol. 2, Geometrie. Berlin: Springer.
———, 1928. Vorlesungen über nicht-euklidische Geometrie, edited by W. Rosemann, 2nd ed. Berlin: Springer.
Lenoir, Timothy, 2006. “Operationalizing Kant: Manifolds, Models, and Mathematics in Helmholtz’s Theories of Perception.” In Friedman and Nordmann (2006), pp. 141–210.
Lie, Sophus, 1893. Theorie der Transformationsgruppen, vol. 3. Leipzig: Teubner. Translated into English as Theory of Transformation Groups: General Properties of Continuous Transformation Groups: A Contemporary Approach and Translation, by J. Merker. Berlin: Springer, 2015.
Neuber, Matthias, 2012. “Helmholtz’s Theory of Space and its Significance for Schlick.” British Journal for the History of Philosophy 20: 163–80.
Patton, Lydia, 2009. “Signs, Toy Models, and the A Priori: From Helmholtz to Wittgenstein.” Studies in History and Philosophy of Science 40: 281–89.
———, 2014. “Hermann von Helmholtz.” Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/hermann-helmholtz/, accessed 3 February 2017.
Poincaré, Henri, 1898. “On the Foundations of Geometry.” The Monist 9: 1–43.
———, 1902. La Science et L’Hypothèse. Paris: Flammarion.
Pulte, Helmut, 2006. “The Space between Helmholtz and Einstein: Moritz Schlick on Spatial Intuition and the Foundations of Geometry.” In Interactions: Mathematics, Physics and Philosophy, 1860–1930, edited by V. F. Hendricks, K. F. Jørgensen, J. Lützen, and S. A. Pedersen, pp. 185–206. Dordrecht: Springer.
Rowe, David, 1992. “Klein, Lie, and the Erlanger Programm.” In 1830–1930: A Century of Geometry, Epistemology, History and Mathematics, edited by L. Boi, D. Flament, and J.-M. Salanskis, pp. 45–54. Berlin: Springer.
Russell, Bertrand, 1897. An Essay on the Foundations of Geometry. Cambridge: Cambridge University Press.
Ryckman, Thomas, 2005. The Reign of Relativity: Philosophy in Physics, 1915–1925. New York: Oxford University Press.
Schlick, Moritz, 1916. “Idealität des Raumes: Introjektion und psychophysisches Problem.” Vierteljahrsschrift für wissenschaftliche Philosophie und Soziologie 40: 230–54.
Torretti, Roberto, 1978. Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel.
Wussing, Hans, 2007. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, translated by A. Shenitzer. Mineola: Dover.
Yaglom, Isaak M., 1988. Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century, translated by S. Sossinsky. Boston: Birkhäuser.
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