Abstract
Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from the spacetimes of classical mechanics and of special relativity to those of general relativity. Not only is it much more complicated to find such invariants in the case of general relativity, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of general relativity seems to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. This paper aims to reconsider these issues by drawing attention to the combination of subtractive and additive strategies (more in line with Riemann’s) in Klein’s own considerations about the determination of the structure of spacetime from 1897 to 1910. I will point out that Cassirer relied on Klein’s argument in some central passages from Substance and Function (1910), and elaborated further on his combined approach in Einstein’s Theory of Relativity (1921), also taking into account the application of Riemannian geometry in general relativity. My suggestion is that an appreciation of Cassirer’s continuing commitment to a variety of geometrical traditions may shed light on his particular understanding of a priori elements of knowledge and avoid some of the classical objections to the idea of a relativization of the a priori.
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Notes
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Cayley in his “Sixth Memoir upon Quantics” (1859) had limited himself to show that his projective system included (Euclidean) metrical. Klein was the first to derive the non-Euclidean cases (i.e., elliptic and hyperbolic geometries).
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Klein specified the second condition in the 1893 version of the paper. In the first version of 1872, he adopted Jordan’s (1870, p. 22) definition, which referred to finite groups of permutations. In that case, the first of the conditions above, along with the existence of a neutral element, is sufficient for the set to form a group. Afterwards, Sophus Lie drew Klein’s attention to the fact that the existence of an inverse operation is required in the case of infinite groups. It is noteworthy that both Galois and Jordan dealt with groups of permutations. Klein showed that the same conditions for permutations to form a group generally apply to groups of operations. In this sense, his Erlangen Program can be considered a fundamental step in the development of the abstract concept of group (Wussing 1984).
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Klein’s first job, before turning to purely mathematical research, was to set up and carry out demonstrations accompanying Julius Plücker’s lectures in experimental physics. Klein worked as Julius Plücker’s assistant at the University of Bonn from 1866 to 1868. When Plücker died, in 1868, Alfred Clebsch appointed Klein at Göttingen to complete the posthumous edition of Plücker’s work on line geometry. This position enabled Klein to move into the circle of algebraic geometers inspired by Clebsch. On the development of Klein’s thought in the early stages of Klein’s career, see Rowe (1992) and Gray (2008).
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All translations from Klein’s texts are my own, unless otherwise indicated.
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Klein’s review appeared in Mathematische Annalen in 1898.
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For a reconstruction of Einstein’s philosophy of science and further references, see esp. Howard and Giovanelli (2019).
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Cf. Biagioli (2020) for a reconstruction of Cassirer’s argument elucidating the dynamical aspect of Cassirer’s notion of coordination.
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See Friedman (2001, Ch. 3) for an account of Habermas’s distinction and for the idea to integrate the relativized a priori with the Habermasian notion of communicative rationality for a comprehensive account of the dynamics of reason throughout the history of the sciences.
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Several studies have shown that Cassirer, and his Marburg teachers before him, did attribute a constitutive function to a priori structures of scientific experience while introducing a dynamical dimension at some level of scientific conceptualization (see esp. Ferrari 2012; Giovanelli 2016; Biagioli 2020).
References
Biagioli, F. 2016. Space, Number, and Geometry from Helmholtz to Cassirer. Cham: Springer.
———. 2020. Ernst Cassirer’s Transcendental Account of Mathematical Reasoning. Studies in History and Philosophy of Science Part A 79 (2020): 30–40.
Cassirer, E. 1910. Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik. Berlin: B. Cassirer. English edition in Cassirer, E. 1923. Substance and Function and Einstein's Theory of Relativity. Trans. M. C. Swabey, and W. C. Swabey. Chicago: Open Court, 1–346.
———. 1921. Zur Einstein’schen Relativitätstheorie: Erkenntnistheoretische Betrachtungen, 347–465. Berlin: B. Cassirer. English edition in Cassirer E. 1923. Substance and Function and Einstein's Theory of Relativity. Trans. M. C. Swabey, and W. C. Swabey. 347–465, Chicago: Open Court.
———. 1936. Determinismus und Indeterminismus in der modernen Physik: Historische und systematische Studien zum Kausalproblem. Göteborg: Göteborgs Högskolas Årsskrift 42. English edition: Cassirer, Ernst. 1956. Determinism and Indeterminism in Modern Physics: Historical and Systematic Studies of the Problem of Causality. New Haven: Yale Univ. Press.
———. 1944. The Concept of Group and the Theory of Perception. Philosophy and Phenomenological Research 5: 1–36.
———. 1950. The Problem of Knowledge: Philosophy, Science, and History since Hegel. Trans. W. H. Woglam, and C. W. Hendel. New Haven: Yale Univ. Press.
Coffa, A. J. 1991. The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press.
DiSalle, R. 2008. Understanding Space-Time: The Philosophical Development of Physics from Newton to Einstein. Cambridge: Cambridge University Press.
Einstein, A. 1916. Die Grundlagen der allgemeinen Relativitätstheorie. Annalen der Physik 49: 769–822. English edition: Einstein, A. 1923. The Foundation of the General Theory of Relativity. Trans. G. B. Jeffrey and W. Perrett. In The Principle of Relativity, ed. H.A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, 111–164. London: Methuen.
Ferrari, M. 1991. Cassirer, Schlick e l’interpretazione ‘kantiana’ della teoria della relatività. Rivista di filosofia 82: 243–278.
———. 2012. Between Cassirer and Kuhn: Some Remarks on Friedman’s Relativized A Priori. Studies in History and Philosophy of Science Part A 43: 18–26.
Friedman, M. 1999. Reconsidering Logical Positivism. Cambridge: Cambridge University Press.
———. 2001. Dynamics of Reason: The 1999 Kant Lectures at Stanford University. Stanford: CSLI Publications.
———. 2002. Geometry as a branch of physics: Background and context for Einstein's `Geometry and experience'. In Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics, ed. D. B. Malament, 193–229. Chicago: Open Court.
———. 2005. Ernst Cassirer and Contemporary Philosophy of Science. Angelaki Journal of Theoretical Humanities 10 (1): 119–128.
Giovanelli, M. 2016. Zwei Bedeutungen des Apriori‘. Hermann Cohens Unterscheidung zwischen metaphysischem und transzendentalem a priori und die Vorgeschichte des relativierten a priori. In Philosophie und Wissenschaft bei Hermann Cohen/Philosophy and Science in Hermann Cohen, ed. C. Damböck, 177–204. Cham: Springer.
Gray, J. J. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton: Princeton University Press.
Hawkins, T. 1984. The Erlanger Program of Felix Klein: Reflections on Its Place in the History of Mathematics. Historia Mathematica 11: 442–470.
Helmholtz, H. v. 1878. Die Tatsachen in der Wahrnehmung. In Schriften zur Erkenntnistheorie, ed. P. Hertz and M. Schlick, 109–152. Berlin: Springer, 1921. English edition as Epistemological writings (Trans: Lowe, M. F., ed. R. S. Cohen and Y. Elkana). Dordrecht: Reidel, 1977.
Howard, D. 2005. “No Crude Surfeit”: Ryckman’s Reign of Relativity. Pacific APA, March 2005.
Howard, D., and M. Giovanelli. 2019. Einstein’s Philosophy of Science. Stanford Encyclopedia of Philosophy, ed. E.N. Zalta and U. Nodelman. Accessed on 22 July 2023: https://plato.stanford.edu/entries/einstein-philscience/
Howard, D., and J. Stachel, eds. 1989. Einstein and the History of General Relativity. Boston: Birkhäuser.
Jordan, C. 1870. Traité des substitutions et des équations algébriques. Paris: Gauthier-Villars.
Klein, F. 1871. Über die sogenannte Nicht-Euklidische Geometrie. Mathematische Annalen 4: 573–625.
———. 1872. Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen: Deichert. English version: Klein, F. (1892). A Comparative Review of Recent Researches in Geometry. Trans. Maella W. Haskell. Bulletin of the New York Mathematical Society 2 (1892–1893): 215–49.
———. 1890. Zur Nicht-Euklidischen Geometrie. Mathematische Annalen 37: 544–572.
———. 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.
———. 1909. Elementarmathematik vom höheren Standpunkte aus. Vol. 2: Geometrie. Leipzig: Teubner.
———. 1911. Über die geometrischen Grundlagen der Lorentzgruppe. Physikalische Zeitschrift 12: 17–27.
———. 1921. Gesammelte mathematische Abhandlungen. Vol. 1. Berlin: Springer.
Norton, J. 1999. Geometries in Collision: Einstein, Klein, and Riemann. In The Symbolic Universe: Geometry and Physics 1890–1930, ed. J. J. Gray, 128–144. Oxford: Oxford University Press.
Poincaré, H. 1902. La science et l’hypothèse. Paris: Flammarion.
Reichenbach, H. 1920. Relativitätstheorie und Erkenntnis apriori. Berlin: Springer. English edition as: The Theory of Relativity and A Priori Knowledge. Trans. M. Reichenbach. Berkeley: University of California Press, 1965.
Rowe, D. 1989. The Early Geometrical Works of Sophus Lie and Felix Klein. In History of Modern Mathematics. Vol. 1: Ideas and Their Reception, ed. D. E. Rowe and J. McCleary, 209–273. Boston: Academic.
———. 1992. Klein, Lie, and the Erlanger Programm. In 1830–1930: A Century of Geometry, Epistemology, History and Mathematics, ed. L. Boi, D. Flament, and J.-M. Salanskis, 45–54. Berlin: Springer.
Ryckman, T. 1991. Conditio Sine qua Non? Zuordnung in the Early Epistemologies of Cassirer and Schlick. Synthese 88: 57–95.
———. 2005. The Reign of Relativity: Philosophy in Physics, 1915–1925. New York: Oxford University Press.
———. 2018. Cassirer and Dirac on the Symbolic Method in Quantum Mechanics: A Confluence of Opposites. In Journal for the History of Analytical Philosophy 6.3: Method, Science, and Mathematics: Neo-Kantianism and Analytic Philosophy, ed. S. Edgar and L. Patton, 213–243.
Schiemer, G. 2020. Transfer Principles, Klein’s Erlangen Program, and Methodological Structuralism. In The Prehistory of Mathematical Structuralism, ed. E. Reck and G. Schiemer, 106–141. Oxford: Oxford University Press.
Schlick, M. 1921. Kritizistische oder empiristische Deutung der neuen Physik? Bemerkungen zu Ernst Cassirers Buch Zur Einstein’schen Relativitätstheorie. Kant-Studien 26: 96–111.
Stump, D. 2015. Conceptual Change and the Philosophy of Science: Alternative Interpretations of the a priori. London: Routledge.
Wussing, H. 1984. The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory. Trans. A. Shenitzer. Cambridge, Mass.: The MIT Press.
Acknowledgments
This research has received funding from the “Rita Levi Montalcini” program granted by the Italian Ministry of University and Research (MUR). I would like to thank the editors of this collection and an anonymous referee for their helpful comments on a previous draft of this paper. I also wish to remark that the current paper is my own work, and no one else is responsible for any mistakes in it.
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Biagioli, F. (2023). Cassirer and Klein on the Geometrical Foundations of Relativistic Physics. In: Russo Krauss, C., Laino, L. (eds) Philosophers and Einstein's Relativity. Boston Studies in the Philosophy and History of Science, vol 342. Springer, Cham. https://doi.org/10.1007/978-3-031-36498-3_4
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