Abstract
The aim of this paper is to analyze Dimitry Gawronsky’s doctrine of actual infinitesimals. I examine the peculiar connection that his critical idealism establishes between transcendental philosophy and mathematics. In particular, I reconstruct the relationship between Gawronsky’s differentials, Cantor’s transfinite numbers, Veronese’s trans-Archimedean numbers and Robinson’s hyperreal numbers. I argue that by means of his doctrine of actual infinitesimals, Gawronsky aims to provide an interpretation of calculus that eliminates any alleged given element in knowledge.
Article Note
The project leading to this paper has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 777786. The investigation is also part of the project CONICYT/FONDECYT Regular Nº 1190965, the project “La deducción trascendental de las categorías: nuevas perspectivas” PR65/19-22446 (Comunidad de Madrid and Universidad Complutense de Madrid) and the project CONICET PIP 11220200101740CO.
References
Bell, J. (2019): The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. Springer: Cham.10.1007/978-3-030-18707-1Search in Google Scholar
Constituting Objectivity: Transcendental Approaches of Modern Physics (2009). Ed. by Bitbol, M., Kerszberg, P. and Petitot, J. Springer: Cham.Search in Google Scholar
Bos, H. (1974): “Differentials, higher order differentials and the derivative in the Leibnizian calculus”. In: Archive for the History of Exact Sciences 14, 1–90.10.1007/BF00327456Search in Google Scholar
Bottazzi, E. ; Kanovei, V. ; Katz, M. ; Mormann, T. and Sherry, D. (2019), “On mathematical realism and the applicability of hyperreals”. In: Mathematical Studies 51 (2), 200–224.10.15330/ms.51.2.200-224Search in Google Scholar
Boyer, C. (1949): The History of the Calculus and its Conceptual Development. New York: Dover.Search in Google Scholar
Cantor, G. (1883): Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: B. G.Teubner. Reprinted in [Cantor 1932, 165–208]. English translation by W. D. Ewald in [Ewald 2007, 881–920].Search in Google Scholar
Cantor, G. (1884), Review of [Cohen 1883]. In: Deutsche Literaturzeitung 5 (Berlin, Feb 23), 266–268.Search in Google Scholar
Cantor, G. (1886), “Über die verschiedenen Standpunkte in Bezug auf das Aktuelle Unendliche,” Zeitschrift für Philosophie und philosophische Kritik 88, 224–233. Reprinted in [Cantor 1932, 370–376].Search in Google Scholar
Cantor, G. (1887): “Mitteilungen zur Lehre vom Transfiniten. I.” In: Zeitschrift für Philosophie und philosophische Kritik 91, 81–125. Reprinted in [Cantor 1890] and [Cantor 1932, 378–439].Search in Google Scholar
Cantor, G. (1890): Zur Lehre vom Transfiniten. Halle: C. E. M. Pfeffer.Search in Google Scholar
Cantor, G., (1932): Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Hrsg. von Ernest Zermelo, nebst einem Lebenslauf Cantors von A. Fraenkel. Berlin; reprinted Hildesheim: Olms, 1966.Search in Google Scholar
Cantor, G. (1991): Briefe, ed. Herbert Meschkowski, Winfried Nilson. Berlin.10.1007/978-3-642-74344-3Search in Google Scholar
Cassirer E. (1902): Leibniz’ System in seinen wissenschaftlichen Grundlagen. Marburg. In: Gesammelte Werke, Bd 1. Hamburg 1998.Search in Google Scholar
Cassirer, E. (1910): Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik. Berlin. In: Gesammelte Werke, Bd 6. Hamburg 2000.Search in Google Scholar
Cantor, Georg: Briefe. Ed. Herbert Meschkowski, Winfried Nilson. Springer, Berlin etc10.1007/978-3-642-74344-3Search in Google Scholar
Cohen, Hermann (1877): Kants Begründung der Ethik. Berlin: F. Dümmlers.Search in Google Scholar
Cohen, H. (1883): Das Princip der Infinitesmal-Methode und seine Geschichte: ein Kapitel zur Grundlegung der Erkenntnisskritik. Berlin.Search in Google Scholar
Cohen, H. (1885): Kants Theorie der Erfahrung. 2nd Edition, Berlin: Cassirer.Search in Google Scholar
Cohen, H. (1902): Logik der reinen Erkenntnis. System der Philosophie, 1. 1st Edition. Berlin.Search in Google Scholar
Cohen, H. (1914): “Einleitung mit kritischem Nachtrag zur Geschichte des Materialismus von F. A. Lange.” 3rd Edition. Reedited by H. Holzhey, Hildesheim, 1984.Search in Google Scholar
Dauben, J. (1979): Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, Massachusetts.Search in Google Scholar
Dauben, J. (1995): Abraham Robinson. The Creation of Nonstandard Analysis: A Personal and Mathematical Odyssey. Princeton, New Jersey.Search in Google Scholar
Edel, G. (1988): Von der Vernunftkritik zur Erkenntnislogik. Die Entwicklung der theoretischen Philosophie Hermann Cohens. Freiburg/München.Search in Google Scholar
Real Numbers, Generalizations of the Reals, and Theories of Continua. (1994) Ed. by P. Ehrlich. Dordrecht, Holland.Search in Google Scholar
Ehrlich, P. (2006): “The Rise of non-Archimedean mathematics and the roots of a misconception I: The emergence of non-Archimedean systems of magnitudes.” In: Arch Hist Exact Sci 60, 1–12110.1007/s00407-005-0102-4Search in Google Scholar
From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II. (2007). Ed. by W. B. Ewald. Oxford.Search in Google Scholar
Ferrari, M. (2011): “Dimitrij Gawronsky und Ernst Cassirer: Zur Geschichte der Marburger Schule zwischen Deutschland und Russland.” In: Gegenstandsbestimmung und Selbstgestaltung. Transzendentalphilosophie im Anschluss an Werner Flach. Ed. by Ch. Krijnen und K. Zeidler. Würzburg 2011, 89–106.Search in Google Scholar
Fisher G. (1994): “Veronese’s Non-Archimedean Linear Continuum.” In: Real Numbers, Generalizations of the Reals, and Theories of Continua (1994). Ed. by P. Ehrlich. Kluwer Academic Publishers. Dordrecht, Holland, pp. 107–145.10.1007/978-94-015-8248-3_4Search in Google Scholar
Fletcher, P. ; Hrbacek, K. ; Kanovei, V. ; Katz, M. ; Lobry, C. ; Sanders, S. (2017), “Approaches to analysis with infinitesimals following Robinson, Nelson, and others,” Real Analysis Exchange 42, no. 2, 193–252.10.14321/realanalexch.42.2.0193Search in Google Scholar
Fraenkel, A. A. (1953): Abstract Set Theory, North-Holland Publishing Company, Amsterdam.Search in Google Scholar
Fraenkel, A. A. (2016): Recollections of a Jewish Mathematician in Germany. Basel.10.1007/978-3-319-30847-0Search in Google Scholar
Frege, G. (1885): Rezension von: H. Cohen, Das Prinzip der Infinitesimal-Methode und seine Geschichte. In: Zeitschrift für Philosophie und philosophische Kritik 87, 324–329.Search in Google Scholar
Gawronsky, D. (1910): Das Urteil der Realität und seine mathematischen Voraussetzungen. Marburg.Search in Google Scholar
Giovanelli, M. (2011): Reality and Negation. Kant’s Principle of Anticipations of Perception. Dordrecht.10.1007/978-94-007-0065-9Search in Google Scholar
Giovanelli, M. (2016): “Hermann Cohen’s Das Princip der Infinitesimal-Methode: The history of an unsuccessful book.” In: Studies in History and Philosophy of Science 58, 9–23.10.1016/j.shpsa.2016.02.002Search in Google Scholar
González Porta, M. (2011): Estudos neokantianos. São Paulo.Search in Google Scholar
Kant and Neo-Kantianism (2020). Kant Yearbook 12. Ed. by D. Heidemann. Boston/Berlin.Search in Google Scholar
Holzhey, H. (1986): Cohen und Natorp, 2 vol. Basel/Stuttgart.Search in Google Scholar
Kanovei, V ; Katz, K. ; Katz, M. ; Mormann, T. (2018): “What makes a theory of infinitesimals useful? A view by Klein and Fraenkel.” In: Journal of Humanistic Mathematics, 8 (1), 108–119.10.5642/jhummath.201801.07Search in Google Scholar
Kauark-Leite, P. (2009): “The Transcendental Role of the Principle of Anticipations of Perception in Quantum Mechanics.” In: Constituting Objectivity: Transcendental Approaches of Modern Physics (2009). Ed. by M. Bitbol, P. Kerszberg, and J. Petitot. Cham, 203–213.10.1007/978-1-4020-9510-8_12Search in Google Scholar
Kerry, B., (1885): “Ueber G. Cantors Mannigfaltigkeitsuntersuchungen.” In: Vierteljahresschrift für wissenschaftliche Philosophie 9, 191–232.Search in Google Scholar
Meschkowski, H. (1965): “Aus den Briefbüchern Georg Cantor.” In: Archive for History of Exact Sciences 2, 503–519.10.1007/BF00324881Search in Google Scholar
Moore, M. (2002), “A Cantorian Argument Against Infinitesimals,” Synthese 133, pp. 305–330.10.1023/A:1021204522829Search in Google Scholar
Mormann, Thomas and Katz, Mikhail (2013): “Infinitesimals as an issue of neo-Kantian philosophy of science.” In: Hopos: The Journal of the International Society for the History of Philosophy of Science 3 (2), 236–280.10.1086/671348Search in Google Scholar
Mormann, Thomas (2018): “Zur Mathematischen Wissenschaftsphilosophie des Marburger Neukantianismus.” In: Damböck, Christian: Philosophie und Wissenschaft bei Hermann Cohen Cham, 101–134.10.1007/978-3-319-58023-4_5Search in Google Scholar
Moynahan, Gregory B. (2018): “The Challenge of Psychology in the Development of Cohen’s System of Philosophy and the Marburg School Project.” In: Damböck, Christian: Philosophie und Wissenschaft bei Hermann Cohen. Cham: Springer, 41–76.10.1007/978-3-319-58023-4_3Search in Google Scholar
Natorp P. (1910): Die logischen Grundlagen der exakten Wissenschaften. Teubner, Leipzig.Search in Google Scholar
Natorp P. (1912): “Kant und die Marburger Schule.” In: Kant-Studien 17, 193–221.10.1515/kant.1912.17.1-3.193Search in Google Scholar
Natorp, Paul (1986): “Zu Cohens Logik.” In: Holzhey, H.: Cohen und Natorp, vol. 2, 43–78.Search in Google Scholar
Peiffer-Reuter, R. (1989): “L’infini relatif chez Veronese et Natorp.” In: (Eds.). (1989). La mathématique nonstandard. Ed. by H. Barreau, J. Harthong. Paris, 117–142.Search in Google Scholar
Petitot, J. (1994): “Esthétique transcendantale et physique mathématique.” In: Neukantianismus. Perspektiven und Probleme. Ed. by H. Holzhey and E.-W. Orth. Würzburg, 185–213.Search in Google Scholar
Pollok, K. (2010): “The ‘Transcendental Method.’ On the Reception of the Critique of Pure Reason in Neo-Kantianism.” In: The Cambridge Companion to Kant’s Critique of Pure Reason. Ed. by P. Guyer. CUP, 2010, 346–379.10.1017/CCOL9780521883863.016Search in Google Scholar
Proietti Carlo (2008): “Natural Numbers and Infinitesimals: A Discussion between Benno Kerry and Georg Cantor.” In: History and Philosophy of Logic 29 (4), 343–359.10.1080/01445340802025768Search in Google Scholar
Robinson, A. (1966): Non-standard Analysis. North-Holland Publishing Company, Amsterdam.Search in Google Scholar
Robinson, A. (1975): “Concerning Progress.” In: The Philosophy of Mathematics. Studies in Logic and the Foundations of Mathematics 80, 41–52.10.1016/S0049-237X(08)71942-6Search in Google Scholar
Russell, Bertrand (1903): The Principles of Mathematics, Cambridge.Search in Google Scholar
Schulthess, Peter (1984): “Introduction to: Das Prinzip der Infinitesimal-Methode und seine Geschichte.” In: Hermann Cohen: Das Prinzip der Infinitesimal-Methode und seine Geschichte. Hildesheim/Zürich/New York, 7–46.Search in Google Scholar
Veit, Bernd (2017): Hermann Cohens Infinitesimal-Logik, Dissertation, Universität Würzburg.Search in Google Scholar
Veronese, G. (1891): Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee esposti in forma elementare. Lezioni per la Scuola di magistero in Matematica. Padova, Tipografia del Seminario.Search in Google Scholar
Veronese, G. (1894): Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten in elementarer Form entwickelt. Mit Genehmigung des Verfassers nach einer neuen Bearbeitung des Originals übersetzt von Adolf Schepp, Leipzig: Teubner.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
