Abstract
The article examines how Salomon Maimon’s concept of number as ratio can be used to demonstrate that arithmetical judgments are analytical. Based on his critique of Kant’s synthetic a priori judgments, I show how this notion of number fulfills Maimon’s requirements for apodictic knowledge. Moreover, I suggest that Maimon was influenced by mathematicians who previously defined number as a ratio, such as Wallis and Newton. Following an analysis of the real definition of this concept, I conclude that within the framework of Maimon’s philosophy, arithmetical judgments cannot be analytical, nor is arithmetic an objectively necessary science, but rather only subjectively necessary. We should also cast doubt on his claim that we can create real objects from pure concepts of the understanding.
Acknowledgements
A significant part of the article is based on my master’s thesis written at the Cohn Institute for the History and Philosophy of Science and Ideas, Tel Aviv University. I am most grateful to Gideon Freudenthal for supervising my work on the thesis and for his intellectual support, as well as to the Cohn Institute and the Ignatz Bubis Jewish Studies Research Grant for offering financial support during my studies. I thank Leo Corry and Ofra Rechter for reading early versions of my work and Michael Friedman (Tel Aviv University) for reading advanced drafts. My gratitude is also extended to Henk J. M. Bos and Sabetai Unguru, who enriched my knowledge on the development of the notion of number. Lastly, I thank the anonymous reviewers for their insightful comments.
References
Andrews, Ethan Allen: A Copious and Critical Latin-English Lexicon. New York 1865.Search in Google Scholar
Bergman, Samuel Hugo: The Philosophy of Solomon Maimon. Tr. by Noah J. Jacobs. Jerusalem 1967.Search in Google Scholar
Buzaglo, Meir: Salomon Maimon: Monism, Skepticism and Mathematics. Pittsburgh, PA 2002.10.2307/j.ctt5hjnkpSearch in Google Scholar
Carson, Emily: “Metaphysics, Mathematics and the Distinction between the Sensible and the Intelligible in Kant’s Inaugural Dissertation”. In: Journal of the History of Philosophy 42 (2), 2004, 165–194. DOI: 10.1353/hph.2004.002610.1353/hph.2004.0026Search in Google Scholar
Chikurel, Idit: Salomon Maimon’s Theory of Invention: Scientific Genius, Analysis and Euclidean Geometry. Berlin/Boston 2020.10.1515/9783110691351Search in Google Scholar
Diderot, Denis and d’Alembert, Jean. le Rond (Eds.): Encyclopédie, ou dictionnaire raisonné des sciences, des arts et des métiers, etc. Ed. by Robert Morrissey and Glenn Roe. University of Chicago: ARTFL Encyclopédie Project. http://encyclopedie.uchicago.edu/.2016,11.202.Search in Google Scholar
Duffy, Simon: “Maimon’s Theory of Differentials As The Elements of Intuitions”. In: International Journal of Philosophical Studies 22 (2), 2014, 228–247. DOI: 10.1080/09672559.2013.86100410.1080/09672559.2013.861004Search in Google Scholar
Engelhard, Kristina and Mittelstaedt, Peter: “Kant’s Theory of Arithmetic: A Constructive Approach?”. In: Journal for General Philosophy of Science 3 (2), 2008, 245–271.10.1007/s10838-008-9072-ySearch in Google Scholar
Franks, Paul: “What Should Kantians Learn from Maimon’s Skepticism?” In: Salomon Maimon: Rational Dogmatist, Empirical Skeptic: Critical Assessments. Ed. by Gideon Freudenthal. Dordrecht/Boston 2003, 200–232.10.1007/978-94-017-2936-9_9Search in Google Scholar
Freudenthal, Gideon: “Maimon’s Subversion of Kant’s Critique of Pure Reason: There Are No Synthetic a priori Judgements in Physics”. In: Salomon Maimon: Rational Dogmatist, Empirical Skeptic: Critical Assessments. Ed. by Gideon Freudenthal. Dordrecht/Boston 2003, 144–175.10.1007/978-94-017-2936-9_7Search in Google Scholar
Freudenthal, Gideon: “Salomon Maimon: Philosophizing in Commentaries”. In: Daat: A Journal of Jewish Philosophy & Kabbalah 53, 2004, 125–160 [in Hebrew].Search in Google Scholar
Freudenthal, Gideon: Definition and Construction. Salomon Maimon’s Philosophy of Geometry. Berlin: Preprint 317 of the Max Planck Institute for the History of Science, 2006.Search in Google Scholar
Freudenthal, Gideon: “Overturning the Narrative: Maimon vs. Kant”. In: Discipline Filosofiche 29 (1), 2019, 47–68. DOI:10.2307/j.ctvj7wp23.610.2307/j.ctvj7wp23.6Search in Google Scholar
Friedman, Michael: Kant and the Exact Sciences. Cambridge, Mass. 1992.Search in Google Scholar
Hanna, Robert: “Mathematics for Humans: Kant’s Philosophy of Arithmetic Revisited”. In: European Journal of Philosophy 10 (3), 2002, 328–353.10.1111/1468-0378.00165Search in Google Scholar
Heath, Thomas Little: The Thirteen Books of Euclid’s Elements. New York. 1956.Search in Google Scholar
Hintikka, Jaakko: “Kant on the Mathematical Method”. In: Kant’s Philosophy of Mathematics: Modern Essays. Ed. by Carl. Jeffrey Posy. Dordrecht/Boston 1992, 21–42.10.1007/978-94-015-8046-5_2Search in Google Scholar
Kitcher, Phillip: “Kant and the Foundation of Mathematics”. In: The Philosophical Review 84 (1), 1975, 23–50.10.1007/978-94-015-8046-5_5Search in Google Scholar
Klein, Jacob: Greek Mathematical Thought and the Origin of Algebra. London 1968.Search in Google Scholar
Lachterman, David R.: “Mathematical Construction, Symbolic Cognition and the Infinite Intellect: Reflections on Maimon and Maimonides”. In: Journal of the History of Philosophy 30 (4), 1992, 497–522. DOI:10.1353/hph.1992.008210.1353/hph.1992.0082Search in Google Scholar
Leibniz, Gottfried Wilhelm: “Nouveaux essaies sur l’entendement humain”. In: Œuvres philosophiques latines & françoises de feu Mr. de Leibnitz: tirées de ses manuscrits qui se conservent dans la Bibliothèque Royale à Hanovre, et publiées par Mr. Rud. Eric Raspe, avec une préface de Mr. Kaestner, Professeur en mathématiques à Göttingue. Ed. by Rudolf Erich Raspe. Amsterdam/Leipzig 1765, 1–496.Search in Google Scholar
Maimon, Salomon: Kritische Untersuchungen über den menschlichen Geist oder das höhere Erkenntniß- und Willensvermögen. Leipzig 1797.Search in Google Scholar
Maimon, Salomon: “Salomon Maimons Lebensgeschichte I & II”. In: Gesammelte Werke. Band I. Ed. by Valerio Verra. Hildesheim 1965. 1–588.Search in Google Scholar
Maimon, Salomon: Giv’at Hamore. Ed. by Samuel Hugo Bergman and Nathan Rotenstreich. Jerusalem 1966 [in Hebrew].Search in Google Scholar
Maimon, Salomon: “Ueber die Progressen der Philosophie veranlaßt durch die Preisfrage der königl. Akademie zu Berlin für das Jahr 1792: Was hat die Metaphysik seit Leibniz und Wolf für Progressen gemacht?”. In: Aetas Kantiana. Vol. 172. Bruxelles 1969.Search in Google Scholar
Maimon, Salomon: “Bacons von Verulam neues Organon”. In: Gesammelte Werke. Band IV. Ed. by Valerio Verra. Hildesheim 1970a, 295–529.Search in Google Scholar
Maimon, Salomon: “Anfangsgründe der Newtonischen Philosophie von Dr. Pemberton: Aus dem Englischen mit Anmerkungen und einer Vorrede von S. Maimon.” In: Gesammelte Werke. Band IV. Ed. by Valerio Verra. Hildesheim 1970b, 531–582.Search in Google Scholar
Maimon, Salomon: “Versuch einer neuen Logik oder Theorie des Denkens, Nebst angehängten Briefen des Philaletes an Aenesidemus”. In: Gesammelte Werke. Band V. Ed. by Valerio Verra. Hildesheim 1970c.Search in Google Scholar
Maimon, Salomon: “Ueber den Gebrauch der Philosophie zur Erweiterung der Erkenntnis”. In: Gesammelte Werke. Band VI. Ed. by Valerio Verra. Hildesheim 1971a, 362–396.Search in Google Scholar
Maimon, Salomon: “Das Genie und der methodische Erfinder”. In: Gesammelte Werke. Band VI. Ed. by Valerio Verra. Hildesheim 1971b, 362–384.Search in Google Scholar
Maimon, Salomon: “Erfindungsmethoden”. In: Ideen und Pläne aus S. M’s hinterlassenen Papieren. In: Gesammelte Werke. Band VII. Ed. by Valerio Verra. Hildesheim 1976, 649–660.Search in Google Scholar
Maimon, Salomon: Essay on Transcendental Philosophy. Tr. by Nick Midgley, Henry Somers-Hall, Alistair Welchman and Merten Reglitz. London/New York 2010.Search in Google Scholar
Neal, Katherine: From Discrete to Continuous: The Broadening of Number Concepts in Early Modern England. Australian Studies in History and Philosophy of Science, 16. Dordrecht 2002.Search in Google Scholar
Newton, Isaac: “Arithmetica Universalis”. In: The Mathematical Papers of Isaac Newton 1683–1684. Vol. 5. Ed. by Derek Thomas Whiteside. Cambridge 1972.Search in Google Scholar
Parsons, Charles: “Kant’s Philosophy of Arithmetic”. In: Kant’s Philosophy of Mathematics: Modern Essays. Ed. by Carl Jeffrey Posy. Dordrecht/Boston 1992, 43–79.10.1007/978-94-015-8046-5_3Search in Google Scholar
Partridge, Eric: Origins: An Etymological Dictionary of Modern English. London/New York 1958.Search in Google Scholar
Potter, Michael: Reason’s Nearest Kin: Philosophies of Arithmetic from Kant to Carnap. New York 2000.10.1093/acprof:oso/9780199252619.001.0001Search in Google Scholar
Pringe, Hernán: “Maimon’s Criticism of Kant’s Doctrine of Mathematical Cognition and the Possibility of Metaphysics as a Science”. In: Studies in History and Philosophy of Science 74, 2018, 35–44. DOI: 10.1016/j.shpsa.2017.07.00610.1016/j.shpsa.2017.07.006Search in Google Scholar
Rechter, Ofra: Syntheticity, Intuition and Sympobic Construction in Kant’s Philosophy of Arithmetic [Dissertation]. New York: Columbia University, 1997.Search in Google Scholar
Senderowicz, Yaron: “Maimon’s ‘Quid Facti’ Argument”. In: Salomon Maimon: Rational Dogmatist, Empirical Skeptic: Critical Assessments. Ed. by Gideon Freudenthal. Dordrecht/Boston 2003, 176–199.10.1007/978-94-017-2936-9_8Search in Google Scholar
Souter, Alexander and Glare, P. G. W: Oxford Latin Dictionary. Oxford 1976.Search in Google Scholar
Thompson, Manley: “Singular Terms and Intuitions in Kant’s Epistemology”. In: Kant’s Philosophy of Mathematics: Modern Essays. Ed. by Carl Jeffrey Posy. Dordrecht/Boston 1992, 81–107.10.1007/978-94-015-8046-5_4Search in Google Scholar
Wolff, Christian: Vollständiges mathematisches Lexicon. Leipzig 1734.Search in Google Scholar
Wolff, Christian: “Mathematisches Lexicon”. In: Christian Wolff: Gesammelte Werke. I. Abteilung. Band II. Hildesheim 1978.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
