Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T17:38:50.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Kant (vs. Leibniz, Wolff and Lambert) on real definitions in geometry

Published online by Cambridge University Press:  01 January 2020

Jeremy Heis
Jeremy Heis*
Affiliation:
Department of Logic and Philosophy of Science, University of California, Irvine, CA, USA
*
*Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper gives a contextualized reading of Kant’s theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid’s definitions. I argue that Kant accepted Euclid’s definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions.

Keywords

KantLambertLeibnizWolffEuclidgeometrydefinition
Type
Research Article
Information
Canadian Journal of Philosophy , Volume 44 , Issue 5-6: Special Issue: Mathematics in Kant’s Critical Philosophy , December 2014 , pp. 605 - 630
Copyright
Copyright © Canadian Journal of Philosophy 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adickes, Erich, ed. 1911. Kants Handschriftlicher Nachlass. Mathematik – Physik und Chemie – Physische Geographie. Volume 14 of Kant, Immanuel. Gesammelte Schriften. Berlin: Walter de Gruyter.Google Scholar
Carson, Emily. 1999. “Kant on the method of mathematics.” Journal of the History of Philosophy 37: 629–652.CrossRefGoogle Scholar
Carson, Emily. 2006. “Locke and Kant on mathematical knowledge.” In Intuition and the Axiomatic Method, edited by Carson, Emily and Huber, Renate, 3–21. Dordrecht: Springer.CrossRefGoogle Scholar
De Risi, Vincenzo. 2007. Geometry and Monadology. Basel: Birkhäuser.CrossRefGoogle Scholar
Dunlop, Katherine. 2009. “Why Euclid’s Geometry Brooked No Doubt: J.H. Lambert on Certainty and the Existence of Models.” Synthese 167: 33–65.CrossRefGoogle Scholar
Dunlop, Katherine. 2012. “Kant and Strawson on the Content of Geometrical Concepts.” Noûs 46: 86–126.CrossRefGoogle Scholar
Euclid. 1925. The Thirteen Books of Euclid’s Elements, Translated from the Text of Heiberg, with Introduction and Commentary. 2nd ed. Translated and edited by Sir Thomas Heath. Cambridge: Cambridge University Press.Google Scholar
Freudenthal, Gideon. 2006. “Definition and Construction: Maimon’s Philosophy of Geometry.”. Preprint 317 of the Max Planck Institute for the History of Science, Berlin. http://www.mpiwg-berlin.mpg.de/Preprints/P317.PDF.Google Scholar
Friedman, Michael. 1992. Kant and the Exact Sciences. Cambridge, MA: Harvard University Press.Google Scholar
Friedman, Michael. 2010. “Synthetic History Reconsidered.” In Discourse on a New Method, edited by Michael, Dickson and Mary, Domski, 571–813. Chicago: Open Court.Google Scholar
Heis, Jeremy. Forthcoming. “Kant on Parallel Lines: Definitions, Postulates, and Axioms.” In Kant’s Philosophy of Mathematics: Modern Essays. Vol. 1: The Critical Philosophy and Its Background, edited by Rechter, Ofra and Posy, Carl. Cambridge: Cambridge University Press.Google Scholar
Hintikka, Hans. 1969. “On Kant’s Notion of Intuition (Anschauung).” In The First Critique, edited by Penelhum, T. and Macintosh, J.J., 38–53. Belmont, CA: Wadsworth.Google Scholar
Kant, Immanuel. 1902. Gesammelte Schriften. Edited by the Königlich Preußischen Akademie der Wissenschaft.29 vols. Berlin: DeGruyter.Google Scholar
Kant, Immanuel. 1967. Philosophical Correspondence, 1759–99. Edited and translated by Arnulf Zweig.Chicago, IL: University of Chicago Press.Google Scholar
Kant, Immanuel. 1992. Lectures on Logic. Translated and edited by J. Michael Young. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kant, Immanuel. 1998. Critique of Pure Reason. Translated by Paul Guyer and Allen Wood. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kant, Immanuel. 2002. “On a Discovery Whereby Any New Critique of Pure Reason is to be Made Superfluous by an Earlier One.” In Theoretical Philosophy After 1781. Translated by Henry Allison. Cambridge: Cambridge University Press.Google Scholar
Kant, Immanuel. 2004. Metaphysical Foundations of Natural Science. Translated and edited by Michael Friedman. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Lambert, Johann Heinrich. 1764. Neues Organon. Band I. Leipzig: Wendler. Partially translated by Eric Watkins in Kant’s Critique of Pure Reason: Background Source Materials. Cambridge: CUP, 2009, 257-274.Google Scholar
Lambert, Johann Heinrich. 1771. Anlage zur Architectonic. Vol.1. Riga: Hartknock.Google Scholar
Lambert, Johann Heinrich. 1895a. “Letter to Holland.” In Die Theorie der Parallellinien von Euklid bis auf Gauss, edited by Engel, Friedrich and Stäckel, Paul, 141–142. Leipzig: Teubner.Google Scholar
Lambert, Johann Heinrich. 1895b. “Theorie der Parallellinien.” In Die Theorie der Parallellinien von Euklid bis auf Gauss, edited by Engel, Friedrich and Stäckel, Paul, 152–207. Leipzig: Teubner. Partially translated by William Ewald in From Kant to Hilbert. Vol. 1. Oxford: Clarendon Press, 158–167.Google Scholar
Lambert, Johann Heinrich. 1915. “Abhandlung vom Criterium Veritatis.” Kant-Studien. Ergänzungsheft 36: 7–64. Partially translated by Erick Watkins in Kant’s Critique of Pure Reason: Background Source Materials (Cambridge: CUP, 2009), 233–257.Google Scholar
Lambert, Johann Heinrich. 1918. “Über die Methode, die Metaphysik, Theologie und Moral richtiger zu beweisen.” Kant-Studien. Ergänzungsheft 42: 7–36.Google Scholar
Laywine, Alison. 1998. “Problems and Postulates: Kant on Reason and Understanding.” Journal of the History of Philosophy 36: 279–309.CrossRefGoogle Scholar
Laywine, Allison. 2001. “Kant in Reply to Lambert on the Ancestry of Metaphysical Concepts.” Kantian Review 5: 1–48.CrossRefGoogle Scholar
Laywine, Alison. 2010. “Kant and Lambert on Geometrical Postulates in the Reform of Metaphysics.” In Discourse on a New Method, edited by Michael, Dickson and Mary, Domski, 113–133. Chicago: Open Court.Google Scholar
Leibniz, Gottfried Wilhelm. 1858. “In Euclidis πρ;ˆτα.” In Leibnizens mathematischen Schriften, edited by Gerhardt, C. I.. Vol. V, 183–211. Halle: Schmidt.Google Scholar
Leibniz, Gottfried Wilhelm. 1970. Philosophical Papers and Letters. 2nd ed.. Translated by L.E. Loemker. Dordrecht: Springer.Google Scholar
Leibniz, Gottfried Wilhelm. 1989. Philosophical Essays. Translated and edited by Roger Ariew and Daniel Garber. Indianapolis, IN: Hackett.Google Scholar
Leibniz, Gottfried Wilhelm. 1996. New Essays on Human Understanding. Edited and translated by Peter Remnant and Jonathan Bennett. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Saccheri, Girolamo. 1920. Euclides Vindicatus. Translated by Bruce Halsted. Chicago, IL: Open Court.Google Scholar
Schultz, Johann. 1784. Entdeckte Theorie der Parallelen. Königsberg: Kanter.Google Scholar
Shabel, Lisa. 2003. Mathematics in Kant’s Critical Philosophy. New York: Routledge.Google Scholar
Sutherland, Daniel. 2005. “Kant on Fundamental Geometrical Relations.” Archiv für Geschichte der Philosophie 87: 117–158.CrossRefGoogle Scholar
Sutherland, Daniel. 2010. “Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early Kant.” In Discourse on a New Method, edited by Michael, Dickson and Mary, Domski, 155–192. Chicago: Open Court.Google Scholar
Webb, Judson. 2006. “Hintikka on Aristotelean Cosntructions, Kantian Intuitions, and Peircean Theorems.” In The Philosophy of Jaakko Hintikka, edited by Auxier, and Hahn, , 195–265. Chicago, IL: Open Court.Google Scholar
Wolff, Christian. 1710. Der Anfangs-Gründe aller mathematischen Wissenschaften. Halle. Reprinted in Wolff, 1962, Gesammelte Werke, I.12.Hildesheim: Olms.Google Scholar
Wolff, Christian. 1716. Mathematisches Lexicon. Leipzig. Reprinted in Wolff, 1962, Gesammelte Werke, I.11.Hildesheim: Olms.Google Scholar
Wolff, Christian. 1740. Philosophia Rationalis sive Logica. Frankfurt and Leipzig. Reprinted in Wolff, 1962, Gesammelte Werke, II.1. Hildesheim: Olms. Partially translated by Richard Blackwell as Preliminary Discourse on Philosophy in General. New York: Merrill, 1963.Google Scholar
Wolff, Christian. 1741. Elementa Matheseos Universae. Halle. Reprinted in Wolff, 1962, Gesammelte Werke, II.29.Hildesheim: Olms.Google Scholar