The Impossible Biangle and the Possibility of Geometry
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Jeffrey L. Wilson
Abstract
Kant repeatedly uses the biangle as an example of an impossible figure. In this paper, I offer an account of these passages and their significance for the possibility of geometry as a science. According to Kant, the constructibility of the biangle would signal the failure of geometry. Whereas Wolff derives the no-biangle proposition from the axiom that between two points there can be only one straight line, Kant gives it axiomatic status as a synthetic a priori principle possessing immediate certainty. Because we are unable to generate a schema for the biangle, the failure of the attempt to construct it is intuitively clear. The parallel between mathematical and empirical concepts is instructive because both involve the synthesis of disparate intuitions into a unity. We do not, strictly speaking, even possess a well-formed concept of the biangle, because its representation cannot fulfill certain basic requirements of concept formation.
Acknowledgements
I am grateful to Nelson Peralta for helpful conversations on Kant’s Table of Nothing, to several anonymous reviewers for comments that substantially improved the paper, and to Riccardo Marchi for translating the Moretto quotation (n. 27) from the Italian.
References
All translations are quoted from The Cambridge Edition of the Works of Immanuel Kant (1992 ff.) and the quotation rules followed are those established by the Akademie-Ausgabe. Kant, Immanuel (1900 ff): Gesammelte Schriften. Hrsg.: Bd. 1 – 22 Preußische Akademie der Wissenschaften, Bd. 23 Deutsche Akademie der Wissenschaften zu Berlin, ab Bd. 24 Akademie der Wissenschaften zu Göttingen. Berlin.
Buroker, Jill Vance (1991). The Role of Incongruent Counterparts in Kant’s Transcendental Idealism. In James Van Cleve and R. E. Frederick (eds.), The Philosophy of Right and Left: Incongruent Counterparts and the Nature of Space. Dordrecht: Springer, 315 – 339.10.1007/978-94-011-3736-2_22Search in Google Scholar
Buroker, Jill Vance (2006). Kant’s ‘Critique of Pure Reason’: An Introduction (Cambridge Introductions to Key Philosophical Texts). Cambridge: Cambridge University Press.10.1017/CBO9780511809545Search in Google Scholar
Capozzi, Mirella. (2006). Biangoli rettilinei e centauri: l’ontologia di Wolff e Meinong. In Rosa M. Calcaterra (ed.), Le ragioni del conoscere e dell’agire. Scritti in onore di Rosaria Egidi. Milano: Franco Angeli, 44 – 56.Search in Google Scholar
Cürsgen, Dirk (2020). Kants Thesen über das Nichts: Überlegungen zur Tafel des Nichts in der “Kritik der reinen Vernunft”. Heidelberg: Cürsgen.Search in Google Scholar
De Risi, Vincenzo (2007). Geometry and Monadology: Leibniz’s Analysis Situs and Philosophy of Space. Basel: Birkhauser.10.1007/978-3-7643-7986-5Search in Google Scholar
Falkenburg, Brigitte (2020). Kant’s Cosmology: From the Pre-Critical System to the Antinomy of Pure Reason. Heidelberg: Springer.10.1007/978-3-030-52290-2Search in Google Scholar
Favaretti Camposampiero, Matteo (2012). Dividing Fiction from Reality: On Christian Wolff’s Metaphysical Project. In Matteo Favaretti Camposampiero and Matteo Plebani (eds.), Existence and Nature: New Perspectives. Heusenstamm: Ontos, 65 – 97.10.1515/9783110321807.65Search in Google Scholar
Friebe, Cord (2017). Kant’s Rejection of Leibniz’s Principle and the Individuality of Quantum Objects. Kant Yearbook, 9, 1 – 18.10.1515/kantyb-2017-0001Search in Google Scholar
Goodwin, William (2018). Conflicting Conceptions of Construction in Kant’s Philosophy of Geometry. Perspectives on Science, 26(1), 97 – 118.10.1162/POSC_a_00269Search in Google Scholar
Grasso, Roberta (2010). Il concetto di nulla nella Critica della ragion pura. Philosophical Readings, II/1, 101 – 149.Search in Google Scholar
Hymers, John (2018). Contradiction and Privation: Baumgarten and Kant on the Concept of Nothing. In Courtney D. Fugate and John Hymers (eds.), Baumgarten and Kant on Metaphysics. Oxford: Oxford University Press, 110 – 130.10.1093/oso/9780198783886.003.0008Search in Google Scholar
Kravanja, Aljoša (2023). Two Models of Kantian Construction. Journal of Transcendental Philosophy, 4 (2), 137 – 155.10.1515/jtph-2022-0013Search in Google Scholar
Kauark-Leite, Patricia (2017). On the Epistemic Status of Absolute Space: Kant’s Directions in Space Read from the Standpoint of his Critical Period. Kant-Studien, 108(2), 175 – 194.10.1515/kant-2017-0015Search in Google Scholar
Marcolungo, Ferdinando L. (2011). Kant e il possibile. In margine al Beweisgrund. In Luigi Cataldi Madonna and Paola Rumore (eds.), Kant und die Aufklärung. (Akten der Kant-Tagung in Sulmona, 24.–28. März, 2010). Hildesheim: Olms, 139 – 153.Search in Google Scholar
Martin, Gottfried (1967). Das gradlinige Zweieck. Ein offener Widerspruch in der Kritik der reinen Vernunft. In Wilhelm Arnold and Hermann Zeltner (eds.), Tradition und Kritik. Festschrift für Rudolf Zocher zum 80. Geburtstag. Stuttgart-Bad Cannstatt: Frommann Holzboog, 229‒235.Search in Google Scholar
Moretto, Antonio (2010). Matematica. In Stefano Besoli, Claudio La Rocca and Riccardo Martinelli (eds.), L’universo kantiano: filosofia, scienze, sapere. Macerata: Quodlibet, 261 – 313.Search in Google Scholar
Peters, Wilhelm Servatius (1964). Zum Begriff der Konstruierbarkeit bei I. Kant. Archive for History of Exact Sciences, 2(2), 153 – 167.10.1007/BF00357653Search in Google Scholar
Peters, Wilhelm Servatius (1966). Widerspruchsfreiheit und Konstruierbarkeit als Kriterien für die mathematische Existenz in Kants Wissenschaftstheorie. Kant-Studien, 57, 178 – 185.10.1515/kant.1966.57.1-4.178Search in Google Scholar
Prauss, Gerold (1994). Kant and the Straight Biangle. In Enno Rudolf and I.-O. Stamatescu (eds.), Philosophy, Mathematics, and Modern Physics: A Dialogue. New York: Springer, 226 – 234.10.1007/978-3-642-78808-6_16Search in Google Scholar
Robert, Gastón (2020). Incongruent counterparts and the absolute nature of space in Kant’s 1768 essay, Directions in Space. Anuario Filosófico, 53(2), 267 – 286.10.15581/009.53.2.002Search in Google Scholar
Schubbach, Arno (2017). Kants Konzeption der geometrischen Darstellung: Zum mathematischen Gebrauch der Anschauung. Kant-Studien, 108(1), 19 – 54.10.1515/kant-2017-0002Search in Google Scholar
Sherry, David (1999). Construction and Reductio Proof. Kant-Studien, 90(1), 23 – 39.10.1515/kant.1998.90.1.23Search in Google Scholar
Smith, Daniel James (2023). How is an Illusion of Reason Possible? The Division of Nothing in the Critique of Pure Reason. Kant-Studien, 114(3), 493 – 512.10.1515/kant-2023-2013Search in Google Scholar
Stang, Nicholas (2021). With What Must Transcendental Philosophy Begin? Kant and Hegel on Nothingness and Indeterminacy. In Gerad Gentry (ed.), Kantian Legacies in German Idealism. London: Routledge, 102 – 134.10.4324/9780429429828-6Search in Google Scholar
Van Cleve, James (2011). Introduction to the Arguments of 1770 and 1783. In James Van Cleve and R. E. Frederick (eds.), The Philosophy of Right and Left: Incongruent Counterparts and the Nature of Space. Dordrecht: Springer, 15 – 26.10.1007/978-94-011-3736-2_2Search in Google Scholar
Vollrath, Ernst (1970). Kants These über das Nichts. Kant-Studien, 61(1), 50 – 65.10.1515/kant.1970.61.1-4.50Search in Google Scholar
Wolff, Christian (1740). Philosophia rationalis sive logica: methodo scientifica pertractata et ad usum scientiarum atque vitae aptata: praemittitur discursus praeliminaris de philosophia in genere. [Latin Logic.] Third edition. Frankfurt and Leipzig: Libraria Rengeriana.Search in Google Scholar
Wolff, Christian (1736). Philosophia prima, sive Ontologia. Frankfurt and Leipzig: Libraria Rengeriana.Search in Google Scholar
Wolff, Christian (1772). Auszug aus den Anfangs-Gründen aller mathematischen Wissenschaften, Halle: Rengerische Buchhandlung.Search in Google Scholar
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Articles in the same Issue
- Titlepages
- Titelseiten
- Titelseiten
- Articles
- On Kant’s Schema of Reality
- Kant’s Prize Essay and Nineteenth Century Formalism
- Reflections on Kant on Reflections
- Magnitude, Matter, and Kant’s Principle of Mechanism
- The Impossible Biangle and the Possibility of Geometry
- Topics of the Kant Yearbook 2025, 2026 and 2027
Articles in the same Issue
- Titlepages
- Titelseiten
- Titelseiten
- Articles
- On Kant’s Schema of Reality
- Kant’s Prize Essay and Nineteenth Century Formalism
- Reflections on Kant on Reflections
- Magnitude, Matter, and Kant’s Principle of Mechanism
- The Impossible Biangle and the Possibility of Geometry
- Topics of the Kant Yearbook 2025, 2026 and 2027