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The Impossible Biangle and the Possibility of Geometry

  • Jeffrey L. Wilson
Veröffentlicht/Copyright: 6. November 2024
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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.

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Published Online: 2024-11-06
Published in Print: 2024-11-06

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 30.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/kantyb-2024-0005/pdf?lang=de
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