Abstract
In the framework of his critical philosophy, Kant, as is well known, felt obliged to adopt a position with regard to the problem of the foundations of geometry. One of the ways in which he did this was to consider whether the concept of a straight biangle was consistent or involved a contradiction. Kant then entangled himself in a striking difficulty, which, however, appears to have been noted by few. To my knowledge, the only other person who has as yet seen it is Gottfried Martin2 and his proposed solution is also the only one that I yet know.
translated from German by Julian Barbour, Oxon, UK.
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References
Das geradlinige Zweieck, ein offener Widerspruch in der Kritik der reinen Vernunft, in: Tradition und Kritik, Festschrift für R. Zocher, W. Arnold ed., H. Zeltner, Stuttgart 1967, p. 229–235
A 220 f. B 268
A 291 B 348
A 163 B 204
see quotation above
A 220 B 268
cf. quotation above, p. 234 f.
cf. Elemente, I, Def. 4
cf., e.g., Archimedes, Werke, Darmstadt 1963, p. 78
cf., e.g. B 16
A 220 f. B 268
A 163 B 204, my italics
B 16
Because for Kant to “construct” a concept means not at all to form a concept, but to connect a (hence already formed) concept with an intuition which corresponds to it (cf., e.g., A 712 ff. B 740 ff.) Thereby it is the intuition which will be “constructed” as such in the sense of formed (cf. Kant himself in A 715 f. B 744 f.), but only namely as the one corresponding to a concept. The concept as such would be correspondingly formed thereby without this intuition, although it would remain completely incomprehensible.
cf. H. Schotten, Inhalt und Methode des planimetrischen Unterrichts, 2 Vols. Leipzig 1890–93, especially Vol. 2, p. 3 ff., and earlier already H. V. Helmholtz, Die Tatsachen in der Wahrnehmung, Darmstadt 1959, p. 58 f. and J. Schultz, Prüfung der Kantischen “Kritik der reinen Vernunft”, Königsberg 1789, Part I, p. 58.
But also such curves which are continuous but not smooth — in one point or even in all points -, do not at all disprove this ansatz of Kant, although this is frequently stated (cf., e.g. M. Friedman, Kant and the Exact Sciences, London 1992, p. 78 if.). Namely, the only positive mark for such a point is that here a curve has infinitely many directions or tangents. Hence it is to be marked positively only by the fact that here a curve changes its direction discontinuously — therefore it possesses basically a direction. Because clearly a curve being continuous must be extension and as such also basically possess a direction.
13th ed. Stuttgart 1987, p. 1 f.
cf. again B 16
cf. G. Frege, Nachgelassene Schriften und Wissenschaftlicher Briefwechsel, Vol. 1, Hamburg 1969, p. 182 ff.
cf., e.g., A 76 f. B 102 f., A 78 f. B 104 f., B 133 f. with commentary
so especially in the Logik, cf. Akademieausgabe Vol. 9, p. 91 ff.
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© 1994 Springer-Verlag Berlin Heidelberg
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Prauss, G. (1994). Kant and the Straight Biangle. In: Rudolph, E., Stamatescu, IO. (eds) Philosophy, Mathematics and Modern Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78808-6_16
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DOI: https://doi.org/10.1007/978-3-642-78808-6_16
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