Abstract

This book investigates the relation between mathematics and philosophy in Kant with special focus on the doctrine of the magnitudes. Without doubt, Moretto, who is himself both a mathematician and a philosopher, achieves final results on this matter, because not only does he provide an immanent interpretation of all parts of Kant’s systematic construction of magnitudes, he also provides a detailed history of Kant’s development. Kant gave courses on mathematics during the first eight years of his teaching at Königsberg and warmly recommended the study of mathematics to his most gifted disciples. He himself expressed a profound admiration for it. He wrote in The Only Possible Argument in Support of a Demonstration of the Existence of God that infinitesimal analysis, or, as he calls it, “higher geometry... in its account of the affinities between various species of curved lines,” reveals many of “the harmonious relations which inhere in the properties of space in general.... All these relations, in addition to exercising the understanding by means of our intellectual comprehension of them, also arouse the emotions, and they do in a manner similar to or even more sublime than that the contingent beauties of nature stir our feelings”. In On the Form and Principles of the Sensible and Intelligible World, Kant wrote that pure mathematics “provides us with a cognition which is in the highest degree true, and, at the same time, it provides us with a paradigm of the highest kind of evidence in other cases”. Besides, the mathematician-philosophers of the Leibniz-Wolff school, J. A. Eberhard, J. C. Schwab, J. G. E. Maaß at Halle, and A. G. Kästner at Göttingen, were among the very first to move critiques against the Critique of Pure Reason by focusing on the problem of the foundation of mathematics. They defended Leibniz against Kant, which prompted Kant to start a mathematical school of his own at Königsberg, and J. Schultz, J. G. K. C. Kiesewetter, and C. G. Zimmermann were Kant’s most notable defenders. Finally, one should not forget that it was in order to introduce a completely new set of ideas concerning the philosophy of mathematics that Kant laid out not only the distinction between analytical and synthetical judgments, but also the whole of the transcendental aesthetics. In fact, Kant proposed giving foundation to rational absolute numbers by means of synthetical a priori judgments; he also considered all arithmetical judgments as synthetical a priori; and by dedicating two antinomies to infinite series he suggested analogies to the representation of infinite series of irrational numbers.