# Moretto, Antonio. Dottrina delle grandezze e filosofia trascendentale in Kant.

Padua: il Poligrafo, 1999. 379 pp. n. p.--This book investigates the relation between mathematics and philosophy in Kant with special focus on the doctrine of the magnitudes (Grossen). 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 Konigsberg (1755-1763) 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 (1763) 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" (Akademic-Ausgabe, vol. 2, p. 95). In On the Form and Principles of the Sensible and Intelligible World (1770), 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" (Akademic-Ausgabe, vol. 2, p. 398). Besides, the mathematician-philosophers of the Leibniz-Wolff school, J. A. Eberhard, J. C. Schwab, J. G. E. Maass at Halle, and A. G. Kastner at Gottingen, 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 Konigsberg, 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.The first chapter of Moretto's book deals with Kant's treatment of divisibility and continuity of magnitudes. In both the precritical and the critical period Kant maintained the infinite divisibility of geometric extension. As regards the physical divisibility of bodies, however, Kant moved from sharing Leibniz's view about the constitution of all bodies by simple elements, the monads, to asserting the infinite divisibility of matter (p. 41). The second chapter is dedicated to extensive and continuous magnitudes. Moretto makes clear that the main difference between Newton, Leibniz, Wolff, and Baumgarten and Kant lies in the fact that while the former always aimed at an "objective" description of magnitudes, Kant chose the standpoint of subjectivity to explain the conditions of possibility of magnitudes and their determination (p. 90). The third chapter proposes a sophisticated reconstruction of Kant's notion of "limit," and the fourth a discussion of the role Kant assigned to transcendental philosophy within the mathematical theories of numbers. Chapter 5 investigates Kant's understanding of intensive magnitudes and his thoughts on the mathematical notion of "function." Finally, chapters 6 and 7 deal, respectively, with infinitely great and infinitely small magnitudes. Moretto's analyses appear at times very technical if seen from the standpoint of a philosopher, which probably means that they can be read with the greatest profit by readers specialized in mathematics. In fact, what Moretto investigates and assesses is not only interesting for Kantian research, it is also interesting for contemporary philosophy of mathematics.--Riccardo Pozzo, The Catholic University of America.

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Author: | Pozzo, Riccardo |
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Publication: | The Review of Metaphysics |

Article Type: | Book Review |

Date: | Mar 1, 2002 |

Words: | 693 |

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