Arithmetic and Combinatorics: Kant and His Contemporaries

Arithmetic and Combinatorics: Kant and His Contemporaries

Arithmetic and Combinatorics: Kant and His Contemporaries

Arithmetic and Combinatorics: Kant and His Contemporaries

Synopsis

This is the only work to provide a historical account of Kant's theory of arithmetic, examining in detail the theories of both his predecessors and his successors.

Until his death, Martin was the editor of Kant-Studien from 1954 , of the general Kant index from 1964, of the Leibniz index from 1968, and coeditor of Leibnizstudien from 1969. This background is used to its fullest as he strives to make clear the historical milieu in which Kant's mathematical contributions developed. He uses Leibniz, Wolff, and others whose work was accomplished before Kant was born as well as Lambert, Mendelssohn, and others roughly contemporary with Kant; and when a point requires it, he refers to Gauss, Grassman, Frege, Russell, and Hilbert.

In her translation Wubnig has approached the original author with an abiding respect. She makes the translation flow in English while preserving as far as possible the flavor of the original. She has added many bibliographical and biographical details to ease the following up of Martin's allusions and suggestions.

Excerpt

In a motion picture directed by Alfred Hitchcock many decades ago, the heroine, having wandered through an inconclusive love affair, now finds her affections thoroughly engaged by a much more prepossessing young man. As she clutches her new lover for the first time, a long corridor barred by a series of doors is shown in montage, and one after another they open, letting in more and more light to flood the hallway. in Gottfried Martin's book something of this sort would happen to the lover of the history of mathematical philosophy; it is not a single illumination, but light arrives in a succession of particular insights supplied for the understanding of Immanuel Kant's peculiar place in the development of modern logic and its relation to arithmetical and geometrical principles.

Ever since Aristotle made the claim for mathematics that it was a science essentially easier than others because its elements were the simplest, men and women have pursued it with the idea that here, at least, the mind could rest in a certainty found nowhere else in the range of human knowledge. On the other hand, just what reasons could be found for this claim to certainty were open to question, and there have been endless disagreements over the foundations of the science. Because of this, parties to the argument have turned to logic or other disciplines, not being content with the seeming arbitrariness of the hypotheses as presented within mathematics alone. This has, then, carried philosophers out of its strict purviews, but they could find comfort in Aristotle's own practice of seeking for the principles and defending them, not in a special mathematical treatise but in several books of the Metaphysics, as well as in important chapters of the Posterior Analytics. in these he does not set up of a list of definitions and axioms, in the style of his quasi follower Euclid; instead one finds continuous prose (in luxuriant thickets, be it said), arguing for or against proposals for considering such-and-such to be the true elements, causes, and principles entering into mathematical science.

Much the same can be said for the work of Kant, who in his immensely . . .

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