Plato's Individuals is rich and rewarding. McCabe's reading will compel us to examine anew the presuppositions we bring to the enterprise of understanding Plato. Her devotion to showing that her thesis is found almost everywhere in the corpus is noteworthy. At times she also seems to strain to assimilate modern and Platonic concerns. If one can accept that Plato's tripartite soul goes over into something we might recognize as the problem of personal identity, it can only be because we are (...) writing off his devotion to reincarnation and transmigration. It is, however, McCabe's novel and energetic defense of individuation that deserves closest scrutiny. Here charges of anachronism and projection will be heard. (shrink)

Available from UMI in association with The British Library. ;Plato's philosophy of mathematics must be a philosophy of 4th century B.C. Greek mathematics, and cannot be understood if one is not aware that the notions involved in this mathematics differ radically from our own notions; particularly, the notion of arithmos is quite different from our notion of number. The development of the post-Renaissance notion of number brought with it a different conception of what mathematics is, and we must be able (...) to see the differences before we can begin to appreciate Plato's views, which must never be seen as applying to a revolutionary notion of mathematics . For this reason the first part of this dissertation must comprise a detailed analysis of these differences, along with some criticisms of those of Plato's modern interpreters who have not taken this point. Much of what will be found here can be found in Jacob Klein's Greek Mathematics and the Origin of Algebra, which will be extensively quoted and interpreted. This book is often quoted, but very seldom understood; or at least, those lessons which should be taken from it are rarely applied to Platonic exegesis. In the second part I attempt a better explanation of Plato's mathematical ontology, beginning with the most extensive passage in the dialogues, the triple image of Sun, Line and Cave in Republic. This reveals a relation between sensibles and forms which is at odds with the traditional view that mathematical objects are never precisely accurate instances of their forms--but this view can be refuted. Plato's view of the nature of mathematical objects will be seen to be a metaphorical way of saying what Aristotle says logically, and is entirely appropriate to the aims and methods of his contemporary mathematicians. Aristotle's criticism of Plato's views is examined and found to make sense and to be appropriate to views which Plato could consistently have held. (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

The authors' novel approach to some interesting mathematical concepts - not normally taught in other courses - places them in a historical and philosophical setting. Although primarily intended for mathematics undergraduates, the book will also appeal to students in the sciences, humanities and education with a strong interest in this subject. The first part proceeds from about 1800 BC to 1800 AD, discussing, for example, the Renaissance method for solving cubic and quartic equations and providing rigorous elementary proof that certain (...) geometrical problems posed by the ancient Greeks cannot be solved by ruler and compass alone. The second part presents some fundamental topics of interest from the past two centuries, including proof of G del's incompleteness theorem, together with a discussion of its implications. (shrink)

Contradicting the long-held belief that Aristotle was the first to discuss individuation systematically, Mary Margaret McCabe argues that Plato was concerned with what makes something a something and that he solved the problem in a ...

Mugler maintains that interpreters have been mistaken in drawing from the pedagogical plan of Republic VII any general conclusion concerning the relative position which Plato assigned to philosophical speculation and mathematics, that in Plato's own intellectual experience the relation of the two was the reverse of that assigned to them there, mathematics being more often the end of metaphysical reflection than its point of departure, and that he recommended mathematical study to his pupils not merely as a propaedeutic for dialectic (...) but primarily because from his own experience he believed that subsequent philosophical meditation upon the fund of knowledge thus acquired would often lead to mathematical discoveries. Mugler does not say how this thesis is to be reconciled with the fact that at the end of the Laws dialectic is still declared to be the supreme science and that in such a late passage as Philebus 57 D ff. as well as in an early one like Euthydemus 290 C, neither of which is "pedagogical," the supremacy of dialectic and its priority to mathematics are affirmed as emphatically as in Republic VII; he simply says that a close interpretation of the "metaphysical" dialogues and the Timaeus will substantiate his contention, and it is therefore necessary to examine his interpretation in detail. (shrink)