Plato's lectures: a hypothesis for an enigma.--Speusippus, Xenocrates, and the polemical method of Aristotle.--The Academy: orthodoxy, heresy, or philosophical interpretation?

An account of how the mathematical sciences turn the mind away from becoming and towards being. There are four main conclusions. 1. The study of numbers, when treated independently of the other sciences, uses a particular conception of the nature of numbers to detach the mind from the influence of perceptible objects. 2. The study of ratios and proportions, explicitly the core of Plato's harmonics, is fundamental also to plane and solid geometry and astronomy. 3. Ratios and proportion form the (...) basis for the synoptic vision of all the mathematical sciences. This vision is an essential preparation for dialectic because it makes mathematicians become aware of the limitations of the mathematics they have been engaged in but it also shows them how they can begin to overcome those limitations. 4. The main purpose of the study of numbers, treated finally in relation to the other sciences, is to calculate ratios. (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)

The old question whether or no the doctrine of ‘intermediate mathematical objects’ ascribed to Plato by Aristotle is to be found in the Divided Line of Republic vi, has been recently raised again in a careful and lucid discussion by Mr. W. F. R. Hardie. I may clear the ground by saying at once that I agree with that part of Mr. Hardie's chapter which deals with those criticisms of the traditional view that have been put forward by Prof. Ferguson (...) and the late Prof. Stocks; nevertheless I do not believe that the objects of the third segment are intermediate between Forms and Sensibles: I propose to argue that they, like the objects of the highest segment, are Forms. (shrink)

The old question whether or no the doctrine of ‘intermediate mathematical objects’ ascribed to Plato by Aristotle is to be found in the Divided Line of Republic vi, has been recently raised again in a careful and lucid discussion by Mr. W. F. R. Hardie. I may clear the ground by saying at once that I agree with that part of Mr. Hardie's chapter which deals with those criticisms of the traditional view that have been put forward by Prof. Ferguson (...) and the late Prof. Stocks; nevertheless I do not believe that the objects of the third segment are intermediate between Forms and Sensibles: I propose to argue that they, like the objects of the highest segment, are Forms. (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.

: Plato’s instructions entail that the line of Republic VI is divided so that the middle two segments are of equal length. Yet I argue that Plato’s elaboration of the significance of this analogy shows he believes that these segments are of unequal length because the domains they represent are not of equally clear mental states, nor perhaps of objects of equal reality. I label this inconsistency between Plato’s instructions and his explanation the “overdetermination problem.” The overdetermination problem has been (...) a perennial concern, and a substantial amount of work has been produced which attempts to deal with it. I offer a classification of approaches to the overdetermination problem as a way of documenting the problem’s significance, and show why these approaches are all inadequate as solutions. My novel resolution of the overdetermination problem rests upon a demonstration that the contradiction is intentional. The later recapitulation of the ratio at 534a reveals that Plato was himself aware that the middle two segments are equal. I argue that this contradiction is a sophisticated device designed to lead the reader of the Republic through the four epistemic stages represented by the line itself. Most significantly, recognition of this mathematical contradiction acts as a goad, spurring independent philosophical reflection just in the way that Plato advocates in the Republic more generally. (shrink)

: The traditional "doctrinal" approach to interpreting Plato's dialogues has been criticized in recent literature on grounds that it can neither account for the structural complexities of the dialogues nor resolve conflicts within or between dialogues. Accordingly, a non-doctrinal, dramatic approach has been offered in its place. In response to this literature, I argue that, though the doctrinal approach is flawed, the non-doctrinal, dramatic approach does not provide a viable alternative. Instead, I offer a revised doctrinal approach based upon Socrates' (...) discussion of "summoners" in Republic 522e–525a and supported by the Meno. (shrink)

In this essy I shall enter into the vexing question of Plato’s „theory of intermediates“, and the relation of this theory to the Sun, Line and Cave section of Republic VI and VII. My thesis is that in the last 75 years or so scholarly opinion has reached a complete impasse, having veered from one extreme to another, rather than in the fashion of an Hegelian thesis and antithesis; this conflict of opinion desperately requires a sythesizing „third“, and in the (...) conclusion of the paper I try to supply this „third“. (shrink)

Many have argued against the claim that Plato posited the mathematical objects that are the subjects of Metaphysics M and N. This paper shifts the burden of proof onto these objectors to show that Plato did not posit these entities. It does so by making two claims: first, that Plato should posit the mathematical Intermediates because Forms and physical objects are ill suited in comparison to Intermediates to serve as the objects of mathematics; second, that their utility, combined with Aristotle’s (...) commentary on Plato’s posit of Intermediates, provides good reason to conclude that Plato did posit them. (shrink)

This book is an anthology of Plato's writings, connected with sections of commentary.

This Companion provides a fresh and comprehensive account of this outstanding work, which remains among the most frequently read works of Greek philosophy, indeed of Classical antiquity in general. The sixteen essays, by authors who represent various academic disciplines, bring a spectrum of interpretive approaches to bear in order to aid the understanding of a wide-ranging audience, from first-time readers of the Republic who require guidance, to more experienced readers who wish to explore contemporary currents in the work’s interpretation. The (...) three initial chapters address aspects of the work as a whole. They are followed by essays that match closely the sequence in which topics are presented in the ten books of the Republic. Since the Republic returns frequently to the same topics by different routes, so do the authors of this volume, who provide the readers with divergent yet complementary perspectives by which to appreciate the Republic’s principal concerns. (shrink)

Anyone who has read Plato’s Republic knows it has a lot to say about mathematics. But why? I shall not be satisfied with the answer that the future rulers of the ideal city are to be educated in mathematics, so Plato is bound to give some space to the subject. I want to know why the rulers are to be educated in mathematics. More pointedly, why are they required to study so much mathematics, for so long?

The second highest level of the divided line in Plato’s Republic (510b-511a) appears to be about the entities of mathematics—entities such as particular (though non-physical) triangles. It differs from the highest level in two respects. It involves reasoning from hypotheses, and it uses visible images. This article defends the traditional view that the passage is indeed about these mathematical ‘intermediates’; and tries to show how the apparently different features of the second level are related, by focussing on Plato’s need to (...) give an account of how we can speak of many particulars of the same kind, without assuming that they are imperfect copies, in the way sensible things can be imperfect copies, of forms. (shrink)

At 435c-d and 504b ff., Socrates indicates that there is a "longer and fuller way" that one must take in order to get "the best possible view" of the soul and its virtues. But Plato does not have him take this "longer way." Instead Socrates restricts himself to an indirect indication of its goals by his images of sun, line, and cave and to a programmatic outline of its first phase, the five mathematical studies. Doesn't this pointed restraint function as (...) a provocation, moving us to want to begin the "longer way" and to make use of its conceptual resources to rethink Socrates' images? I begin by finding a double movement in the complex trajectory of the five studies: they both guide the soul in the "turn away from what is coming to be ... [to] what is" (518c) and, at the same time, lead the soul back, albeit in the medium of pure intelligibility, to the sensible world; for the pure figures and ratios that they disclose constitute the core structures of sensible things. I then draw on what Socrates says about geometry and harmonics to address three fundamental questions that he leaves open: the nature of the Good in its responsibility for truth and for the being of the forms; the relations of forms, mathematicals, and sensibles as these are disclosed by dialectic; and the bearing of the philosopher's discovery of the Good on his disposition towards his community and the task of ruling. I close by marking six sets of further questions that these reflections bequeath for dialogues to come. (shrink)