"A discussion of Plato's evaluation of mathematics as an intellectual discipline, and his reasons for training his philosopher-rulers to be mathematical experts"--

The author takes the general reader on a tour of the mathematical puzzles and paradoxes inherent in voting systems, such as the Alabama Paradox, in which an increase in the number of seats in the Congress could actually lead to a reduced number of representatives for a state, and the Condorcet Paradox, which demonstrates that the winner of elections featuring more than two candidates does not necessarily reflect majority preferences. Szpiro takes a roughly chronological approach to the topic, traveling from ancient Greece to the present and, in addition to offering explanations of the various mathematical conundrums of elections and voting, also offers biographical details on the mathematicians and other thinkers who thought about them, including Plato, Pliny the Younger, Pierre Simon Laplace, Thomas Jefferson, John von Neumann, and Kenneth Arrow.

The first of two volumes collecting the published work of one of the greatest living ancient philosophers, M.F. Burnyeat.

This book presents a reinterpretation of early Greek mathematics, one of the most tantalizing intellectual subjects of the last 2,000 years. The first part offers several new interpretations of the idea of ratio in early Greek mathematics and illustrates them in detailed discussion of several texts. Part Two discusses the historical context of the subject--what we know of Plato's academy during his lifetime, the origin of our text of Euclid's Elements, and what we know of early Greek numerical practice. The book finishes with an account of the theory of continued fractions and its history since the 17th century.

Plato's Republic is one of the most well-known and widely discussed texts in the history of philosophy, but how might we get to the heart of this work today, 2500 years after it was originally composed? Alain Badiou invents a new genre in order to breathe fresh life into Plato's text and restore its universality. Rather than producing yet another critical commentary, he has retranslated the work from the original Greek and, by making various changes, adapted it for our times. In this innovative reimagining of a classic text, Badiou has removed all references specific to ancient Greek society, from the endless exchanges about the moral courage of poets to those political considerations that were only of interest to the aristocratic elite. On the other hand, Badiou has expanded the range of cultural references: here philosophy is firing on all cylinders, and Socrates and his companions are joined by Beckett, Pessoa, Freud and Hegel. They demonstrate the enduring nature of true philosophy, always ready to move with the times. Moreover, Badiou the dramatist has made the Socratic dialogue a true oratorial contest: in his version of the Republic, the interlocutors have more in mind than merely agreeing with the Master. They stand up to him, put him on the spot and thereby show thought in motion. Through this work of writing, scholarship and philosophy, we are able, for the first time, to read a version of Plato's text which is alive, stimulating and directly relevant to our world today.