This is the most extensive examination of Euclid's Elements in English since the last revision of T. L. Heath's monumental three-volume translation and commentary appeared over fifty years ago. While the present work augments and updates Heath's in the light of subsequent scholarship, its principal concern is to apply the resources of modern logic and philosophy of mathematics to the Elements in order to provide an understanding of the distinctively Greek conception of the foundations of mathematics.  Mueller probes the internal logic and development of the Elements, giving a careful and full account of the independence and interdependence of its various books and of their mathematical and logical foundations. By considering alternative paths Euclid might have followed he clarifies the motivations underlying Euclid's actual choices. The results of his analyses are encapsulated in analytic diagrams. Appendixes listing all the propositions and presuppositions of the Elements make it easy for a reader to focus on particular parts of the Elements and to study the exact role of individual propositions.  In order to bring out distinctive aspects of Greek mathematics the author makes frequent comparisons and contrasts with later mathematical developments. For example, he gives a detailed explanation of the differences between Euclid's axiomatic method and Hilbert's, between Euclid's development of arithmetic and modern ones based on Peano's axioms, and between Euclid's theory of proportion and Dedekind's theory of real numbers. The result is not only a clarification of Greek mathematics but of the mathematical enterprise itself.

Philosophy of Mathematics and Deductive Structure in Euclid's Elements

Science & Math

Mueller, Ian

History & Philosophy

Mit Pr

A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics. It offers a well-rounded perspective, examining similarities to modern views as well as differences. Rather than focusing strictly on historical and mathematical issues, the book examines philosophical, foundational, and logical questions. Although comprehensive in its treatment, this study represents a less cumbersome, more streamlined approach than the classic three-volume reference by Sir Thomas L. Heath (also available from Dover Publications). To make reading easier and to facilitate access to individual analyses and discussions, the author has included helpful appendixes. These list special symbols and additional propositions, along with all of the assumptions and propositions of the Elements and notations of their discussion within this volume.

Philosophy of Mathematics and Deductive Structure in Euclid's Elements (Dover Books on Mathematics)

Dover Publications

The commentary of Alexander of Aphrodisias on Aristotle's Prior Analytics 1.8-22 is a very important text, being the main ancient commentary with chapters in which Aristotle invented modal logic - the logic of propositions about what is necessary or contingent (possible). The first volume of Ian Mueller's translation covered chapters 1.8-13, and reached as far as the chapter in which Aristotle discussed the notion of contingency. In this, the second volume, the 'greatest' commentator, Alexander, concludes his discussion of Aristotle's modal logic. Aristotle also invented the syllogism, a style of argument involving two premises and a conclusion. Modal propositions can be deployed in syllogisms, and in the chapters included in this volume Aristotle discusses all the syllogisms containing at least one contingent premiss. In each volume, Ian Mueller provides a comprehensive explanation of Alexander's commentary on modal logic as a whole.

Alexander of Aphrodisias: On Aristotle Prior Analytics 1.14-22

Politics & Social Sciences

Philosophy

Bristol Classical Press

Aristotle believed that the outermost stars are carried round us on a transparent sphere. There are directions in the universe and a preferred direction of rotation. The sun, moon and planets are carried on different revolving spheres. The spheres and celestial bodies are composed of an everlasting fifth element, which has none of the ordinary contrary properties like heat and cold which could destroy it, but only the facility for uniform rotation. But this creates problems as to how the heavenly bodies create light, and, in the case of the sun, heat. The topics covered in this part of Simplicius' commentary are: the speeds and distances of the stars; that the stars are spherical; why the sun and moon have fewer motions than the other five planets; why the sphere of the fixed stars contains so many stars whereas the other heavenly spheres contain no more than one (Simplicius has a long excursus on planetary theory in his commentary on this chapter); discussion of people's views on the position, motion or rest, shape, and size of the earth; that the earth is a relatively small sphere at rest in the centre of the cosmos.

On Aristotle on the Heavens 2.10-14 (Ancient Commentators on Aristotle)

Simplicius Of

Astronomy & Space Science

The commentary of Alexander of Aphrodisias on Aristotle's Prior Analytics 1.8-22 is a very important text, being the main ancient commentary with chapters in which Aristotle invented modal logic - the logic of propositions about what is necessary or contingent (possible). The first volume of Ian Mueller's translation covered chapters 1.8-13, and reached as far as the chapter in which Aristotle discussed the notion of contingency. In this, the second volume, the 'greatest' commentator, Alexander, concludes his discussion of Aristotle's modal logic. Aristotle also invented the syllogism, a style of argument involving two premises and a conclusion. Modal propositions can be deployed in syllogisms, and in the chapters included in this volume Aristotle discusses all the syllogisms containing at least one contingent premiss. In each volume, Ian Mueller provides a comprehensive explanation of Alexander's commentary on modal logic as a whole.

Alexander of Aphrodisias: On Aristotle Prior Analytics 1.14-22 (Ancient Commentators on Aristotle)

Bloomsbury Academic