Philosophy of Mathematics and Deductive Structure in Euclid's Elements

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.