The Thomist: A Speculative Quarterly Review

EUCLID AND ARISTOTLE HE TECHNICAL and philosophical interpretation of Greek mathematics is dominated by the opposite views of Plato and Aristotle, which have influenced even modern mathematical philosophy. Briefly, Plato teaches that mathematical objects are intermediate between the Ideas and the sensible world: as such, they reflect the eternal relations of their own Ideas are separate from the external world; they are discovered and not invented; they are expressed in assertorical and not in problematical propositions. On the other hand, Aristotle thinks also that mathematical objects are intermediate between Being and the sensible world: but they cannot be deduced from the analysis of Being and have no separate existence; indeed, they are abstracted from the sensible world, which accounts for their being applicable to it. Hence Plato accounts mathematically, and Aristotle logically, for the rational character of the world. These doctrines, which were widely known in the Greek world, have influenced most thinkers in one way or another. Is it possible at all to determine which conception is paramount in the work of Euclid the mathematician? The problem is made difficult by the absence of any direct information from Euclid himself, and also by the cosmopolitan and eclectic character of the intellectual climate of Alexandria where Euclid lived. While scholars grant that the author of the Elements has widely used in his work the Aristotelian theory of demonstration , most o£ them would insist that his ultimate vision was Platonic indeed. We believe, however, that the Euclidian systematization of mathematics is an application of the rational theories of both the Academy and the Lyceum; and that the Aristotelian trends in Euclid are stronger than any other. But 374 EUCLID AND ARISTOTLE 375 this debate must be prefaced with an account of the Alexandrian atmosphere and of the work Euclid left to posterity. THE CLIMATE OF ALEXANDRIA AND EucLm's Wmucs The conquest of Egypt by Alexander the Great in 332 B. C. had important consequences for the development of learning. While Athens still remained the seat of literary interests and speculative philosophy, Alexandria became the dynamic centre of scientific pursuits and of the technical arts. Owing to its geographical remoteness and to the political rivalry between Rome and Carthage, the Nile Delta was less open to an effective external pressure than Sicily or Greece itself. The death of Alexander in 323 B. C. and the wisdom and might of the Diadochs, gradually robbed Athens of its political preponderance . }-,ounded at a point where East meets West, Alexandria soon rose to the status of a great intellectual and commercial metropolis. With their broad cultural horizon widened the expeditions of the lVIacedonian conqueror, the Alexandrians came into contact with the esoteric as well as the empirical doctrines of the Persians, the Babylonians, the Assyrians, and the Phoenicians. Moreover they had at their very door the ancient and complex knowledge of the Egyptian priests who had been the first masters of the Greeks and had still many things to teach in spite of thei:r declining influence. The fusion of the Hellenic and of the Eastern civilizations made Alexandrian thought more cosmopolitan, but also more specialized and more mystical. But on the other hand, these circumstances also favoured frequent exchanges between Greece and Egypt. The great philosophical schools of Athens had already given their best to the world: Platonists, Peripatetics, Stoics, Epicureans, Sophists and Sceptics could cross over the Eastern Mediterranean and widen the circle of their disciples. Yet the attraction of so many different and contradictory doctrines was tempered by the mystical interests and practical minds of the Alex- 876 THOMAS GREENWOOD andrians themselves. It is obvious that all these mutual influ- .ences gave a great stimulus to intellectual pursuits, which could not be ignored by the leaders of the country. Learning was greatly encouraged by the Greek rulers of Egypt, and especially by Ptolemy II Philadelphus and Arsinoe his wife. The Ptolemies were responsible for the foundation and maintenance of the Museum and the Library, which were housed in the royal citadel. In these two famous scientific institutions, eminent thinkers, scientists, engineers, alchemists and physicians were able to carry out research and experiments; while the wise discussed philosophy and religious speculations. They...