I offer a new interpretation of Aristotle's philosophy of geometry, which he presents in greatest detail in Metaphysics M 3. On my interpretation, Aristotle holds that the points, lines, planes, and solids of geometry belong to the sensible realm, but not in a straightforward way. Rather, by considering Aristotle's second attempt to solve Zeno's Runner Paradox in Book VIII of the Physics , I explain how such objects exist in the sensibles in a special way. I conclude by considering the (...) passages that lead Jonathan Lear to his fictionalist reading of Met . M3,1 and I argue that Aristotle is here describing useful heuristics for the teaching of geometry; he is not pronouncing on the meaning of mathematical talk. (shrink)

Cleary discusses the origin, development, and use of the many senses of priority as a central thesis in Aristotle’s metaphysics. Cleary contends that one of the most revealing problems for the ambiguity of Aristotle’s relationship to Platonism is that of the ontological status of mathematical objects. In support of his claim, Cleary analyzes a curious passage from Aristotle’s _Topics, _where he appears to accept a schema of priorities that makes mathematical entities more substantial than sensible things. How does Aristotle try (...) to reconcile the ordering of things dictated by sciences like mathematics and dialectic with the ordering of sense experience upon which his own physics and metaphysics are based? To find the answer, Cleary reviews three different outlines of the many senses of priority given by Aristotle himself and found in _Categories _12-13, _Metaphysics _Delta 11, and _Metaphysics _Theta 8. Cleary suggests there is an implicit hierarchy for Aristotle that leads him to posit the Prime Mover at its apex as complete actuality and, therefore, as the focus for the concept of priority. Having reviewed Aristotle’s treatment of the many uses of priority, Cleary demonstrates how the concept is used in some typical arguments by Aristotle for his mature metaphysical positions. (shrink)

In Proclus' penetrating exposition of Euclid's method's and principles, the only one of its kind extant, we are afforded a unique vantage point for understanding the structure and strenght of the Euclidean system. A primary source for the history and philosophy of mathematics, Proclus' treatise contains much priceless information about the mathematics and mathematicians of the previous seven or eight centuries that has not been preserved elsewhere.

This book examines Aristotle's critical reaction to the mathematical cosmology of Plato's Academy, and traces the aporetic method by which he developed his own ...