| Histories of mathematics commonly credit the Greeks with developing a unique notion of mathematics as a body of demonstrated propositions ultimately derived from a set of undemonstrated first principles. Euclid's Elements (third century B.C.) became the paradigm of this ideal of mathematical rigor, and is rightly celebrated as a remarkable text in classical Greek mathematics. What is perhaps equally remarkable, however, is the way in which antiquity drew geometry into the realm of philosophical debate. Proclus' Commentary on the First Book of Euclid's Elements (fifth century A.D.) provides a rich picture of a tradition in which the principles, proofs, and purposes of geometry were all actively explored and widely debated. A familiar threefold philosophical approach to mathematics is evident in Proclus' Commentary: What is the ontology of mathematical objects? How do we know and reason securely with these objects? Why is mathematics useful? Yet, despite the familiarity of the questions, many of the ancient answers today seem surprisingly foreign. Aside from antiquarian interest, this strangeness recalls us to a reflexive analysis of the presuppositions driving modern philosophical debates on ancient philosophical issues: this, it might be said, is part of the point of engaging seriously with ancient philosophies of mathematics. |
