Proclus’ conception of geometric space and its actuality

Abstract

The main aim of this paper is to present Proclus ’ philosophy of geometric extension not so much from the point of view of what he says about it as what he does with it. I will henceforth pay particular attention to the role of spatial configurations in the practice which he describes. My motivations are twofold. First, although Proclus ’ philosophy of geometry has received quite a lot of attention in the scholarship, this attention has remained mainly inspired by the philosophical doctrine expounded in the Prologues. It did not engage much, at least in a systematic way, with the material given in the actual commentary of Euclid ’s propositions and the mathematical practice there described. As a consequence, the complexity and the flexibility of Proclus ’ views on the geometric imagination were not always well rendered. I would like to complete existing descriptions by paying more attention to these details, although, as I will indicate, they may sometimes introduce important nuances, if not tensions, in the philosophical system. Second, Proclus provides indications throughout his commentary about the geometric practice which go far beyond his own specific philosophical agenda. He deals, for example, with objections that other mathematicians and philosophers raised against Euclid ’s proofs. These objections, which sometimes stem from views opposite to his own (typically, Epicurean objections of an “empiricist” flavor), were taken seriously enough to ask for answers which Proclus also mentions (or sometimes even initiates). For the modern reader, these passages are precious because they provide, by contrast, testimonies about certain conditions of the practice which seem to have been accepted by the various interlocutors (whatever their philosophical background may have been). I would like to reconstitute some of these conditions and compare these reconstructions with recent works dealing with ancient Greek geometric practice, especially as regards the use of diagrams.