Mathematics as Paideia in Proclus :: Philosophy Philosophical Essays

Mathematics as Paideia in Proclus ABSTRACT: I examine one aspect of the central role which mathematics plays in Proclus's ontology and epistemology, with particular reference to his Elements of Theology. I focus on his peculiar views about the ontological status of mathematical objects and the special faculties of the soul that are involved in understanding them. If they are merely abstract objects that are "stripped away" from sensible things, then they are unlikely to reorient the mind towards the intelligible realm, as envisioned by Plato in the Republic. Thus, in order to defend the function of mathematics as a prodaideutic to dialectic, Proclus rejects Aristotelian abstractionism in favor of an elaborate account in terms of Nous projecting images of its Forms through the medium of the imagination. In metaphorical terms, he replaces the Aristotelian image of the soul as a blank tablet with that of a tablet that has always been inscribed and is always writing itself, while also being written on by Nous. The mediat ing function of mathematics for understanding the higher realities is grounded in the fact that its central principles of Limit and Unlimited have a universal provenance in Proclus's whole system of reality. Introduction Alexander of Aphrodisias established abstractionism as an Aristotelian dogma about mathematical objects, but for later Neoplatonists this proved difficult to reconcile with the educational function of mathematics in Plato's philosophical curriculum. Thus Proclus, for example, rejected abstractionism as a basis for the ascent to the realm of Forms, and proposed an alternative view based on the typical Neoplatonic hierarchy of Nous, Soul, and Nature. At the highest noetic level, geometrical Forms are unextended and indivisible, so that only at the level of Soul can they become available for study by the geometer when they are embodied in the intelligible matter supplied by the imagination. Proclus also accepted that geometrical forms can be embodied in sensible matter, though they never have the exactitude necessary for science, nor could they ever acquire it through abstraction. Thus the diagrams used by the geometer are products of the imagination, which are really projections by the higher intellect onto a lower level so as to facilitate the study of geometrical objects. Proclus seems to accept that the human intellect can never attain the Platonic goal of studying geometrical Forms in their pure and unextended form as paradigms. Although such a goal can be achieved only by divine Nous, yet it becomes for Proclus the guiding rationale for his whole system.