The divided line, which forms the foundation of Platonic dialectic, is an important aspect of Proclean epistemology. For example, he uses this passage as a basis for his introduction to his Commentary of Euclid’s Elements and it therefore serves as a good introduction to his philosophy in general. To give a brief background of his rather complex form of Neoplatonism, one can characterize as an eminently holistic philosophy based on what could be termed a dynamic multi-modal hierarchical structure of reality. His opening to his Commentary on Plato’s Parmenides can serve as a brief overview of this philosophy:
I pray to all the gods and goddesses to guide my mind in this study that I have undertaken-to kindle in me a shining light of truth and enlarge my understanding for the genuine science of being; to open the gates of my soul to receive the inspired guidance of Plato; and in anchoring my thought in the full splendour of reality to hold me back from too much conceit of wisdom and from the paths of error by keeping me in intellectual converse with those realities from which alone the eye of the soul is refreshed and nourished, as Plato says in the Phaedrus. I ask from the intelligible gods, fullness and wisdom, from the intellectual gods the power to rise aloft, from the supercelestial gods guiding the universe and activity free and unconcerned with material inquiries, from the gods to whom the cosmos is assigned a winged life, from the angelic choruses a true revelation of the divine, from the good daemons an abundant filling of divine inspiration, and from the heroes a generous, solemn, and lofty disposition. (618, 19)
Perhaps more than any other Neoplatonist, Proclus was interested in developing the educational program, often referred to as the quadrivium and trivium, which Plato introduced in the Republic. Therefore he was very much concerned with mathematics, geometry and astronomy. For example, he presents the following paraphrase of Plato’s divided line in his presentation of a general theory of science:
In the Republic he sets on one side the objects of knowledge and over against them the forms of knowing, and pairs the forms of knowing with the types of knowable things. Some things he posits as intelligibles (noeta) others as perceptibles (aistheta); and then makes a further distinction among intelligbles between intelligibles and understandables (dianoeta) and among perceptibles and likenesses (eikasta). To the intelligibles, the highest of the classes, he assigns intellection (noesis) as its mode of knowing, to understandables understanding (dianoia), to perceptibles belief (pistis), and to likenesses conjecture (eikasia). He shows that conjecture has the relation to perception that understanding has to intellection; for conjecture apprehends the images of sense objects in water or other reflecting surfaces, which, as they are really only images of images, occupy almost the lowest rank in the scale of kinds, while understanding studies the likenesses of intelligibles that have descended from their primary simple and indivisible forms into plurality and division. For this reason the knowledge that understanding has is dependent on other and prior hypotheses, whereas intellection attains to the unhypothetical principle itself. (11, 9)\
Moreover, in the two prologues to his Commentary on Euclid’s Elements, Proclus outlines a full epistemological program centered around the central books of Plato’s Republic in an eclectic synthesis of elements from Pythagorean, Platonic, Aristotelian, and Stoic sources. Influenced by Pythagoreanism, he outlines a metaphysical system based on mathematical and geometric-based concepts, presented as binary and ternary concepts, the highest being the notions of the limit and unlimited (peiras, apeiron) (based on Plato’s Philebus 16cff and 23cff.): ‘’To find the principles of mathematical being as a whole, we must ascend to those all-pervading principles that generate everything from themselves: namely, the Limit and the Unlimited. For these, the two highest principles after the indescribable and utterly incomprehensible causation of the One, give rise to everything else, including mathematical beings’’(6). These principles determine the aspects of the divisible and indivisible, establishing a limit/indivisible, unlimited/divisible correspondence:
For this reason both in Nous and in the intermediate orders of souls-that is, those natures that directly breathe life into bodies- the limiting factors have an essential priority over the things that are limited, as being less divisible, more uniform, and more sovereign; for among immaterial forms unity is more perfect than plurality, the partless more perfect than what proceeds in and away from it, and what bounds more perfect that what gets its limit from something other than itself. (86)
The images of the straight line and the circle are further related to cosmological functions, the line being related to the soul and the circle to the intellect and are also related to the fundamental ontological Neo-platonic dynamic of procession, related to the line, and reversion, related to the circle: ‘‘the straight line makes clear the procession [πρόοδος] of psychical life from superior things, and the bending-back [κατάκαμψις] into a circle makes clear its intellectual turning [στροϕή]’.] (InTimaeum [2.248,11–23; cp. 2.244,15–17]).
This leads to his theory of the ontological status of mathematicals are situated at an intermediary between the indivisible and the divisible, containing aspects of both. This theory forms the basis of his theory of projection. It is through the imagination that mathematical concepts rooted in the soul become articulated in human understanding, through a process of collection and division. The imagination transforms archetypal forms into concrete images. It is considered an intermediary faculty between understanding and sensation: ‘’To this difficulty we reply that the imagination in its activity is not divisible only, neither is it indivisible. Rather it moves from the undivided to the divided, from the unformed, to what is formed’’ (In Euclid 95).