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, 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, which 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).
