Plato distinguishes between four levels of reality and four corresponding faculties of the soul. The lowest level is the level of eikones (likenesses, semblances), which are representations of orata (visible things). Examples include shadows and reflections. The faculty that deals with eikones is termed eikasis (conjecture), while the faculty that deals with orata is termed pistis (faith or conviction). These two levels make up the sensible world, and are the object of doxa (opinion or belief) as opposed to gnosis (knowledge in the proper sense of the word). In particular, empirical science would fall under doxa. The next two levels make up the world of eidos (forms or ideas) or the intelligible world. The lower level of the two corresponds to objects that have likenesses in the world of sensible things and the higher corresponds to all forms. In particular, mathematical objects belong to the former level. The faculty of the soul that deals with mathematical objects Plato calls dianoia, or discursive reasoning. Dianoia uses the forms as hypotheses and derives conclusions from these hypotheses (compare this to the modern axiomatic method in mathematics where statements are assumed as axioms and conclusions derived from logic). Dianoia can thus be characterized as purely formal reasoning. This is in contrast to noesis, or intellectual intuition, which treats hypotheses not as first principles, but as hypotheses, and moves from the hypotheses to the principles upon which the hypotheses depend, all the way to the first principles of thought. This is done through the use of the dialectic, which in contrast to discursive reasoning, is a method that does not treat hypotheses as static truths, but instead attacks them in order to refine them and ascend higher to first principles.
There is a relation of representation between each of the levels. Reflections are representations of visible things, which are representations of the forms of visible things, which in turn “participate” in higher-level forms (forms may partake of forms, as each form is one form, and so partakes of oneness; an important question is whether a form can partake of itself). A two-world interpretation of Plato holds that reality is essentially bifurcated into the sensible world and the world of forms. Such an interpretation of Plato is naive, as it does not take into account the notion of participation. Just as a reflection of something is unreal when compared to the thing itself, so too are visible objects unreal when compared to the forms they participate in. What we have is not a bifurcation, but a gradation. The world of particulars subsists, but only as a mere reflection of the world of forms, having a derived rather than independent existence (this is what the allegory of the cave intends to show). This gradation carries on into the world of forms itself, with some forms being more real than others (in fact, Neoplatonists distinguished the reality of a form with how many other forms partake of it).
Plato Plato’s Forms Neoplatonism Dianoia Nous Doxa Gnosis Dialectic Participation