There’s something strange going on with Plato’s divided line. It is a complicated “something strange,” as it often is with Plato, and requires some elucidation. There is an analogy going on between shadow, thing, idea, and form. If we call these S, T, I, and F, we are told, “S:T::I:F::(S:T::I:F)”—that is, that not only do shadow and thing bear the same relationship to each other as idea and form, but that this is the same relationship as between the sensory and the intellectual. That “S:T::I:F” I can accept, but why must the parts of this equation be proportionate to its whole? It results in a number of odd claims, foremost, that T=I. In what sense are things and ideas the same?
Let us leave aside this question for a moment. The above equations allow us to construct another geometrical shape, not a divided line but a divided square, which will serve much the same purpose. Plato actually does this, in the Laws, when talking about things divine, images of things divine, things human, and images of things human. As examples of these, he gives mountains, shadows of mountains, houses, and pictures of houses, but it is easy to see how they could be reinterpreted to be analogous to form, idea, thing, and shadow. So let us look at this square:
FORM THING IDEA SHADOW
We can see that S:T::I:F::(S:T::I:F). Additionally, T=I, insofar as the area of the rectangle THING equals that of the rectangle IDEA. Granted, this portrayal ignores the human half of the divided line—noesis, dianoia, pistis, eikasia—for to include those would require a divided cube. But for our purposes it is enough. The geometric reason for T=I is more clear now; S:T::I:F, but also S:I::T:F. S is two steps removed from F either way. One wonders, what are the philosophical implications of this?
The concept of mimesis, seems to recur here as well—as should perhaps not surprise us, for Plato was discussing art when he described the square in the first place. Recalling earlier, when mimesis was divided into reflection and representation, it seems that we can associate each with one of the two identical elements, T and I. Reflection seems associated with T; a mirror attempts to show us things, and Plato’s complaint is that it does a poor job of it. Representation, on the other hand, can be associated with I; a representation of a separate reality, a heterocosm, can offer nothing to our understanding of reality save general laws that we infer from our comparison of the world portrayed with our own, and Plato’s complaint is that the laws inferred are false. Mimesis begins in SHADOW—in fictions—and tries to bring us into THING and IDEA; Plato says that, without the guidance of philosophy at least, it fails. But worse, it seems, is that it cannot bring us from THING or IDEA towards FORM. Even when mimesis works perfectly, reflection can only bring us from the top of the bottom, and representation from the right to the left; it is not clear that they can build on each other, that together they can bring us from SHADOW to FORM.
Because I enjoy diagrams, and because I like to play with words, I like to label the rows and columns in this divided square. I do so as follows:
But, of course, I could not defend the claim that FACT=FACT.