Analogy of the divided line
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Plato, in his dialogue The Republic Book 6 (509D–513E), has Socrates explain the literary device of a divided line to teach basic philosophical views about four levels of existence (especially "the intelligible" world of the forms, universals, and "the visible" world we see around us) and the corresponding ways we come to know what exists.
Socrates asks us to imagine a line divided into two parts. The larger part (segment CE) represents the intelligible world and the smaller (segment AC), the visible world. Then, he says, imagine each part of the line further divided. As it turns out, the divisions in the segment for the intelligible world represent higher (DE) and lower (CD) forms, respectively. Moreover, the divisions in the segment for the visible world represent ordinary visible objects (BC), on the one hand, and their shadows, reflections, and other representations (AB), on the other.
It is important to note that the line segments are said to be unequal: the proportions of their lengths is said to represent "their comparative clearness and obscurity" and their comparative "reality and truth," as well as whether we have knowledge or instead mere opinion of the objects. It can be shown with simple mathematics that any line divided in the way Socrates describes that the two middle sections, BC and CD, are necessarily the same length. Hence, we are said to have relatively clear knowledge of something that is more real and "true" when we attend to ordinary perceptual objects like rocks and trees; by comparison, if we merely attend to their shadows and reflections, we have relatively obscure opinion of something not quite real.
Socrates uses this familiar relationship, between ordinary objects and their representations or images, in order to illustrate the relationship between the visual world as a whole (visual objects and their images) and the world of forms as a whole. The former is made up of a series of passing, particular reflections of the latter, which is eternal, more real and "true." Moreover, the knowledge that we have of the forms--when indeed we do have it--is of a higher order than knowledge of the mere particulars in the perceptual world.
Consider next the difference between the two parts of the intelligible world, represented by segments CD and DE. Plato's discussion of this is apt to seem obscure. The basic idea is that the lower forms (represented by CD) are the real items of which the ordinary particular objects around us are merely reflections or images. The higher forms, by contrast--of which the so-called Form of the Good is the "highest"--are known only by what has come to be called a priori reasoning, so that strictly speaking, knowledge of them does not depend upon experience of particulars or even on ideas (forms) of perceptually-known particulars.
This can be explained a bit further. In geometry and arithmetic, we often use particular figures to fix our ideas and make demonstrations clear. Moreover, in these sciences, we make certain postulates and draw conclusions that are only as trustworthy as the postulates. By contrast, the intelligible is "that which the reason itself," rather than image-assisted imagination, lays hold of by the power of dialectic, treating its assumptions not as absolute beginnings but literally as hypotheses, underpinnings, footings, and springboards so to speak, to enable it to rise to that which requires no assumption and is the starting point of all, and after attaining to that again taking hold of the first dependencies from it, so to proceed downward to the conclusion, making no use whatever of any object of sense but only of pure ideas moving on through ideas to ideas and ending with ideas. (511b-c)
What all this might mean is essentially to ask, "What are the details of Plato's and Socrates' rationalism?" The reference to and idolization of "pure ideas," as well as deduction as it were without assumptions (or with one grand assumption or principle, as The Form of the Good is sometimes portrayed), is something reflected again and again in later rationalists. The above text finds later echoes in Descartes' interest in pure, a priori deduction and Kant's transcendental arguments.
Plato, through Socrates, explicitly names four sorts of cognition associated with each level of being:
[A]nswering to these four sections, assume these four affections occurring in the soul--understanding (noesis) for the highest, reasoning (dianoia) for the second, belief (pistis) for the third, and for the last, picture thinking or conjecture (eikasia)--and arrange them in a proportion, considering that they participate in clearness and precision in the same degree as their objects partake of truth and reality. (511d-e)
Not too much weight should be put on the English (or Greek) meanings of the words here, however. Any significant meaning that these words have, when used as technical terms for Plato, needs to be informed by the metaphysical and epistemological edifice that supports them.
The metaphor of the divided line immediately follows another Platonic metaphor, that of the sun: see Plato's metaphor of the sun. It is immediately followed by the famous allegory of the cave.de:Liniengleichnis fi:Jaettu viiva
