Analogy of the divided line

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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 ideas about the four levels of existence (especially the intelligible world and the visible world) and the corresponding ways we come to knowledge about what exists, or come to mere opinions about what exists.

Imagine a line divided into two parts. The longer part (segment CE) represents the visible world and the smaller (segment AC) represents the intelligible world. Next, imagine each part of the line divided in the same ratio as the first division. The divisions in the segment for the intelligible world represent higher (AB) and lower (BC) forms. The divisions in the segment for the visible world represent ordinary visible objects (CD) and other representations such as their shadows or reflections (DE).

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 readily verified that, for any line divided in the way Socrates prescribes, the two middle sections, BC and CD, are necessarily of the same length.^[1] 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.

The first Visible segment is the largest because it represents "clarity and opacity" (Republic, 509e) This is because Plato assumes that the people who look at reflections and other non-substantial images take them at value and do not question them. Because it is so simple to do this, it makes up the largest chunk. As we move up the line, the beliefs and trust makes up the second part of the Visible section. These are the objects that make reflections; the puppets in the cave analogy for example. These things are substantial images and are either natural or artificial.

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 that, again taking hold of the first dependencies from it, so as to proceed downward to the conclusion, making no use whatsoever 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 "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--intellection or reasoning (noesis) for the highest, understanding (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. (trans. Shorey 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.

^ Let the length of AE be equal to $\ 1$ and that of AC equal to $\ x$ , where $\ 0 < x < 1$ (following Socrates, however, $\ 0 < x < 1/2;$ insofar as the equality of the lengths of BC and CD is concerned, the latter restriction is of no significance). The length of CE is thus equal to $\ 1-x$ . It follows that the length of BC must be equal to $x - x \times x \equiv (1-x)\times x$ , which is seen to be equal to the length of CD. Quod erat demonstrandum.