Both the analogy of the Divided Line and the allegory of the Cave are designed by Plato to address the same issue. The Divided Line analogy is introduced first, and is almost immediately followed by the allegory of the Cave. (Plato, The Republic, Book VI 507 thru Book VII 520.) They are both intended to provide the audience (personalized in the character of Glaucon) with guidance in understanding the role that Plato's Theory of Forms plays in his ontology.
Plato uses the analogy of the Divided Line to illustrate some of his basic ideas about existence and knowledge. In particular, through this analogy, Plato emphasizes the relationship between image and object as a means of elucidating his theory of the relationship between perceptual objects and the realm of Forms. According to his Theory of Forms, objects in the perceptual world gain their existence through participation in the Forms. Hence they are not themselves real in the same sense as Forms are real, but exist as mere imperfect reflections or images of the Forms. To Plato, the intelligible world, the world of forms, is the reality, the object, and the perceptual world is the image or reflection of those forms. Both the divided line and the cave are meant by Plato to illustrate and emphasize this.
Imagine a line divided into two unequal parts. The unequal proportions Plato intends to represent the relative degree of clearness, precision, truth, and reality versus obscurity, vagueness, and unreality. The shorter part (less clear, more obscure) represents the visible (perceptual) world -- the "world of becoming or passing away". The longer part (clearer, less obscure) represents the intelligible world -- "the domain where truth and reality shine resplendent".
At the next step, each of the two major parts of the line is in turn divided in the same proportion. The shorter part of the perceptual world represents images, shadows, reflections, and things of that sort. Things that are not themselves the actual things of the world. The longer part of the perceptual world represents the actual things of the world of which the shorter part are imperfect likenesses. Thus as before, the shorter part is the less clear, less real, and the longer part the more clear and more real.
Likewise for the division of the intelligible world, the shorter part of the intelligible world represents images, and the longer part the objects of which those images are imperfect likenesses. The intelligible world is thus to be divided into "mental images"on the one hand, and the pure Forms on the other, with the shorter (and hence less clear) segment of the intelligible world representing the "mental images". Plato uses the example of geometric diagrams to illuminate the transition from the objects of the perceptual world to the images of the intelligible world. A diagram drawn in the sand of a circle or a triangle is a perceptual object. But the geometer converts the physically drawn diagram of a particular circle or triangle into a mental image of a generalized circle or triangle. And that mental generalization is an intelligible image of the Form which is the idea of the circle or triangle.
[In one part] "the mind uses as images those actual things which themselves had images in the visible world; and is compelled to pursue its inquiry by starting from assumptions and travelling, not up to principle, but down to conclusion. In the second, the mind moves in the other direction, from assumption up to principle which is not hypothetical; and it makes no use of the images employed in the other section, but only of Forms, and conducts its enquiry solely by their means."Plato, Republic VI, 510b.
As a consequence of the proportionality into which the line is divided, the segment that represents the objects of the perceptual world is equal in length to the segment that represents the images of the intelligible world. Plato does not specifically comment on this equivalence. But it is usually interpreted to mean that the mental images reflect the run of the mill Forms of which the objects of the perceptual world partake. Hence the one-to-one relationship in the degree of clarity and reality for the objects of the perceptual world and the "mental images"by which we recognize the Forms in which those objects partake. Of course, the actual intelligible objects of which these are merely images, are the Forms themselves in the fourth and final line segment. The Forms are accessible only to a priori reasoning -- like the "Form of the Good". Hence they are the most clear and the most real.
The same two-level four-fold division of the world into object-image pairs described by the analogy of the divided line, is repeated by Plato in the allegory of the cave. In the Allegory of the Cave, Plato describes prisoners (representing "most people") chained so that all they can see are shadows on the wall before them. These shadows they take to be reality. When a prisoner is released, the first thing he sees is that the shadows are cast on the wall by statues and images carried between the prisoners and a great fire. When the prisoner is brought out of the cave into the daylight, he is at first blinded by the bright sun. As his eyes slowly adjust, the first things he can see are the less bright shadows and reflections in water cast by things in the bright sunlight. Then as his eyes adjust to the bright sun, he can finally see the real objects that cast the shadows and are reflected in the water -- and that are the original bases for the statues and other images carried before the fire to cast shadows on the wall of the cave. Finally, the prisoner descends back into the cave, bringing his knowledge of the reality behind the shadows back to his fellow prisoners, and suffering all of the prejudice that effort would entail.
Clearly, the two stories fit very tightly together. The first-level division of the divided line becomes the division between the cave (the perceptual world) and the outside (the intelligible world). At the second level, the first segment of the divided line that represents images and representations in the perceptual world becomes the shadows on the wall of the cave. The second segment of the line that represents the objects of the perceptual world becomes the statues (themselves images of the objects in the outside world) that are held aloft between the fire and the wall to create the shadows on the wall. The third segment of the divided line that represents the images of the Forms being partaken in by objects becomes the shadows and reflections in water that the released sun-blind prisoner is first able to see. While the final segment of the divided line representing the Forms themselves becomes the objects in the outside world, all brightly illuminated and energized by the Sun (the Form of the Good).
The analogy of the divided line is used by Plato to elucidate his ontology. As such, it is a static picture of how he divides things and thinking up into different realms. Intellection or reasoning Plato reserves to the realm of Forms, the realm where Forms are revealed through dialectic and a priori reasoning alone. Understanding he restricts to the realm of mental images, the realm where we make postulates and draw conclusions. Belief he applies to the realm of perceptual objects. And conjecture dominates the realm of the images of perceptual objects. When people do not have the knowledge to look beyond the things one can see by taking only a superficial look (the shadows on the wall of the cave), they do not know how to being a search for the intelligible, the real truths.
The allegory of the cave, on the other hand, is used by Plato to elucidate his notions of how to gain knowledge of the Forms. It is thus a dynamic picture of education and learning. Most people are chained by their ignorance, seeing only the shadows on the wall. The student in search of enlightenment progresses "up"the scale - from the shadows on the wall that is the entire world for most people; to the statues paraded before the fire as one begins to appreciate that there are things "behind"the shadows that are the cause and source of the things we daily experience; to the shadows and reflections of things on the "outside"as one begins to grasp the Theory of Forms and understand the shape and manner of Forms; to seeing the objects in the outside world as one gains an understanding of Forms and the role that they play in our ontology and epistemology; to the light of the Sun as one gains an understanding of the Form of the Good; and finally the return to the cave as one accepts the challenge of teaching others the wisdom of the Forms and the Form of the Good.
We can therefore see that the analogy of the Divided Line and the Allegory of the Cave fit very well indeed. The fit is not perfect, of course. Any analogy breaks down if the details are pushed too far. The prisoner's trip back down into the cave has no correspondence in the analogy of the divided line, for example. But because the topic is the same (Plato's Theory of Forms), and the intent is the same (to illustrate Plato's division of the world into Object-Image pairs, the one being real while the other only an imperfect image), the two fit together quite well. Any apparent discrepancies can be attributed to the slightly different focus the two have (ontology versus education).
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