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Part of a series on Plato Early life · Works Platonism · Epistemology Idealism / Realism Theory of Forms Form of the Good Third man argument Euthyphro dilemma Five regimes Philosopher king Allegories and metaphors Ring of Gyges · The cave The divided line · The sun Ship of state · Myth of Er The chariot Related articles The Academy in Athens Socratic problem Commentaries on Plato Middle Platonism Neoplatonism Neoplatonism and Christianity v • d • e 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 to which knowledge is acquired about what exists, or come to mere opinions about what exists. The analogy of the divided line immediately follows another Platonic metaphor, that of the sun, and is in turn followed by the Allegory of the Cave. Contents 1 Imagine a line divided into two parts 1.1 The intelligible world 2 Notes 3 External links // Imagine a line divided into two parts The Divided Line – The shorter segment (AC) is generally taken as representing the physical world and the longer segment (CE) as representing the intelligible world.[1] In The Republic, Plato describes the Divided Line this way: Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion, and suppose the two main divisions to answer, one to the visible and the other to the intelligible, and then compare the subdivisions in respect of their clearness and want of clearness, and you will find that the first section in the sphere of the visible consists of images. And by images I mean, in the first place, shadows, and in the second place, reflections in water and in solid, smooth and polished bodies and the like: Do you understand? Yes, I understand. Imagine, now, the other section, of which this is only the resemblance, to include the animals which we see, and everything that grows or is made.[2] Thus AB represents shadows and reflections of physical things, and BC the physical things themselves. These correspond to two kinds of knowledge, namely belief (πίστις pistis) about physical things and illusion (εἰκασία eikasia) about "shadows," which do not really exist.[3] In the Timaeus, Plato includes the natural sciences in the category of belief, while the category of illusion includes all the "opinions of which the minds of ordinary people are full."[3] The intelligible world The segment CE, representing the intelligible world, is divided into the same ratio as AB, giving the subdivisions CD and DE (it can be readily verified that CD must have the same length as BC[4]): There are two subdivisions, in the lower of which the soul uses the figures given by the former division as images; the enquiry can only be hypothetical, and instead of going upwards to a principle descends to the other end; in the higher of the two, the soul passes out of hypotheses, and goes up to a principle which is above hypotheses, making no use of images as in the former case, but proceeding only in and through the ideas themselves.[2] Plato describes CD, the "lower" of these, as involving mathematical reasoning (διάνοια dianoia),[3] where abstract mathematical objects such as geometric lines are discussed. Such objects are outside the physical world (and are not to be confused with the drawings of those lines, which fall within the physical world BC). However, they are less important to Plato than the subjects of philosophical reasoning (νόησις noesis), the "higher" of these two subdivisions (DE): And when I speak of the other division of the intelligible, you will understand me to speak of that other sort of knowledge which reason herself attains by the power of dialectic, using the hypotheses not as first principles, but only as hypotheses — that is to say, as steps and points of departure into a world which is above hypotheses, in order that she may soar beyond them to the first principle of the whole.[2] Plato here is using the familiar relationship between ordinary objects and their shadows or reflections in order to illustrate the relationship between the physical world as a whole and the world of forms as a whole. The former is made up of a series of passing 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 physical world. In particular, knowledge of the forms leads to a knowledge of the Form of the Good.[1] The equality in length of the segments BC and CD hints at an interdependence between mathematical objects and the physical world.[5] Apart from the fact that geometers use physical diagrams to describe the abstract objects which they study (something Plato explicitly mentions), there may perhaps also be a deeper connection. In the Timaeus, for example, Plato suggests a connection between the Platonic solids and the four classical elements (earth, air, water, and fire). The following table summarises the analogy of the divided line: Segment Type of knowledge Type of object DE (highest) Philosophical knowledge (noesis) Forms, especially the Form of the Good CD Mathematical knowledge (dianoia) Abstract mathematical objects, such as numbers and lines BC Beliefs about physical things (pistis), including scientific knowledge Physical things AB (lowest) Opinions, illusions (eikasia) "Shadows" and things which do not really exist Notes ^ a b "divided line," The Cambridge Dictionary of Philosophy, 2nd edition, Cambridge University Press, 1999, ISBN 0521637228. ^ a b c Plato, The Republic, Book 6, translated by Benjamin Jowett, online ^ a b c Desmond Lee and Rachana Kamtekar, The Republic, Notes to Book 6, Penguin, 1987, ISBN 0140449140. ^ Let the length of AE be equal to and that of AC equal to , where (following Socrates, however, ; 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 . It follows that the length of BC must be equal to , which is seen to be equal to the length of CD. ^ Diskin Clay, Platonic Questions: Dialogues with the Silent Philosopher, Penn State Press, 2000, p 236, ISBN 027102044X. External links At MIT.edu: Plato's Republic: Translated by Benjamin Jowett At Perseus Project: Plato's Republic: Translated by Paul Shorey (1935) annotated and hyperlinked text (English and Greek) Plato's Analogy of the Divided Line: A read at the Eastern Division Meetings of the American Philosophical Association, December 1988.