Analogy of the divided line
The analogy of the divided line () is presented by the Greek philosopher Plato in the "Republic" (509d–511e). It is written as a dialogue between Glaucon and Socrates, in which the latter further elaborates upon the immediately preceding analogy of the sun at the former's request. Socrates asks Glaucon to not only envision this unequally bisected line but to imagine further bisecting each of the two segments. Socrates explains that the four resulting segments represent four separate 'affections' (παθήματα) of the psyche. The lower two sections are said to represent the visible while the higher  two are said to represent the intelligible. These affections are described in succession as corresponding to increasing levels of reality and truth from conjecture (εἰκασία) to belief () to thought (διάνοια) and finally to understanding (). Furthermore, this analogy not only elaborates a theory of the psyche but also presents metaphysical and epistemological views. Description. In "The Republic" (509d–510a), Plato describes the divided line this way: The visible world. Thus AB represents shadows and reflections of physical things, and BC the physical things themselves. These correspond to two kinds of knowledge, the illusion (εἰκασία "eikasia") of our ordinary, everyday experience, and belief  (πίστις "pistis") about discrete physical objects which cast their shadows. In the "Timaeus", the category of illusion includes all the "opinions of which the minds of ordinary people are full," while the natural sciences are included in the category of belief. The intelligible world. According to some translations, the segment CE, representing the intelligible world, is divided into the same ratio as AC, giving the subdivisions CD and DE (it can be readily verified that CD must have the same length as BC: Plato describes CD, the "lower" of these, as involving mathematical reasoning (διάνοια "dianoia"), 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 understanding (νόησις "noesis"), the "higher" of these two subdivisions (DE): 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 Ideas (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 Ideas – 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 Idea (Form) of the Good. Metaphysical importance. The analogy of the divided line is the cornerstone of Plato's metaphysical framework. This structure illustrates the grand picture of Plato's metaphysics, epistemology, and ethics, all in one. It is not enough for the philosopher to understand the Ideas (Forms ), he must also understand the relation of Ideas to all four levels of the structure to be able to know anything at all. In the "Republic", the philosopher must understand the Idea of Justice to live a just life or to organize and govern a just state. The divided line also serves as our guide for most past and future metaphysics. The lowest level, which represents "the world of becoming and passing away" ("Republic", 508d), is the metaphysical model for a Heraclitean philosophy of constant flux and for Protagorean philosophy of appearance and opinion. The second level, a world of  fixed physical objects, also became Aristotle's metaphysical model. The third level might be a Pythagorean level of mathematics. The fourth level is Plato's ideal Parmenidean reality, the world of highest level Ideas. Epistemological meaning. Plato holds a very strict notion of knowledge. For example, he does not accept expertise about a subject, nor direct perception (see "Theaetetus"), nor true belief about the physical world (the "Meno") as knowledge. It is not enough for the philosopher to understand the Ideas (Forms), he must also understand the relation of Ideas to all four levels of the structure to be able  to know anything at all. For this reason, in most of the earlier Socratic dialogues, Socrates denies knowledge both to himself and others. For the first level, "the world of becoming and passing away," Plato expressly denies the possibility of knowledge. Constant change never stays the same, therefore, properties of objects must refer to different Ideas at different times. Note that for knowledge to be possible, which Plato believed, the other three levels must be unchanging. The third and fourth level, mathematics and Ideas, are already eternal and unchanging. However, to ensure that the second level, the objective, physical world , is also unchanging, Plato, in the "Republic", Book 4 introduces empirically derived axiomatic restrictions that prohibit both motion and shifting perspectives.