Editing Analogy of the divided line (section)

== The intelligible world ==
According to some translations,<ref name="unequal" /> 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'''<!-- Syntax error: parentheses not closed.  -->:<ref>Let the length of '''AE''' be equal to <math>\scriptstyle 1</math> and that of '''AC''' equal to <math>\scriptstyle x</math>, where <math>\scriptstyle 0 < x < 1</math> (following Socrates, however, <math>\scriptstyle 0 < x < 1/2</math>; 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 <math>\scriptstyle 1-x</math>. It follows that the length of '''BC''' must be equal to <math>\scriptstyle x - x \times x \equiv (1-x)\times x</math>, which is seen to be equal to the length of '''CD'''.</ref>

{{quotation|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 (510b).<ref name="Republic" />}}

Plato describes '''CD''', the "lower" of these, as involving '''mathematical reasoning''' (διάνοια ''[[dianoia]]''),<ref name="PenguinNotes" /> where abstract [[Philosophy of mathematics#Platonism|mathematical objects]] such as [[Line (geometry)|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'''):

{{quotation|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 (511b).<ref name="Republic" />}}

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 [[Theory of Forms|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 [[Form of the Good|Idea (Form) of the Good]].<ref name="CDP" />