Analogy of the divided line

Description

In The Republic (509d–510a), Plato describes the divided line to Glaucon 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 (eikasía) 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 term eikasía (), meaning conjecture in Greek, was used by Plato to refer to a human way of dealing with appearances. Particularly, it is identified as the lower subsection of the visible segment and represents images, which Plato described as "first shadows, then reflections in water and in all compacted, smooth, and shiny materials". According to the philosopher, eikasia and pistis add up to doxa, which is concerned with genesis (becoming).

Eikasia has several interpretations. For instance, it is the inability to perceive whether a perception is an image of something else. It therefore prevents us from perceiving that a dream or memory or a reflection in a mirror is not reality as such. Another variation posited by scholars such Yancey Dominick, explains that it is a way of understanding the originals that generate the objects that are considered as eikasia. This allows one to distinguish the image from reality such as the way one can avoid mistaking a reflection of a tree in a puddle for a tree.

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.

Tabular summary of the divided line

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 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.

Notes

References

1. Older Greek texts do not differentiate unequal from an equal (ανίσα, αν ίσα)
2. [[Plato]], ''[[Plato's Republic. The Republic]]'', Book 6, translated by [[Benjamin Jowett]], [http://www.filepedia.org/files/Plato%20-%20The%20Republic.pdf online] {{webarchive. link. (18 April 2009)
3. "divided line," ''The Cambridge Dictionary of Philosophy'', 2nd edition, Cambridge University Press, 1999, {{ISBN. 0-521-63722-8, p. 239.
4. Desmond Lee and [[Rachana Kamtekar]], ''[[Plato's Republic. The Republic]]'', Notes to Book 6, Penguin, 1987, {{ISBN. 0-14-044914-0.
5. "eikasia". FOLDOC.
6. Dorter, Kenneth. (2006). "The Transformation of Plato's Republic". Lexington Books.
7. Crombie, I. M.. (2012). "An Examination of Plato's Doctrines: Plato on Knowledge and Reality, Volume 7". Routledge.
8. Payne, Andrew. (2017-10-13). "The Teleology of Action in Plato's Republic". Oxford University Press.
9. Let the length of '''AE''' be equal to \scriptstyle 1 and that of '''AC''' equal to \scriptstyle x, where \scriptstyle 0 < x < 1 (following Socrates, however, \scriptstyle 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 \scriptstyle 1-x. It follows that the length of '''BC''' must be equal to \scriptstyle x - x \times x \equiv (1-x)\times x, which is seen to be equal to the length of '''CD'''.
10. Gail Fine, ''Knowledge and Belief in Republic V-VII'', in G. Fine (ed.) ''Plato I'' (1990), also in S. Everson (ed.) ''Cambridge Companions to Ancient Thought I: Epistemology'' (Cambridge University Press: New York, 1990), pp. 85–115.
11. Nicholas Denyer, ''Sun and line: the role of the Good'', in G. R. F. Ferrari (ed.) The Cambridge Companion to Plato's Republic (Cambridge University Press: New York, 2007), pp. 284–309.
12. ''Republic'' [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Abook%3D7%3Asection%3D520c 520c]: "For once habituated ... you will know what each of the 'idols' is and whereof it is a semblance, because you have seen the reality of the beautiful, the just and the good."
13. [http://rachelsingpurwalla.files.wordpress.com/2010/01/platos-defense-of-justice.pdf Rachel G.K. Singpurwalla, ''Plato’s Defense of Justice in the Republic''] "justice is a virtue appropriate to both cities and individuals, and the nature or form of justice is the same in both (''Republic'', 435a)."
14. [http://www.the-philosopher.co.uk/lawsofthought.htm James Danaher, ''The Laws of Thought''] "The restrictions Plato places on the laws of thought (i.e., "in the same respect," and "at the same time,") are an attempt to isolate the object of thought by removing it from all other time but the present and all respects but one."
15. ''Cratylus'' [https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Crat.+439d&fromdoc=Perseus%3Atext%3A1999.01.0172 439d-e] "For if it is ''ever'' in the same state, then obviously ''at that time'' it is not changing (Plato's realism); and if it is ''always'' in the same state and is always the same, how can it ever change or move without relinquishing its own form (Aristotle's realism)"
16. ''Republic'' [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Abook%3D7%3Apage%3D520 520c] "For once habituated you will discern them infinitely better than the dwellers there, and you will ''know'' what each of the 'idols' is and whereof it is a semblance, because you have seen the reality of the beautiful, the just and the good."
17. ''Cratylus'' [https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Crat.+439d&fromdoc=Perseus%3Atext%3A1999.01.0172 439d-e] "How, then, can that which is never in the same state be anything? ... nor can it be known by anyone."
18. ''Republic'' [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Abook%3D4%3Asection%3D436b 4.436b] "It is obvious that the same thing will never do or suffer opposites in the same respect in relation to the same thing and at the same time"
19. ''Republic'' [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Abook%3D4%3Asection%3D437a 4.437a] "let us proceed on the ''hypothesis'' that this is so, with the understanding that, if it ever appear otherwise, everything that results from the assumption shall be invalidated"
20. Also see the ''Timaeus'' [https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0180%3Atext%3DTim.%3Asection%3D29c 29c] on empirical "likelihood"
21. It is interesting to note that modern logical analysis claims to prove that the potentially falsifiable "empirical content of a theory is exactly captured by ... axiomatization ... that uses axioms which are universal negations of conjunctions of atomic formulas" [http://www.hss.caltech.edu/~fede/wp/empirical.pdf (C. Chambers, ''The Axiomatic Structure of Empirical Content'')] {{webarchive. link. (27 June 2010)
22. Plato's [[Principle of Non-Contradiction]] (''Republic'', 4.436b) for the objective, physical world is presented with three ''axiomatic'' restrictions: The same thing ... cannot act or be acted upon ... in contrary ways ... (1) in the same part (2) in relation to the same thing (3) at the same time.