Analogy of the divided line
Encyclopedia
Plato
, in his dialogue The Republic Book 6 (509D–513E), has Socrates
explain through the literary device of a divided line his fundamental metaphysical
ideas as four separate but logically connected model
s of the world. The four models are arranged into a first pair for the visible
world, and a second pair for the purely intelligible world. The models are described in succession as corresponding to increasing levels of reality from common illusion, to belief, to reasoning, and then to philosophical understanding
.
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
.
In The Republic (509d-510a), Plato
describes the Divided Line this way:
, the illusion (εἰκασία eikasia
) of our ordinarily, 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.
Plato describes BC, 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 CD). However, they are less important to Plato than the subjects of philosophical understanding (νόυσις noesis), the "higher" of these two subdivisions (AB):
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.
The following table summarizes the analogy of the divided line:
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. The second level, the world of fixed physical objects, became Aristotle's model. The third level might be a Pythagorean level of mathematics. The fourth level is Plato's own reality, the world of highest level Ideas.
,482d), nor direct perception (the Theatetus
), nor true belief about the physical world (the Meno
) as knowledge. For knowledge, he also requires philosophical understanding of the relevant Ideas (Forms), as a basis for proper justification at all other levels of the Divided Line. Thus, for this reason, in most of the "early, 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 objective, physical world is also unchanging, Plato, in the Republic, Book 4 introduces axiomatic
restrictions that prohibit both motion and shifting perspectives.
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
, in his dialogue The Republic Book 6 (509D–513E), has Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
explain through the literary device of a divided line his fundamental metaphysical
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
ideas as four separate but logically connected model
Conceptual model
In the most general sense, a model is anything used in any way to represent anything else. Some models are physical objects, for instance, a toy model which may be assembled, and may even be made to work like the object it represents. They are used to help us know and understand the subject matter...
s of the world. The four models are arranged into a first pair for the visible
Sense
Senses are physiological capacities of organisms that provide inputs for perception. The senses and their operation, classification, and theory are overlapping topics studied by a variety of fields, most notably neuroscience, cognitive psychology , and philosophy of perception...
world, and a second pair for the purely intelligible world. The models are described in succession as corresponding to increasing levels of reality from common illusion, to belief, to reasoning, and then to philosophical understanding
Platonic epistemology
Platonic epistemology holds that knowledge is innate, so that learning is the development of ideas buried deep in the soul, often under the midwife-like guidance of an interrogator. In several dialogues by Plato, Socrates presents the view that each soul existed before birth with "The Form of the...
.
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
Allegory of the cave
The Allegory of the Cave—also known as the Analogy of the Cave, Plato's Cave, or the Parable of the Cave—is an allegory used by the Greek philosopher Plato in his work The Republic to illustrate "our nature in its education and want of education"...
.
Imagine a line divided
In The Republic (509d-510a), Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
describes the Divided Line this way:
The visible world
Thus DE represents shadows and reflections of physical things, and CD the physical things themselves. These correspond to two kinds of knowledgePlatonic epistemology
Platonic epistemology holds that knowledge is innate, so that learning is the development of ideas buried deep in the soul, often under the midwife-like guidance of an interrogator. In several dialogues by Plato, Socrates presents the view that each soul existed before birth with "The Form of the...
, the illusion (εἰκασία eikasia
Eikasia
Eikasía was used by Plato to refer to a human way of dealing with appearances.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....
) of our ordinarily, everyday experience, and belief
Belief
Belief is the psychological state in which an individual holds a proposition or premise to be true.-Belief, knowledge and epistemology:The terms belief and knowledge are used differently in philosophy....
(πίστις pistis) about discrete physical objects which cast their shadows. In the Timaeus
Timaeus (dialogue)
Timaeus is one of Plato's dialogues, mostly in the form of a long monologue given by the title character, written circa 360 BC. The work puts forward speculation on the nature of the physical world and human beings. It is followed by the dialogue Critias.Speakers of the dialogue are Socrates,...
, 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 AC, representing the intelligible world, is divided into the same ratio as CE, giving the subdivisions BC and CD (it can be readily verified that CD must have the same length as BC):Plato describes BC, the "lower" of these, as involving mathematical reasoning (διάνοια dianoia
Dianoia
Dianoia is a term used by Plato for a type of thinking, specifically about mathematical and technical subjects. It is the capacity for, process of, or result of discursive thinking, in contrast with the immediate apprehension that is characteristic of noesis...
), where abstract mathematical objects such as geometric lines
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
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 CD). However, they are less important to Plato than the subjects of philosophical understanding (νόυσις noesis), the "higher" of these two subdivisions (AB):
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)
Theory of Forms
Plato's theory of Forms or theory of Ideas asserts that non-material abstract forms , and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form is often capitalized...
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.
The following table summarizes the analogy of the divided line:
| Segment | Type of knowledge | Type of object |
|---|---|---|
| AB (highest) | Philosophical understanding (noesis) | Ideas (Forms) Theory of Forms Plato's theory of Forms or theory of Ideas asserts that non-material abstract forms , and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form is often capitalized... , especially the Idea (Form) of the Good |
| BC | Mathematical reasoning (dianoia), including theoretical science | Abstract mathematical objects, such as numbers Natural number In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively... and lines Line (geometry) The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects... |
| CD | Beliefs about physical things (pistis), including empirical science | Physical objects |
| DE (lowest) | Opinions, illusions (eikasia) | "Shadows" and "reflections" of physical objects |
Metaphysical importance
The allegory of the Divided Line is the cornerstone of Plato's metaphysical framework. This structure, well hidden in the middle of the Republic, a complex, multi-layered dialogue, 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. The second level, the world of fixed physical objects, became Aristotle's model. The third level might be a Pythagorean level of mathematics. The fourth level is Plato's own 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 (the GorgiasGorgias (dialogue)
Gorgias is a Socratic dialogue written by Plato around 380 BC. In this dialogue, Socrates seeks the true definition of rhetoric, attempting to pinpoint the essence of rhetoric and unveil the flaws of the sophistic oratory popular in Athens at this time...
,482d), nor direct perception (the Theatetus
Theaetetus (dialogue)
The Theaetetus is one of Plato's dialogues concerning the nature of knowledge. The framing of the dialogue begins when Euclides tells his friend Terpsion that he had written a book many years ago based on what Socrates had told him of a conversation he'd had with Theaetetus when Theaetetus was...
), nor true belief about the physical world (the Meno
Meno
Meno is a Socratic dialogue written by Plato. It attempts to determine the definition of virtue, or arete, meaning virtue in general, rather than particular virtues, such as justice or temperance. The first part of the work is written in the Socratic dialectical style and Meno is reduced to...
) as knowledge. For knowledge, he also requires philosophical understanding of the relevant Ideas (Forms), as a basis for proper justification at all other levels of the Divided Line. Thus, for this reason, in most of the "early, 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 objective, physical world is also unchanging, Plato, in the Republic, Book 4 introduces axiomatic
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
restrictions that prohibit both motion and shifting perspectives.
External links
- At MIT.edu: Plato's Republic: Translated by Benjamin JowettBenjamin JowettBenjamin Jowett was renowned as an influential tutor and administrative reformer in the University of Oxford, a theologian and translator of Plato. He was Master of Balliol College, Oxford.-Early career:...
- At Perseus ProjectPerseus ProjectThe Perseus Project is a digital library project of Tufts University that assembles digital collections of humanities resources. It is hosted by the Department of Classics. It has suffered at times from computer hardware problems, and its resources are occasionally unavailable...
: Plato's Republic: Translated by Paul ShoreyPaul ShoreyPaul Shorey, Ph.D., LL.D., Litt.D. was an American classical scholar, born at Davenport, Iowa After graduating from Harvard in 1878 he studied in Europe at Leipzig, Bonn, Athens, and Munich . He was a professor at several institutions from 1885 onward...
(1935) annotated and hyperlinked text (English and Greek) - James Danaher, "The Laws of Thought", The Philosopher, Volume LXXXXII No. 1
- Plato's Analogy of the Divided Line: A read at the Eastern Division Meetings of the American Philosophical Association, December 1988.