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Friday, December 4, 2015

Plato's Divided Line

Plato's Divided Line

By The Information Philosopher
Friday, December 4, 2015

At the end of Book VI of the Republic (509D-513E), Plato describes
the visible world of perceived physical objects and the images
we make of them (in our minds and in our drawings, for example).

The sun, he said, not only provides the visibility of the objects,
but also generates them and is the source of their growth and

Many primitive religions identify the sun with God,
for good reason.

Beyond this visible world, which later philosophers (esp. Immanuel
Kant) would call the phenomenal world, lies an intelligible world
(that Kant will call noumenal.

The intelligible world is (metaphorically) illuminated by "the
Good" (τον ἀγαθὸν), just as the visible world is illuminated by
the sun.

The division of Plato's Line between Visible and Intelligible is
then a divide between the Material and the Ideal, the foundation
of most Dualisms.

Plato may have coined the word "idea" (ἰδέα), using it somewhat
interchangeably with the Greek word for shape or form (εἶδος ).

The word idea derives from the Greek for, "to have seen."

Plato's Line is also a division between Body and Mind.

The upper half of the divided line is usually called Intelligible as
opposed to Visible, meaning that it is "seen" by the mind (510E),
by the Greek Nous (νοῦς), rather than by the eye.

In most modern Indo-European languages there are two words
that correspond to the English, "to know."

One of these derives from, "to be cognizant of" or "to be acquainted
with" the other from, "to have seen."

The first is the cognate (sic) of English, "know." e.g, Greek gnosis
(γνῶσις), meaning knowledge.

For knowledge the Greeks also used epistέme (ἐπιστήμη), the root
for our word epistemology.

Examples of European words that we also translate "know," but
which derive from "see," are savoir (Fr.) and wissen (Ger.).

In Republic, Book VI, 507C, Plato describes these two classes of
things, those that can be seen but not thought, and those that can
be thought but not seen: καὶ τὰ μὲν δὴ ὁρᾶσθαί φαμεν, νοεῖσθαι δ᾽
οὔ, τὰς δ᾽ αὖ ἰδέας νοεῖσθαι μέν, ὁρᾶσθαι δ᾽ οὔ.

Plato's Divided Line

At Republic, Book VI, 508B-C, Plato makes an analogy between
the role of the sun, whose light gives us our vision to see (ὄψις)
and visible things to be seen (ὁρώμενα) and the role of the Good (τἀγαθὸν).

The sun rules over our vision and the things we see.

The Good rules over our (hypothetical) knowledge and the (real)
objects of our knowledge (the forms, the ideas):

“This, then, you must understand that I meant by the offspring of
the good which the good begot to stand in a proportion with itself:
as the good is in the intelligible region to reason [CD] and the
objects of reason [DE], so is this (sc. the sun) in the visible world
to vision [AB] and the objects of vision [BC].”

φάναι με λέγειν τὸν τοῦ ἀγαθοῦ ἔκγονον, ὃν τἀγαθὸν ἐγέννησεν
ἀνάλογον ἑαυτῷ, ὅτιπερ αὐτὸἐν τῷ νοητῷ τόπῳ πρός τε νοῦν καὶ τὰ
νοούμενα, τοῦτο τοῦτον ἐν τῷ ὁρατῷ πρός τε ὄψιν καὶ τὰ ὁρώμενα.

At 509D-510A, Plato describes the line as divided into two sections
that are not the same (ἄνισα) size.

Most modern versions represent the Intelligible section as larger
than the Visible.

But there are strong reasons to think that for Plato the Intelligible
(being unitary abstraction) is to the Visible (with its many concrete
particulars) as the One is to the Many.

We don't know whether Plato imagined the Intelligible or the Visible
to be the larger section, but it seems clear that he pictures the Intelligible
section above the Visible, so we use a vertical line.

Plato then further divides each of the Intelligible and the Visible
sections into two. He argues that the new divisions are in the same
ratio as the fundamental division.

“Represent them then, as it were, by a line divided into two
unequal sections and cut each section again in the same ratio (the
section, that is, of the visible and that of the intelligible order),
and then as an expression of the ratio of their comparative
clearness and obscurity you will have, as one of the sections of the
visible world, images.

By images I mean, first, shadows, and then reflections in water and
on surfaces of dense, smooth and bright texture, and everything of
that kind, if you apprehend.”

“I do.”

“As the second section assume that of which this is a likeness or an
image, that is, the animals about us and all plants and the whole
class of objects made by man.”

“I so assume it,” he said.

“Would you be willing to say,” said I, “that the division in respect
of reality and truth or the opposite is expressed by the proportion:
as is the opinable (δόξα) to the knowable (γνῶσις) so is the likeness
(ὁμοιωθὲν) to that of which it is a likeness (ὡμοιώθη)?”

ὥσπερ τοίνυν γραμμὴν δίχα τετμημένην λαβὼν ἄνισα τμήματα, πάλιν τέμνε ἑκάτερον τὸ τμῆμα ἀνὰ τὸν αὐτὸν λόγον, τό τε τοῦ ὁρωμένου γένους καὶ τὸ τοῦ νοουμένου, καί σοι ἔσται σαφηνείᾳ καὶ ἀσαφείᾳ πρὸς ἄλληλα ἐν μὲν τῷ ὁρωμένῳ τὸ μὲν ἕτερον τμῆμα εἰκόνες — λέγω δὲ τὰς εἰκόνας πρῶτον μὲν τὰς σκιάς, ἔπειτα τὰ ἐν τοῖς ὕδασι φαντάσματα καὶ ἐν τοῖς ὅσα πυκνά τε καὶ λεῖα καὶ φανὰ συνέστηκεν, καὶ πᾶν τὸ τοιοῦτον, εἰ κατανοεῖς.

ἀλλὰ κατανοῶ.

τὸ τοίνυν ἕτερον τίθει ᾧ τοῦτο ἔοικεν, τά τε περὶ ἡμᾶς ζῷς καὶ πᾶν τὸ φυτευτὸν καὶ τὸ σκευαστὸν ὅλον γένος. τίθημι, ἔφη. ἦ καὶ ἐθέλοις ἂν αὐτὸ φάναι, ἦν δ᾽ ἐγώ, διῃρῆσθαι ἀληθείᾳ τε καὶ μή, ὡς τὸ δοξαστὸν πρὸς τὸ γνωστόν, οὕτω τὸ ὁμοιωθὲν πρὸς τὸ ᾧ ὡμοιώθη;

Later, at 511D-E, Plato summarizes the four sections of the divided

“Your interpretation is quite sufficient,” I said; “and now,
answering to these four sections, assume these four affections
occurring in the soul: intellection (νόησιν) or reason for the highest
[DE], understanding (διάνοια) for the second [CD]; assign belief
(πίστις) to the third [BC], and to the last picture-thinking or
conjecture (εἰκασία) [AB], and arrange them in a proportion,
considering that they participate in clearness and precision in the
same degree as their objects partake of truth and reality.”

ἱκανώτατα, ἦν δ᾽ ἐγώ, ἀπεδέξω. καί μοι ἐπὶ τοῖς τέτταρσι τμήμασι τέτταρα ταῦτα παθήματα ἐν τῇ ψυχῇ γιγνόμενα λαβέ, νόησιν μὲν ἐπὶ τῷ ἀνωτάτω, διάνοιανδὲ ἐπὶ τῷ δευτέρῳ, τῷ τρίτῳ δὲ πίστιν ἀπόδος καὶ τῷ τελευταίῳ εἰκασίαν, καὶ τάξον αὐτὰ ἀνὰ λόγον, ὥσπερ ἐφ᾽ οἷς ἐστιν ἀληθείας μετέχει, οὕτω ταῦτα σαφηνείας ἡγησάμενος μετέχειν.

We can collect the various terms that Plato has used to describe
the components of his divided line. Some terms are ontological,
describing the contents of the four sections. Some are
epistemological, describing how it is we know those contents.

509D-510A likenesses, images, shadows, imitations, our vision
(ὄψις, ὁμοιωθὲν) the physical things that we see/perceive with
our senses (ὁρώμενα, ὁμοιωθὲν) opinion, beliefs (δόξα, νοῦν)
knowledge (γνῶσις, νοούμενα)

511D-E conjectures, images, (εἰκασία) trust, confidence, belief
(πίστις) understanding, hypothesis (διανόια) intellection, the
objects of reason (νόησις, ἰδέαι, ἐπιστήμην)

By insisting that the ratio or proportion of the division of the
visibles (AB:BC) and the division of the intelligibles (CD:DE)
are in the same ratio or proportion as the visibles to the
intelligibles (AC:CE), Plato has made BC = CD.

And indeed Plato at one point identifies the contents of these
two sections.

He says (510B) that in CD the soul is compelled to investigate by
treating as images the things imitated in the former division (BC).

“Consider then again the way in which we are to make the division
of the intelligible section.”

“In what way?”

“By the distinction that there is one section of it [CD] which the
soul is compelled to investigate by treating as images the things
imitated in the former division [BC], and by means of assumptions
from which it proceeds not up to a first principle but down to
a conclusion, while there is another section in which it advances
from its assumption to a beginning or principle that transcends
assumption (ἀρχὴν ἀνυπόθετον), and in which it makes no use of
the images employed by the other section, relying on ideas only
and progressing systematically through ideas.”

σκόπει δὴ αὖ καὶ τὴν τοῦ νοητοῦ τομὴν ᾗ τμητέον.


ἧι τὸ μὲν αὐτοῦ τοῖς τότε μιμηθεῖσιν ὡς εἰκόσιν χρωμένη ψυχὴ ζητεῖν ἀναγκάζεται ἐξ ὑποθέσεων, οὐκ ἐπ᾽ ἀρχὴν πορευομένη ἀλλ᾽ ἐπὶ τελευτήν, τὸ δ᾽ αὖ ἕτερον—τὸ ἐπ᾽ ἀρχὴν ἀνυπόθετον—ἐξ ὑποθέσεως ἰοῦσα καὶ ἄνευ τῶν περὶ ἐκεῖνο εἰκόνων, αὐτοῖς εἴδεσι δι᾽ αὐτῶν τὴν μέθοδον ποιουμένη.

Plato distinguishes two methods here.

The first (the method of the mathematician or scientist) starts
with assumptions or hypotheses (ὑποθέσεων) - Aristotle called
them axioms - and proceeds to a conclusion (τελευτήν) which
remains dependent on the hypotheses or axioms.

The second (the dialectician or philosopher) advances from
assumptions to a beginning or first principle (ἀρχὴν) that
transcends the hypotheses (ἀνυπόθετον), relying on ideas
only and progressing systematically through ideas.

The question whether and in what way such first principles exist
is the subject of "first philosophy" that Aristotle's editors called

Ιn the traditional order of Aristotle's works, these books came
after the Physics and were so titled - ΤΟΝ ΜΕΤΑ ΤΑ ΦΥΣΙΚΑ.

The Neoplatonist philosopher Porphyry is said to have asked the
"fateful question" of whether Plato's ideas (forms or essences)
really exist.

Aristotle unequivocally thought that the Platonic ideas were
simply abstractions from concrete existents.

In the twentieth-century jargon of Existentialism, "existence
precedes essence."

Platonism (sometimes called Platonic Realism or Platonic Idealism)
is the view that the ideas exist, but since they are not physical,
their existence is outside space and time.

In the nineteenth and early twentieth centuries a view called
psychologism (or conceptualism) was developed, arguing that
the ideas exist as concepts only in human minds.

Anti-Realism (also called Nominalism) denies the separate abstract
existence of the ideas, claiming that the number 3, for example,
exists only in all the concrete sets of three things.

The abstract concept of redness exists in the set of all red things.

This was Aristotle's view.

Plato claims that the dialectical method is more holistic and
capable of reaching a higher form of knowledge, possibly related
to his notion that the eternal soul has "seen" all these truths in past
lives (incarnations) and before (we might call it the before life).

Aristotle's divided line!

Plato's idea that the Good (τον ἀγαθὸν) is the creator (δεμιύργος)
of all the ideas (the forms of the Good) resembles our information
philosophy view that ergodic processes are the source of everything
of positive value (our Ergo).

Plato does not identify the Good with material things or even with
the abstract ideas and forms. The Good is the creative process that
generates the forms.

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