Plato's Divided Line
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 nurture. 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 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.
“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].”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.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.
“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.”Later, at 511D-E, Plato summarizes the four sections of the divided line
“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.”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.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.
“Consider then again the way in which we are to make the division of the intelligible section.”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 Μetaphysics. 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 beforelife). 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.