(4th tetralogy : The Soul - 2nd dialogue of trilogy)

Foreword (August 7, 2026): This page is now completed and includes the translation / adaptation, not only of the introduction, but also of all the notes of the French version. The translation of the notes has led to a few changes to the translation of the analogy itself. Some of the notes link to notes in other pages of the site (Allegory of the cave, Parallel between sun and good..., whose notes have not been translated yet. These links are there in anticipation and will work properly only when these other pages will have been fully translated into English, notes included.
(April 8, 2026): At this point in time, this page, aside from the translation from the Greek of the analogy, includes only a translation in English of the Introduction, which is common to both the analogy of the line and the allegory of the cave, an shows how both "images" should be read in parallel, each one shedding light on the other. The translation of the notes available in the French version is in progress and will come later.

Another page of this site, titled "The Vocabulary of the Line", that can be accessed by clicking on this link, provides a table listing all the occurrences of a number of important words of this analogy, and also of Republic VII, 533e7-534a8, which recalls the analogy, along with their translation in English in the various editions of the Repuplic I had access to. This table, along with the commentaries that accompany it, including the list of the 9 English words used by the 7 quoted translators to translate the 7 occurrences of the Greek word eidos in the analogy of the line, gives an idea of how difficult it is to translate such a text without betraying it, showing that the same Greek word may be translated as different English words, not only from a translator to another, but also by the same translator within a few lines' interval, and that the same English word may be used to translate several Greek words, here again sometimes by the same translator.

(direct access to translation)

Introduction and setting

This section of the Republic is one of the most difficult to properly understand, insofar as it describes in a static way through a geometric analogy the four increasingly rich and faithful modes of apprehension of reality, first through sight (the "seen", the first two modes), then through intelligence (the "perceived by intelligence", the last two modes), which are accessible to human beings (anthrôpoi) because of their nature and that, obviously, to understand it completely, one must have reached the fourth mode, the richest, which is not given to everyone. In doing so, it tries to make us understand the power, but also the limits, of logos*, which specifies us as human beings, by means of logoi, that is, to free ourselves from the grip of words by means of words, using images for this purpose. It is complemented by another image, the allegory of the cave, which follows it and sheds light on it by illustrating the progression over time through these different modes of apprehension for each human being through "education" (paideia, 514a2) in a broad sense not limited to "scholastic" education, but considered as potentially continuing throughout life (as Socrates examplifies throughout the dialogues). But the allegory in turn must be understood against the backdrop of the analogy of the line, since it presents in a dynamic perspective the progression through the different modes of apprehension of reality that the analogy of the line presents to us in a static perspective, up to the ultimate mode of apprehension, culminating, in the allegory, in the contemplation at leisure of the sun itself, which everyone, starting with Socrates himself, knows to be almost impossible in this life without blinding oneself, a mode of apprehension that is problematic because it is beyond words. And these two images must be understood in the light of what immediately preceded them, the parallel between the good and the sun, which makes the idea of the good (hè tou agathou idea) the analogue for intelligence of the sun for sight, and in a more distant way, the discussion on knowledge and opinion that opened the long discussion that responds to what Socrates called the "third wave" of objections (cf. 472a1-7), the one that the principle of the "philosopher king" that he stated at 473c11-e2 cannot fail to arouse, and also from what follows, with the reminder of these two images towards the end of Book VII in reverse order in the discussion on dialektikè as the ultimate knowledge that the philosopher must master, which is not a mere summary but offers important clarifications, and finally the discussion on the different kinds of beds at the beginning of Book X, which sheds light on the meaning of the terms eidos and idea and the differences Plato makes between both.
So rather than artificially maintaining in my notes and comments a suspense of no great value, I prefer to take into account in the notes both what preceded and what follows the analogy to avoid the reader getting lost in false leads which will complicate one's understanding of this text, all the more so since there are many false leads in the secondary literature on this analogy, which most commentators have misunderstood, among other things because they did not try to reconcile the second part of this analogy, the one dealing with the perceived by intelligence, with the detailed description of what happens outside the cave in the allegory that follows, preferring to read these two images in the light of a supposed "theory of forms / ideas" accepted a prioiri even if it does not fit with what they suggest, and by giving too much importance to the geometric dressing of the analogy.

 * For the multiplicity of meanings of the Greek word logos (logoi in the plural), which I prefer to leave untranslated, please refer to the page of this site where I have reproduced the entry in the LSJ Greek-English Lexicon dedicated to it. Let it be said in short that the range of meaning of this word is organized around two main registers: on the one hand, everything that has to do with human speech and the various uses that can be made of it (speech, conversation, discourse, report, narrative, etc.), and on the other hand, everything that revolves around the notion of reason, the faculty to which speech gives us access and what it allows us to do (reason, reasoning, justification, relationship, proportion, etc.). The link between the two registers of meaning is made through the idea that logos designates speech as bearing meaning, that is to say, as reasonable speech. Plato's dialogues as a whole can be seen as a long exploration of what it means and implies to be an anthrôpos ("human being"), i.e. an animal endowed with logos, in the line of the gnôthi sauton ("learn to know thyself", not only as an individual, but as a member of the species anthrôpos ("human being")).
In the translation of the analogy, I do not translate the word logos, only specifying in a note, on occasion, in what more specific sense it is used in this instance, especially when it participates in ready-made expressions. In the introduction and notes, I use the Greek word, as here, without specifying the relevant translation(s) in each case.(<==)

Summary of the analogy of the line

To make what following understandable to someone who has not yet read the analogy of the line, it is therefore necessary to give a brief summary of it (those who prefer to read the text to get a first feel of it by themselves before reading this commentary can go directly to the translation and come back here later, keeping in mind that any translation, mine included, is necessarily biased by the translator's choices). This analogy, as I just said, immediately follows the parallel between the good and the sun, and uses another imagery to complement it. Socrates, in the continuity of what was said before, begins by equating the good and the sun with two sovereigns reigning (basileuein, 509d2) each on what he calls by various names so as not to freeze a "technical" vocabulary to speak with necessarily unsuitable words of "ensembles" which have at least as many differences as resemblances and to which one would not spontaneously associate the same words: he begins by speaking simultaneously of genos ("birth, origin, family, nation, people, genus, species...") and topos ("place, position, region, district... "), which he qualifies respectively as "intelligible" (noèton), i.e. open to understanding by the intellect (noûs), for what the good "reigns" over, and "visible" (horaton) for what the sun "reigns" over, before introducing the term eidos ("appearance (for sight), form, shape, figure, kind, sort, class, species", a word derived from a root meaning "to see"), also associated with the same qualifiers, "visible" (horaton) and "intelligible" (noèton), but repeated in reverse order.
Socrates then asked Glaucon to imagine a line segmented into two unequal segments, which he described in 509d8 as "that of the seen family and that of the though / perceived by intelligence one" (to te tou horômenou genous kai to tou nooumenou), returning to the word genos ("birth, origin, family, nation, people, genus, species...") but this time using new qualifiers, horômenon ("seen") and nooumenon ("perceived by the noûs" ("mind / intelligence"). Then he asks him to segment each of these two segments into two subsegments "according to the same logos" (ana ton auton logon), a key expression to which I will return later in this presentation and in which, because of this geometric dressing which implies the lengths of segments, logon is naturally understood as meaning "(numerical) ratio", one of the possible meanings of this word rich in multiple meanings in Greek. He then describes the division of the segment of the seen, saying that it is done "based on the [relative] clearness (saphèneia) and lack of clearness (asaphèia) of the ones compared to the others, contrasting images (eikones), of which he gives as examples shadows and reflections, that is to say, moving images that occur spontaneously, without the intervention of man, unlike paintings or sculptures, which are otherwise objects in their own right, with the "originals" of these images, that is to say the living beings, the plants and everything that is produced by human craftsmanship (including statues and painted images), designated by the expression to skeuaston holon genos. By way of transition between the description of the division of the segment of the seen and the description of the division of the segment of the perceived by intelligence, he suggests that "it is divided with regard to truth (alètheia) and its absence [according to the following ratio(nale)]: as the opined [is] to the known, so what is made similar (to homoiôthèn) [to something is] to what it has been made similar to (to hôi hômoiôthè)". Socrates then comes to the division of the segment of the perceived by intelligence made "according to the same logos" and, while, in the realm of the seen, he characterized each of the two sub-segments by the "objects" that are perceived by sight, in the realm of the perceived by intelligence, he characterizes each of the two sub-segments by approaches (methodoi, 510b8) that are used there. He thus opposes:
- an approach that could be described as "problem-solving oriented" in that it is the one used by people who seek to solve specific problems, practical or theoretical, starting from "supports" (hupotheseis, a word that should not be understood in the modern sense of "hypotheses", but in the etymological sense of "what is laid under", that is, "basis, foundation, support (of reasoning), starting point") chosen according to the problem to be solved (the examples he gives when he illustrates it with the example of geometers are the even, the odd, angles, figures) and admitted without asking questions about them, designated by words whose meaning is supposed to be obvious, and using as "images" of what they are talking about with these words "objects" (for example, drawings of geometric figures) that are part of what populates the second sub-segment of the seen and that have been discussed before, even though they admit that what they reason upon are "things" "that cannot be seen otherwise than by thought (dianoia)" (511a1), and satisfied as soon as, the problem being delt with having been solved, they have reached the end (teleutè, 510b6) of their specific reasoning and found the solution they were looking for;
- to an approach, which could be described as "comprehensive" (both in the sense of "showing extensive understanding" and in that of "encompassing everything") in that it is the one used by people who, before rushing in search of solutions, question the "supports" (hupotheseis) which they might have to rely upon to do so, one by one, in order to test their "solidity / stability", that is to say, they seek to understand what is hidden behind the words they use, trying to go back to a unifying principle (archè) that "enlightens" everything and gives meaning and coherence to all these "supports" without being itself the "support" for something else (anupotheton, 510b7) of greater "value" (which is none other than the idea of the good, the good being that which everyone seeks for itself and not for the sake of something else (cf. 505d5-506a2), even if Socrates leaves it to us to understand it), no longer relying on "images" (eikones) (visible "things"), but on what Socrates calls eidè, and which designates for him what we suppose to be common to everything to which we attribute the same name, as he will specify in the prelude to the discussion on the different kinds of seats( / beds) at the beginning of Book X, at 596a6-7, and which refers to what is described in the first approach as what "cannot be seen otherwise than by thought (dianoia)" but is apprehended through the "images" that visible / material realities are supposed to give of them, which makes them horômena eidè (horômenois eidesi, "seen appearances", 510d5), but which must now become a noèton ("intelligible", in the sense of "percetible by intelligence") eidos and must be apprehended by abstracting from all the "images" (eikones) that one would be tempted to associate with them and above all as distinct both from the words that are associated with them and from what they are only "appearances" of for a human mind.
In both cases, the tool used is the logos, but only the second approach fully implements the power of the dialegesthai (hè tou dialegesthai dunamis, 511b4), that is to say, assumes that we master the power and limits of logos by having understood how it can give us access to something other than the words that compose it, provided that one relies on the sharing of experiences in the practice of dialogue (to dialegesthai), that is to say, by having become dialektikos by mastering dialektikè ("dialectic", but in a sense that has yet to be specified), which Socrates will describe as the ultimate knowledge that the philosopher must possess and in which we must see, not a particular technique, but the ability to properly use logos (which, as the allegory of the cave will show, is available at all stages of the progression through the segments since this is what distinguishes the anthrôpoi from other animals), to access what is beyond words without letting oneself be trapped by them and of which the eidè that we associate with these words are still only "reflections". A good example of the conflict between these two approaches, in a non-geometric register (but where geometry also plays an exemplary role that resonates with what Socrates is saying here), is given by the Meno, who opposes an interlocutor who wants us to answer his question (can excellence (aretè) for a human being be taught?) to a Socrates who refuses to answer it until he and Meno have agreed on what they mean by "excellence" (aretè) in the case of human beings.
At the end of the analogy, Socrates asks Glaucon to associate each of the four segments with an "affection / state of mind" (pathèma, a noun derived from the verb paschein ("to suffer / have sometning done to oneself / be affected by"), which is opposed to pragma ("fact / thing"), derived from prattein ("to do / act / accomplish"), as paschein is opposed to prattein), proposing the following names for them, which should not be considered as perfectly adequate, exclusive and definitive insofar as Socrates introduces a new analysis for which he does not have pre-existing names, names which would anyhow not convey the full import of what he has in mind, and the names he proposes highlight only an aspect of what he is talking about without exhausting all its richness, as shown by the fact that, when he returns to the analogy in the discussion on dialektikè towards the end of Book VII (533e7-534a3), he changes one of these names, and not the least, since it is that of the pathèma ("affection / state of mind") associated with the second sub-segment of the perceived by intelligence, the one corresponding to the final stage of the progression toward knowledge: to the first sub-segment of the seen, that of "images", he associates eikasia, a word having the same root as eikôn, the word he used to speak of images; to the second sub-segment of the seen, that of material realities, he associates pistis, a word that may be translated as "trust, faith" and which puts forward the idea that at this point, we have become aware of the fact that our knowledge of the world around us is based more on the trust we have in the people around us and their words about things that we have never seen for ourselves and perhaps never will be able to see (including everything that belongs to the past before our birth, such as Achilles and Ulysses) than on direct view with our own eyes of these things, what we see for ourselves representing only a very small part of what we admit as "existing"; to the first sub-segment of the perceived by intelligence, he associates dianoia, a word derived from noûs ("mind / intelligence") via the verb dianoeisthai ("to think"), which can therefore be translated as "thought", or, to be precise, as "discursive thought" to highlight the fact that it manifests itself through discourse, speech, logoi, or else by "wandering thought" by privileging one of the possible meanings of the prefix dia, which may mean in composition "here and there", to emphasize the fact that it is a thought that does not have the compass that the idea of the good constitutes, precisely because it has not taken the time to examine the hupotheseis ("supports") that it uses in the light of this principle; to the second sub-segment of the perceived by intelligence, he associates noèsis, a word having a meaning close to that of dianoia but from which the prefix dia- ("through / to the end", or, as we have just seen, "here and there") is absent), which is replaced by epistèmè ("knowledge") in the recalling of the analogy toward the end of Book VII. And he asks Glaucon to order them "being led by this logos (ana logon) that as the ***s upon which it is (eph' hois estin) partake of truth (aletheia), so these partake of clearness (saphèneia)", suggesting that the degree of "clearness / evidence" (saphèneia) of these different affections / states of mind depends on the level of participation in the truth (aletheia) of that upon which it is (eph' hois estin), an expression that opens the door to all sorts of misunderstandings, which must be unnderstood in the light of the previous discussion on knowledge and opinion at the end of Book V (provided it too has properly been understood) and to which I will return in the notes. Let us say here only for the sake of brevity that "the *** upon which it is" of sight is not the object we see, which could then just as well be "the *** upon which it is" of hearing if it produces sounds, "the *** upon which it is" of smell if it produces odors, or "the *** upon which it is" of opinion if we form an opinion about it, but that which is specifically produced in our mind by sight, and which the mind interprets as patches of colour to which it will give one meaning or another, and that "the ***s upon which it is" of opinion is not the "object" or event about which one forms an opinion, which might as well be an object of knowledge for another, or for the same person at another time in one's life, but that which is formed in the mind about it and expresses by means of thought or uttered words the apprehension of it by the one whose opinion it is, a formulation which, in the case of opinion, is only a sequence of words which the one who thinks or says them is not able to justify.

Summary of the allegory of the cave

After this summary of the analogy of the line, let us proceed with a summary of the allegory of the cave, which, as Socrates tells us in its opening, illustrates "our nature regarding education (paideia) and the fact of not being educated (apaideusia)". He depicts human beings (anthrôpoi) who have been chained in a cave since birth, caught in bonds that prevent them from any kind of movement, including simply moving their heads, so that they can only see the wall of the cave in front of them. Behind them, and therefore invisible to them, there is a road, hidden by a wall, on which human beings (anthrôpoi) walk, bearing at arm's length all kinds of implements as well as statues of men (andriantai, a word formed on the root anèr, andros which means "man" as opposed to "woman", and therefore implies the difference between sexes, unlike anthrôpos) and other animals, which are the only things that protrude from the wall (the anthrôpoi that carry them are completely hidden by the wall). Farther away behind the wall, the road and the bearers, and high up, there is a fire whose light casts on the wall of the cave facing the prisoners the shadows of the utensils and statues protruding from the wall. These shadows (one of the examples of images used by Socrates in the analogy of the line about the first subsegment of the seen) are therefore the only things they can see as long as they remain caught in the bonds that prevent them from moving. But, if they cannot move, they can talk with one another (dialegesthai) and, in order to do this, must give names to the shadows they see by grouping them into families on the basis of similarities of appearance and by assigning these names to the families thus identified (the allegory merely says that, "the same [things] being around [again], they would take the habit of giving names to those [things] they see", 515b4-5), and they hear the echo (sound reflection, reflections being the other example of images used by Socrates in the analogy of the line) of the words spoken by the bearers behind the wall when they happen to produce sounds (phtheggesthai), which makes them believe that these sounds come from the shadows (let us not forget that they can neither move, nor turn their heads and therefore see each other as long as they remain in bonds). In such a situation, when dialoguing with one another, they "would hold as the true (to alethes) nothing but the shadows of the implements (515c1-2). Socrates then imagines that one of these prisoners is freed "naturally" (phusei) from one's bonds and forced to turn the head towards the fire, the wall and the objects that protrude from it and of which so far, he has only seen the shadows. At this point, he imagines a dialogue between the prisoner freed from his bonds and still dazzled by the light of the fire and therefore having difficulty distinguishing the objects of which he could only see the shadows until now and an interlocutor about whom Socrates says nothing but who is obviously not another prisoner immobilized in bonds, in which, when asked about what he sees now as compared to what he was seeing before, the prisoner who was released from his bonds shortly before (but who is still in the cave), "would deem the [things] he formerly used to see (the shadows) truer (alethestera) than those now pointed at (the objects protruding from the wall)" (515d6-7) and turns back toward what he used to see, considering it "clearer" (saphestera) than what he is now forced to look at. Later, he is pulled out of the cave through a side opening (not behind the fire) toward the outside, without having to walk toward the fire and pass the wall (Socrates does not specify this explicitly, but his silence on this point is a "proof" of it: considering the time he spent describing the turning around and reactions of the prisoner freed from his bonds when looking towards the fire, if he had had to go to the other side of the wall and discover the porters, he would not have failed to describe the prisoner's reactions to this new stage of progression.) Once out, the prisonner is at first dazzled by the light of the sun and therefore "unable to see a single one of the [things] now called true" (tôn nun legomenôn alèthôn, 516a2-3) and a period of habituation is required to become able to see what is outside in the bright light of the sun. In the course of this habituation, the freed prisoner at first sees only the shadows and reflections (again the two examples of images used by Socrates in the analogy of the line) of human beings (anthrôpoi) and everything else (of what was visible in the cave) before being able to see them "themselves" (auta) and finally to discover the heavens and the stars, including the moon, first at night, and of course the sun, first through its reflections (another reference to the images of the analogy of the line), then by looking directly at it. Socrates then describes the reasoning (sullogizein) that the prisoner who has finally been able to see the sun at leisure (at least this is what the allegory suggests) holds to oneself: "he would by this time conclude by way of reasoning (sullogizoito) about it that it is the one providing the seasons and the years and supervising all the [things] in the seen place and, of those [things] they themselves used to see, responsible (aitios) in some way of all [of them]" (516b9-c2). Then, he remembers his former life as a prisoner in the cave and the way in which those who live there form opinions (doxazein) and "the privileges for the one most sharply observing the [things] passing by and best at remembering which ones among them used to be carried before or after or simultaneously, and as a result of this, most capable indeed of foretelling what would come" (516c9-d2) and he pities his former comrades in captivity, preferring everything rather than living such a life based exclusively on opinion. And yet, he has to go back and the allegory ends with the return of the freed prisoner to the cave, where, at first, he has difficulty seeing the shadows (a symmetrical vision problem of the one caused by the passage from darkness to light, caused this time by the passage from light to darkness), and where he is the object of ridicule by the other prisoners who have remained in their bonds when he pretends to share his discoveries with them when he is no longer even able to see clearly what, for them, is the only "reality", so much so that, tired of his dissenting speeches, they would no doubt end up killing him if given the opportunity (a transparent allusion to the trial and death sentence of Socrates).

The reminder

In the discussion on dialektikè as the ultimate skill that the philosopher must master, towards the end of Book VII, Socrates returns to the allegory of the cave and the analogy of the line, recalling them in reverse order, and he uses this reminder to add important information along the way. Concerning the allegory of the cave, he twice clarifies what was meant by the expression "the shadows and after that the images on waters of men and those of the other [***]" (tas skias... kai meta touto en tois hudasi ta te tôn anthrôpôn kai ta tôn allôn eidôla) used at 516a6-7 to describe what the prisoner who has just exited the cave can see: first in 532a1-5, when he speaks of "the song / law (nomos) itself that the dialegesthai executes / fulfills, the one which, despite its being intelligible (noèton), the power of sight would mimic, which we described as attempting to look first toward the living [creatures] themselves (auta ta zôia), then toward the stars themselves (auta) and in the very end toward the sun itself (auton)", where it is obvious that the reference to sight which would imitate a "song / law" in the intelligible realm refers to the allegory where sight is indeed meant to represents analogically intellectual activity outside the cave, which means that what is evoked here is the aim of the freed prisoner's progress out of the cave, the "*** themselves" (ta auta) visible only outside the cave, which now includes, before he turns toward the stars and the sun, "the living [creatures]" (ta zôia), and not only the anthrôpoi; then a few lines later, when he evokes the previous stages of this progress and in particular the exiting from the cave, speaking of "the ascent out of the subterranean [place] toward the sun, and there, regarding the living [creatures] and plants (ta zôia te kai phuta) and the light of the sun, inability to yet look [at them]" (532b7-c1). As can be seen, each new mention is an opportunity to expand the whole of what he has in mind, which confirms that it is ultimately everything that is visible in the cave that also has an intelligible dimension outside the cave. And the discussion about tables and beds at the beginning of Book X will show that this includes even the artefacts produced by human activity, the skeuè of 514c1. Concerning the analogy of the line, he begins by providing some clarifications on what he means by dianoia, the affection he associated with the first sub-segment of the perceived by intelligence when he speaks of the subject-matters that are parts of the training program of the future philosopher kings that he has just described (arithmetic, plane and solid geometry (he calls the former geomerty and the later stereometry), astronomy, harmony), describing them as "the arts (technai) we have been reviewing, which we often called 'sciences' (epistèmas) out of habit, but would require another name, connoting more clearness than 'opinion' (doxa), but more obscurity than 'science /knowledge' (epistèmè); '(discursive / wandering) thought' (dianoia), I think, [is how] we defined it earlier" (533d4-7), then he resumes, again in reverse order, the list of the four affections / states of mind (pathèmata), which he now calls moirai, "parts / lots", with an idea of destiny / fate which suggests that not all are necessarily destined to reach all these "lots", by changing his vocabulary, introducing new notions and adding some decisive details on the different relationships between all these elements, saying that "it is satisfactory then, as before, to call 'knowledge' (epistèmè, which here takes the place of what was called noèsis in the analogy) the first part, the second one '(discursive / wandering) thought' (dianoia), the third one 'trust' (pistis) and 'imagery' (eikasia) the fourth one, and those two together 'opinion' (doxa) and the two other together 'intellection' (noèsis, which, in the analogy, designated only the second sub-segment of the perceived by intelligence); and opinion (doxa) on the one hand [is] about becoming (genesis), intellection (noèsis) on the other hand about beingness (ousia); and what beingness (ousia) [is] with regard to becoming (genesis), intellection (noèsis) [is] with regard to opinion (doxa), and what intellection (noèsis) [is] with regard to opinion (doxa), knowledge (epistèmè) [is] with regard to trust (pistis) and (discursive / wandering) thought (dianoia) with regard to imagery (eikasia)" and immediately adding "but the relation of analogy (analogian) between what those [are] upon (eph' hois), and the division into two parts of each one of these two, that which is in the realm of opinion (doxaston) and that which is in the realm of intelligence (noèton), let us drop [that], so that it doesn’t fill us full with discussions many times longer than those having preceded" (533d7-534a8), confirming in particular by these last words that he intends to say nothing more about the ratio(nale) (logos) used to divide in two "according to the same logos" (ana ton auton logon) each of the two segments that he no longer calls that of the seen (horômenon) and that of the perceived by intelligence (nooumenon), but opinable / that which is in the realm of opinion (doxaston) and intelligible / that which is in the realm of intelligence (noèton). It should be noted at this point that all these fluctuations in vocabulary are not due to inaccuracies on Plato's part manifesting a lack of mastery of his subject, but on the contrary result of a deliberate choice on his part not to freeze in each case on a single word, which would thus become a "technical" term, to invite us to look for what he is talking about beyond the words he uses to talk about them, which, for him, is precisely the mark of one being dialektikos.

The limits of the use of images (analogy and allegory)

Before undertaking the "decoding" of these two "images", one should be conscious of the limits imposed on an author by the use of images. Plato is perfectly aware of the fact that an image is never identical in all respects to what it is only an image of, as shown by the words he puts in the mouth of his Socrates speaking to Cratylus in Cratylus, 432b1-d10 about names envisioned as "images" (eikones) of what they designate: "images are far from possessing the same [things / attributes / properties / characteristics...] as what they are images of" (endeousin hai eikones ta auta echein ekeinois hôn eikones eisin, Cratylus, 432d2-3). Thus, for instance, when he chooses to represent, in the allegory of the cave, the idea of the good (hè tou agathou idea) by the sun, in keeping with the choice already made earlier in the parallel between the good and the sun, he is more or less "constrained", to be consistent with this choice, to represent all the other ideas (ideai) by the stars, but then, if this choice is fitting for suggesting that ideas, like the stars in heaven, can only be distinguished from one another by their relative position in relation to one another, that is to say, by the relations they maintain with one another in a whole (heaven for the stars) which has meaning only as a whole, the implied consequence of this choice is that, in the allegory, he cannot describe a situation where one sees at the same time, directly or through reflections, both the sun and the other stars (that is to say, all the ideas, including the idea of the good), and he has to make do with the alternation of day and night, thus suggesting an excessive difference between the idea of the good and all the other ideas (even if this primacy of the idea of the good is emphasized by Socrates just before the analogy of the line, at 509b8-10, when he says that "the good is not beingness (ousia), but still beyond beingness (ousia), standing above them (the [fact of] being (intelligible) and the beingness (to einai te kai tèn ousian)) owing to its seniority and power"), which will not prevent him from using this day / night alternation to represent another aspect of the separation between the two sub-segments of the perceived by intelligence, thought (dianoia) without the compass of the sun / idea of the good for the night and the apprehension of the world around us in the full light of this sun / idea of the good for the day. Nor can he speak of shadows in relation to the stars, including the sun, but only of reflections, when he proposes to take up outside the cave the reference to the shadows and reflections evoked in connection with the first sub-segment of the line, as he did inside the cave by mentioning the shadows on the wall of the cave and the echo / reflection of the words of the bearers, to suggest a correspondence between the two images (line and cave) and to try to illustrate in the allegory the similarity of the relationship (logos) which serves to divide the two segments of the line. Similarly, by choosing to represent the whole of the "real", visible as well as intelligible, by a line, while it emphasizes the continuity between the two realms, he also directs the reader's mind towards an arithmetical understanding of the word logos that he uses in connection with the division of the two segments resulting from the first segmentation (logos in the sense of "numerical ratio "), and thus towards a "population" of the different segments by countable elements / "units" supposed different for each segment (each such "unit" can be in one segment only), which ended up polluting the understanding of the analogy until today.
If it is almost certain that Plato chose these "images" with the greatest care and composed his text in such a way as to make the best use of their potentialities in order to make clear what he was trying to have his readers understand, we must never, in interpreting them, seek perfect concordism, nor want at all costs to give too precise a meaning to the slightest detail of the text, nor, above all, expect that a single one of these images would say everything he tries to make readers understand about the "real", forgetting that, if he proposed several "images", it is precisely because none could alone represent everything he was trying to make readers understand. It is therefore recommended to read in parallel both the analogy of the line and the allegory of the cave, in order to interpret them in the light of one another, and in the light of other sections of the Republic and other dialogues on the same themes, and to move back and forth between them in order to progressively reach an overall view of their common model, which will have to be clarified and refined over the course of these comings and goings until arriving at an understanding of the "original" that allows to understand how each of the proposed images is a relevant, though partial, image of it, not only roughly, but up to a certain level of detail, so long as as one doesn't want it to say everything.
What Plato seeks to describe are the different ways in which our senses, and mainly sight, and our mind / intelligence (noûs), apprehend a "real" that acts on them (the pragmata, a word derived, let us remember, from prattein, "to do / act / accomplish"), and of which we are a part. In this quest, three terms / expressions play a major role: (to) auto / (ta) auta ("(the) *** itself / themselves"), eidos ("appearance", etymologically "for sight", evolving toward "sort, kind") and idea ("idea"), not so much in the images, since this is precisely what he seeks to "image" analogically, but in the explanation that is given of these images. The main issue here is to figure out if, as Socrates asks at the beginning of the Sophist with regard to the three words sophistès ("sophist"), politikos ("politician") and philosophos ("philosopher"), "[we] deem all these to be one [and the same thing] or two or, as is the case for the names, three and, distinguishing the families / classes (genè) exactly as a single name, [we] attach a family / class (genos) to each" (Sophist, 217a7-9), in short, if these three terms / expressions refer to the same thing, to two distinct things (and in this case which are the two that are synonymous), or to three distinct things. Neither of these two images alone can answer this question since:
- the analogy of the line speaks of eidè ("appearances / kinds"), never of ideai ("ideas") and evokes auta ("the *** themselves") when Socrates mentions, in an allusion to the conversation with Meno's slave in the eponymous dialogue, "the square itself" (tou tetragônou autou, genitive singular) and "the diagonal itself" (diametrou autès, genitive singular) at 510d7-8, specifying that it is on this that geometers conduct their reasoning by "attempting to see (idein) what cannot be seen otherwise than by thought (dianoia)" (510e3-511a1) even if they make images of it to help themselves in their reasoning.
- the allegory of the cave suggests viewing the stars as images of ideai ("ideas") rather than eidè ("appearances / kinds") by clearly implying that the sun represents the idea (and not the eidos) of the good (Plato never uses the words to tou agathou eidos ("the eidos of the good")), and refers to the auta ("the *** themselves") in a way that clearly distinguishes them from the stars / ideai ("ideas") since at 516a8 he talks about finally seeing, once out of the cave ( therefore in the intelligible), men (anthrôpoi, in the plural) and all the rest auta ("themselves"), and no longer only their shadows and reflections ("the images on waters of men and those of the other [***]" (en tois hudasi ta te tôn anthrôpôn kai ta tôn allôn eidôla, 516a7), the only things visible at first, when exiting the cave, due to lack of habituation to the light of the sun.
From this standpoint, what any serious interpretation of the allegory of the cave must take into account and explain concerning what takes place outside the cave, that is, in the intelligible, is:
- that we can see two categories of "objects", on the one hand men (in the plural, which excludes that it is the unique "form / idea of Man") and all the rest of what was also visible inside the cave (each "object" in its individuality, as is hinted at by the fact that, for men, Socrates uses the plural), and on the other hand the heaven and the stars, invisible from inside the cave, which can be called respectively "terrestrial objects" and "celestial objects";
- that the "objects" of these two categories are perceived first through their shadows (for "terrestrial objects" only, and more specifically for human beings) and their reflections (for both categories of "objects"), then in themselves (if shadows concern only human beings (which Plato doen't explicitely says) and reflections only are mentioned about the stars, aside from the fact that, as I said above, the stars do not produce shadows of themselves, it is because, in the intelligible, shadows stand for the words people produce as telling other people something about the person uttering them, and reflections stand for the logoi produced by people about anything as saying something about what they talk about, and that only human beings (thus, not the stars) talk (produce "shadows"), but that stars may be talked about (be seen through "reflections" in the speeches of people));
- that the "earthly objects", men and all the rest, are only listed one after another, while the "celestial objects", the stars, are apprehended as a whole, heaven (ouranos), explicitly mentioned in the words "the [***] in heaven and heaven itself" (ta en tôi ouranôi kai auton ton ouranon, 516a8-9), for which the succession of shadows and reflections / originals is replaced by the succession of night and day, the reflections being mentioned only in relation to the sun (even if it is quite possible to see reflections of stars in waters);
- that the prisoner, at the moment he exits the cave, can at first only see shadows and reflections of the "earthly objects", those that he already saw and heard inside the cave, and that therefore, what he sees in this first of four stages outside (two for the "earthly objects", two for "celestial objects", where, moreover, the case of the sun is treated separately from that of the other stars, but, here again, in two distinct stages) cannot be the ultimate "reality", but he cannot know it since he is precisely not able to see the originals of what he sees only images and reflections of and is therefore in the same situation in relation to these new "objects" of apprehension as he was inside the cave when he could only see the shadows of objects protruding from the wall and "would hold as the true nothing but the shadows of the implements" (515c1-2);
- that even with regard to the sun, image in the allegory of the idea of the good (or, to be more specific, image of the good itself, what plays in the allegory the role of the idea of the good being the image of the sun perceived by sight, itself image of the intelligence in the allegory once outside the cave, by human beings remaining on earth and thus unable to see it as it is in itself in it's own place), there are, outside the cave, therefore, two levels of apprehension, first through reflections, then directly, which implies that what can be seen outside the cave is not limited to "forms / ideas" (a reflection of the sun is not the analogue in the allegory of the "idea of the good", since it is the image of the sun directly formed in the eyes that plays this role), or that these "forms / ideas" come in two flavors, ideai (the sun and stars, or rather the light they emit perceived directly by human beings) and eidè (their "reflections" in the minds of human beings).
To refuse to seek an analogical meaning for these elements is to suppose that Plato, carried away in a poetic delirium by the enthusiasm induced in him by the image he had chosen, would have diluted it at pleasure by stuffing it with useless details (insofar as they had no analogical meaning), at the risk of compromising its understanding, which is contrary to everything that a careful reading of the dialogues leads us to suppose about Plato as a writer, philosopher all the time before being poet on occasion. Why in particular, if this were the case, would he have taken care to create resonances between the allegory of the cave and the analogy of the line, and in the allegory between the descriptions of the interior and exterior of the cave with references to shadows and reflections in each case? We cannot therefore claim to have understood Plato if we limit ourselves to saying that what "populates" the outside of the cave, that is to say "the intelligible place" (to noèton topos, cf. 509d2 and 517b5), are the "forms / ideas", without specifying whether we are talking about eidè or ideai, and most often by considering these two terms as synonymous in this context, and designating the auta ("the *** themselves"), i.e. the ultimate "realities", and without saying which of the four families of "objects" mentioned by Socrates in his description of the exterior of the cave (shadows and reflections of "terrestrial objects", the "earthly objects" themselves, reflections of "celestial objects", the "celestial objects" themselves) corresponds to these "forms / ideas / *** themselves".
Similarly, concerning the analogy of the line, it is necessary to explain:
- what Socrates means when he asks to divide the two segments "according to the same logos" (ana ton auton logon) and which logos he has in mind;
- why he speaks only of eidè ("appearances / kinds"), never of ideai ("ideas");
- why he describes "objects" in the division of the segment of the seen and intellectual processes in the division of the segment of the perceived by intelligence;
- what he means by "seen eidè" (horômenois eidesi, plural dative) at 510d5 and by "intelligible eidos" (noèton eidos, singular) at 511a3;
- what are the "images" (eikones) that he mentions at 510b8 and opposes to the eidè ("appearances / kinds"), the only kind of "things" the approach (methodos) that characterizes the second segment of the perceived by intelligence must deal with;
- What is meant by an approach that relies only on eidè ("appearances / kinds").

Additional insights

As I have already suggested, the decoding of the two images provided by the analogy of the line and the allegory of the cave, must take into account other passages of the dialogues dealing with the same subjects, both in the Republic and in other dialogues that precede or follow them in the order that I claim to be that of the dialogues organized in tetralogies. Particularly relevant from this point of view, on top of the "decoding" of the allegory by Socrates who immediately follows it (517a8-c5) and the recalling of the two images in the discussion on dialektikè already mentioned, are:
- the experience with Meno's slave in the eponymous dialogue (to which the mentions of the square and the diagonal in the analogy of the line refer) which shows how, on the same subject, the same person can move from an opinion (doxa), in this case false, to knowledge, and, at the end of the dialogue, the discussion on the difference between right / true opinion (orthè / alèthès doxa) and knowledge (epistèmè) (Meno, 97a9-98b6), where, with regard to the same thing, in this case the road to Larissa, that is to say, something that concerns the sensible world, one person may have knowledge and another may have a mere opinion;
- the myth of the soul compared to a winged chariot in the Phaedrus (Phaedrus, 246a3, ff.) and more specifically the section that describes the procession of human souls following those of the gods who pass to the other side of the vault of heaven where "they contemplate the [realities that are] beyond heaven" (theôrousi ta exô tou ouranou, 247c1-2) while human souls can only glimpse at them from a distance without being able to access the "supracelestial place" (huperouranion topon, 247c3) where are "the colorless and formless and intangible beingness really being, that can be contemplated only by the intelligence, pilot of the soul, around which [takes place] the birth of true knowledge" (hè gar achrômatos te kai aschèmatistos kai anaphès ousia ontôs ousa, psuchès kubernètèi monèi theatè nôi, peri hèn to tès alèthous epistèmès genos, 247c6-8) and where the auta ("the *** themselves"", cf. autèn dikaiosunèn, "justice itself", at 247d6) are visible, which only a few, privileged, can no more than glimpse at from afar;
- later, in the "dialogical" part of the same dialogue, the description, at Phaedrus, 265d3-e4, of the two processes favored by Socrates that characterize the one he qualifies as dialektikos ("skilled in the art of dialogue", 266c1): on the one hand, a process of synthesis aimed at leading what is initially dispersed in multiple ways "towards a single idea" (eis mian te idean), on the other hand, a process of analysis aimed at "carving according to eidè (kat' eidè) while respecting the natural joints (kat' arthra hèi pephuken, word for word "according to the joints where it is disposed by nature"), and taking care not to break any part by dealing [with them] in the manner of a bad butcher" (265e1-3), wordings in which it should be noted that idea is used to speak of what is a target (eis) towards which one tends without being sure of reaching it, as one can look at the stars to find one's way, and eidè is used to speak of carving with the risk of making a mistake if one does not do it according to (kata) the natural joints (arthra hèi pephuken), which suggests a certain distance from the ideai that one is not certain to reach and an uncertainty about the eidè, which are the result of our activity of analysis of a "real" which is not these ideai and in relation to which we can be mistaken;
- the discussion on knowledge (epistèmè) and opinion (doxa) at the end of Book V of the Republic, where Socrates seeks the "upon what it is" (eph' hôi esti) of both and finds it, not in the "objects" on which knowledge or opinion is formed, which, as the Meno has shown and as everybody knows, can be the same, but, regarding knowledge, in "the being / what is" (to on, 477a9), and regarding opinion, in the "received ideas" (nomima, 479d4) of the many, that is to say, regarding knowledge, in something that is external to the logos and serves as a regulator by implying the notion of true and false, and for opinion, in the words and logoi borrowed from others by people who do not seek to test their relevance themselves and are content to accept them based on the greater or lesser trust they have in those they borrow them from, or more precisely in what is induced in the minds of the people whose opinion it is by the logoi of other people whose (true or false) opinion it is also or who have knowledge about the subject (which loses its status as knowledge for anyone who accepts it without verification and without really "understanding" it and being able to justify it);
- the discussion on the different kinds of beds in Book X of the Republic, which clearly marks the difference between eidè, human constructs that vary from one person to another and evolve for each person over the course of one's life due to growing experience, answering a problem of naming and intended to give meaning to the words one uses, and ideai, non-human constructs, principles of intelligibility, the same for all, distant targets of the eidè having an "objectivity" which is "proven" by the fact that we can more or less understand one another, which are, in the example chosen by Socrates among the creations of human craftsmanship of which we all have daily experience to facilitate our understanding, what craftsmen look toward (with the "eyes" of the mind) to make what they want to make (a table or a bed in the example), and which are not "models" for their instantiations, as he tries to make us understand, on the one hand by never presenting, in this discussion on imitation in the most general sense, the craftsman as an "imitator" (of the idea that inspires him), contrary to what he does for the painter, who is indeed an imitator of what he takes as a model, and on the other hand by attributing their authorship to a "creator" whom he successively qualifies as "a god" (theos, 597b6), to make it clear that the ideai are not human creations, but are the creations of a being endowed with intelligence, and even with an intelligence superior to that of men; as a "gardener / planter" (phutourgos, 597d5) sowing his creations "in nature" (en tèi phusei), to make it clear that the ideai do not constitute a separate "world" but are an integral part of our world that they make intelligible to us and that they have more to do with "seeds" from which multitudes of "shoots" can germinate, all different from one another from the same seed (all the leaves of the same oak tree are different from one another and have nothing to do in their appearance with the acorn from which they grew, but are nevertheless all oak leaves, not willow leaves or pine needles, and even less horses or men) than with a "model" that would have to be reproduced as accurately as possible; and finally as "king" (basileus, 597e7), to make it clear that these ideai that he "sows" are akin to laws, the works par excellence of a king, for the nature in which he sows them, thus complementing each image with other images highlighting other aspects of this divine creator of whom we can only speak in myths (the Timaeus) or images;
-Phaedo, 99d4-100c8, where Socrates on the one hand warns against the risk of blindness if we try to look at the sun in direct view rather than through reflections in water, even and especially during an eclipse, which should lead us to be wary of what he proposes as the final stage of the progression of the prisoner freed from his bonds and out of the cave in words that their redundancy and emphasis should suffice to invite us to question the idea that anybody might "see clearly and contemplate the sun itself by itself in its own place as it is" (ton helion... auton kath' hauton en tèi autou chôrai ... katidein kai theasasthai hoios estin, 516b4-7), and on the other hand, after explaining that, fearing that his soul would suffer the same blindness by looking at facts / things (pragmata) with the eyes and the other senses, he had decided to seek "the truth of beings" (tôn ontôn tèn alètheian, 99e6) in the logoi, adding that he did not consider "that one who examines in logoi the beings (ta onta), examines them more in images (en eikosi) than the one [who examines them] in acts / facts (en ergois)" (100a1-3), thereby suggesting that apprehension by the senses is just as much an apprehension through images as apprehension by logoi, simply though images that are of a different kind in each case, speaks of auta ("the *** themselves") by saying that he reasons "taking as a support that beautiful itself as such is something, as well as good and great and all the others" (hupotemenos einai ti kalon auto kath' hauto kai agathon kai mega kai talla panta, 100b5-7), which he makes "the kind of cause" (tès aitias to eidos, 100b3-4) which satisfies him to explain that something is beautiful or good or great, etc., these qualifiers resulting for him from the fact that what is thus qualified "partakes" (metechei, 100c5) in the auton ("the *** itself") from which it receives a qualification, beautiful, good, great... "itself as such" (auto kath' hauto), and not in any eidos or idea of that (the word eidè ("appearances / kinds") is used a little later, but it is by Phaedo retelling to Pythagoreans in Phlius years after the event the death of Socrates, not in the words he attributes to Socrates himself, and more precisely in a summary of which he is the author, not in a restitution of the verbatim of the discussion with Socrates, when he summarizes his story after an interruption by Echecrates by saying "after... it had been agreed that each of the eidè is something... (epei... hômologeito einai ti hekaston tôn eidôn...,102a11-b1), so that it is Phaedo, far from Athens and outside the control of Socrates, who has been dead for some time, who equates the auta ("the *** themselves") that he believes Socate spoke of with the eidè ("appearances"), not Plato's Socrates);
- Sophist, 259e4-6, where the Elean Stranger who leads the dialogue with Theaetetus, at the moment when he is about to investigate "what logos can be" (logon... ti pot' estin, 260a7-8), preludes to this investigation by saying that "[it is] by means of the intertwining of eidè with one another [that] logos happens for us" (dia tèn allèlôn tôn eidôn sumplokèn ho logos gegonen hèmin), a key statement that makes logos an intertwining, and not a mere juxtaposition, not of words, but of eidè with one another, and which can help us to understand why, in the second process of the perceived by intelligence, the one that must use the logos by making use of the power of dialegesthai, one must find one's way through the eidè and the eidè alone, since they, and they alone, are what carry the meaning we give to words, which are in themselves no more than modulations of sounds with no relation with what they purport to designate other than through mere convention, and can change from one language to another without the meaning of the words changing.

The limits of logos

But in this decoding, we must also take into account the limits of logos, the only tool we have for this work, and understand that, if Plato had recourse to images, analogy and allegory, and in other places such as the Phaedrus, myth, it is because he was perfectly aware of the fact that these things cannot be explained in a totally satisfactory way by reasoning, rational discourses, purely explanatory logoi that do not have recourse to analogies and other forms of spoken images. To attempt to put back into purely explanatory discourse what Plato has wittingly chosen to present through images is to believe that we are smarter than him and simply not to have understood what he is trying to make us understand about the power, and the limits, of logos, and more specifically of the dialegesthai, which is the use of it through dialogue (hè tou dialegesthai dunamis, 511b4), which precisely conditions access to the second sub-segment of the perceived by intelligence. If, as he implies in the passage from the Phaedo quoted above, the logoi only give us images of beings, any apprehension of beings that would go beyond the stage of the "images" provided by the words that constitute the logos, or rather by the eidè that we associate with them to give them meaning, whatever form it takes, reasoning, as well as myth or allegory, would be incommunicable by nature, since communicating it would bring us back to the level of words and logoi. As a matter of fact, logoi are not the prerogative of the segment of the perceived by intelligence, and the allegory of the cave is clear on the fact that prisoners who remain immobilized in bonds since childhood are capable of dialegesthai ("practice dialogue") and, to do so, give names to the shadows they see on the basis of similarities and differences in their visual appearance (cf. 515b4-5). This suggests that a form of logoi is present in all four sub-segments of the line, potentially different in each case, but it is precisely these differences that distinguish the segments from one another. If we accept the relation mentioned by Socrates at the beginning of the discussion on the different kinds of beds between names and eidè, if there is logos, there are eidè, and eidè in a collective, not individual sense, referring to something common to several "beings" sharing the same name or attribute. The whole question is then to know what these eidè are based upon in each case. In the first sub-segment of the seen, corresponding to the initial state of the men (anthrôpoi) staged in the allegory when they are still prisoners immobilized in bonds, they are based exclusively on the visual appearance, represented by the shadows on the wall of the cave, the only things that the prisoners can see at this initial stage. In the second sub-segment of the seen, represented in the allegory by the turning around towards the objects projecting these shadows, they are based on the whole three-dimensional material dimension of the "beings" and the data of the five senses that can be taken into consideration to enrich the eidè associated with the names, whether inherited from the first stage or new, given to these "beings", not only those that we have been able to collect ourselves on what we have personally experienced, but also those that we have been able to gather from logoi of people we trust on what we have not personally witnessed. The move to the first sub-segment of the perceived by intelligence, that is to say the exit from the cave, presupposes that we have become aware of the fact that these "beings" are not only material, but also have a part of intelligibility resulting from the fact that they are the work of a craftsman (dèmiourgos), divine or human, endowed with intelligence and who had a purpose in creating them, and that the names by which they are designated no longer refer only to a simple visual appearance or to material properties, but to something "that cannot be seen otherwise than by thought (dianoia)" (511a1), and moreover that certain "beings" that were not visible in the cave (because they were immaterial) are visible outside the cave (the stars in heaven). The eidè associated with the names of all these "beings", material and immaterial, are then made explicit by relations between "beings" (translated into logoi, thus in an "intertwinning of eidè" which, according to the Elean Stranger at Sophist, 259e4-6, constitutes logos) which can take, in their most rudimentary form, the form of "definitions" and in a more elaborate form, the form of "Socratic" dialogues) rather than by references to visual and more generally sensible properties. The last stage is the one where we have understood that words, no more than the eidè that are associated with them and which change as each one progresses towards the stars / ideai that constitute the target for each one of them and the sun / idea of the good that "enlightens" them all and gives them meaning, are the "beings" themselves (auta) and that the best we can hope for is to "map" as closely as possible the starry heaven that is offered to us, that is to say, to master the relations that exist between all the stars / ideai, and above all between each of them and the sun / idea of the good, by expressing them in logoi / "intertwinning of eidè" whose strongness will be tested through dialegesthai, that is to say, the confrontation of points of view in interpersonal dialogue. In short, the ideai are the distant target towards which the eidè tend, which evolve for each one as one progresses in and out of the cave, but they are not yet the auta, who, as the myth of the Phaedrus mentioned above says, are on the other side of heaven and are accessible only to the gods, and perhaps to a few privileged humans who will not be able to benefit others directly since it would require to fall back to the level of words and logoi.

Keys to understanding the allegory of the cave

A few keys now to understand the allegory of the cave: the anthrôpoi ("men" in the sense of human beings regardless of sex) are human souls, in accordance with what Socrates says to Alcibiades at Alcibiades, 130c1-6, that, "since (as the preceding discussion has shown) neither the body, nor the whole of the two (body and soul) is man (anthrôpos), there remains that either it is nothing, or, if it is something, the [word] 'man' (anthrôpos) can be attributed to nothing else but the soul" (mèden allo ton anthrôpon sumbainein è psuchèn), and therefore that "the soul is man" (hè psuchè estin anthrôpos). These anthrôpoi ("human beings") as prisoners before and after being freed from their bonds are these human souls as capable of education and knowledge that will indeed free them form those bonds; as bearers hidden by the wall, they are these same human souls as objects of knowledge for one another as "animators" of their bodies perceptible by the senses, but invisible so long as one remains inside the cave since they are immaterial (which is depicted by the fact that they are hidden by the wall), but becoming "visible", that is to say, in the imagery used by the allegory, intelligible, when one leaves the cave, like all the rest of what was visible in the cave.
The bonds (desmoi) that have prevented these anthrôpoi from moving and turning their heads since childhood, allowing them to see only the shadows on the wall of the cave facing them, i.e. the images provided by sight, and thus make them prisoners (desmôtai), represent the prejudices inherited from their environment and the prevailing opinion that it is sight and sight alone that reveals the world around us and that it gives us an adequate and exhaustive apprehension of everything that "exists", leading us to admit without prior reflection on the way sight and human beings as a whole function that things are as we see them ("they [the prisoners caught in these bonds] would hold as the true nothing but the shadows of the implements, 515c1-2). All the spontaneous and unconscious work that takes place naturally in each person from the first moments of one's life and allows one to learn little by little to interpret the data from sight with the help of movement and the data from the other senses, in particular touch and hearing, leading one to individualize the patches of color that sight provides in a two-dimensional space, to associate "forms" (eidè) with them and to become aware of the three-dimensional character of the space in which one evolves, and to realize that the world is made up of a multitude of "things" that are more or less autonomous from one another and are not limited to what one sees at a given moment, that the "things" that one sees do not cease to "exist" because one closes one's eyes or they disappear from one's field of vision, that they do not change size depending on the distance at which they are from us, contrary to what the images of them that form in our eyes might lead us to believe, in short, everything that allows one to give meaning to the raw data provided by the eyes which are limited to patches of color, all this work, precisely because it was done unconsciously before we began to think and became automatic and remained unconscious, does not induces a specific reflection in most people that would make them aware of the fact that all this work of interpretation is done by the mind (noûs) and not by the eyes, and which could challenge the received idea that things are as we see them and that only the things that can be seen "exist". In other words, the word "bonds" (desmoi) used in the allegory refers, in the real world of which it is an image, to something that can still be described as "bonds", provided that this word is understood in a figurative sense that it can have in Greek (as in English), close to that which it has at Laws, VII, 793b4, in the mention of the "bonds of every constitution" (desmoi pasès politeias) constituted by the "unwritten customs" (agrapha nomima), described with the help of a word, nomima, which is found in the mouth of Socrates at Republic, V, 479d4, towards the end of the discussion on knowledge and opinion, where he makes them the "upon what it is" (eph' hôi esti) of opinion and where I have translated it as "received ideas". This understanding implies that the liberation from these bonds / prejudices cannot be the work of a third party but is purely internal to those who free themselves from them: it is the personal awareness of the fact that sight can be deceptive and above all that there are a multitude of things whose "existence" we admit without ever having seen them ourselves, such as the inside of our own body, or animals that do not live in the region we inhabit. The allegory limits itself to illustrating this by contrasting two-dimensional shadows and three-dimensional objects of which they are shadows, statues of men and animals representing their material bodies and manufactured objects, whose interior these shadows do not reveal to us, but we could broaden the issue by introducing the dimension of time and speak of everything that we admit without having been eyewitnesses, persons as well as events. In other words, our apprehension of the world is based more on "belief" than on actual sight. It is mainly the result of the "trust" (pistis, the affection (pathèma) that Socrates associates with the second sub-segment of the seen) that we have in the people around us and in their words and in those who have preceded us and have transmitted to us oral histories or writings on things and facts from elsewhere or from the past. It is therefore, at least initially and forever for some of them (those that belong to the past in particular), in the range of opinion, not of knowledge, and reaches us through logoi more than through the senses and opportunities to see for ourselves what we believe "exists" (whatever this word means) or "has existed". And, as a matter of fact, opinion is based on the trust (pistis) we place in those who transmit to us information about the world by means of logoi, and not on a personal experience and even less on the result of a personal search for knowledge by means of reason. The liberation of the prisoner that Socrates evokes is therefore indeed a turning around that compels ("he would be compelled", anagkazoito, 515c6) from the inside those who realize that sight is at the origin of only a small part of what one claims to know and is not always reliable, and we understand, with this way of understanding the "bonds" that bind prisoners, why he can say that this liberation takes place "naturally" (phusei). It is only the result of an inner work of the mind / intelligence (noûs) with which each human being (anthrôpos) is endowed, which other people can arouse by their words, but which they cannot do in the place of those who want to be freed from their "blinders". The prisoner thus freed moves from the illusion that it is sight that guarantees the existence of things for him and reveals them to him as they are to a state in which he is obliged to trust others to apprehend the world in its complexity (and in its duration). And it is not difficult to understand that this awareness does not take place without hesitation, without going back and forth (the "sparklings" (marmarugas) mentioned by Socrates at 515c9) and that whoever frees oneself in this way "deems the [things] he formerly used to see (ta tote horômena) truer than those now pointed at (deiknumena) (515d6-7) since the former (the shadows), one could see them, and the others, one does not see them with one's eyes but they are "shown / depicted" by means of words and logoi: thus, for instance, it is by trusting people who have examined disembowelled corpses on a battlefield or decaying bodies showing the skeleton, or doctors who have dissected cadavers, and supposing that all men are constituted in more or less the same way, that one can admit without seeing it that there are inside one's own bodies, made invisible by the skin that surrounds them, a heart, lungs, brain, bones, etc., which one will never see as long as one remain alive (X-rays, ultrasounds and other MRIs did not exist in Plato's time, and even this still provides only images whose interpretation is based on reasoning and scientific theories that are not within everyone's reach), and will at most be able to palpate (the bones) or hear (the heart). But this "turning around" does not yet take us out of the cave and, as long as it remains in the register of trust and not of the personal practice of reasoning validating the opinions that we have initially accepted without proof, leaves us in the material world of the objects protruding from the wall, and the path is still long and difficult toward admitting that there can also "exist" "beings" that are not accessible to any of our senses, but only to our mind / intelligence (noûs), that is to say, are "visible" (figuratively speaking) only outside the cave, as is the case for the "souls" that move the bodies / statues of which we see, in the literal sense, only the shadows (the bearers hidden by the wall).
The stars, which are only visible outside the cave, are, as I have already said, the immaterial ideai ("ideas") associated not only with abstract notions such as "beautiful", "just", "good" (the sun, which, by Socrates' own admission in his "decoding" of the allegory in 517b8-c1, represents the idea of the good, hè tou agathou idea), but also with everything that is sensible / visible and has a name, as "man" (the moon?), or even "table" or "bed" (cf. the discussion on the different kinds of beds in Republic X, 596a10, ff.). They represent the "objective" target, the same for all, in our search for knowledge (gnôsis in the sense of "inquiry"), that which is accessible to human reason about what they are the ideai of, the auta ("the *** themselves") which are on the other side of heaven, as the myth of the Phaedrus says, and therefore out of reach by human logos. On the other hand, what is not pictured in the allegory are the eidè, which exist at all stages of the prisoner's progress, accompanying the words one uses and evolving as one progresses from a simple visual appearance associated with the shadows that are the only thing one sees when one is still caught in bonds to logoi locating the star / idea corresponding in relation to the neighboring stars and to heaven as a whole, leaving aside any reference to the sensible "images" associated with the specific instances of this idea, accessible as sensible in the cave and as intelligible outside the cave thanks to these ideai, and especially thanks to the light projected on them by the sun / idea of the good.
What must be deduced from all this is that there are two categories of "things" in this allegory, those that can be apprehended both in the cave through the senses (as visible, and more generally sensible) and outside the cave through intelligence and reasoning (as intelligible), the example of which is detailed by Socrates in the case of the anthrôpoi ("human beings"), and those that can only be apprehended outside the cave (as intelligible) because they do not activate any of our senses, the stars in heaven and the brightest of them, the sun. But in each case (the sensible in the cave, the intelligible part outside the cave of the sensible which are only visible in the cave, the purely intelligible), there are always two stages: that of shadows and reflections and that of the "originals" of these shadows and reflections. Thus, if we go into detail in the case of humans, their bodies are represented in the allegory by the andriantai ("statues of men" with a reference to sex) carried by the bearers who circulate hidden by the wall (their souls in their role as invisible animators of these bodies); their shadows that prisoners see while immobilized in bonds are the visual appearance of their bodies (the statues of men protruding from the wall), and more precisely the image of them formed in the eyes; the sound reflections that constitute the echo of the words spoken by the bearers and that seem to come from the shadows, are their words, considered at this point as exclusively sensible phenomena, sounds (hence the use to describe them of the verb phtheggesthai, whose general meaning is "to produce sounds", whether for humans or animals or anything else); outside the cave, this time it is the anthrôpoi themselves, that is to say their immaterial souls, which can end up being apprehended, but at first, as long as one is not yet accustomed to the light of the sun / idea of the good, it is still only possible to "see", in this intelligible register that manifests itself through logos, their shadows (the words they say) and their reflections (the words said by others about them), before possibly being able, after habituation, to see them "themselves" (auta), in a vision that can only be individual and of which nothing can be said, since to speak of them supposes falling back to the level of words and logoi. And in all this, what is perceived, whether by the senses or by intelligence, is always anthrôpoi (in the plural) considered and understood individually, individual souls understood each as principle of intelligibility of a particular human being. If we wish to find in the allegory an "idea of man," we must look for it in heaven, which can only be understood as a whole of which the individual elements, the stars, spots of light all more or less identical with one another, images in the allegory of intelligible ideai, as I have already said, are only understood and can be identified by their position in relation to each others on the basis of a "cartography" that it is our responsibility to establish as precisely as possible (just as intelligible ideai have meaning only through the relationships they have with each others and that we seek to represent through logoi), probably in the moon, the one that takes up the most space there and the only heavenly body aside from the sun that is explicitly mentioned by name, just as anthrôpoi are the only "objects" in the cave that are explicitly mentioned by name (at 516a7) in what is to be seen by the freed prisoner just out of the cave.
What is true of humans is also true of all natural material creatures and all human creations as intelligible, for which these four stages are depicted. But the fact is that their names (such as for instance "anthrôpos") were given by the prisoners immobilized in bonds on the basis of eidè limited to only the visual appearances that the shadows constitute, and that these names do not change for those who are freed from their bonds and may eventually exit the cave, so much so that the same names are used in the real world, associated with eidè that evolve as they progress, to talk about what the allegory represents by things that it distinguishes by different names: in the case of the word anthrôpos, shadows (skiai) of statues of men, statues of men (andriantai), "shadows" (skiai) and "reflections" (eidôla) in waters outside the cave of these men invisible in the cave, but present outside, and finally these men (anthrôpoi) themselves (auta) and the idea that corresponds to them in heaven (the moon?), and that it is very difficult not to spontaneously associate with them the "shadows", the visual images from which they originated, and that it is difficult not to try to establish relations between the original and the image between these different levels of apprehension, a particularly risky practice as long as one has not reached the end of the progression and undertakes it in the progression from the interior to the exterior of the cave before having mastered the apprehension of what is outside as a whole, where it leads to mistaking mere reflections, the eidè that we associate at this point with the words we use and that we naively think are the same for everyone, for the ultimate reality and the statues seen inside the cave for "images" of these eidè: this is what happens with the supposed "theory of forms / ideas" attributed to Plato even though he warns us against such an understanding by means of the images completely understood of the analogy of the line and allegory of the cave; but to completely understand them, one must have reached the end of the progress he describes in them, have understood that these eidè, different for each one according to one's degree of progression towards the stars / ideai which are their "objective" but distant target, inaccessible to most, if not all, are not yet these stars and even less the auta ("the *** themselves") which, as the myth of the Phaedrus tells us, are in the supercelestial place, which is not given to everybody*. The last step is therefore to have become aware of the fact that any "cartography" of the stars / ideai that does not take into account the preeminent role played in relation to them by the sun / idea of the good as the "(leading) principle [which is] support of nothing" (archèn anupotheton, 510b7) leaves us in the night, where it does not matter whether we see the stars themselves or their reflections (the logoi held on them) (which explains why Socrates doesn't mention reflections of the stars, but only of the sun) since we do not see the one who "illuminates" them all and whose light during the day allows us to see / understand everything, including what is inside the cave made "visible" / understandable by it outside the cave. And here, contrary to what is the case for the stars of the night, which can only become comprehensible in the light of the sun, even if it makes them "invisible" in broad daylight, it is important to determine whether the representation we have of the sun / idea of the good is still only a reflection of it more or less resembling and more or less fluctuating (the logoi held over it by the city) or an image of itself in direct sight. But at the same time, we must not presume our strength and believe that we can, in this life, "see distinctly and contemplate the sun itself by itself in its own place as it is." The teleutaion that introduces the line from which these words are taken means "in the end / finally" without specifying what "end" is meant and teleutè ("end, accomplishment, completion"), from which it derives, may mean "death", which leaves open the possibility that the "end" that Socrates is thinking of here is death, which is probably the only means, if there is one for us, to be able to see the god sun "in its own place". The real conclusion of this whole process of education is not the enduring contemplation of the sun / idea of the good itself, which everyone knows is practically impossible, and which Socrates does not present in an assertoric way, but qualified by an oimai ("I suppose"), and which moreover would teach us nothing about anything (it is not by looking at the sun that we can see the world around us, but by taking advantage of its light to look at the world around us), but the sullogizesthai that follows in the allegory (516b9-c2), the reasoning that leads to the understanding of the role of the sun / idea of the good as the "light" of intelligence, as Socrates explained just before the two images in the paralleling of the good and the sun, introduced by the expression kai meta tauta, which everybody understands as meaning "and after that", i.e. in the sense of a temporal succession (which the preposition meta can indeed have), but can also be understood from the primary meaning of meta, which is "among, in the midst of", in the sense of "and in the midst of these [reflections]", that is, as suggesting that the conclusions it presents can be reached in the context of attempts to better see the sun / idea of the good and its light illuminating everything else, without this implying having seen it "itself by itself in its own place as it is".

* To temper the presumptuous character that such an assertion can have, I will simply add that my first serious work on Plato dates back to the end of the 70s, that the first version of my annotated translation of the analogy of the line dates from 2001 and that it is only during a fourth update of it in 2022, at the age of 78, that I really understood everything I present here about this analogy, in particular about the interpretation of the division of the segment of the perceived by intelligence, even if I had found most of the keys to reading the allegory of the cave about ten years earlier. Plato does not give himself up without effort and the education of the man he describes in the allegory of the cave is never finished... (<==)

Keys to reading the analogy of the line

The first mistake that must be avoided about the analogy of the line is that of believing that Socrates opposes two kinds of "realities", those that are "visible", qualified by a property that refers only to the sense of sight, but in fact includes everything that is accessible to the senses (aistheton, a word used later in the analogy, at 511c1), that is to say, all material "realities" that are sensible and subject to becoming, and those that are "intelligible", and only intelligible, therefore immaterial, inaccessible to the senses and not subject to becoming, which leads in fact to consider that all "reality" is either one or the other, but not both at the same time, and that therefore the material world is unintelligible and cannot be an object of knowledge, whereas precisely the allegory of the cave, properly understood, shows us that visible (and by extension sensible) "realities" can also be intelligible ("visible", in the analogical sense relating to the "sight" of the mind, outside the cave). What Socrates opposes here are, not two categories of "realities", but two modes of apprehension of the "realities" that surround us and are accessible to us by the senses and / or intelligence (noûs), apprehension by sight (and only sight at this point) and apprehension by intelligence. If sight is thus emphasized, it is because it is the sense to which we attach the most importance and trust most for our apprehension of the world around us, to the extent that, as I said above, for the Greeks of the time, one of the verbs meaning "to know" (eidenai) is none other than the perfect of the verb meaning "to see" (idein, which replaces certain tenses of the verb horan, another verb meaning "to see", from which horaton ("visible") comes), as if to better confirm that to have seen is to know. And so we should not immediately generalize the term "visible" to assume that it is used instead of "sensible" to refer to the material world as a whole (as I myself have sometimes done in other pages of this site and in previous versions of this page in French) in a synecdoche that would designate the whole by a part, effectively inviting the reader to think more about the "objects" that can be apprehended by the senses than about the modes of apprehension of these "objects" compared to the mode of apprehension by intelligence. Once we have understood this, there is no longer any reason to suppose a priori that these two modes of apprehension are exclusive of each other for the same "object" of apprehension, even though we have no difficulty in admitting that the same "object" can be at the same time visible, audible, tangible..., in short, apprehensible also by other senses, and that therefore visible "objects" can be intelligible as well as they can be tangible and / or audible and / or fragrant and / or sapient.
If we take a closer look at the text with these remarks in mind, we see that Socrates distinguishes three levels for which he uses a different vocabulary:
- the level of "objective realities", of the pragmata, which act (prattein) on our eyes and our mind / intelligence (noûs) and which are therefore qualified as "visible" (horaton) when they have the power to activate sight and "intelligible" (noèton, in the sense of "perceivable by intelligence") when they are capable of activating the mind / intelligence (noûs) without necessarily passing through the mediation of the senses, the one being no more exclusive of the other than the fact of being visible is exclusive of the fact of being tangible, or audible, or perceptible by other senses (in other words, a "reality" qualified as "visible" can also be "intelligible", and a "reality" qualified as "intelligible" may or may not be "visible", and may even not be perceptible by any of the senses): at this level, he uses verbal adjectives ending in -tos (-ton in the neuter) which indicate in this case a simple possibility, equivalent in Greek to the adjectives ending in -able or -ible in English, as precisely visible, intelligible, thinkable which serve to translate them;
- the level of the "affections / states of mind", of the pathèmata (cf. 511d7), produced in us by these "objective realities" (pragmata) visible or only accessible to intelligence but not to sight: he uses the terms "seen" (horômenon) and "perceived by intelligence" (nooumenon) in connection with them, passive present participles that refer, not to what activates sight or intelligence, but to what is produced in our mind, which "is born" (the primary meaning of genos), under the effect of these activators of sight or intelligence during effective occurrences of what is implied by the verbs used, "to see" (horan) and "to think" (noein);
- the level of the different kinds of logoi, expressed or simply thought, that these "affections" (pathèmata) arouse in us: he speaks of them as "opined" (doxaston), i.e. expressed as a mere opinion, and as "known" (gnôston), i.e. being the end result of a personal search for knowledge (gnôsis in the sense of "inquiry").
By qualifying at 509d8 the two segments of the first division of the line as "that (the segment) of the genos seen and that (the genos) of the perceived by intelligence" (to te tou horômenou genous kai to tou nooumenou), he shows that he locates the line at the level of the second of these three levels, the intermediate level of the "affections / states of mind" (pathèmata) produced in us by the "objective realities", the pragmata ("facts / things") that act (prattein, from which pragma derives, which is the opposite of paschein "to suffer / be affected by", from which pathèma derives) on our eyes and our mind / intelligence, and not at the level of these pragmata, these "objective realities". This is suggested by the use of the word genos, which must be taken here in a sense that evokes "origin" ("that which originates in sight / perception by intelligence") with an idea of filiation that my translation by "family" seeks to preserve. And this is confirmed by the preference for the words "seen" (horômenou) and "perceived by intelligence" (nooumenou) rather than "visible" (horaton) and "intelligible" (noèton) which emphasize the intrinsic properties of the "activators" (pragmata) of sight and intelligence and thus orient the mind towards a classification of "objects" rather than of the "affections" produced by these "objects". Talking about "visible" (horaton) and "intelligible" (noèton) makes it easier to suppose that they are disjoint subsets (a "thing" is either visible or intelligible, not both), while talking about "seen" and "perceived by intelligence" facilitates the acceptance of the fact that the same "thing" can be both seen and perceived by intelligence, which is what Plato's Socrates has in mind, as confirmed by the allegory of the cave. This should therefore avoid an almost general error in the understanding of the analogy, the one that suggests that the division proposed by the analogy of the line is a division of the "real" in which each element of this whole is in one and only one of the four sub-segments described by Socrates, with in the background the idea of opposing what is visible (and by extension sensible / material) and, as such, supposedly unintelligible because it is in perpetual becoming and therefore can only be the object of opinions, with what is only intelligible and therefore escapes becoming and can only be the object of knowledge. Exit then, for instance, the thesis that the "proper objects" of the first segment of the perceived by intelligence (that of the dianoia) are who knows what "mathematical objects", not that such "objects" do not exist (between the square itself (to tetragonon auton, cf. 510d7), which has neither dimensions nor spatial position and can be represented by the logos "a closed plane figure which has four equal rectilinear sides forming between them four right angles", and a square drawing made by a surveyor on a material support, which will never be strictly a "real" square, there is the square whose sides measure two feet, to which Socrates refers in his discussion with Meno's slave, the square whose sides measure ninety feet drawn on all baseball fields, and all those which have a specific dimension, considered in the abstract and not in a particular material figure, and which are immaterial and in infinite number), but because they are not the exclusive "population" of the first sub-segment of the perceived by intelligence, the proper object of dianoia, which can consider with its own modes of reasoning, which, alone, characterize it, any "object" that is sensible / material or only intelligible. A symptom of this error of interpretation is to speak of the segment of the visible (horaton) and of the segment of the intelligible (noèton) and not of the segment of the seen (horômenon) and of the segment of the perceived by intelligence (nooumenon), ignoring the change of vocabulary that takes place at 509d8 compared to 509d2-3, a change in vocabulary that the vast majority of translators indeed ignore, translating the two series of words (horaton and horômenon on the one hand, noèton and nooumenon on the other) by the same words ("visible" and "intelligible" respectively) Now, as I have already said, the allegory of the cave confirms that this is not the case and that the analogy does not seek to classify in distinct "boxes" with no possibility of moving from one to another the "things" (pragmata) with which we are confronted in our lives, depending on whether they affect our sight or our mind / intelligence (noûs), but to highlight the different ways in which we react to the "affections" (pathèmata) that they cause on them, and that it is in fact the same "things" that are apprehended in the four (if they are visible) or only two (if they are purely intelligible) sub-segments, simply under different "appearances" (eidè) conditioned by what gives us access to them, sight or mind / intelligence (noûs). But, by the second division of each segment, Socrates shows us that the "affections" (pathèmata) that he has in mind are not limited to the "gross" production resulting from the action of the pragmata on the "organs" that they can act upon (eyes for sight, ears as receptors of logoi for intellingence or thought as merely capable of producting inner logoi), that is to say, to the image produced in the eyes (no matter where exactly and by what physical process) for the seen, or to the words heard or simply thought for the perceived by intelligence, but include the way in which intelligence interprets them, either straightforwardly without becoming aware that they are only images (in the seen) or words (in the perceived by intelligence), or by becoming aware that they are only "representations" that do not show us what they represent as it is. It is indeed when we are not aware of the fact that sight gives us only "images" (eikones) of what it allows us to see, that is, in the terms of the allegory of the cave, that we "hold as the true nothing but the shadows of the implements" (515c1-2), that we live in a world of images, of "shadows" (the first segment of the seen), and only when we become aware of that that we can begin to gain access to what sight gives us only an image of, as indeed distinct from the image we perceive of it (the second segment of the seen). And in the same way, we cannot consider what we speak of and think about with words as having an autonomous "reality", and therefore know it (the second segment of the perceived by intelligence), so long as we do not think of that as distinct from the words we associate with it, as do those for whom "beautiful", "just", "good" for instance, are only words (the first segment of the perceived by intelligence).
But there is more. Once we have understood that the divisions that Socrates invites us to make on the line do not seek to define "boxes" in which to locate distinct "beings", each one in one and only one of these "boxes", but to distinguish different ways of apprehending the same "beings", we can finally understand why, if the division of the segment of the seen inventories indeed perceptible "things" (images and originals), the division of the segment of the perceived by intelligence inventories, not "things", but intellectual processes, modes of reasoning, and, only as "objects" of these approaches, "beings" distinct from those of the segment of the seen (words and eidè). Sight, with the help of a first stage of interpretation by the mind (Plato would say "by the soul (psuchè)") of the raw data it provides, can only identify and distinguish individual "objects" from one another, whereas intelligence as such can only function through reasoning that establishes relations between these "objects", or rather between the "abstractions" that it associates with them, precisely to make them intelligible, the eidè gradually rid of their references to the sensible. In the visible, the defining criterion is the "status" that the soul grants to what sight allows it to apprehend, that of adequate and exhaustive representation of what activates it (which amounts to taking a simple image for reality) or that of a simple partial image of it; in the intelligible, the defining criterion is the "status" intelligence grants to that by means of which it reasons, words and the logoi it constructs with them, that is, whether or not it is aware of the fact that the eidè that it associates, unconsciously or consciously, with these words, are the very things to which these words claim to refer (the auta) or simple "images" of them adapted to the human mind and its limits. In both cases, therefore, the division is indeed made "according to the same logos": in each case, the difference is made according to whether or not one is aware that what the involved organ, the eyes in one case, the intelligence in the other, is working on is only an "image / representation" of what activates this organ and not the "thing" itself.
The second mistake that must be avoided is that of misunderstanding the expression ana ton auton logon ("according to the same logos") used by Socrates to tell Glaucon how to divide the two segments, that of the seen and that of the perceived by intelligence, resulting from the first division of the line, about which he said only one thing, which is that it had to be segmented into two unequal (anisa) segments without giving any reason for this request and, what's more, without specifying a proportion (one of the meanings of logos) between these two segments. Concerning the first division of the line into two segments, the anisa ("unequal") reading has been disputed since antiquity, but no one to my knowledge has ever found an acceptable reason to justify them being unequal or on the contrary equal (isa), while the explanation is implicitely provided by the allegory of the cave that follows: if one finds outside the cave, that is to say, in the segment of the perceived by intelligence, all that is visible / sensible in the cave, that is to say, in the segment of the seen, plus the stars of heaven invisible from within the cave, that is to say the purely intelligible beings with no action on our senses, it is therefore that the segment of the perceived by intelligence must be greater than that of the seen, and Socrates can say this without knowing what the size of the one and the other are (which would require first having said what they represent that is measurable and having defined an appropriate unit of measurement) and without even knowing what is the numerical ratio between the two, since he does not specify how to measure them. If, then, Socrates knows that he does not know the ratio (logos) between the two segments, at least not as a numerical ratio, how could he ask Glaucon to segment each of the two segments according to that same ratio?! ... But what the commentators have not noticed is that the "same" (auton) of "according to the same logos" (ana ton auton logon) can be understood in two ways: (1) according to the same logos as the one used to divide the line into two segments (the way the vast majority, if not all, of translators and commentators understanding it), or (2) both according to the same logos as the other, without reference to the first division. Now, if (1) implies (2) (the two ratios which serve to divide each of the two segments being both equal to a same third, the one which served to divide the line into two segments, are equal to each other), the converse is not true. Now, nothing in the analogy (except the weight of tradition) invites us to think that the division of each of the two segments, that of the seen and that of the perceived by intelligence, must be done according to the logos (numerical ratio or logical reason) which served to divide the segment the first time around, logos about which, as I have already said, Socrates says nothing except that the two segments must be unequal. And when he returns to these divisions in the discussion on dialektikè, towards the end of Book VII, using a vocabulary that takes us from the level of "affections / states of mind" (pathèmata), which is the one where he had initially located the line, to the level of logoi, the vocabulary of opinion (doxa) and intelligence (noèsis), by proposing to call "these two together ("trust" (pistis) and "imagery" (eikasia), the two "affections / states of mind" (pathèmata) associated with the segment of the seen), 'opinion' (doxa), and the other two ("knowledge" (epistèmè, which here replaces what was called noèsis in the analogy) and "(discursive / wandering) thought" (dianoia), the two "affections / states of mind" (pathèmata) associated with the segment of the perceived by intelligence), 'intellection' (noèsis, which, in the analogy, designated only the second sub-segment of the perceived by intelligence)", it is, after associating "opinion" (doxa) with "becoming" (genesis) and "intellection" (noèsis) with "beingness" (ousia), to propose a number of equivalences of relations between these different notions: " what beingness (ousia) [is] with regard to becoming (genesis), intellection (noèsis) [is] with regard to opinion (doxa), and what intellection (noèsis) [is] with regard to opinion (doxa), science / knowledge (epistèmè) [is] with regard to trust / faith (pistis) and (discursive / wandering) thought (dianoia) with regard to imagery (eikasia). However, none of these parallels concerns the relationship between "knowledge" (epistèmè) and "discursive / wandering thought" (dianoia), the "affections / states of mind" (pathèmata) associated with the two sub-segments of the perceived by intelligence, and that between "trust " (pistis) and "imagery" (eikasia), the "affections / states of mind" (pathèmata) associated with the two sub-segments of the seen, which he said at the outset should be equal (division "according to the same logos"). The relationships he evokes here are those between the homologous sub-segments of the two segments: the relationship between "knowledge" (epistèmè) and "trust" (pistis) (the second sub-segment of each segment) and the relationship between "(discursive / wandering) thought" (dianoia) and "imagery" (eikasia) (the first sub-segment of each segment), to say that they are analogous to the relationship between "intelligence" (noèsis) and "opinion" (doxa), which is itself analogous to the relationship between "beingness" (ousia) and "becoming" (genesis). Now, it should be noted that this equality of relations (the ratio of the first sub-segment of the seen to the first sub-segment of the perceived by intelligence is equal to the ratio of the second sub-segment of the seen to the second sub-segment of the perceived by intelligence, and they are both equal to the ratio of the segment of the seen as a whole to the segment of the perceived by intelligence as a whole) is mathematically true whatever the ratio that has been used to segment the two segments so long as it is the same, even if it is not the one that was used to segment the line into two segments, in other words, is true in the understanding (1) as well as in the understanding (2) of ana ton auton logon, since its truth depends only on the equality of the relations used for the second division, which holds both in understanding (1) and in understanding (2). So, if Socrates had also wanted the ratio between "knowledge" (epistèmè) and "(discursive / wandering) thought" (dianoia) on the one hand, and between "trust" (pistis) and "imagery" (eikasia) on the other hand, equal by hypothesis since the division must be made "according to the same logos" (ana ton auton logon), to be also equal to the ratio between "intelligence" (noèsis) and "opinion" (doxa), designating here the segment of the perceived by intelligence and that of the seen, he would not have failed to say so. One may also wonder whether the clarifications he gives on the relations of homologous sub-segments, which only complicates his already dense sentence without adding much to what he has said before, have not been added by him to make us understand that, if he goes into such detail and does not add the equality of the ratio between the two sub-segments of each segment and that between the two segments as a whole, it's indeed because this equality is not true. And this is confirmed by what he says immediately afterwards, precisely of these two relations, when he adds: '"but the relation of analogy between what those [are] about, and the division into two parts of each one of these two, opinable (doxaston) and intelligible (noèton), let us drop [that], Glaucon, so that it doesn’t fill us full with discussions many times longer than those having preceded" (533d7-534a8), that is to say that he does not want to say more about the identical ratio that served to divide the two initial segments "according to the same logos" (ana ton auton logon), which implies that it is equal to none of those he has already mentioned, including the ratio between "intelligence" (noèsis) and "opinion" (doxa), the new names by which he refers to the two segments of the fisrt segmentation. We must therefore understand ana ton auton logon ("according to the same logos") in sense (2) (the two segments are segmented both according to the same logos as the other, which is not the logos that was used to segment the line into two segments).

Clearness (saphèneia) and truth (alètheia)

Concerning the division of each segment, that of the seen and that of the perceived by intelligence, "according to the same logos" (ana ton auton logon), even if, as we have just seen, Socrates ends up saying that he does not want to discuss it, we can nonetheless try to understand the scattered indications he gives about it. In the course of the analogy, Socrates evokes two "criteria" that could be relevant: the notion of "clearness" (saphèneiai) and its opposite (asapheia), which he mentions at 509d9 at the beginning of his description of the division of the segment of the seen, and that of "truth" (aletheia) and its absence, which appears at the transition between the description of the division of the segment of the seen and that of the division of the segment of the perceived by intelligence, at 510a9. If these two notions are so close that saphes, the adjective from which sapheneia is derived, from the primary meaning of "clear, manifest, obvious", also has the derived meaning of "sure, true", it is necessary to make a clear distinction between the two and above all to see what role truth plays in the analogy.
Saphenia, "clearness", only makes sense in relation to an observer and the perception of what one observes. It is therefore a relevant property for affections (pathèmata) and everyone would agree that, in most cases at least, an image is less clear, or if not less clear, less exhaustive, than its original, since it does not reveal everything. But it should also be noted that, as the allegory of the cave highlights, "clearness" being a partly subjective notion in that it depends on the generic specificities and the individual quality of the observer's organs, the lack of clearness can be caused by a lack as well as by an excess of light, as it happens to the prisoner when he turns around in the cave and, dazzled by the fire when he tries to see the objects protruding from the wall, considers the shadows to which he was accustomed "clearer" (saphestera, 515e4) than the objects protruding from the wall seen in the full light of the fire, the same phenomenon being reproduced when he exits the cave with regard to the objects present outside ("men and the other [***]"), 516a7), and, again, the other way around, when he returns to the cave and can no longer distinguish the shadows after becoming accustomed to the brightness of the sun outside the cave. And since the allegory makes us understand that the "objects" that are found in the different segments under different "appearances" are the same, we must conclude that this notion of saphèneiai ("clearness") is not what permits to sort and classify different "objects" into different segments, but a property of the perception we have of the same "object", whatever it is, in the different segments. The perception to which the second segment of the seen gives access (the statues of men, in the case of the anthrôpoi, i.e. their material body) will always, once one has become accustomed to this mode of perception, be clearer than that to which the first segment of the seen gives access (the shadows, i.e. the mere two-dimensional visual appearance), and so on until the fourth segment, the second segment of the perceived by intelligence, which gives access to a perception which, complemented by those obtained in the other three segments put in their appropriate place, will give the clearest, because it is the most complete, apprehension of what is submitted to examination, whatever it may be.
Alètheia ("truth"), designated by a noun derived from the adjective alèthès whose primary meaning is "not hidden", concerns the greater or lesser accuracy in the relationship that exists between the perception by an observer of something, fact or "thing", which acts on that observer (a pragma) and what it is the perception of. It therefore also presupposes an observer, but where saphenia ("clearness") had a partly subjective character depending on the observer and his organs of observation, aletheia ("truth") has an "objective" character in that it is assessed by comparison to what is observed and more or less "revealed". And since an observer alone cannot appraise this accuracy, being unable to apprehend the observed "object" otherwise than by means of one's organs of perception (senses and intelligence), whose accuracy is precisely what must be appraised, and since no one can read in the thoughts of someone else, this appraisal can only take place between observers confronting their perceptions expressed in the form of logoi by means of dialogues. And indeed, in the analogy of the line, aletheia ("truth") is mentioned in a context (510a8-10) dealing with opined (doxaston) and known (gnôston), terms that refer to logoi expressing opinions or manifesting knowledge, and in the allegory of the cave, reference is made to aletheia ("truth") only in contexts of logoi or dialegesthai: at 515c1-2 when Socrates says that prisoners immobilized in bonds " holds as the true nothing but the shadows of the implements" shortly after having said that "if they were able to dialogue (dialegesthai) with one another... the same [things] being around [again], they would take the habit of giving names to those [things] they see" (515b4-5); at 515d1-7, in the context of a dialogue between the prisoner who has just been freed from his bonds and an unidentified character ("if someone told him...", "ei tis autôi legoi, 515d1-2) but who is clearly not one of the prisoners immobilized in bonds, where Socrates suggests that the freed prisoner would come to "deem the [things] he formerly used to see (the shadows) truer (alèthestera) than those now pointed at (the objects protruding from the wall)" (515d6-7); at 515e6-516a3, when Socrates says about the prisoner exiting the cave that, dazzled by the light of the sun, he would be "unable to see a single one of the [things] now called true" (tôn nun legomenôn alèthôn, 516a2-3).
To properly understand the role that Socrates attributes to the criterion of "truth" in the division of the line, one must be very attentive to the formulation he uses and not let oneself be fouled by preconceived ideas or influenced by words from him that will only come later and in another context (what he says about the relationship between opinion and knowledge in the discussion on dialektikè towards the end of Book VII): at the transition between the description of the division of the segment of the seen in images and originals of these images and that of the division of the segment of the perceived by intelligence, he asks Glaucon if he agrees that "it (the segment of the seen, whose mode of division he has just described) is divided with regard to truth (alètheia) and its absence [according to the following ratio(nale):] as the opined [is] to the known, so what is made similar (to homoiôthèn) [to something is] to what it has been made similar to (to hôi hômoiôthè)" (510a8-10). What Socrates proposes here is to extend to the relationship between image and model, generalized here, in the perspective of its imminent extension to the segment of the perceived by intelligence, by the use of the expressions "what has been made similar" (to homôiothen) and "what it has been made similar to" (to hôi hômoiôthè), a distinction supposedly accepted between opined and known with regard to truth. Thus, the question is to figure out what is this distinction between opined and known that can be extended to the case of the relationship between an image / resemblance and its model / original and, since it is supposed to have been accepted, to look for it in a distinction which has been discussed earlier. And what is suggested here cannot be to consider opinion as an "image" of knowledge, since that would amount to making the parallel suggested by Socrates work the other way around and, moreover, ignores the problem of "truth" that is at the start of this parallel. The distinction between opined and known from the standpoint of truth supposedly accepted that Socrates proposes to Glaucon to extend to the image / resemblance with respect to its model / original, is the one that Glaucon himself stated at Republic, V, 477e6-7, in the discussion on the difference between knowledge and opinion, when he described knowledge as "infallible" (anamartèton) and opinion as the opposite ("that which is not infallible"), which, reformulated in terms of "truth", amounts to saying that knowledge is always true, whereas opinion, as we have known since the Meno, can be either true or false. Transposed to the image in relation to the original, what Socrates wants us to admit here is that an original, as an original, is always "true", whereas an image / reproduction / similarity of this original can be true or false in relation to it, that is to say, give a more or less accurate and more or less exhaustive image / reproduction / similarity of it. In other words, the problem of "truth" arises only for "images / reproductions / similarities" with regard to their "original / model". And, from this point of view, the "truth" is not appraised in terms of all or nothing, but in the perspective of a progressive "unveiling" that can admit of multiple stages when we are confronted with multiple "images / reproductions / similarities" of the same "original / model", some in the form of images provided by sight, others in the form of logoi, which is precisely the case for us in our apprehension of what surrounds us, as the allegory of the cave makes clear. Socrates therefore does not invoke truth to help us sort out things that, as images / reproductions, would be false in relation to others that would be their originals / models and, as such, would be true, such as visible / material objects in relation to who knows what immaterial "ideas", but to draw our attention to the fact that everything that has the status of "image / reproduction" in relation to something else poses a problem of "truth" / accuracy when compared to its "model / original" which cannot be reduced to the dilemma "true / false", true for the original, false for the reproduction / image, and it is indeed the case for opinion in relation to knowledge, which can be totally or partially true, or completely false.

Knowledge and opinion

It is only when the analogy is reminded in the discussion on dialektikè towards the end of Book VII that Socrates evokes, not the distinction between opined (doxaston) and known (gnôston), but that between opinion (doxa) and intelligence (noèsis), by grouping together the two affections / states of mind (pathèmata) associated with the segment of the seen under the name of opinion (doxa) and the two of the perceived by intelligence (noûs, as an "organ") under the name of intellection (noèsis, as an activity), and will clarify his thought on these subjects in terms that I have already quoted in the discussion on the meaning of "according to the same logos" (ana ton auton logon), but which should be examined here from another standpoint: "It is satisfactory then, said I, as before, to call “knowledge” (epistèmè) the first part, the second one “(discursive / wandering) thought” (dianoia), the third one “trust” (pistis) and “imagery” (eikasia) the fourth one, and those two together “opinion” (doxa) and the two other together “intellection” (noèsis); and opinion (doxa) on the one hand [is] about becoming (genesis), intellection (noèsis) on the other hand about beingness (ousia); and what beingness (ousia) [is] with regard to becoming (genesis), intellection (noèsis) [is] with regard to opinion (doxa)". But here again, it is necessary to understand what Socrates is saying, in the light of a proper understanding of the two images of the analogy of the line and the allegory of the cave. When he says that "opinion (doxa) [is] about becoming (genesis)" and that "intellection (noèsis) [is] about beingness (ousia)", he does not mean that everything that is of the visible / sensible / material order and therefore subject to becoming can only be the object of opinion and that only that which is of an exclusively intelligible order can be attributed a "beingness" (ousia), since the allegory of the cave has shown us that everything that is in the cave as visible is found outside the cave as intelligible. It simply means that as long as we remain in the affections / states of mind associated with the segment of the seen, eikasia and pistis, that is to say, that we remain inside the cave, we can only arrive at opinions and that it is only when we agree to leave the cave and therefore to reach at least dianoia, that we can begin to have an understanding of what we are interested in, whether it is sensible or purely intelligible, and that we may hope to reach some day knowledge (epistèmè) by gaining access to the last sub-segment. In other words, it is each one of us who, by accepting or refusing to admit that even those things that partake of becoming can have a share of intelligibility and that there can be "beings" that do not affect any of our senses (the stars of the allegory of the cave), that is to say, to leave the cave, either forces oneself to remain at the level of opinions or opens the door to intelligence, including intelligence of "realities" subject to becoming. If we do not accept that things subject to becoming can nonetheless, through the power of logos, be assigned "attributes" by means of words saying that "it is" this or that and thus describing their "beingness" (ousia), it is we ourselves who condemn ourselves to remain in the realm of mere opinions. And it is also the case if we refuse to admit that these "attributes" are more than mere words.

The same logos

If one wants to maintain at all costs that the logos used to segment each segment of the line in two is the same as the one used to segment the line the first time around (as I have said that all commentators do), supposing in a simplistic way that this relationship is that of image to original, even though, as we have seen above, Socrates refuses to discuss this relationship for fear of endless discussions (which shows that it is not as simple as that), arguing that, in the discussion on dialektikè, he associates the first segment with opinion (doxa) and the second with intelligence (noèsis), we may try to liken the relationship between opinion and knowledge to that between image and original. And indeed, we may consider a particular opinion, something "opined" (doxaston), that is to say, an opinion expressed by specific words spoken or thought, as an "image" of knowledge, of something "known" (gnôston), when it is a true opinion, insofar as it can reproduce exactly the words that would be used by someone who possesses knowledge about what this opinion is about, and as the difference between the two is not in the wording used, the words which express it, but in the ability or lack of ability of the person who uses these combinations of words to account for them and to justify their relevance, in the fact that for one of them these words are justified only by the fact of having heard them, or words with a similar meaning, used by others, while for the other they are the result of a personal work of investigation (one of the meanings of gnôsis) whose unfolding he is able to reproduce, but Socrates never says this explicitly and, as we have seen above, this is not what he says in the analogy at 510a8-10 when he draws a parallel between "opined" (doxaston) and "known" (gnôston) on the one hand, "image / reproduction / similitude" ("that which has been made similar" (to homôiothen)) and "original / model" ("that to which it has been made similar" (to hôi hômoiôthè)) on the other hand. And, in any case, opinion as an abstraction (doxa), which encompasses both true and false opinions, is not the same as a particular opinion present in the thought of a particular person (a doxaston), just as intelligence (in the sense of understanding) as abstraction (noèsis) is not the same thing as a particular knowledge (a gnôston) present in the mind of a particular person, and it is not because certain particular opinions constituted by sequences of words may, when they are true, be "images / reproductions / similarities" of knowledge on the same subject, capable of being expressed by the same sequence of words, that opinion as such, an abstraction which is no particular sequence of words, can be considered as an "image / reproduction / similarity" of knowledge as such, an abstraction which is also no particular sequence of words. It cannot therefore be deduced from the fact that Socrates associates "opinion" with the two sub-segments of the seen and "knowledge" with the two sub-segments of the perceived by intelligence that the relationship between the segment of the seen as a whole and that of the perceived by intelligence as a whole is a relation of image to original and is therefore the same as that which presided over the division of each of these two segments into two sub-segments "according to the the same logos" (ana ton auton logon), which, by the way, is not itself a mere relationship between image and original. In fact, the difference between the affection / state of mind (pathèma) incured by prisoners who have only ever seen shadows (the first sub-segment of the seen), eikasia, and that incured by the prisoners who have been released but remain in the cave (the second sub-segment of the seen), pistis, is not so much the fact that, in one case, they see images and in the other their originals, since, among the originals, there may be fabricated objects (skeuè) that are themselves images (paintings and statues, for instance), but the fact that if, in the first case, they do indeed see images (the shadows), they are not aware of that and take these images for "reality", that, as Socrates says in the allegory, "[they] hold as the true (to alethes) nothing but the shadows of the implements" (515c1-2), whereas in the second case, when they have become able to "see" both shadows ("see" in the literal sense) and manufactured objects of which they are shadows ("see" in the figurative sense), they can no longer ignore that shadows are only shadows, that is to say, some kind of images of something else that they also "see", but this time with the eyes of the mind, through the words (logoi) of people they trust (pistis).
And just as, in the seen, the notion of "image" (eikôn) used to distinguish the second sub-segment from the first is a "false lead" that should lead us to generalize the notion of "image" to everything that is perceived by sight, so, in the perceived by intelligence, the notion of "image" (eikôn) used to distinguish the second sub-segment from the first, the one according to which some of the words we use would serve to designate "things" which are "images" of who knows what "realities" accessible only by "thought" (dianoia) and which will end up being called eidè when Socrates comes to speak of the second sub-segment, is a "false lead" that should lead us to generalize this notion of "image" to everything that is expressed by logoi, that is, to understand, so as to be able to make the best possible use of the whole power of dialegesthai (hè tou dialegesthai dunamis, 515b4), that it is all the logoi that give access only to "images" of "beings" (ta onta) through the mediation of the eidè the words refer to, provided we stop associating them with visual images (eikones in the primary sense) in order to try to reach the ideai they point toward, that is to say, the principles of intelligibility that allow them to be understood in the light of the good (thus, for instance, one will not understand what a bed is by looking for what is shared, from a visual, or even material, standpoint, between all the beds one has seen before, that is to say, by limiting oneself to the horômenon eidos ("seen appearance"), but only when one understands that a bed (klinè) is a piece of furniture intended to allow one or more people (in general no more than two) to lie down (klinein) on it to sleep (noèton eidos ("intelligible appearance") which points toward the idea of bed); or, to give another example, in order to become a philosopher, it is not enough to physically resemble Socrates, as is the case for Theaetetus (cf. Theaetetus, 143e8-9), who may well be a brilliant mathematician, but is unable to give birth to a meaningful logos on what "knowledge" means, because he has not started with an inquiry on logos, through which this "knowledge" is manifested (to use the wording of the analogy, he posits logos as a hupothesis for which he does not feel the need to give an explanation), an inquiry he ends up starting, but will leave unfinished and unsuccessful, only at the end of the dialogue when all his previous attempts to define knowledge have failed, ending with a definition of knowledge as true opinion accompanied with logos, even though shortly before, Socrates had him admit that opinion is a kind of logos, which means that his definition can be reformulated as true logos accompanied with logos, one must resemble him at the level of the logos, as is symbolically the case for the young Socrates, who bears the same name, and who takes the place of Theaetetus to dialogue in the Politics with the Elean Stranger, who took Socrates' place in the Sophist to make Theaetetus understand what logos is and how it can be true or false, on a subject of much greater importance for a future philosopher ideally destined to become "king", the proper function of the one who should govern the city).
In both cases, it is the allegory of the cave that makes it possible to set things right: for the seen, by presenting as the "proper object" of the first sub-segment the shadows of everything that protrudes from the wall, whether they are "originals" (the "statues of men" (andriantai) that are part of it are not there as statues, but as representing the bodies animated by the anthrôpoi / souls hidden by the wall) or "images" in the usual sense ( which are part, as manufactured objects, of the skeuè of 514c1), to lead us to understand that it is all that sight reveals us that is only "images" of what activates it; for the perceived by intelligence, by evoking the logoi uttered in the cave by the bearers hidden by the wall, considered at this point as purely physical phenomena, through their echo, which makes them sound "reflections", to prepare us to understand that outside the cave, it is all the logoi uttered by the anthrôpoi that constitute intelligible "shadows" and "reflections" of those who utter them and what they speak of, that is to say, of the "beings" (ta onta), of the "*** themselves" (ta auta) that act on our senses and our mind / intelligence (noûs) and are only perceptible to us through eidè, visible / sensible or intelligible. More specifically, logoi are made with words that only take on meaning for us through the eidè that we associate with them, consciously or unconsciously, and these eidè, even when they are purely intelligible and freed from the sensible images that gave rise to them, are still only "images", indeed intelligible, but nevertheless always "images", of "beings" that we seek to understand but cannot apprehend as they are in themselves (the auta). In other words, what is at stakes there is the way we understand the role of words in relation to what is not them and of which we claim to speak, and moreover, to speak as "knowing", that is to say as possessing knowledge (epistèmè) about them. These words are what serve as a "support" for the logos, for the reasoning that we carry out to try to reach knowledge, and it is these words that Socrates has in mind when, in the description of the division of the perceived by intelligence, he uses the word hupothesis, whose etymological meaning is "placing (thesis) under (hupo)", and the difference between the two sub-segments is that between the reasoning of those who accept the words of the language they speak and give them the meaning(s) resulting from usage without asking questions about them (to know the name is to know what it is the name of, just as, in the cave, to see the shadow is to see what it is the shadow of) and about something else, distinct from them, to which they would refer, and who therefore construct reasoning possibly exhibiting perfect logical rigour on "foundations" (one of the meanings of hupothesis) which are ultimately only opinions inherited from usage (the meaning of the words they use), which prohibits the result of their reasoning, no matter how rigorous they are, from being "knowledge" since it is built on opinions, and the reasoning of those who seek to know what is hidden behind the words before embarking on reasoning with them and who have become aware of the fact that what matters is not the words, but the intelligible eidè that we associate with them in our minds, free of any reference to specific visual images and capable of being associated with several different words, which are still only the "appearance" that can be taken for our intelligence (noûs) by the auta, the pragmata ("facts / things") themselves, which we seek to know, which play in the perceived by intelligence the role of objects protruding from the wall in the cave.
So, the two divisions were indeed made "according to the same logos", according to a similar justification / rationale (possible meaning of logos), which is not the one that was used to segment the line the first time around, but, as Socrates himself suggests at the end, this logos is not easy to formulate in a succinct way and is not simply the relationship between image and original, of "what has been made similar in relation to what it has been made similar to", but is more a difference than a relationship, the difference between perceiving something that, as a mere perception (by sight and the other senses or by the mind), is only an "image / resemblance" without being aware that it is only that, an "image / resemblance", and the fact of having become aware of it. In short, the goal is ultimately to understand that, in the perceived by intelligence as well as in the seen, we only have access to "images / resemblances / likenesses" of the "*** themselves" (auta) that act upon our senses and intelligence, not to the "*** " themselves (or only in a way that cannot be shared since any sharing passes through words).

Conclusion: Is Theaetetus sitting?

Let us try to summarize all that has just been said in order to bring out the main lines of what Plato seeks to make us understand through the analogy of the line and the allegory of the cave illuminating each other and illuminated by other sections of the Republic and other dialogues. Our senses and our intelligence (noûs) are subjected to "affections" (pathèmata) under the effect of "things" (pragmata) that may or may not affect one or more of our senses, but always affect our intelligence (noûs), either through the mediation of the senses, or directly, the one not being exclusive of the other (for instance, we can both see something with the eyes and conceive it by thought). Each of these "organs" (sense organs and intelligence insofar as it apprehends what affects it) has its own specifications and constraints and apprehends a different "aspect" (eidos in its original meaning in the case of sight, also used by analogy in the case of intelligence) of what may affect it (colors, sounds, smells, etc., for the senses, criteria of intelligibility for intelligence directly affected without the mediation of the senses). In the analogy of the line and the allegory of the cave, Socrates limits himself, as far as the senses are concerned, to that of sight, because it is the most pregnant, the richest in "practical" information to allow us to find our way in the world and the one that, for this reason, we have the most trust (pistis) in, in short, the one that most "binds / chains" us, and therefore he distinguishes between "seen appearances" (horômena eidè) and "intelligible appearances" (noèta eidè), but wants us to understand that everything that reveals itself to us under a "seen appearance" (horômenon eidos), that is to say, that is visible (horaton) as capable of affecting the organs of sight, also has an "intelligible appearance" (noèton eidos), but that certain "things" can have only an "intelligible appearance" (noèton eidos) and therefore not be "visible" (nor perceivable by the other senses). On the basis of resemblances ("same", tauton) and differences ("other", thateron) and disregarding the location in time and space in the case of sensible perceptions, each person associates names (most often pre-existing) with these eidè, and uses them to produce logoi, simply thought or audible. But Socrates does not stop there and is mainly interested in the different ways in which we interpret these "affections" and conceive what is at their origin, since we cannot know the "things" (pragmata) that affect us as they are "in and by themselves" (auta), but only through the "affections" (pathèmata) that they induce in us "filtered" by the "organs" that they affect, intelligence (noûs) included. From this perspective, he differentiates "affections" (pathèmata) not so much based on what is at their origin, but by the different manners in which the person who is affected envisions the "thing" (pragma) that is at the origin of this affection and expresses it, explicitly or implicitly, into logoi, in short, by the "state of mind" (one of the possible meanings of pathèma) of that person with regard to what affects him / her. And he distinguishes four ways in which a person can apprehend what affects him / her. Either the person considers that "things" (pragmata) are only what (s)he (or others) can see (segment of the seen / inside the cave), or one admits that there are also "things" "that cannot be seen otherwise than by thought (dianoia)" (511a1), such as for instance the square about which geometers demonstrate theorems, and that the role of intelligence (noûs, from which dianoia derives) is not so much to inventory the "things" (pragmata) that we see (with our eyes) in order to give them names as to bring to light the relationships between "things" (pragmata), whether seen or not, which allow us to progress toward understanding them (segment of the perceived by intelligence / outside the cave). For those who are in the first case, inside the cave, the words are of two kinds, those which refer to "things" (pragmata) seen, or at least visible, that is to say, that others have seen and that one could see by oneself if the circumstances allowed it, and those which refer to nothing that can be seen, therefore for them to no "thing" (pragma), and which therefore, for them, have meaning only through the various uses that are made of them, what Socrates calls nomima at Republic, V, 479d4, towards the end of the discussion on the distinction between knowledge (epistèmè) and opinion (doxa) and which he considers the proper object (in the Aristotelean sense) of opinion. Thus, if such a person hears the sentence "Theaetetus sits" (Theaitetos kathètai, Sophist, 263a2) which the Elean Stranger uses in the Sophist as an example of a minimal sentence carrying meaning whose truth or falsity can be put to the test among the people who witness the scene in which it is supposed to be spoken, for such a person, the word "Theaetetus" refers to a "thing" (pragma) that one can see, while the word "sits" does not refer to any "thing" (pragma) that one can see independently of the person to whom this qualifier is applied. This word therefore has for this person only the meaning(s) learned from others when learning to speak or, in later life, through discussions, but in all cases resulting from usage. And since some of the words one uses have meaning for that person only through the way they are used by others in their logoi or by logoi defining them in a dictionary, one cannot reach knowledge (epistèmè) when using them, but only express opinions (doxa) that might be called into question if usage were to change, since they refer only to logoi, not to "facts" (the primary meaning of the word pragma), to "beings" (onta), and this not because of the intrinsic nature of the "things" (pragmata) he speaks about as being subject to becoming, but because of that person's attitude towards them, one's refusal to consider as "things" (pragmata) what one cannot see. Such a person therefore does not see in the logos "Theaetetus sits" the connection between two distinct "things" (pragmata), "Theaetetus" and "sit", but the mention of a single "fact / thing" (pragma) "Theaetussitting", so that in the end he never sees "Theaetetus", but only "Theetetetussitting" or "Theaetetusstanding" or "Theaetetuslying" or "Theaetetusasleep" or..., and therefore cannot "know" what "sitting" is, but cannot "know" either what "Theaetetus" is apart from everything that participates in the sight of him at a given moment when seeing him, whether it is a permanent feature of him, such as the color of his eyes or hair (which can change with age), or an "accident" (in the Aristotelean sense), such as sitting or standing, or wearing a white or red garment that replaces the color of his skin in some places and hides his nakedness. In this first category of people, Socrates distinguishes two subgroups (sub-segments on the line / situations in the cave) depending on whether they consider that sight gives them a perfectly adequate apprehension of what they see and that therefore what they consider to be "things" (pragmata) is exactly as what sight reveals to them, without realizing that sight only gives them an "image" of it (first sub-segment of the seen, that of eikasia ("imagery") / prisoners immobilized in bonds "that [can] hold for the true nothing but the shadows of the implements" (515c1-2)), or that they admit that sight gives them only a partial and potentially misleading "image" of what they see in certain situations, reveals to them only the outer envelope, and above all that they cannot see with their own eyes everything they admit as "existing" and have to trust (pistis) other people to give them a more complete idea of the world around them (second sub-segment of the seen, that of pistis / prisoner freed from one's bonds but remaining in the cave). Among those who admit that there are also "things" (pragmata) "which cannot be seen otherwise than by reflection (dianoia)" (511a1) (segment of the perceived by intelligence / outside the cave), the distinction into two groups is made on the basis of their attitude toward words. To highlight these two possible attitudes, let us return to our minimal sentence "Theaetetus sits": for all these people, "sitting" refers to a pragma ("fact / thing") regardless of whether it is applied to Theaetetus or to anyone else, but to a pragma ("fact / thing") that is accessible only to intelligence (noûs). When we read these words in the Sophist, we imagine the scene: Socrates and his young namesake, Theodore, Theaetetus and the Elean Stranger are in a gymnasium, sitting on one or more benches and discussing. But let's imagine a different setting where this small group is in a modern hospital room where four of them have come to visit Theaetetus who is being treated there and that Theaetetus is in a hospital bed with a reclineable upper part. Does "Theaetetus sit" mean that Theaetetus has raised the reclining part of his bed to be in a position closer to sitting than lying? But from what degree of inclination should we say "sitting" rather than "lying"? In other words, what exactly does "sitting" mean? From there, two attitudes are possible: either we will try to define "sitting" more precisely by asking ourselves questions about the feet (can they still touch the ground when we are "sitting"?), the angle that the legs make with the thighs (are we still sitting when the legs are horizontal in the extension of the thighs, for instance, on an articulated hospital bed, or on a beach chair that can be reclined to the horizontal?), the angle that the thighs make with the trunk (at what angle are we no longer standing, but sitting, thinking for example of a sit-stand stool?), the part of the body weight that may still rest on the legs, the minimum surface area of the buttocks which must be supported on a load-bearing support, etc. (and it is easy to imagine a "Socratic dialogue" on this important question ending in aporia!), or we will try to get around the difficulty by reformulating the statement in a less problematic way that will allow us to obtain everyone's agreement on the de facto situation (pragma) of Theaetetus, for instance "Theaetetus is in his hospital bed". The first approach (to look for a more precise definition of "sitting"), which would be that of an Aristotle, is an approach that focuses on words and thinks that to understand is to make a rigorous reasoning on the right words, so that the word ends up taking the place of the "thing" (pragma) with which it is associated in the thought of a person having this approach (first sub-segment of the perceived by intelligence, that of dianoia / first stage of the prisoner exiting the cave, where one still cares only for the shadows and reflections that logoi are). The second approach, which is that of Plato's Socrates, takes note of the imprecision of language by understanding that it is made up of words that are not the "things" (pragmata) that they designate, but simple "labels" more or less arbitrarily chosen to which we give meaning by attaching them to fluctuating eidè that each person constructs and adjusts for this purpose in one's head and which are not even yet the "things" (pragmata) of which one seeks to speak and that it is therefore better not to obsess over words but always focus on the "facts" (pragmata), admitting that in many cases the "boundaries" between words, in fact between eidè, are blurred rather than seeking an inaccessible rigor in the fixing of these "boundaries" and forcing the "facts" (pragmata), in fact to freeze the eidè that we associate with the words with which we try to describe them, under labels that do not quite suit them, the "truth" not being in the words and logoi, but in the pragmata (facts / things) (the second segment of the perceived by intelligence, that of noèsis / epistèmè / the prisoner looking toward the *** themselves (ta auta) of the "things" seen in the cave and found in their intelligible dimension outside along with the stars in heaven). But above all, the person who is in the second approach knows that the logos was not given to humans simply to make an inventory of the shadows in the cave and stick labels (names) on them, nor even to reason outside the cave according to the specific problems we are faced with, but to find a path toward the good and that it is this idea of the good that must serve as a compass in all one's reasoning. Thus, in the case of the words "Theaetetus sits" used by the Elean Strange, his purpose in employing them was not to provide circumstantial information on the position of Theaetetus during the discussion, which is perfectly neutral with regard to the good and has no bearing whatsoever on the discussion itself, but to give a minimal example of logos whose truth could be acknowledged by all the participants in the discussion, as opposed to another example, "Theaetetus flies", the falsity of which could also be acknowledged by all present, in order to show that it is possible to hold a false speech (pseudo logos), a fundamental demonstration to make good use of logos, an essential requirement to be a good human being since it is precisely the fact of being endowed with logos that specifies one as a human being. And under these conditions, in the modern setting of a hospital that I proposed for the discussion, in any case imaginary, invented by Plato for his dialogue, if the sentence "Theaetetus sits" had raised objections on the part of one or more of the participants preferring to consider it as lying, he would have circumvented the difficulty by changing his example rather than embarking into a long discussion on the precise meaning of "sitting" and the "boundary" between "sitting" and "lying" which would not have brought anything good to the ongoing discussion on false speech, for instance by replacing it with "Theaetetus is hospitalized" which, in the supposed context, would have been considered true by all present.
We are therefore far from the simplistic image that opposes a visible / sensible world in perpetual becoming, not object of knowledge but only of opinion, to an intelligible world of pure ideas escaping becoming, which would be the only object of knowledge and of which the visible "world" would be only a pale "image". It is within the intelligible realm, that is to say, outside the cave, that the distinction between what is subject to becoming and what is not and the thought process that allows the intelligibility of both is made by intelligence. And the question of whether the sensible "world" is an "image" of an intelligible "world" can no longer be posed in these terms. To put it in the imagery of the allegory of the cave, do we mean that the statues of men (andriantai) that represent human bodies in the allegory are "images" of the moon or the star that represents in the heaven of the allegory the "idea of Man" (as suggested by the "simplistic" understanding conveyed by the supposed "theory of ideas"), or that they are each the "image" of the bearer / soul that animates it, represented in the allegory by the anthrôpos hidden by the wall that carries it, invisible in the cave and which becomes "visible", i.e., intelligible, only outside the cave, or that it is these human souls, only "visible" outside the cave, that is to say, in the intelligible, who are an "image" of the moon or the star which represents in the heaven of the allegory the "idea of Man", in which case the relationship between image and original, if there is one, is no longer between the visible and the intelligible, but within the intelligible itself?...
With these preliminaries in mind, it is now time to grapple with the text of the analogy.

(to previous section: The Good and the Sun)

[509c]...
And no way indeed, he said, (2) you shall stop here but at least, this likeness concerning the sun, expound it again, in case you omit [something] one way or another.
But of course, I said, I’m certainly omitting many [things]!
Well then, he said, don’t leave aside the smallest bit of it.
I suppose, said I, [it will] still [be] a lot. Nevertheless, in so far as [it is] presently possible, I won’t willingly omit anything.
Don’t indeed, he said.
[509d] Then think them, said I, as we were saying, to be two and to reign, one over an intelligible family and domain, the other in turn over a visible [family and domain] (3) – [I say “visible"] so that I don’t seem to you, in saying “heaven”, to behave like a sophist about the word. (4) But then, do you grasp these two appearances, (5) visible, intelligible? (6)
I grasp.
Well then, taking for instance a line (7) segmented into two unequal (8) segments, segment anew each one of the two segments (9) according to the same logos, (10) that of the seen family and that of the perceived by intelligence one, (11) and you will have, based on the [relative] clearness and lack of clearness (12) of the ones compared to the others, on the one hand in the seen, [509e] on the one hand one of the two segments: images (13) –I call indeed images, first [510a] shadows, then reflections on waters and on [other things] insofar as they are by design at the same time compact, smooth and bright, and everything of that kind, (14) if you understand [what I mean]. (15)
But [of course] I understand!
Then place the other one, to which this one resembles: (16) the living creatures around us, and all that is planted and the whole family of what is fabricated. (17)
I place it, he said.
Would you then be willing to say about it, said I, that it is divided with regard to truth and its absence [according to the following ratio(nale):] as the opined [is] to the known, (18) so what is made similar [to something is] to what it has been made similar to? (19)
[510b] Yes indeed, he said, absolutely.
Consider then also in turn the segmentation of the intelligible, (20) how it must be segmented.
How?
In such a way that one part of it [is where], using the [things] formerly imitated as images, a soul is constrained / constrains itself to investigate from supports, driven / advancing, not toward a (leading) principle, but toward an end, the other on the contrary [is] the one [where it is] by going toward a (leading) principle [which is] support of nothing from a support and without the images [revolving] around that, making for oneself with the appearances themselves the road map through them. (21)
Those [things] you say, he said, I don’t fully grasp. (22)
[510c] One more time, then! Said I. This way, you’ll grasp more easily what was said before. I think indeed that you know (23) that those who busy themselves in geometry and calculation (24) and the like, setting as supports for themselves  (25) the odd and the even and the [various] figures (26) and three appearances of angles (27) and other [things] akin to these according to each road map (28) those [things] on the one hand, [behaving] as [people] knowing, (29) making them supports for themselves, they don’t think fit to give a logos about them (30) either to themselves or to others, [510d] as if [it were] completely evident, taking them as starting principles on the other hand, going from beginning to end through all the rest, they end up (31) consistently (32) on what they had set their investigation in motion upon. (33)
Of course indeed, he said, that at least I know!
Thus also that besides, they also make use of the seen appearances (34) and make for themselves their logoi (35) on them thinking (36) not about them but about those they resemble, (37) making their logoi for the sake of the tetragon (square) itself, (38) of the diagonal itself, (39) and not [510e] of the one they draw, and same [thing] in the other cases, those very [things] they mold and draw, (40) of which there are shadows and reflections in waters, (41) using them in turn as images, but attempting [511a] to see (42) what cannot be seen otherwise than by thought. (43)
You tell the truth, he said.
So I indeed said intelligible this appearance, but the soul constrained / constraining itself to make use of supports in its investigation about it, (44) not going toward a (leading) principle, as if unable to rise higher than the supports, but using as images the very [things] that are copied by the ones below, and those in comparison with those because, [being] visible, [they are] held in esteem and honored. (45)
[511b] I grasp, he said, that you are talking about that [which falls] under the scope of geometry and the arts akin to it. (46)
Grasp then about the other segment of the intelligible, when I talk about what logos itself reaches through the power of the dialegesthai, (47) making (48) the supports not (first) principles, but really supports, like stepping stones and springboards, (49) so that, going all the way to that [which is] support of nothing, toward the (leading) principle of the whole, (50) having taken hold of it, (51) reaching further in turn to what can be reached from it, (52) it thus follows a downward path toward an end (53) [511c] without making also use in any way of anything perceptible by the senses, (54) but with appearances themselves, through them, into them, it ends also into appearances. (55)
I grasp it, he said, though definitely not sufficiently, for you seem to me to be talking about a long-term work, (56) but at least that you want to declare (57) as being clearer (58) what, among what is and also [is] intelligible, (59) is observed (60) under the guidance of the science of dialegesthai (61) from what [is observed] under that of what is called “arts”, (62) where supports [are] (first) principles and those who consider are indeed constrained / indeed constrain themselves (63) to consider these through thought (64) rather than through senses, but because they don’t investigate (65) by going back up to a (guiding) principle [511d] but from supports, they seem to you not to have intelligence (66) about those [things], despite their being intelligible with the help of a (guiding) principle. (67) And it seems to me that you are calling “thought" the state of mind (68) of those dealing with geometry (69) and that of those dealing with similar [things] rather than “intelligence”, thus thought being something intermediate between opinion (70) and intelligence. (71)
You have most sufficiently followed, (72) said I. And now, take with me (73) upon the four segments, those four affections / states of mind produced in the soul, (74) apprehension by intelligence (noèsis) first upon the highest, (discursive / wandering) thought (dianoia) [511e] then upon the second, to the third one then assign trust (pistis) and to the last imagery (eikasia), (75) and order them being led by this logos (76) that as the ***s upon which it is partake of truth, so these partake of clearness. (77)
I grasp it, he said, and I concur and order them as you say.

(to next section : The Allegory of the Cave)

(1) This page is an adapted translation of the French version dated April 17, 2023 (V4.1). The translation of Plato's Greek has been reworked directly from the Greek. For the translation of the introduction (completed April 8, 2026) and notes (completed August 7, 2026) in English, I worked with the help of Microsoft Word "Translate" function, but only as a starting point to be reworked and adjusted to the English. When the French version quoted other translations in French of the Republic, they have been replaced by quotations of translations in English and the comments had to be adjusted accordingly.
For a few comments of what I am attempting to achieve in my translations, see the page listing the translated excerpts of the Republic. There are certain words, such as logos, eidos, idea, that I prefer not to translate because they may have different meanings in different contexts, and thus warrant different English words to translate them in each case, but, when translating them, the reader no longer sees that they are the same word in Greek, a fact that Plato often plays with. (<==)

(2) The speaker here is Glaucon, answering Socrates who just accused him, half jockingly, to have been the reason for his verbal "hyperboles" when comparing the good and the sun (see the end of the previous section). (<==)

(3) The "two" referred to in this reply of Socrates are the good and the sun which was used in the previous section to give an image of it.
"To reign, one over an intelligible family and domain, the other in turn over a visible [family and domain]]" translates the Greek basileuein to men noètou genous te kai topou, to d' au horatou. Genos, of which genous is the accusative singular, is the noun derived from the verb gignesthai ("to be born / become"), which in Plato's time had a range of meanings spreading from the primary meaning of "birth" evolving towards that of "that which is born / becomes", individually ("offspring") or collectively ("family / race / tribe"), to more abstract meanings such as those of "class", "kind", "genus" or "species". If I translate it here as "family" rather than "class" or "species, it is because it is the translation that best preserves in English the link with the idea of birth without losing the abstract meaning no longer implying the idea of "birth" in the biological sense that it has when we speak for instance of "family of words" or "family of musical instruments". And we will see in the following that this extent of the range of meaning is important for understanding what Plato is talking about.
It should also be noted that the Greek has no articles before noètou genous te kai topou ("a family and an intelligible domain"), nor before horatou ("visible"), and that therefore not only are the words noètou and horatou not substantized (the intelligible, the visible), but are only qualifications of names, genous ("family, class") and topou ("place, domain"), which characterize them without individualizing them. Socrates is therefore not talking about the place / domain or the family / class, which would be, one of them, supposedly unique, intelligible, the other, also supposedly unique, visible, but of two qualifiers, noèton (« intelligible ») et horaton (« visible »), which can be applied to something that can be considered as a place or as a family / class over which "reigns" either the good, or the sun. This formulation is more open than the formulation with definite articles, which invites opposing the two places / families thus defined as being distinct from each other, the intelligible domain or family being distinct from the visible domain or family, while the absence of an article, rendered in English by indefinite articles, is more open to the possibility that it be the same domain or the same family considered from different points of view which imply different qualifiers.
With these words, Socrates continues to exploit the analogy between the good and the sun in an attempt to make us better understand in relation to what the good plays the role that the sun plays in relation to the visible "domain / place" that it illuminates, and which he qualified as noètos at 508b12-c2, using for the first time in the Republic this verbal adjective of the verb noein ("to perceive by the mind (noûs), think, reflect, conceive") which had never been used in previous dialogues (earlier in the order of the tetralogies that I present on this site), a word that I have translated, according to the usual translation, as "intelligible" but which could also be translated as "thinkable" taken in the sense of "capable of being apprehended by thought" (on this word and its translation, cf. note 79 to my translation of the section entitled "The Good and the Sun"). In this context, the most general term that had come to his mind to designate that which is illuminated by the sun and which, as a result, becomes "visible (horaton)", was the word topos, whose primary meaning is "place", and which I had then translated as "domain", and he had therefore spoken of a noètos topos, of an "intelligible domain" as being in relation to the good what the horatos topos, the "visible domain" is in relation to the sun, though he was aware that, if the notion of topos ("place") is fitting for the visible, it can only be used by analogy when talking of the intelligible, just as the verb "to see" is used by analogy to speak of "views" of the mind, of "ideas", a word that derives precisely from the Greek idea, a noun derived from idein, the aorist infinitive of a Greek verb meaning "to see", when a little later in the analogy, Socrates says about the "square itself" (to tetragonon auton), "the diagonal itself" (hè diamètros autè), that these are realities "what cannot be seen otherwise than by thought" (ha ouk an allôs idoi tis è tèi dianoiai, 511a1). And he had arranged his sentence in such a way that topos is explicit when it is qualified as noètos, where it is precisely problematic, and only implied when it is qualified as horatos, to which it is more adapted, inviting us to question the relevance of this term in relation to what is only abstractions indeed without links with time and space, in short, to arouse our wonderment, he who will tell us in the Theaetetus that the thaumazein ("to wonder") is the beginning of philosophy (cf. Theaetetus, 155d).
It should be noted, however, that, while it is not too difficult to understand that when Socrates speaks of a "visible place" (horatos topos), he refers to the space in which the things that sight allows us to apprehend are located, to the totality of what the sun illuminates and makes visible, the fact remains that a "place" (topos) as such is not "visible", not only because what we see are objects or people located in one or another place and not the place as a container, but also because it is not sight by itself that can give us the notion of a three-dimensional space, since what our eyes strictly speaking "see" are only patches of color in a two-dimensional "space" (in the same way as the film or the sensor of a camera, which is but a portion of a plane surface) and that it is only thanks to an effort of our mind that has become unconscious to us but which has required years of learning in the early days of our life that, thanks to the fact that we have two eyes and with the help of our other senses, touch in particular, neither of which perceives the "place" as such, and also to our ability to move, we have learned not to "see", but to "estimate" the distance that separates us from each of the objects we see and their respective positions in relation to one another, and therefore to become aware of the "depth" of the space that surrounds us and its three-dimensional character. In other words, the notion of "place" (topos) is itself a purely intelligible abstraction to which habit has led us to give a quasi-material "reality" that makes it impossible for us to think of anything other than as situated in a "place", even though it is only by abstracting from the position in space (and in time) that we can give names to what we see.
Here, Socrates, starting from the "dominant" position occupied respectively by the good and the sun in their own "domain", introduces a new image by means of the verb basileuein ("to reign (over)") which refers to the idea of kingship and assimilates the good and the sun to two sovereigns each reigning over his own "country". And this new image allows him to introduce another term to speak of what is under the dominion in one case of the good, in the other of the sun, that of genos, whose range of meaning I mentioned at the beginning of this note and which I translate as "family", so as to avoid us absolutizing a single way of speaking of it. Indeed, what most often characterizes a kingdom is the "territory" (topos, which can also be translated in such a context as "region") it covers and the "people" (genos, of which this is one of the possible meanings in such a context) that inhabits it. Now, as far as the visible /sensible is concerned, just as we ended up "objectifying" the notion of "place (topos)" to the point of no longer being able to imagine something that would not be in a "place", we have also "objectified" what "populates" this place to the point of having great difficulty in imagining that anything other than the material "objects" that "populate" it can exist.
Here again, Socrates, as was the case at 508b12-c2 with topos ("region"), constructs his sentence in such a way that topos ("place / region") and genos ("family / people / species") are explicit about the noèton ("intelligible") and implied about the horaton ("visible"). But the situation is not quite the same because if topos cannot be easily transposed into the realm of the intelligible, the same is not true for genos, which, as I said at the beginning of this note, already had, in Socrates' time, abstract meanings such as that of "class", "genus" or "species". Socrates thus makes us realize once again, if we take the trouble to think about it, that, as was already the case with topos ("place") in a less obvious way, we speak of the sensible and material world with the help of concepts, such as those of place and class, which are not data of our senses, but already purely intelligible abstractions, which belong to the "kingdom" of the good. And, in this parallel, he invites us to question the way in which, in line with what he did previously for sight in the light of the sun and knowledge in the "light" of the good, we could understand in an analogical way the concepts of place (topos) and "population" (genos), that is to say, in a way, of container and content, in the case of "elements" of the intelligible realm. But, as the ensuing discussion will show, if he evokes this question, which is spontaneously in everyone's mind so indissolubly linked for us to the question of "existence", it is to quickly do away with it to return to what was already his primary concern in the parallel between the good and the sun, which is not the existential perspective underlying these questions about the what and the where, but the epistemological one based on the fact that our perception of these two orders of "facts / things" (pragmata), the fact that we see and we think, is a given of experience. The main problem is then to understand how this can contribute to what is our only concern: to do what is good (agathon) for us, properly understood as composite beings made up of body and a tripartite soul (gnôthi sauton, "learn to know thyself"), in order to reach excellence (aretè) and true happiness. And it is not certain that knowing "where" the pure concepts that our mind perceives are located and what "matter" they are made of is a prerequisite to an appropriate answer to this concern, assuming that our intelligence is able to conceive it.
But, for those who are interested in these questions, we can note in conclusion that the fact that Socrates supposes an analogy between the "kingdom" of the sun and the "kingdom" of the good, if he does not tell us "where", in which topos ("place"), is the "kingdom" of the intelligible, suggests something to us about where it is not: by locating them too in a "kingdom", like the "objects" offered to view, it implies that they too have an "objective" reality and are therefore not solely in our heads, are not mere creations of our minds. But "objective" does not mean "material" and Plato's Socrates, like us, has an easier time saying what they are not than saying what they are. In the end, as everything that is to come will confirm, for Socrates, they "are" realities that solicit our intelligence, pragmata ("facts / things") that act (prattein, the verb from which pragma is derived) on it, which is, at least in a first time, passive in relation to them as sight is passive in relation to the visible realities that solicit it (hence the name pathèmata, "affections", which Socrates gives to what they produce in us, using the noun derived from the verb paschein, "to suffer" as opposed to prattein, "to act"). (<==)

(4) "[I say 'visible'] so that I don’t seem to you, in saying “heaven”, to behave like a sophist about the word" translates the Greek hina mè ouranou eipôn doxô soi sophizesthai peri to onoma. This part of the sentence coming immediately after "... on the visible (horatou)", which is an parenthesis offering a comment on a choice of word, implies something that calls for the hina ("so that"), which I have made explicit by adding in square brackets the words "[I say "visible"]" which are not in the Greek.
"To behave like a sophist" translates the Greek sophizesthai, a verb derived from the noun sophistès, of which the English "sophist" derives, and which was the qualifier by which the likes of Protagoras, Gorgias, Hippias, Thrasymachus and other Prodicoses were designated in the time of Socrates and Plato. The verb had a pejorative connotation, implying an idea of bad faith, fraud, in short of recourse to what are still called "sophistic arguments" today.
Socrates' remark concerns the two Greek words translated respectively as "visible" and "heaven": "visible" is horatou in Greek (genitive here as a complement of basileuein, "to reign over"), while "heaven" is ouranou (genitive for the same reason). Considering the sun, since it is the sun we are talking about here, as the "king" of heaven was not absurd in Socrates' time, and the two words horatou and ouranou are sufficiently similar (the h in horatou only renders in the Latin alphabet the rough breathing on the initial omicron of horatou, which did not exist in Plato's time, when breathings had not yet been invented to mark the aspiration or lack of aspiration of a vowel at the beginning of a word) to explain why Socrates, at Cratylus, 396b7-c3, derives the word ouranos ("heaven") from the verb horan, "to see" and this resemblance is at the origin of what could indeed appear to be no more than a pedantic pun (in the manner of a "sophist").
This being said, if it is normal from a certain point of view to make the sun the king of heaven, we may also think that what the qualification of "celestial" suggested by the association with heaven would more appropriately apply to is the "intelligible" rather than the "visible". Indeed, for most people, "celestial" evokes the divine since heaven is the abode of the gods, and, as such, is more related to the immutable, the eternal, and therefore the intelligible than to the constantly changing visible. This is what even the fanciful etymology of the Cratylus suggests, since by deriving the qualifier ourania, "celestial" (which is also the name of one of the nine Muses), which is given to the view one has "es to anô" (upwards), from "horôsa ta anô" (looking at things from above), Socrates associates it with the "pure mind (katharon noun)" that this sight provides. And indeed, the allegory of the cave that follows will use, as we have seen in the introduction, heaven where the sun is enthroned as an image, not of the noètos topos (the "intelligible domain") as a whole, but of the noblest part of it, the one where the ideai, that is to say the pure intelligibles without a sensible / material counterpart, are located and the anô (upwards) movement to describe the educational process that gives us access to them (thus, at 516a5, we find the expression ta anô to refer to what the prisoner leaving the cave sees, and we find it again at 517b4, and the adverb is used several times to refer to the goal of the ascension of the freed prisoner).
In fact, this mention of heaven is undoubtedly there to remind us of another image that Socrates gave, in the dialogue that precedes the Republic, of the "place" where we can contemplate what he did not yet call the noèta, the one he developed in his second discourse of the Phaedrus, describing the procession of souls who ascend to the top (pros akrôi) of heaven and, when it is that of a god, "having passed outside, rises on the back of heaven" (exo poreutheisai, estèsan epi tôi tou ouranou nôtôi, Phaedrus, 247b7-8) and can contemplate "the colorless and formless and impalpable beingness (ousia; on the difficulties of translating ousia, and the different possible translations, see note 103 to my translation of the previous section) which really is, to be seen by intelligence (nôi), pilot of the soul, alone, the origin (genos) of true knowledge" (hè achrômatos te kai aschèmatistos kai anaphès ousia ontôs ousa, psuchès kubernètèi monôi theatè nôi, to tès alèthous epistèmès genos, Phaedrus, 247c6-8) in "the supercelestial place" (ton huperouranion topon, Phaedrus, 247c3) which is its "domain" (touton echei ton topon, "it (the colorless, etc.) occupies this place", Phaedrus, 247d1), that is to say, to see in it justice itself (autèn dikaiosunèn), moderation (sôphrosunèn) and a knowledge that is not subject to becoming and dispersion in the multitude of beings in the world that is ours (Phaedrus, 247d7-e3). We find the same spatial vocabulary as here, with the use of topos (and also that of genos, which I have translated here as "origin", another possible meaning of the word), but with a much more developed imagery, which uses the other side of heaven, that is to say a "place" where we are no longer subject to the revolution of the stars measuring time and which is no longer illuminated by the sun, therefore outside the horatos topos, as an image of what is unrelated to time and space, and shows us that if we had to use an adjective evoking heaven to qualify the "place" of the noèton, it would be huperouranion ("supercelestial") and not ouranion ("celestial"). In other words, for Plato's Socrates, heaven is still part of the "visible". (<==)

(5) "Do you grasp these two appearances" translates the Greek echeis tauta ditta eidè. After topos and genos, Socrates introduces here a third term to which the qualifiers horaton ("visible") and noèton ("intelligible/thinkable") can be associated, and not the least, eidos (of which eidè is the nominative plural). Eidos, like idea, which has a very similar meaning, is a word derived from a root meaning "to see" which refers originally to "appearance" for sight of something considered individually, with meanings such a "form, shape, figure", evolving from there toward a collective sense (the "appearance" or "form", visual or otherwise, shared by a number of persons or things) to signify "kind, sort, type," senses in which it becomes almost synonymous with some of the meanings of genos, as shown by their almost interchangeable use by the Elean Stranger in the divisions of the Sophist and the Statesman. Another range of meaning, to which Plato may have contributed in part, is that of "form" and finally of "idea" as an abstract "form" in the mind. In Aristotle's "technical" vocabulary, genos translates as "genus" and eidos as "species", the species being considered as a subset of the "genus" whose members are distinguished from those of other species belonging to the same genus by a so-called "specific" difference (a word derived from the same root as "species"), precisely the one that is common to all the members of the species (thus, the genus "animal" includes the species "man", specified by the distinctive character "endowed with logos (logikos)").
Faced with the multiplicity of meaning of this word, the question arises as to the meaning in which it should be understood here and whether it should be understood as having the same or a different meaning from the one it takes in its other occurrences in the rest of the analogy (510b8, 510c4, 510d5, 511a3, 511c1, 511c2). The temptation is indeed great to understand it here in a "non-technical" sense close to that of genos that preceded it in the same sentence, and to translate it with a word like "type" (Shorey) or "kind" (Grube, Reeve) (Cornford and Lee translate it as "order of things"). The first problem with these translations, as well as with the translation by "form" used by Bloom, is that nothing is left in these words of the link with sight which is at the root of the Greek word, which may be important in a context where Socrates uses the analogy with sight to talk about the intelligible. But the biggest problem is that these translations, if they may be appropriate here, are harder to accept for other occurrences of eidos in the rest of the analogy, occurrences where most translators assume a "technical" meaning to this word and translate it as "form" or "Form" with a capital letter, or "idea". The result is that it is the translator who chooses when the word is supposed to have the "technical" meaning of the supposed "theory of forms/ideas" attributed to Plato and when it is used in a "usual" sense, to be defined in each case, and who imposes his choice on the reader (one can get an idea of the result of this way of proceeding by looking at the page of this site on "The vocabulary of the line" mentioned in the introductory note, where one will find, on a light pink background in the table, the translation of each of the 7 occurrences of eidos in the analogy of the line in the different translations I had access to). As far as I am concerned, I decided to use the same translation for all the occurrences of eidos in the analogy of the line and to choose a word that keeps a trace in English of the origin of the Greek word in relation with sight, and I therefore used the translation by "appearance" (derived from a verb whose first meaning is "to become visible") which is the primary meaning of eidos, from which the others are derived, and it so happens that, as I progressed in my translation and its commentary, this choice, which will seem heretical to "politically correct" Platonists for at least some of the occurrences of eidos towards the end of the analogy, opened up perspectives for me and ended up imposing itself on me and completely renewing my understanding of what Plato seeks to make us understand through this word.
What is happening here, it seems to me, is that with the word eidos, easier to understand in both realms, so much so that the Greeks of the time of Socrates and Plato were used to employing analogies with sight and the language of vision in the realm of thought, as we still do today (for instance with such expressions as "I see what you mean"), Socrates wants to focus the discussion on the problem of the different modes of perception by us of what is offered to our apprehension, that is, of the different "appearances" (eidè) under which what acts on us manifests itself to us, and more specifically of the two that play the greatest role in our path toward knowledge, sight and thought, in short, to return to the epistemological rather than existential perspective that was already his at the end of the parallel between the good and the sun. As I already hinted in note 3, this is what Socrates is most interested in, because this is what we all have personal experience of and it is this that must remain our main concern, to get to know ourselves, and therefore in particular to understand the power and limits which are ours as human beings, both in the realm of sight and in the realm of thought and intelligence.
This move from the "objective" point of view of visible or thinkable realities supposed to be located in a "place" (topos) and belonging to a "people" (genos) to the "subjective" point of view of the perception that we have of them by sight or by thought, is moreover marked grammatically by the difference in the tense of the verbs associated with these different terms and by the very choice of these verbs: the topos and the genos are mentioned as object complements in an infinitive clause in the present tense whose verb is basileuein ("to reign") and the subject autô ("(both of) them", a dual neuter accusative) complement of noèson ("think", imperative) which opens the line, that is to say in a grammatical context that hides as much as possible any notion of time and adopts the point of view of the sovereigns that the sun and the good are, while the eidè are introduced as a complement to the verb echeis ("grasp"), a verb conjugated in the present tense, in the second person singular, in an interrogative form that implicates the interlocutor by questioning him about his understanding of what Socrates is saying to him. Moreover, this echeis, a form of the verb echein, one of the meanings of which is "to have", is opposed to the einai ("to be") of duo auto einai kai basileuein... ("them... to be two and to reign...") and takes us from the sphere of being to the sphere of having, of the "possession" by our mind of something of these "appearances". But there is still more, regarding this verb! Indeed, echein, aside from its usual meaning of "to have", has a very broad range of meanings spreading from the very concrete meaning of holding in one's hands, as may be the case for what is visible, to the abstract meaning of grasping with one's mind, as may be the case for what is purely intelligible (a range of meanings which is preserved by the translation as "grasp" that I retained, the verb "to grasp" having this same range of meanings in English). Plato is not here simply stressing a difference between visible and intelligible; he, at the same time, by choosing a verb having such a broad range of meanings, wants to suggest in still another manner the continuity there is in the ways we "grasp" both "ranges" of perception through our senses and mind, a continuity which is also suggested, as I said earlier in this note, by the use of the single word eidos for both as opposed to the two words used at first, genos ("family / class") and topos ("place / domain"), one of which, genos, being more fit for the intelligible, the other, topos, more fit for the visible. And this continuity, which will soon be stressed by the use of a single line to represent both eidè, and later illustrated by the allegory of the cave, describing a continuous progress from grasping by sight inside the cave to grasping by the mind outside the cave, should preclude our understanding of these two realms (to return to the image suggested by the verb basileuein ("to reign over")) as being two distinct "worlds".
Socrates is therefore not speaking here of two "species / kinds / sorts" of objective "realities", one of which would be visible, the other intelligible, in an "existential" perspective of inventory and classification of these realities, but of two "appearances" that the realities take (the same or different realities, that is not the question for the moment, but the allegory of the cave will show that they can be the same) that offer themselves to our apprehension according to whether we perceive / grasp them by sight or by intelligence. We will soon see that, in the rest of the analogy, he will speak a few lines apart of horômena eidè ("seen appearances"; horômenois eidesi, plural dative, at 510d5) and noèton eidos ("intelligible appearance", at 511a3), which confirms this reading (and it is interesting to note in this connection that all the translators I had access to except Bloom translate these two successive occurrences of the same Greek word eidos as two different English words, thus depriving the reader of the means of seeing that Plato used the same word both times).
But then, an immediate consequence of this reading is that it is no longer possible to make eidè in the supposedly "technical" sense the ultimate realities, the only ones that really "exist". If, in fact, there is continuity of meaning when we move from the visible to the intelligible realm, if, in both "realms", eidè remain "appearances", for sight in one case, for the thinking mind in the other, then the noèta eidè, the "intelligible appearances" are no more the reality of what they are only "appearances" for our mind than the horômena eidè, the "seen appearances", are! We find here among commentators and translators over the ages a phenomenon similar to what happened with the word ousia, as I described it in note 103 to my translation of the parallel between the good and the sun: seeing Plato use a word that has one or more "usual" meanings in a supposedly "technical" sense, they establish watertight boundaries between this "Platonic" meaning and the "usual" meaning(s) of the word and they no longer seek to see how it was possible to pass from one meaning to the other and to preserve the slightest trace of these usual meanings in the supposedly "technical" sense. Here, when you think about it and forget twenty-five centuries of Platonism, the result is particularly disconcerting! It would have to be assumed, as I have already said at the end of note 86 to my translation of the parallel between the good and the sun, that Plato, seeking to make us understand by analogy with the visible the functioning of thought and the kind of realities it gives us access to, would have decided to use the same word, eidos, in the two realms by giving it diametrically opposed meanings in each of them: what has the least reality in the visible realm (the simple "appearance", insisting on its artificial and deceptive character) and what has the most in the intelligible realm (reality "itself")! No! It is more reasonable to forget all that we have learned about Plato and his supposed "theory of ideas", and to accept the idea that Plato uses eidos with a continuity of meaning between the visible and the intelligible: in both cases, it refers to an "appearance", that is to say, not to "reality" considered "as such in itself" (auto to ***) but to what is perceived by sight in one case, by the human mind in the other, each of these perceptual functions having its specific characteristics and limits as such (sight cannot be activated by sounds, smell by colors, etc.), but also more restrictive particular limits in each individual at a given moment of his existence (who may be or have become myopic as he grows older, or be colorblind, for instance, in the case of sight, narrow-minded, or downright mentally handicapped, for instance, in the case of thought). The important distinction between the different meanings of eidos is not between visible eidos and intelligible eidos, but between the individual meaning concerning a single person or thing, which leads us toward the idea of "image" (eikôn), and the collective meaning in which the word designates something which is common to a plurality, whether this something is of a sensible or intelligible nature.
In fact, to see more clearly what Socrates means by eidos, it is necessary to refer to what he says as a prelude to the discussion on the different kinds of couches/beds at the beginning of Book X of the Republic, that is to say as a prelude to a discussion supposed to shed light on the meaning he gives to eidos and idea and the differences he makes between these two terms: "We are, methinks, in the habit of positing for our own use some eidos, unique in each case, for each of the pluralities upon which we impose the same name" (eidos gar pou ti hen hekaston eiôthamen tithesthai peri hekasta ta polla hois tauton onoma epipheromen, 596a6-7). In what Socrates presents as not his but "the usual way" of proceeding (ek tès eiôthuias methodou, 596a5-6), he establishes a close connection between eidos and onoma ("name / word"): an eidos is something that is common to a plurality to which we associate a name. And it is necessary to pay attention to all the words of this sentence. "We are in the habit": Socrates is not talking about something that would be the exclusive practice of a few rare individuals who created the words we use, one of those that, in the Cratylus, he would occasionally call onomatourgos ("name-maker", Cratylus, 389a1), onomatôn thetès ("giver of names", Cratylus, 389d8), dèmiourgos onomatôn, ("crafter of names", Cratylus, 390e2, 431e1-2), or ho themenos prôtos ta onomata ("the first to have given names", Cratylus, 436b5), ton tithemenon ta onomata ("he who gives the names", Cratylus, 438a4-5), and most often nomothetès, (20 occurrences in the Cratylus) which means in the etymological sense "custom / law (nomos) giver (thetès)", an allusion to the role of "custom / law" that language plays for us, but of something that is a habit for "all of us" over time, since the verb eiôthamen is in the first person plural of the perfect of the active indicative, the perfect being the tense of the action begun in the past which continues in the present. It is not the occasional activity of someone who invents a new word from time to time, invents a neologism (as Plato occasionally does, in particular in the Sophist, and no doubt also in the Cratylus, in particular with the word onomatourgos ("name-maker", Cratylus, 389a1)), but of the most often unconscious activity that we all practice since childhood every time we learn a new word and continue throughout our lives for the words we know as their meaning becomes clearer or adjusted for us, and therefore the eidos we associate with it evolves in our mind. For this eidos, Socrates does not say that we receive it once and for all, the same for all, as if it fell from heaven coming from some transcendent "reality" that would impose itself on us and that we would only have to "look at" to grasp it in a perfect and definitive way, but that we are used to "positing" it (tithesthai), by using the middle (the voice that indicates the active involvement of the subject in the action) infinitive of the verb tithenai, whose primary meaning is "to put / set / place", and hence "to institute / establish", or else "to give (a law)", which is the verb that is found in almost all the expressions used by Socrates in the Cratylus to speak of those who "give" names, and in particular in the -thetes of nomothetes ("lawgiver"). In short, it is each one of us who not so much constantly "invents" names as "posits for oneself" eidè that he associates with the names that, most often, he learns from others, and who adjusts these eidè during all one's life according to one's growing experience.
Should we deduce from this that these eidè are purely subjective and different for each person? Yes and no. Yes, in that, indeed, for each one, the eidos that one "posits for oneself" (the meaning of the middle form tithesthai) in relation to any name at a given time in his life depends on one's own experience up to that point, which is personal and different for each one, and which also evolves for that person, which explains why it may happen that we don't understand one another even when we speak the same language. No, insofar as this "appearance" (eidos), whether visible / sensible or intelligible, is that of "beings" (onta) which "are" what they are independently of the way in which each person perceives them and that they therefore "condition" what each person may more or less accurately perceive at a given time in one's life according to the capabilities of one's sensory organs and one's intelligence at that time, organs and intelligence that are "built" on the same model for all. In short, these eidè have the same objective origin, the pragmata ("activities" in the primary sense, thus presupposing "activators / actors / agents") and are apprehended by the same "organs" (sense and intelligence), different from one to another only by the degree of perfection they have reached, not by the principles of functioning that determine their proper object, color, sound, etc.), which ensures that, in at least a large number of cases, we are able to understand one another and cooperate productively through dialegesthai ("the practice of dialogue"). And if we no longer understand one another in some cases and cannot agree on the meaning of certain words, especially in the case of abstract words such as "beautiful", "just", "good", etc., and we do not manage to agree on the good (to agathon), which is the light that must enlighten us on everything else (beautiful, just, etc.), even if we admit, at least implicitly, its objectivity (cf. Republic VI, 505d5-506a7), this does not mean that, for these words, there is nothing behind them, but that the effort to understand what they cover is greater, precisely because they do not act on the senses, but on the mind (noûs) only. There is therefore in each case a target that may be called "objective", which is what a perfect human being, whose sensory organs and intelligence would be at their point of perfection, could apprehend, and this target is what Plato calls idea and represents by the stars in heaven in the allegory of the cave. But for each one of us, the eventual access to this remote and inaccessible target by mere words presupposes a progression in time, from a purely visual and sensitive apprehension to a purely intelligible apprehension freed from all references to the sensible and therefore to (visual) images. For Plato, the idea is the principle of intelligibility of what it is an idea of, but remains an "appearance" for each one of us, and not yet the "*** itself" (auto to ***) of which the idea is only the intelligible appearance for a human mind, who cannot apprehend "beings" as they are, or at least cannot, by nature, know if this is the case: if human beings were not endowed with the sense of smell, we could not know that we do not perceive certain characteristics of flowers. The human mind can only reason with the help of words, which do not guarantee that we can name everything and therefore understand everything: if we did not have a sense of smell, we would not have invented words to talk about smells and therefore we would not be able to understand this aspect of reality. It follows, therefore, in particular that hè tou agathou idea ("the idea of the good") is not auto to agathon ("the good itself"), as I explained in note 2 to my translation of the parallel between the good and the sun. What follows will show us the new and fruitful perspectives that this way of understanding eidos et idea opens up for a better understanding of what Plato seeks to make us understand.
That eidè have to do with the problem of naming is confirmed by what the Elean Stranger says at Sophist, 259e4-6, when he states that "[it is] by means of the intertwining of eidè with one another [that] logos happens for us" (dia tèn allèlôn tôn eidôn sumplokèn ho logos gegonen hèmin). In saying this, in fact, he wants to make it clear that it is the eidè that we associate, unconsciously or consciously, with the words we use that carry the meaning we give them, and not the words themselves, which are in the end only phonic (spoken language) or graphic (written language) phenomena chosen by mere convention to designate what they designate, as shown by the fact that they change from one language to another to designate the same thing. It is therefore indeed relations between eidè that are represented by the assemby of words that make up our logoi, which are only simple linear juxtapositions and not "intertwinings" (to use the word used by the Stranger), and determine their meaning or absence of meaning, that is to say their status with regard to truth. The eidè that each person associates at some point in one's life with the words one uses may change over time along with one's increased experience, in particular as a result of one's practice of dialegesthai, allowing one to confront one's understanding with that of others and to make it evolve accordingly, but the ideai that are the targets do not change, no more than the starry heaven which is their image in the allegory of the cave, in which the position of each star in relation to all the others does not change, with a few rare exceptions. And it is these immutable relations between ideai / stars that the relations we establish between eidè seek to represent by juxtaposing words one after the other in logoi.
This analysis of the terms chosen by Plato to put them in the mouth of his Socrates makes us perceive that, if Plato is opposed to any "technical" vocabulary because he knows that, as a matter of fact, many words have several meanings and very often several words can be used to designate the same thing, or very similar concepts, and that it is more realistic to make do with this state of affairs than to seek to impose one's law on words and to restrict their meaning in order to make them unequivocal and exclusive of one another (see on this point the remark made by the Elean Stranger to Socrates the Younger at Stateman, 261e, or what Socrates says to Theaetetus in Theaetetus, 184c), this does not prevent him from taking extreme care in the choice of words he uses to best play with their evocative power and their ability to raise images, making the work of translators particularly difficult, at least when they are aware of this state of affairs, as the case of eidos in particular shows. (<==)

(6) By moving from the "objective" point of view, which is that of the topos ("place / domain") and genos ("family / people"), to the "subjective" point of view of eidè ("appearances") which refer to the mode of apprehension by the observer (see previous note), Socrates reverses the order in which he mentions the two realms of reality: as long as Socrates was considering "objectively" the "kingdom" of the good and using that of the sun only as an image of the first one, he was talking about the noèton ("intelligible") before talking about the horaton ("visible"), but as soon as he assumes the point of view of the perceiving subject, he starts from the most accessible to go towards the most problematic, that is to say from the visible, easier to apprehend, to progress towards the intelligible, which will be the order in which all the rest of the analogy will develop, and also the whole allegory of the cave.
If we now take an overview of this reply of Socrates, we can note that it began in Greek with the word noèson ("conceive, perceive with your mind, think", an active aorist imperative in the second person singular of the verb noein), which invited Glaucon to make use of his noûs, of his mind, of his thought, of his intelligence, to try to understand what the intelligible / thinkable is, that is to say to practice what he is trying to apprehend, and ends with the word noèton ("intelligible / thinkable", verbal adjective of the same verb) because of the choice made by Plato's Socrates to invert the order in which he has so far mentioned the two domains (it is to make perceptible in my translation this assonance between the first and the last word of Socrates' line, which differ in Greek only by one letter, sigma in noèson, tau in noèton, which I have translated here noèton as "intelligible / thinkable" and not simply as "intelligible", the added "thinkable" echoing the initial "think").
By enclosing, so to speak, this entire reply in thought, Plato's Socrates wants us to realize that, in the end, it is through noûs, thought, intelligence, that we can apprehend even the visible and "understand" it. And indeed, the eidè in the primary sense, the "(external visible) forms" which we think of as apprehended directly by the eyes, are in fact not given to us by sight, which, as I have already implied in note 3 about the notion of "visible place", gives us access only to constantly changing patches of colour, but by an operation of the mind (noûs) working on the raw data of sight with the help of other senses such as touch to help discern "forms", schèmata (another Greek word with a meaning close to that of eidos and idea, which will be found later in the analogy about geometers, in the specialized sense of "figure"), around these patches of diverse colours, so that the real split is not between visible, quite real and tangible eidè on the one hand and abstract eidè on the other hand, but between the raw data of the senses which have no meaning in themselves on the one hand and the eidè abstracted by our noûs ("mind, intelligence, thought") from these data from the senses or from some other source on the other, but which are indeed eidè in both cases (on this question, see my article (in French) "De la couleur avant tout chose. Les schemas invisibles du Ménon", published in 2010 in issue 14 of the online philosophical journal Klèsis, also accessible on this site by clicking here). This is what is in the background of Socrates' question asking Glaucon if he has indeed "grasped" these two eidè, the two "appearances" that form in both cases in our minds, and not some in our eyes and others in our mind. And it is not certain that Glaucon saw all the implications of this seemingly innocuous question, but which is ambiguous in Greek because of the different meanings that can be given to eidè (see previous note), an ambiguity that is lost in English as soon as the translator makes a choice between these different meanings: it is in fact not the same thing to say "Do you understand these two kinds [of things], visible, intelligible / thinkable?" or to say "Do you understand these two appearances, visible, intelligible / thinkable?" because, in the first case, we are in the opposition between two groups / families / categories / kinds of "things" taken as a whole, the visible kind as a whole and the intelligible kind as a whole, and the word eidè has a minimal weight to the point that it could almost be deleted without practically changing the meaning of the question (as indeed does Jowett), while the second formulation puts all the weight on the word eidè and invites us to consider each eidos individually in order to know what qualifier is appropriate for it, without excluding the possibility that the same "thing" may give rise to both a visible eidos and an intelligible / thinkable eidos. In short, in the first case, we have two qualifiers, visible (horaton) and intelligible / thinkable (noèton) that we associate with two orders of realities that we try to distinguish without getting too attached to the word that designates them collectively (in Greek, genos could just as well have done the trick, and in English, words as neutral as "kind [of things]" are quite appropriate), to the point that one could almost leave it implied so as not to have to choose, whereas in the second case, one is specifically interested in the term eidos and wants to make it clear that it can cover a wider set than its origin and its primary meaning of exclusively visible appearance suggests, without implying that the eidè of each category necessarily come from distinct realities, and even less so that eidos might be the term designating "realities" belonging to one of the two categories to the exclusion of the other. If indeed eidos is the important term here, to which either the qualifier horaton or the qualifier noèton can be applied, it means that eidos is not the specific term to speak of realities in the realm of the noèton, since, in the realm of the visible, Plato's Socrates is able to make the difference between an eidos in the original sense of "appearance perceived by sight" and what gives rise to it, and that he would therefore not have chosen this term to speak of the realities of the intelligible realm themselves rather than the perceptions that we have of them through our mind, supposedly distinct from the realities not visible to the eye that are at the origin of them. In short, either eidè is used here in a sense that calls for a complement, left implicit here ("kinds / sorts / species of what?"), which, by default, is replaced by as general a term as possible ("two kinds / genera / sorts / species of things"), and this means that a few lines apart, Plato makes his Socrates use the same term to designate a class (eidos understood here in the sense of "kind, sort [of thing]") and the elements of which it is composed (eidos used in the explanations that follow about the division of the segment of the noèton to designate the individual perceptions on which our mind reasons), which doesn't facilitate understanding, or else eidè is taken here in a sense that suffices in itself and does not call for an implied complement, the primary meaning of "appearance" and the question concerns the different ways in which something, anything, can appear to us, a meaning that is consistent with its other uses in the rest of the analogy, son long as we do not try to give it a "technical" meaning handed over by tradition in total contradiction with its usual meanings.
One may then wonder if Socrates' incise on the "puns" that sophists could make on horatou by comparing it to ouranou, is not in fact intended to preemptively draw the attention of Glaucon, the rest of the audience, and us readers, to a much more "dangerous" word that he will use immediately afterwards, namely the word eidè. The considerations of the Elean Sranger in the Sophist on the "fight of giants" (gigantomachia, Sophist, 246a4) between those whom he calls respectively "friends of eidè" ("tous tôn eidôn philous", 248a4) and "sons of the earth" ("gegeneis", 248c2) suggest that, in Socrates' time, the word eidos might have been used by certain thinkers and sophists in the sense that some would like it to have today in the supposed "theory of forms / ideas" that they believe is to be found in Plato, precisely those whom the foreigner qualifies as eidôn philous, "friends of eidè" (eidôn is the genitive plural of eidos). Socrates would then be here whispering to Glaucon something like "I warn you, my dear Glaucon, I am not playing with words like those sophists who play with horatou / ouranou, nor like those who distort the meaning of eidos to make it say the opposite of what it means when they turn toward "heaven"!"
Having come to the end of this reply of Socrates, and in continuity with the previous note, I would like to show how difficult it can be, for the reader who has access only to translations, to perceive all the subtleties that Plato has slipped into his text and some of which I have presented in the preceding notes. Indeed, most translators don't realize how careful Plato is in choosing his words, even though he does not freeze them in a "technical" sense, and no doubt precisely because of this, and in constructings his sentences, using all the resources of grammar and adjusting the progressiveness of the insights he projects on the subjects under discussion, which makes them take liberties in translating with what they see only as stylistic effects, as will be seen by examining the translation of these two sentences by the various translators I had access to, the complete Greek text of which is:
Noèson toinun, èn d' egô, hôsper legomen, duo autô einai, kai basileuein to men noètou genous te kai topou, to d'a au horatou, hina mè ouranou eipôn doxô soi sophizesthai peri to onoma. All' oun echeis tauta ditta eidè, horaton, noèton;
- Jowett: "You have to imagine, then, that there are two ruling powers, and that one of them is set over the intellectual world, the other over the visible. I do not say heaven, lest you should fancy that I am playing upon the name (ouranos, horatos). May I suppose that you have this distinction of the visible and intelligible fixed in your mind ?" Jowett "translates" the expression genous te kai topou as a single word, "world", which is a possible translation of neither one, and strongly orients the mind toward the idea that the visible and the intelligible (adding two definite articles which are not in the Greek) are two completely distinct "worlds". But saying that the two are in different "places", "positions" or "domains" (some of the possible meanings of topos) don't imply that they are in different "worlds" with no communication between both. And in its translation of the second part, he doesn't even translate eidè but, here again, adds articles before "visible" and "intelligible", which reinforce the idea that "the (explicit) visible" and "(the, implicit) intelligible" between which a "distinction" (a word which doesn't translate any Greek word in Plato's text) should be made, are indeed two mutually exclusive "worlds", thus erasing the ambiguity of Plato's wording. The word "distinction" which I just mentioned stems from his translation of echeis as "you have the distinction of, rather than a mere translation as "you have", but this completely misses the point Plato was trying to implicitely make by carefully choosing this specific word here due to its broad range of meanings spreading from an "objective / factual" "you have" to a "subjective" "you grasp", implying an active participation of the subject being addressed. By talking of having a distinction, Jowett sides with an "objective / factual" understanding of the word, and thus, with a distinction between two separate "worlds".
- Shorey: "Conceive then, said I, as we were saying, that there are these two entities, and that one of them is sovereign over the intelligible order and region and the other over the world of the eye-ball, not to say the sky-ball, but let that pass. You surely apprehend the two types, the visible and the intelligible.", with a note on "sky-ball" which reads "Cf. the similar etymological pun in Cratyl. 396 B-C. Here, as often, the translator must choose between over-translating for some tastes, or not translating at all." Shorey seems to have been more interested in finding an English equivalent of the pun alluded to by Socrates than in being accurate in the important part of the sentence: he translates topos as "region", which is acceptable, and genos as "order", which is very remotely a possible translation of it, completely erasing the link of the word to the notions of birth and family, and reverse the order of the two words in Greek, and translates the single Greek word horatou by "over the world of the eye-ball" in order to prepare for his translation of ouranou by "the sky-ball", so as to propose an English equivalent of the supposed Greek pun. But in so doing, he introduces concerning the visible the word "world", supposed to be another translation of the word topos, which is only implied here in the Greek, and thus implies that the important word in this description is topos, understood as "world", thus implying that the intelligible and the visible (with definite articles which are not in the Greek) are two distinct "worlds". Besides, he introduces the word "entities", which translates no Greek word, but only explicits what is implied by the pronoun autô and thus refers to the good and the sun, but without any added value. And in the second part, he translates eidè as "types", implying that he doesn't understand it in a supposedly "platonic" technical sense but in one of its "usual" meanings, and adds, here again, definite articles before "visible" and "intelligible", reinforcing the idea that they are two distinct "worlds" and erasing the ambiguity of Plato's wording between an understanding of eidè as refering to the different ways we perceive what acts upon our senses and mind, which may be the same in some cases, and an understanding which implies that what we "perceive" is of different "types" for the visble and for the intlligible.
- Cornford: "Conceive, then, that there are these two powers I speak of, the Good reigning over the domain of all that is intelligible, the Sun over the visible world-or the heaven as I might call it; only you would think I was showing off my skill in etymology. At any rate you have these two orders of things clearly before your mind: the visible and the intelligible?", with a note on "my skill in etymology" which reads "Some connected the word for heaven (ouranos) with horan 'to see' (Cratylus, 396 B), It is sometimes used for the whole of the visible universe." Cornford doesn't translate genos and translates topos as "domain" with regard to the intelligible, but explicits it regarding the visible with a different translation, talking about the visible world, adding in both cases a definite article which is not in the Greek. Besides, he breaks the symmetry between the two "families and domains" by talking in the first case of ""all that is intelligible" and in the second of "the visible", "all that is" translating no Greek word or words. And his translation of eidè as "orders of things" compounded with the addition of definite articles before "visible" ("the visible") and "intelligible" ("the intelligible"), and his translation of echeis by the factual "you have before your mind" (these last three words added to the Greek), makes it impossible to envision that some "things" might be both visible and intelligible.
- Lee: "'You must suppose, then,' I went on, 'that there are these two powers (with a note saying: "The form of the good and the sun") of which I have spoken, and that one of them is supreme over everything in the intelligible order or region, the other over everything in the visible region – I won't say in the physical universe or you will think I am playing with words. At any rate, you have before your mind these two orders of things, the visible and the intelligible?'", with a note on "playing with words" saying: "The Greek words for 'visible' and for 'physical univers' (or more literally 'heaven') bear some resemblance to each other, and it has been suggested that there was some connection between them", and another note on "orders" saying: "Eidos: good example of Plato's non-technical use of the term, to mean 'kind', 'sort', 'type' (as also at 511a, 'type of thing'). The technical (theory of 'forms') use is a natural sequel because things of a particular kind have a particular form." Lee transforms the "and" (te kai) between "order" (his translation of genos) and "region" (his translation of topos) into an "or" and explicits "region" only after "visible", and he forces the understanding of the second part of the line toward the identification of two distinct "orders" (his translation of eidos) of things, the (added) visible and the (added) intelligible, that we "have before [us]", a translation of echeis which precludes a more subjective understanding of the word in the register of perception. And eventually, he explicits in the note the distinction between a "technical" and a "non-technical" understanding of the word eidos and forces the reader to understand it, here and at 511a3, in the "non-technical" sense, even though he attempts to explain how the "technical" meaning derives from the "non-technical".
- Bloom: "Well, then, I said, conceive that, as we say, these two things are, and that the one is king of the intelligible class and region, while the other is king of the visible. I don't say "of the heaven" so as mot to seem to you to be playing the sophist with the name. Now, do you have these two forms, visible and intelligible?", with a note on "to be playing the sophist with the name" which reads: "The genitive form of the word for visible, horatou, is, in the Greek orthography, very close to that for heaven, ouranou. The last syllable of ouranou is identical with the genitive of the word for intelligence (noûs), which is the root of the word for intelligible (noèton). (Cf. 494d and note 13; 614b and note 13). There is, hence, a play on the words for heaven·and visible; and, in addition, the question is raised, on the basis of the words, as to the relation of the heavens to the intelligible world (cf. 528e ff.). According to Cratylus (396c), the astronomers say that ouranos received its name from intelligence, and rightly so." Bloom explains in his preface that he has made a rule of "in so far as possible always us[e] the same English equivalent for the same Greek word", which is a mixed blessing regarding a problem that has no good solution, as the case of the word logos demonstrates, for which he can't abide by his rule, translating it variously as "speech", "argument", "reason", "ratio", "account"... As a result, here, he translates eidè as "forms", as he does for all its occurrences in the analogy, and, more generally, in the whole Republic. He stays very close to the Greek except that he adds definite articles that are not present in the Greek when he write "the intelligible class and region" and "the visible" in the first part of the line (but he doesn't do it at the end). And his translation of duo autô einai as "these two things are", with a stress on "are" used absolutely forces an existential flavor where there is none if the words are more naturally translated as "them to be two".
- Grube: "Understand then, I said, that, as we say, there are those two, one reigning over the intelligible kind and realm, the other over the visible (not to say heaven, that I may not appear to play the sophist about the name! ). So you have two kinds, the visible and the intelligible.", with a note on "play the sophist about the name", which reads "He means play on the similarity of sound between ouranos, the sky, and horaton, visible." Grube adds definite articles that are not present in the Greek when he write "the intelligible kind and realm" and "the visible" in the first part of the line, and also when he writes at the end "the visible and the intelligible", and he translates both genos and eidè as "kind(s)".
- Grube/Reeve: "Understand, then, that, as we said, there are these two things, one sovereign of the intelligible kind and place, the other of the visible (l don't say "of heaven" so as not to seem to you to be playing the sophist with the name). In any case, you have two kinds of thing, visible and intelligible", with a note on "playing the sophist with the name", which reads "The play may be on the similarity of sound between ouranou ("of heaven") and horatou ("of the visible"). More likely, Socrates is referring to the fact that ouranou seems to contain the word nou, the genitive case of noûs ("understanding"), and relative of noètou ("of the intelligible"). If he said that the sun was sovereign of heaven, he might be taken to suggest in sophistical fashion that it was sovereign of the intelligible and that there was no real difference between the good and the sun." The most noticeable changes introduced by Reeve on Grube translation regarding the problems we are dealing with here are the addition of "things" after « those / these two" and after "two kinds" at the end and the replacement of "realm" by "place" as the translation of topos, all of them pulling the translation toward a more "concrete / materialistic" understanding.
- Reeve: "Then you should think, as we said, that there are these two things, one sovereign of the intelligible kind and place, the other of the visible-- I do not say "of the heavens," so as not to seem to you to be playing the sophist with the name. In any case, do you understand these two kinds, visible and intelligible?", with a note on "playing the sophist with the name", which reads: "The play seems to be on the similarity of sound between orano (sic) ("the heavens") and orato ("visible")." In his translation of 2004 under his sole name, Reeve gets rid of the definite articles before "visible" and "intelligible" at the end (but not in the first part of the line: "the intelligible kind and place", "the visible") and translates echeis as as "you understand" rather than "you have", giving this verb a meaning associating it with the activity of the mind in a context where what is in discussion is precisely the distinction between the activity of the senses on material things, or rather, on the sensible dimension of material things, and the activity of the mind on everything it can grasp, whether sensible or purely intelligible. (<==)

(7) To make it clear that the problem of "where?" is not the one that interests him, Socrates will build his analogy from the most minimalist "place" that can be imagined, a line, the simplest and most rudimentary geometric (and therefore "schematic") figure that allows him to distinguish different positions (the point, the only figure that is even simpler does not allow it): not even a plane and still less a volume, but a figure having only one dimension, and which, as a result, poses as many problems to "fit" the three-dimensional visible into it as to "align" the dimensionless intelligible into it!
The only thing that this choice tells us, but it is important, is that there is continuity between the two orders since we start from a single line. As for the geometrical appearance that he will thus give to his analogy, it should not deceive us: it is there less to suggest a certain rigour in the reasoning than to make it easy to grasp for the listeners starting from a schema ("figure", in particular in the geometric sense) as simple and as easy to represent mentally as possible, even for children, that is to say of the "embryo" of eidos reduced to its simplest expression (on the idea that the schemata, in the sense of "geometric figures", are in some way the most elementary versions of eidè, which explains why Socrates, in his discussion with Meno, in an attempt to make him understand what he is looking for when he speaks of a single eidos common to all the aretai ("forms of excellence", rather than "virtues") of man, chooses the word schèma to give him an example of a definition, see my article (in French) already cited in the previous note "De la couleur avant tout chose. Les schemas invisibles du Ménon", published in 2010 in issue 14 of the online philosophical journal Klèsis, also accessible on this site by clicking here). We will also see in the continuation of the Republic, in Book VII, that, in the training program proposed by Socrates to select the future leaders of the city, geometry comes in second place, just after arithmetic and far before dialectics. To use an analogy that appeals to the simplest of geometric "figures" is therefore to consider his young interlocutors as mere beginners, except that he does not, strictly speaking, practice geometry, but merely uses a geometric analogy in an attempt to make them understand something of what is the ultimate object of dialectics, the idea of the good, that is to say, tries to make them understand by example how this object of study that is geometry can lead to what is the goal of this whole training program (cf. Republic VII, 531c9-d4, at the beginning of the section devoted to dialectics). While putting himself within their reach, he thus gives a chance to his interlocutors (and Plato to his readers) to rise above the (geometric) dressing of the analogy in order to progress in the understanding of the idea of the good. The multi-millennial experience seems to prove that it is not easy not to be trapped by appearances. (<==)

(8) I explained in the introduction what was the problem with this anisa ("unequals") which has been disputed since antiquity, but for which no one to my knowledge has given a satisfactory explanation any more than the proponents of the isa ("equal") reading have given a satisfactory justification for their choice. For those who are interested, I keep here the translation of a slightly modified version of the note that appeared in the previous version of this page in French, which states the problem and the different proposed readings before examining the reasons that could justify chosing one or the other for the commentators who did not see that the key to the problem was given by the allegory of the cave properly understood, that is to say, all of them.
The Greek text translated as "unequal" is anisa. This is the lesson given by manuscript A (Parisinus), one of the best for the establishment of the Greek text of the dialogues; but another manuscript, the Vindobonensis F, gives the lesson an, isa, the an giving a hypothetical character to the verb that precedes it, and the isa qualifying as "equal" the segments which, in the other lesson, are unequal. This lesson is more difficult to accept, not so much because it reverses the meaning, passing from unequal to equal segments, but because it isolates between commas the part of the phrase isa tmèmata, "equal segments", whose role in relation to what precedes and what follows becomes difficult to determine. On the other hand, another possible division (let us not forget that in Plato's time, texts were written without punctuation and without spaces between words, like an unbroken sequence of letters), adopted by Stallbaum, is an' isa, where an' is an ana elided before the initial vowel of isa. In this reading, an' isa tmèmata would be translated as "according to equal segments", or, with the context, "taking, for example, a line segmented in two, according to equal segments". Some editors (Ast) even go so far as to remove the an altogether, which leads to "taking for example a line segmented into two equal segments".
The scolia (marginal notes by scholars dating back to the first centuries of our era and which, from a certain point on, began to be copied in the margin of the text from manuscript to manuscript, at the same time as the text itself) on this passage show that the debate on this word goes back to antiquity (see the note ad loc. in Shorey's edition for Loeb and the notes in French in R. Baccou's translation, Garnier, GF90, and in E. Chambry's edition for Budé); thus, for instance, among the Neoplatonists, Iamblichus was reading isa, and Proclus, anisa.
If we admit that the text written by Plato did indeed bear the letters alpha, nu, iota, sigma, and alpha, the question is whether we should make it a single word, anisa, or two, an' isa. If, in order to assess the linguistic plausibility of the second option, we examine the uses of ana, of which an' is the elided form, in the rest of the dialogues, we find that this word is rare and that, apart from its use in a quotation from the Odyssey at Republic, III, 387a7, it is not to be found, in the 28 dialogues that I include in my tetralogies, only eleven times, and always in one of the following two groups of words: ana logon (at the end of our section, in 511e2, and also at Phaedo, 110d3, Timaeus, 29c2, 37a9, 53e4, 56c7 and 82b3, and Laws, X, 893d1; for the still rarer uses of analogon in a single word, used in particular by Socrates at 508b13 in the parallel between the good and the sun which immediately precedes the analogy of the line, see note 78 in my translation of the previous section) or ana ton auton logon (in the line that follows, at 509d7-8, and also at Phaedo, 110d5 and Timaeus, 32b5), i.e. associated with logon used in a sense that evokes "reason" in the mathematical sense, i.e. "ratio" of a proportion or "reason" of a series. This makes the reading an' isa less likely, but does not make it totally implausible: ana always intervenes in an expression that suggests an idea of proportionality, which would remain the case here with the reading an' isa, except that, this time, the proportion would be fixed and would be equality.
If textual criticism and grammar cannot decide for one reading or another, can we find criteria of coherence with the context and intentions that can be detected in Socrates' words that would argue in favor of one or the other lesson? Before answering this question, it is worth asking ourselves what importance Plato's Socrates gives to the mathematical turn of his analogy. If, in fact, as I have tried to show in the introduction and in the preceding notes, and as the rest of the analogy will show, Socrates is not interested in the question of "where?" (the spatial "location" of each of the two "realms") but rather by that of our different modes of apprehension of reality, and that he only "spatializes", and this, in as rudimentary a way as possible, the image he proposes to fix the minds of his young listeners, knowing full well that the intelligibles are not in time and space, then, the question of how to segment the line quantitatively is probably not paramount in his mind. In order to establish a purely quantitative relationship between visible and intelligible realities, it would first be necessary to be able to count the "population" of each of the two "kingdoms", to remain in the image he has just used, which is obviously impossible, and Socrates is perfectly aware of this. And moreover, if we were able to do so, what would it do us on the road toward the good? And the rest of the analogy will show us that the "relations" that Socrates will establish between the different segments have nothing numerical about them, just as the "analogy" he established between the (non-numerical) relationships existing between the sun, light and sight and those existing between the good, truth, and the quest for knowledge had nothing numerical.
There is, however, one aspect of this problem of division that is important, but it is not strictly speaking quantitative, but qualitative, and it is not the analogy of the line that gives us the key, but the allegory of the cave, as I explained in the introduction. The allegory of the cave will in fact make us understand that everything that is found inside the cave (image in the allegory of the visible / sensible), starting with the anthrôpoi ("human beings"), has its counterpart outside the cave (image of the intelligible), but that there are also "realities" outside the cave that have no counterpart inside: the sun and the stars in heaven (image in the allegory of the pure intelligibles such as the good, the beautiful, the just, but not only since Socrates will make us understand at the beginning of book X that there are also ideai, that is to say purely intelligible apprehensions of things like "table" and "bed", which implies "stars" representing these ideai in the allegory), which is translated in the allegory by the fact that, where Socrates describes inside the cave a single "division" in two stages for the prisoner's progress, the time when he is caught in bonds and sees only shadows and hears only echoes and the time when he is freed from these bonds and turns back to what the shadows were shadows of, he describes two successive progressions of the prisoner out of the cave, first a progression from the "shadows" and "reflections" (no longer on the wall of the cave, but on the ground or in the waters on the surface of the earth) to the originals of these shadows and reflections, in relation to the anthrôpoi ("human beings") and the other "things" that are on the surface of the earth, then again a similar progression in relation to the stars and the sun. But this does not mean that there should be six segments and not four in the analogy of the line, for the process of dividing the segment of the noèton ("perceived by intelligence") into two sub-segments is the same with regard to what is on earth and with regard to what is in heaven, shadows and reflections on the one hand, original of these shadows and reflections on the other (the one that was already used inside the cave for the horaton ("seen")), so that, if we go about segmenting the noèton ("perceived by intelligence") in two stages, there are in the end only two sub-segments there too: shadows and reflections of terrestrial as well as celestial realities (for celestial realities, there are only reflections, for the stars do not produce shadows of themselves, and reflections are mentioned only for the sun, for reasons which I explain in note 51 to my translation of the allegory of the cave), original either terrestrial or celestial of these shadows and reflections, just as, inside the cave, on the one hand, there were shadows and echoes ("sound reflections") and on the other hand the sources of these shadows and echoes / reflections. To put it plainly: all sensible "realities" have an intelligible dimension, but there are also intelligible realities that are just that and are not the intelligible dimension of individual sensible realities (these are the men and other things that the prisoner out of the cave can see outside, first through their shadows and reflections, and then in themselves). It is therefore necessary to read anisa ("unequal") in a single word.
As to what the numerical value of this ratio might be, if I may be allowed the expression, Plato's Socrates does not give a damn about it, for the very question is meaningless: as I said above in the analogical language previously used by him, it is impossible to number the population of the two kingdoms, of which there is nothing to prove that it is fixed! The justification of anisa ("unequal") did not need to count anything, and yet it is quite certain. (<==)

(9) In the beginning of this sentence, we find twice the verb temnein, "to cut," and twice the noun tmèma, which derives from it and means "part cut off, section, piece" and, in mathematics, "segment." It is to make these assonances perceptible that I have translated temnein as "to segment" rather than as "to cut". (<==)

(10) "According to the same logos" translates (partially, since I do not translate logos) the Greek ana ton auton logon. I prefer not to translate the word logos because, as I said in the introduction (see the note on the prologue to the introduction), a proper understanding of the analogy depends in large part on the meaning(s) that should be given this word. Jowett, Cornford and Reeve translate it as "porportion", while Shorey, Lee, Bloom, Grube, Grube/Reeve translate it as "ratio". The problem with both translations is that they are limited to the numerical meaning of logos and preclude any other understanding of the word among the multitude of meanings it may have, which is most regrettable since Plato precisely plays with this multitude of meaning here, and more specifically, with the meaning of "ratio" in the mathematical register suggested by the analogy and the meaning of "rationale" relevant for what the image is attempting to illustrate. As I said at the end of note 8, Socrates doesn't care what the respective "size" of the "seen" and the "perceived by intelligence" are for the simple reason that he knows he has no way of measuring / numbering them. The result is that numbers are irrelevant in the analogy, despite its geometrical guise. And in Greek, using the word logos here is a way of preserving that guise while at the same time providing a hint about the way the analogy should be understood when moving from the image if offers to what it purports to be the image of and to make us understand through it, so that if the translation only preserves one of the meanings conveyed by Plato's Greek, it defeats his whole purpose in choosing this specific image by giving more weight to the geometrical guise than Plato intended. When Plato's Socrates tells us to segment the line into two unequal segments, it is his way of telling us that we can segment it any way we like except the very specific case where they would be equal, which nothing would justify. In other words, he doesn't care what the ratio between the sizes of the two segments might be, because it has no meaning regarding what the image purports to make us understand!.
This being the case, we may now focus on the word "same" (auton). "Same" as what? For all the commentators to my knowledge, the answer is simple: "same as the ratio that was used to segment the line the first time around" of which I just showed that Socrates doesn't care and leave it to us to choose (provided we avoid the very special case where they would be equal), not even telling us which one should be the greater to better convince us that their size is irrelevant. Now, if one reads this text without prejudice and ignoring the "evidence" served by the commentators, there are two ways of understanding it: (1) according to the same logos that served to segment the line into sensible and intelligible (the understanding of all the commentators) or (2) according to the same logos as each other, without this logos necessarily being the one used for the first segmentation. And if (1) implies (2) since, in this case, the three ratios are identical, the converse is not true. Moreover, it may be thought that, since Socrates did not say anything about any logos regarding the first division (for which the word logos does not appear), it is somewhat premature to understand this "according to the same logos" as referring to something that has not been mentioned previously and that therefore each listener of Socrates (and reader of Plato) could only fix at random in the absence of clarification on his part. Under these conditions, if this was indeed what he had in mind, he would probably have been more explicit and would have said instead "according to the same logos as the one used to segment the line the first time around". But above all, since we know from the allegory of the cave what justified the anisa ("unequal"), namely, the presence in the intelligible of purely intelligible "realities", and therefore not accessible to sight and the other senses, in addition to the intelligible component of all the visible "realities", which are therefore equal in number to these "realities" of the segment of the seen, whatever this number may be, which we do not need to know and which we will never be able to know, it is not difficult to understand that there is no reason why the numerical ratio that we do not know between the size of the segment of the seen and that of the segment of the perceived by intelligence should serve precisely to segment each segment in two. What meaning ("rationale") would this have? We must therefore understand the group of words ana ton auton logon ("according to the same logos") in sense (2) and wait for the rest of Socrates' explanations to know what this logos is, not so much in numerical terms as in terms of what these "relations" are supposed to represent in what the analogy proposes an analogy of. It should be noted that this way of understanding the words allows the listener / reader to mentally form a representation of the line that would almost fully respect Socrates' instructions even if one has fixed the two numerical relationships, that between the segment of the sensible and that of the perceived by intelligence and that between the two sub-segments of each of the two segments, in an arbitrary and no doubt different way from one listener / reader to another, except for one detail, the question of whether the segment of the perceived by intelligence is greater or smaller than that of the sensible, a question to which the answer will only be suggested by the allegory of the cave that immediately follows the analogy. And if Socrates is not more precise at this point, it is one more indication that the numerical values of the lengths of these segments and those of their relations to each other are not what interests him and have no figurative meaning in the analogy. And if he did not specify here the only point that had a figurative meaning, the sense of the inequality between the segment of the seen and the segment of the perceived by intelligence, it is because he would have had to launch into explanations that it was too early to give and that he preferred to let the listeners / readers find it for themselves from the images that he was going to successively propose.
One last (mathematical) remark on this question of logos / relationship between the different segments and sub-segments of the line. When Socrates recalls the analogy towards the end of Book VII, in the discussion on dialektikè as the culmination of the training of philosophers, in 533e8-534a8, he proposes equalities of relations that he had not specified in the analogy, namely, that if we designate by I the segment of the perceived by intelligence and by V the segment of the seen, and moreover by I1 and I2 the two sub-segments of the perceived by intelligence and by V1 and V2 the two sub-segments of the seen located on the line in the order V1, V2, I1, I2, it being understood that, if we follow the instructions given here by Socrates, we have I1 + I2 = I (the two sub-segments of the perceived by intelligence put end to end reconstitute this segment), V1 + V2 = V (the two sub-segments of the seen put end to end reconstitute this segment) and I2/I1 = V2/V1 (the two segments have been segmented in two "according to the same numerical ratio"), then we also have I/V = I1/V1 = I2/V2, and this, regardless of the ratio I2/I1, provided only that it is equal to V2/V1, and without it being necessary that it should also be equal to I/V (a detailed demonstration of this will be found in note 52 in my translation of this section of the Republic under the title "Definition of dialektikè"). Now, in the particular case where I2/I1 and V2/V1, equal to each other, are also equal to I/V (understanding (1) of the ana ton auton logon which we have rejected, but which all commentators retain), it follows that I2/V2 = I/V = I2/I1 and I1/V1 = I/V = V2/V1, from which we deduce that I1 = V2, that is to say, the two median sub-segments, the second sub-segment of the seen and the first sub-segment of the perceived by intelligence, which follow each other on the line, are equal, which greatly embarrasses the commentators, who do not know what meaning is given to this equality, of which Socrates says nothing. This is one more reason to prefer understanding (2) of the ana ton auton logon, which does not imply this equality, rather than to epilogue on the question of whether Socrates (or rather Plato who holds the pen) was aware or not of this implication of his instructions (if, and only if, we understand them in the sense (1)), seeking to solve a problem that one has created for oneself by a choice of understanding that nothing imposes when another choice, just as acceptable, makes the problem disappear and other reasons encourage one to opt for this second choice. (<==)

(11) "That of the seen family and that of the perceived by intelligence one" translates the Greek to te tou horômenou genous kai to tou nooumenou, an expression that refers to the "each one of the two segments" (hekateron to tmèma) which preceded the ana ton auton logon commented on in the previous note. It is here, and only here, that Socrates specifies what he associates each of the two segments of the first division with. And he does so by returning to the term genos, which evokes the idea of "population" suggested by the analogy of the two "kingdoms", but also and above all, that of "origin" (the primary meaning of genos), but by changing the qualifiers associated with each "family" and the order in which he mentions them. Indeed, he no longer speaks of horaton genos ("visible family") and noèton genos ("intelligible family"), but of horômenon genos ("seen family") and nooumenon genos ("perceived by intelligence family"), replacing the verbal adjectives with the present participles passive of the same verbs horan ("to see") and noein ("to think, reflect"). This change in terminology to qualify each of the two genes clearly shows that it has moved from the "objective" problematic of the characteristics of what "populates" each kingdom (being "intelligible" (noèton) or "visible" (horaton)) to the "subjective" problematic of the modes of perception, and therefore of the "origins" of these perceptions (in the sense of "organs" producing them, not "objects" acting on these "organs"). And it is this change of perspective that explains why he reverses the order between these two "families" to begin with that of the seen, which is easier for us to grasp, as he had done at the end of the previous line when he spoke of the two eidè ("appearances") starting with the visible for reasons that I specify in note 6 above. The result of this inversion is that now, the term genos is made explicit for the horômenon, the seen, and implied for the nooumenon, the perceived by intelligence, so that this time we find ourselves with a relatively abstract term in the sense in which it is taken here, genos ("family" tending towards the meaning of "genus / species / kind"), explicitly associated with the one of the two orders which is on the side of the concrete, the "seen", and implied for the most abstract, the "perceived by intelligence", where previously, the concrete term topos was explicitly associated with the term qualifying the more abstract of the two orders, the intelligible, and implied for the one for which it was best suited, the visible. In short, the order of enunciation has been reversed between visible / seen and intelligible / perceived by intelligence, but in both cases, Plato manages to make completely explicit the one of the two associations of terms which poses problems, leaving only implicit the one that seems to be self-evident.
In the section Keys to understanding the analogy of the line of the Introduction, I underlined the decisive importance that this change in vocabulary has for the understanding of the analogy. It is, under these conditions, regrettable that most translators attach little importance to these fluctuations in vocabulary, not suspecting that they could have a precise meaning, and conceal them from those who do not have access to the Greek text, as shown by the fact that, of all those whose translations I have consulted, only Pachet (Folio) in French and Bloom (Basic Books) in English take into account the changes in terminology of the original using different words to translate noèton and nooumenon on the one hand, horaton and horômenon on the other (Jowett, in English, stays halfway, using "intellectual world" for noèton, and "intelligible" for nooumenon, but "visible" for both horaton and horômenon, which breaks the symmetries of Plato's text). (<==)

(12) "Clearness and lack of clearness" translates the Greek saphèneiai kai asapheiai. The adjective saphès, of which saphèneia is the substantive, and asaphès, of which asapheia is the substantive, its antonym, means "clear, manifest, evident," and consequently, "true, sure." The criterion of clearness is indeed the most relevant criterion in the "kingdom" of the sun and the light it spreads, which we seek to segment in two. But by introducing this criterion, Socrates confirms that what concerns him is not so much the content, the "population" of the segments, regardless of the fact that these contents are apprehended by "observers," as the perception we may have of what populates them, and that the logon ("relation") that he has in mind for the division of the segments does not take the path of being a numerical relation. And this impression is reinforced by the fact that the Greek words that I have translated as "you will have" are soi estai, word for word "will be for you", a way of involving Glaucon a little more in the division and of bringing in his own perception. It should also be noted that the clause "based on the [relative] clearness and lack of clearness of the ones compared to the others" appears, in the Greek as in my translation, before Socrates adds, in addition to the "you will have" that preceded it, "on the one hand in the seen" (en men tôi horômenôi), which suggests that the criterion of clarity (saphèneia) is not reserved only for the segment of the seen and will also intervene in the division of the segment of the perceived by intelligence. And in fact, the word can be used, in Greek as well as in English (as when, for example, it is said of someone that one does not have a clear understanding of something), in an analogical way (analogon) in relation to reasoning, thoughts, in short perceptions in the realm of the intelligible.
This being so, the greater or lesser brightness of an object, even if we limit ourselves to sight and take this word in the primary sense, is not an intrinsic property of this object, but presupposes an observer endowed with sight and depends on the light under which one looks at it. And on the other hand, the degree of clarity varies continuously, just like light, and is not in itself a means of clearly and indisputably dividing the whole of the seen, let alone the perceived by intelligence. In fact, the ensuing discussion, and in particular, once again, the allegory of the cave, will make us understand that Socrates is not trying here to give us a criterion that would allow us to associate to each "element" of the line, that is to say of the whole of what is seen and perceived by intelligence, a precise numerical value (here, its "clarity", measured in who knows what unit, which could be relevant both for the seen and for the perceived by intelligence) making it possible to locate this "element" in one and only one of the four sub-segments, since each element of the seen will be found in the four sub-segments under different perceptions: its visual appearance, represented in the allegory by the shadows seen by the prisoners caught in bonds on the wall of the cave facing them; its purely material component not limited to what is perceived by sight alone, represented in the allegory by the objects protruding from the wall and projecting these shadows, and in particular in the case of human beings (anthrôpoi), the "statues of men" (andriantas, 514c1); his shadows and intelligible reflections outside the cave, i.e. the logoi pronounced by him (in the case of the anthrôpoi) and on him (in all cases); and finally, its intelligible "reality", represented in the allegory by the anthrôpoi (which strictly speaking designates human souls) and all the rest which, in the case of souls, are present but invisible because they are hidden by the wall inside the cave, but become visible outside the cave. What Socrates means here is that, for any element, the perception we have of it is all the more "clear", in the analogical sense, as we progress through the segments, from minimal in the first sub-segment of the seen, where we perceive only its visual appearance, that is to say only what is perceptible by sight, to maximum in the second sub-segment of the perceived by intelligence, when we understand everything that is accessible to the human intelligence (of which nothing allows us to know if this exhausts the "reality" of this element, since we have no other organ of perception beyond our intelligence). (<==)

(13) "Images" translates the Greek eikones, plural of eikôn, substantive derived from the defective verb eoika, which means "to be like, look like", and also "to seem, to beseem, to fit", and whose participle used as an adjective, eikôs, can mean "seeming like" as well as "fitting" or even "likely, probable" (all of Timaeus' speech in the Timaeus is described by him at the beginning as an eikota muthon, a "probable myth", Timaeus, 29d2). But, beyond the word chosen to translate eikôn, and even the Greek word retained by Plato, it is through the examples that follow, and especially through the use he will make of it in the allegory of the cave, that we must try to understand what he has in mind when he speaks of eikones before speculating on what could be inspired by a preconception of the meaning of eikôn in Greek or "image" in English.
In introducing this word to describe the first sub-segment, Socrates chooses a term that does indeed imply a "relationship" since "image" implies "model", or "original", which an image is an image of, and this image-to-model relationship effectively makes it possible to distinguish two types of visual perceptions. On the other hand, the link between the concepts of image and original and the greater or lesser sapheneiai ("clarity") of one in relation to the other is not immediately obvious, especially as long as Socrates has not specified what he puts under the term "image", which he will hasten to do in the rest of the line. (<==)

(14) The two examples of "images / resemblances" given by Socrates, shadows (skiai) and reflections, designated here by the word phantasmata, a generic term to designate visions without substance, "apparitions, phantoms, visions, dreams", or extraordinary celestial "phenomena" (this is the word at the root of the English "phantasm"), the meaning of which is perfectly specified here by the words "reflections on waters and on [other things] insofar as they are by design at the same time compact, smooth and bright (i.e., reflective surfaces)" that follows the word, direct thought, not towards images made by men, paintings or sculptures for example, which are objects in their own right and which, as such, fall into the category of what Socrates will call the whole family of what is fabricated (to skeuaston holon genos)" (510a6) in the description of the second sub-segment of the seen, but towards "images" that form naturally and are animated, some of which (reflections on waters without wrinkles or on a high-quality mirror) are difficult to distinguish from the original of which they are an image without involving any senses other than the sense of sight (touch in particular) and without a habit acquired in the first years of life that ends up dispensing with having to touch each time to remove the doubt. We will find this vocabulary in the allegory of the cave, at different stages of the prisoner's progression, first caught in bonds, then freed, except that the word phantasmata alternates with the word eidôla, a word with a meaning similar to phantasmata (eidôla at 516a7, phantasmata at 516b5), but always accompanied by the words "on waters" (en tois hudasi) to make us understand that with different words, it is indeed the same thing that he is talking about (an implicit exercise intended for the reader, of which we will find other examples in the analogy and its recalling in Book VII, where, precisely, in the recalling that is also made of the allegory, Socrates will return to the word phantasmata (532c1), but always with the precision "on waters"). It is therefore from the use that is made of these notions in the allegory that we must try to understand what Socrates has in mind when he speaks of eikones ("images"), and not from any preconceptions that we might have of what images are. Concretely, in the allegory of the cave, he mentions shadows (skias, 515a7) in the first phase, the one describing the prisoner as caught in bonds that allow him only to see the shadows, which corresponds to the first sub-segment of the seen, and it is clear that these shadows, which are the only thing that the bound prisoners see, correspond to all the data of sight, or in other words, to the natural "images" that are formed in the eyes as part of the process of vision, whether of images in the usual sense, shadows, reflections or images made by men, or of their originals. In other words, what Socrates seeks to make us understand is that EVERYTHING we see is never more than images of the material reality that surrounds us, images formed by the biological processes associated with vision involving the eyes and the soul that interprets them, but also what acts on the eyes, the material "objects" that are in our field of vision, and the light that makes them visible to us. In this first stage, "reflections" also are mentioned, except that in this case, they are sound "reflections", the echo of the voices of the bearers hidden by the wall. Shadows and reflections are again mentioned in the first phase of the release from the cave of the freed prisoner, which corresponds to the first sub-segment of the perceived by intelligence about the material realities already present in the cave, but apprehended this time in their intelligible dimension, at 516a6, where Socrates says that the latter will have to get used to the external light by starting by looking at "the shadows and after that the images on waters of men and those of the other [***]" before being able to see "[those ***] themselves" (auta), in what corresponds to the second sub-segment of the perceived by intelligence. In these intelligible shadows, we must see, in the case of the anthrôpoi ("men" as a species), their words (logoi), and in these intelligible reflections the words (logoi) made by men about these men and all the rest. There is a last mention of reflections when the prisoner turns towards heaven and the stars that represent in the allegory, the pure intelligibles without a sensible component, when Socrates describes the last stage of this progression (second sub-segment of the perceived by intelligence about the pure intelligibles without material / sensible counterpart) by saying that the prisoner will see the sun itself and no longer "reflections of it on waters or some other place" (516b4-5), reflections that represent, in the case of the sun, the discourses made by men on the good and correspond to the first sub-segment of the perceived by intelligence about the pure intelligibles (Socrates does not speak of shadows at this stage, since the stars and the sun do not produce shadows of themselves, which is fitting with the symbolism of the allegory since the stars do not speak and therefore do not make themselves known by logoi emanating from them). (<==)

(15) "You understand" translates the Greek katanoeis, from the verb katanoein, which adds to noein, "to perceive by the mind, apprehend, think, conceive" (the activity of the noûs), a notion of completeness introduced by the prefix kata-, which adds to the mere notion of "thinking" that of "understanding", that is, of conduction that thinking all the way to understanding. (<==)

(16) "Resembles" translates the Greek eoiken, a form of the verb from which eikôn ("image") comes. The second segment includes the "originals", the "models", of the images which populate the first segment, that is to say, in the light of what the allegory of the cave makes us understand, all that which can form an image in human eyes. (<==)

(17) "the living creatures (zôia) around us, and all that is planted (pan to phuteuton) and the whole family of what is fabricated (to skeuaston holon genos)": Socrates' description of the second segment is what most resembles, in all the analogy of the line, the census of a "population" and this is what may have prompted commentators, especially when they consider that visible creatures, and more generally sensible / material creatures, have no place in the intelligible, to make up for Socrates' silence on the "population" of the two sub-segments of the perceived by intelligence with similar descriptions, and to suppose that the description of what he had just "populated" the first sub-segment of the seen with had to be taken at face value, understanding "shadows", "reflections" and other similar "images"of what "populates" the second sub-segment of the seen, here described, in the literal sense, thus wondering whether or not man-made images (paintings and sculptures, for instance) were part of it, without being able to decide. But once again, it is the allegory of the cave that must guide our interpretation of the analogy. And in the description of what protrudes from the wall and casts shadows visible to the prisoners caught in bonds, we find a wording very close to that found here. Socrates speaks of "men carrying implements (skeuè) of all kinds rising above the wall and statues of men (andriantas) and other living animals (zôia) made of wood and stone and fashioned in all possible ways" (514b8-515a1). In both cases we find:
- Man-made objects, here described at the end of the list by the words "the whole family of what is fabricated", which translates the Greek to skeuaston holon genos, and described in the allegory at the beginning of the list by the expression "implements of all kinds", an expression by which I translate the Greek word skeuè. Skeuaston is the verbal adjective of the verb skeuazein, "to prepare, make ready" (accessories, meals, remedies), or "to arrange, dress up", a verb derived from the word skeuè, which means "equipment, attire, apparel, harness, equipment of a soldier", itself derived from skeuos (of which skeuè, used in the allegory, is the neuter plural), which can designate any object of equipment, furniture, tool, instrument, weapon, etc., and whose meaning is sufficiently general for Protagoras, according to Aristotle (cf. Rhetoric, III, 1407b6-8), to have used it in a grammatical sense to designate the gender of the neuter, as opposed to both the masculine (arrèn) and the feminine (thèlus). Skeuaston is, in a general way, the "artificial", the "manufactured", as opposed to the "natural", to that which "grows" (phuein) on its own;
- Living beings (zôia), i.e. the products of nature endowed with a specific reproducible organization, described here at the beginning of the list by two expressions: ta peri hèmas zôia ("the living creatures around us") and pan to phuteuton ("all that is planted") which can be taken each in a specialized sense ("animals" for the first one and "plants" for the second one) leading to two distinct subsets, or in a broad sense ("alive" for the first one, "begotten" for the second one) leading to a certain redundancy between the two expressions, justified by two distinct points of view on the same "creatures", the fact that they are alive on the one hand, since the primary meaning of zèn, the verb from which zôios is derived, from which zôia is the plural neuter, is "to live", the fact that they are begotten / reproduce naturally, since one of the meanings of the verb phuein, from which phuteuton via phuteuein is derived, is "to be born", which leads for phuteuein to the meaning "to beget / procreate" (next to the meaning "to plant", derived from the meaning "to grow" for phuein), and described in the allegory at the end of the list by the same term zôia ("living") after having made a special case of men by the use of the word andriantas, which means in the primary sense "statue of a man" by derivation from anèr, andros, which means "man" as opposed to "woman", called by the fact that what Socrates describes at the time are manufactured objects that protrude from the wall and are animated by men (anthrôpoi) invisible because they are hidden by the wall.
The reversal of the order between the two lists is probably due to the fact that, as in the allegory, everything that protrudes from the wall is described as manufactured objects, statues or other handicrafts, it was necessary to start with the objects made by men (skeuè) for their personal use before reducing to the rank of manufactured objects (by the creator god) the material bodies of all living beings, represented by mere statues. But in both cases, what is listed are all the products of a creation / manufacture that manifest a certain organization responding to a finality, whatever the producer, god or man, and it is this idea of finality that makes them potentially intelligible and therefore apt to reappear as "objects" of other modes of apprehension in the segment of the perceived by intelligence, that is to say, in the imagery of the allegory, outside the cave.
It is interesting to note that, when Socrates describes the genos that "populates" the second segment, while he mentions all animal and vegetable life on the one hand, and all the artificial manufactured on the other, it is to the second, the artificial, and to it alone, that he explicitly associates the word genos, whereas one would rather have expected it to be associated with the living, since the primary meaning of this word is "birth" (cf. note 3). Once again, for a word that he takes in a broader sense, Plato's Socrates manages to associate it with what we least expect it with.
Let us note that in taking the trouble to specify that he has in mind "the whole (holon) family of what is fabricated", Socrates, as I have already pointed out in note 14, invites us to include painted pictures and statues, which are indeed also manufactured objects, and even, as the ensuing discussion will show, the figures that geometers "mold and draw, of which there are shadows and reflections in waters" (cf. 510e1-3). The reason is that, once we have understood that the notion of "image" that he had in mind is the one that the sight of anything material produces within us, it does not matter whether what we see is an original or is itself an image, since, for sight, it is an "original" of which it gives us an "image".
In a different context, in which Socrates will be more interested in the producers of the different kinds of images, that of the discussion on the three kinds of beds, at the beginning of Book X (cf. Republic, X, 596b-597e, translated in the page entitled "The three kinds (eidè) of seats (beds)"), where the problem will be to specify what "imitation (mimesis)" means, he will put forward the images of beds produced by men, and more specifically painters, and no longer images that occur naturally, without human intervention. This is because here, Socrates is concerned with two modes of man's apprehension of the same visible reality, one direct, the other indirect, whereas, when he speaks of the bed, he will seek to distinguish two modes of reproduction by man of the same unique idea (cf. 596b3, b7, b9) whose paternity he attributes to "a god (theon)" (597b6), one which preserves the functional purpose implied by this idea (to allow people to lie on it), the other that reproduces only a possible external appearance in an object that is nevertheless man-made, but unfit for the function of a bed.
The logos that presides over the division of the segment of the seen thus seems to be the (non-numerical) relationship that exists between an image (in this case the one produced in the eyes) and what it is an image of (in this case everything that is visible, whether it is an "original" or an "image"). But this way of seeing things is still partial, because it only takes into account the "objective" part of what is at stake, the "fact" (pragma) that, from a certain point of view at least, it is an "image" (the one formed on the retina, to put it in modern terms) or its "original". If we want to better define the "relationship" (logos) between what Socrates will end up calling two pathèmata ("affections / states of mind"), we must also take into account the state of mind of the one who is thus "affected", and distinguish, as the ensuing discussion will show, not simply two categories of "realities" in the context of a partition of the seen where each visible element would be on one side or the other, but two attitudes towards what is offered to the eye: to take it for "reality" without realizing that it is only an image or to become aware of the fact that it is only an image and draw the consequences of that. And it is therefore the combination of this relationship between image and original, and the state of mind of the person who perceives the "image", depending on whether or not he is aware that it is only an "image / representation" that should guide us in trying to understand the division of the perceived by intelligence ana ton auton logon ("according to the same logos").
After having mentioned what is listed in these two inventories, it may be interesting to look at what is not listed. The two categories he lists here, the living (animal and plants), encompassing everything that "grows" (phuein in Greek) under the effect of its own "nature" and therefore constitutes phusis in the original sense of the term (and not in the broader sense that we give today to the word "nature", which is its usual translation), and what is manufactured, that is to say, the product of man's creative activity, does not cover all that we would expect to find in a description of the visible order: it lacks everything that we would call today the mineral order (rocks, stones, earth, etc.) and more generally everything that constitutes the framework in which animals and plants grow and the products of human art are made: soil, plains, mountains, lakes, rivers, seas, etc. Now this omission may help us understand what Socrates puts behind the word eidos ("appearance"), for it is not difficult to see that what he retains in this second segment is only that whose apparent unity (that of its "appearance") results from an organizational principle that is not perceptible by sight or by the other senses, but presupposes an effort of reflection of the intellect in order to be discovered. In living beings, this principle is precisely the vital principle that determines the way in which each living being will "grow", phuein, that is to say its phusis, its "nature", and in man-made objects, it is the intention, the finality, which has presided over the work of the craftsman who has made that object. And what characterizes the organizational principles that interest Socrates here is that they do not allow one to do anything one wants with what they organize: if I take a stone, the unity of which is perceptible by sight and touch, and I break it in two, I will end up with two stones, certainly different from the original stone, but which will be as much stones as the stone I broke; the same thing happens if I saw a piece of dead wood to make two pieces of wood where I had one at the beginning, two pieces of wood which, except for the size, will be just as much pieces of wood, logs or beams, as the piece I saw. On the contrary, if I take a dog and cut it in two, I will not end up with two dogs, but at best a dog that is still alive but amputated of a limb or a non-vital part of itself and a piece of dead flesh, at worst two pieces of bloody dead flesh and no dog at all; in the same way, if I cut down a tree, I shall have on one side a stump which may or may not start again and begin to produce branches and leaves again, and on the other a heap of wood and leaves which will soon dry up, but certainly not two trees; and if I saw a bed in half, I will not have two beds after this operation (except in special cases involving the "art" of the one who does it taking into account the specificities of the original bed, which can be considered a new creation, for instance to transform a double bed into two single beds), but two piles of building materials incapable of being used as beds. In fact, this is at the heart of the distinction between "matter" and "form", and the example of the tree is particularly enlightening in helping us to understand how the Greek word hulè ("wood", and more specifically "timber for building", as opposed to dendron, which means "tree planted, growing and bearing fruit") could have come to mean "matter" in the most general philosophical sense through the meaning of "building material" of whatever it may be: the wood of a tree is, in fact, of all that constitutes the "matter" of living beings, that which undergoes the least transformations in its visible appearance by passing from the "living" state (the trunk of the planted tree which continues to grow, phuein) to the "dead" state (the trunk of the felled tree which is dried in order to make it possible to cut it into planks later); We can therefore think that it is in it that we grasp most closely by sight what constitutes the "matter" of living beings, reusable as building material by men (the other materials used by men in their "manufacturing" actiivities (ta skeuasta), minerals or metals, exist in nature only in this state of "matter", never in a state where they are "alive", at least not in a visible way where they are what gives its shape to the living being of which they are one of the components).
The eidos, the "appearance", which interests Plato's Socrates, is therefore this visible "form" that suggests a principle of organization of the raw "matter" seen by the eye and perceptible to touch that only intelligence can apprehend behind the raw data of the senses produced by this matter. But human intelligence can analyze this eidos in two opposite directions: if we stick to the visible appearance, the eidos we arrive at is limited to a more or less complex "form" that we can try to analyze in terms of a schema, a "figure", by reducing it to elementary schemata such as the triangle, the circle and the square (or even more elementary, the line, which serves as a support for the analogy that Socrates is developing, precisely because it is the most elementary of the eidè, and therefore the one that is least likely to contaminate what he is talking about), and we find ourselves on the side of geometry, seeking to put into equations the more or less complex curves that constitute the contours of what we are interested in (the only thing that is preserved in the shadows), and possibly the volumes that constitute it, once we have become aware (through intelligence) of the fact that, contrary to what we might believe based on sight alone, this object of our attention is not two-dimensional, but three-dimensional; in this strictly "materialist" perspective, there is, in the end, no difference between living and manufactured beings on the one hand, raw material on the other, between the trunk of the standing tree and the same trunk drying at the sawmill, or more generally, between a turtle and a pebble on the beach, since both lend themselves to an analysis of their visible "form" that can lead to similar figures. It is this path that leads, pushed to the limit, to a modeling / mythologization of matter from elementary triangles in the Timaeus and to all the more complex models in the form of equations that are in use nowadays. If, on the other hand, we are interested in what, behind the visible appearance, aside from the schema that intelligence detects in the data provided by sight, confers its unity, its specificity and its relative permanence on what is at the source of our perception, we arrive at perceiving an eidos through reasoning, through logos, an "appearance", of an exclusively intelligible kind that points for us toward the idea that expresses the principles of organization of this living creature or this human construction. And it is this idea, the principle of intelligibility, much more than the visible schema alone, that will allow us to classify in a relevant way, that is to say "respecting the natural joints " to use Socrates' words at Phaedrus, 265e1-2, all our perceptions according to the eidè ("species") into which they can be grouped (a tree is not a horse, a horse is not a man, nor a chariot, and a coyote is neither a wolf nor a dog), when we have understood that it is not the mere "form" / schema, reproducible to the point of error in marble or bronze by a Daedalus or some other talented sculptor, which makes the horse or the man or the coyote, but something else that "animates" this "form" and the material of which it is made, nor is it a particular "shape" / schema that makes an animal a coyote rather than a wolf or a dog, or an assembly of pieces of wood a bed rather than a table, since there are an infinite number of different ways of assembling various materials to make a bed, some of which can be confused with a table.
What this inventory shows is that what interests Socrates here in the first place are the realities that we will be able to apprehend at the four levels represented by the four segments, that is to say, realities perceptible by the senses, but whose sensible materiality responds to a purely intelligible idea that provides its principle of intelligibility. And not a purely conjunctural idea that would belong to it at times as an attribute, like the fact of being beautiful, or fair, or great, but an idea that is inseparable from it, that reveals its own nature, like being a man or a horse or a coyote or an oak tree or a statue of Athena. And this is what the allegory of the cave will confirm by taking men as a privileged example throughout the progression of the freed prisoner. Of course, what we apprehend at each level is not the same thing: the shadow is not the statue of a man of which it is a shadow, and the statue is no more the man (the soul) of which it is the "body" than the reflection of this man / soul on the surface of a lake outside the cave (i.e. of the words said about him), but behind these different "appearances" (eidè) hides the same ultimate reality: Socrates, or Plato, or you or me.
And it is precisely because these realities can make us "come out of the cave" that they are mainly the ones that interest Plato's Socrates. Men begin to think from the data of their senses, and mainly from those of sight. And that fact must be taken into account if one wants to make them progress toward a better understanding. Not that other realities, such as a pebble, a pile of sand or a mountain, do not have a "form", a pattern, an appearance (eidos), which allows them to be given a name; but because their schema is purely conjunctural and does not derive from a principle of intelligibility that is intrinsic to them, even if there is indeed something common to everything to which the same name is given, expressible by logoi. The cause of the conical shape of a pile of sand results from physical laws that can be discovered, but not from the principles of the grains of sand that constitute it, which have been "assembled" by chance and could, under the action of external forces, take the form of a beach or a dune. These laws would therefore not tell us much of importance about ourselves, the main subject of our quest, while understanding what makes the difference between a horse and a man tells us something about the animal that we are and understanding why a bed is built the way it is helps us to create an appropriate environment to live "well". Told in the language of the allegory of the cave, Socrates has no use for the shadow of the wall behind which the bearers of statues of men, animals or manufactured objects walk and the wall of the cave on which these shadows are projected; What interests him are the shadows of the statues protruding from the wall and these statues, especially if they drive us to go and see what is happening outside the cave. (<==)

(18) Doxaston et gnôston are the verbal adjectives ending in -tos (-ton in the neuter) of the verbs doxazein ("to think, imagin, hold an opinion", derived from doxa, "opinion, conjecture, judgement (well grounded or not)", which I translate as "to opine" to make the common root with doxa ("opinion") apparent in English) and gignôskein (formed from the pure root gnô, which is found without repetition in the active aorist infinitive gnônai ; "to seek / to learn to know") respectively. In general, verbal adjectives express the possible (Greek equivalent of the English suffixes "-able" or "-ible"), which would lead to the translations as "opinable" and "knowable". But things are not always so straightforward in Greek as far as the verbal adjective is concerned, and sometimes it is equivalent to a simple passive past participle, that is, in our case, "opined" and "known." And, regarding gnôstos, more frequent than doxaston (of which only 11 occurrences are listed in the Greek Classics available at Perseus, 10 of which in Plato's dialogues), it is precisely the meaning of "known" that is the first meaning given by the LSJ, before the meaning of "knowable". These two new verbal adjectives complement a list of similar terms used in previous lines that already includes noèton ("intelligible"), horaton ("visible"), phuteuton ("planted") and skeuaston ("fabricated"). The fact that, regarding noèton and horaton, Socrates also uses nooumenon ("thought") and horômenon ("seen") for the passive meaning militates in favour of the translation of verbal adjectives by words in -able / -ible, which would lead to phuteuton being translated as "plantable" and skeuaston as "fabricable". Unfortunately, dictionaries do not give these meanings for these two words, but only the passive meanings of "planted" and "prepared by art, artificial" respectively. The result is that, if we leave it at that, we are no further advanced to decide whether doxaston should be translated by "opinable" or by "opined," and gnôston by "knowable" or by "known." But if we take into account the rest of the comparison, which compares the relationship between the doxaston and the gnôston and the relationship between "what is made similar" (to homoiôthen) and "what it has been made similar to" (to hôi hômoiôthè), it is clear that, in the second part, Socrates does not have in mind "things" that would have the possibility of being represented by images and the possible images that could be made of them, but rather actual images and their specific models, which invites us to understand that, in the first part, he refers to opinions expressed as opposed to actual knowledge by identifiable persons, which confirms the translation of doxaston as "opined" and of gnôston as "known". (<==)

(19) "It is divided with regard to truth and its absence [according to the following ratio(nale):] as the opined [is] to the known, so what is made similar [to something is] to what it has been made similar to" translates the Greek auto dièirèsthai alètheiai te kai mè, hôs to doxaston pros to gnôston, houtô to homoiôthen pros to hôi hômoiôthè (word for word "this to be-divided by_truth also and not, like the opined with_regard_to the known, so the being_made_similar with_regard_to the to_wich it_is_made_similar") while respecting the conciseness of the original text. After introducing the criterion of saphenia ("clearness / evidence") and when moving from the segment "of the seen family" (to (tmèna) tou horômenou genous, 509d8) to the segment "of the perceived by intelligence (family)" to (tmèna) tou nooumenou (genous), id.), Socrates introduces another criterion, that of "truth or not" (alètheiai te kai mè), which only makes sense in the context of a logos, by referring to the relationship between doxaston ("opined / expressed as an opinion") and gnôston ("known / result of a search for knowledge"), which brings us back to the discussion on knowledge (epistèmè) and opinion (doxa) at the end of Book V, where he had made nomima (Republic, VI, 497d4), the "preconceived ideas" conveyed by usage in simple logoi, the proper object of opinion, as opposed to the "beings" (onta), the proper object of knowledge, and to the distinction he had made between knowledge and opinion at 508d4-9, when, in the parallel between the good and the sun, he said: "So now, the [case] of the soul, conceive [it] this way: when what truth lights from above and that which is, it relies upon this, it conceives and gets to know this and appears to have intelligence; but when it’s upon what is diluted in darkness, what becomes and perishes, it produces opinions and sees dimly, turning those opinions up and down and then it seems not to have intelligence" (in this sentence, "gets to know" translates egnô, aorist of gignôskein ("to learn to know" or "to recognize" at the end of an apprenticeship), from which the verbal adjective gnôston used here by Socrates derives, and "it produces opinions" translates doxazei, present tense from the verb doxazein ("to have, to express an opinion"), from which the verbal adjective doxaston coupled here with gnôston by Socrates is derived), in a context where Socrates seemed to equate truth (aletheia) with the "light" produced by the idea of the good (hè tou agathou idea) (cf. notes 86 and 94 in my translation of the parallel between the good and the sun).
It is normal for Socrates to introduce the issue of truth into this analogy, since at Republic, V, 475e4, as a prelude to the discussion on knowledge and opinion that closes Book V, he defined philosophers as "those [who are] in love with the spectacle of truth" (tous tès alètheias philotheamonas) and it is not surprising that he does so at the articulation between the examination of the segment of the seen, for which the criterion of "clarity" (saphèneia) was more suitable, and the examination of the segment of the perceived by intelligence, where everything happens in logoi and where therefore the criterion of truth is decisive. What Socrates seeks to do here is to show that the criterion of truth is not completely inoperative in the order of the seen, not, as one might think, to suggest that the opined is an "image" of the known, which would imply making the parallel he uses work the other way around, saying that the opinion is to the known what the image is to the original. And in fact, Socrates intends to transpose to the relationship between the image, described by the general wording "what is made similar" (to homoiôthen), and the original, described by the general symmetrical wording "what it has been made similar to" (to hôi hômoiôthè), a characteristic of the relationship between opined and known relating to the truth. It remains to be found which one. And for this, we must go back to the discussion on the difference between knowledge (epistèmè) and opinion (doxa) mentioned at the beginning of this note, where it was already Glaucon who was Socrates' interlocutor and where he refused to equate knowledge and opinion insofar as knowledge is for him "infallible" (anamartèton) while opinion is not, which amounts to saying, in terms of "truth", that knowledge is always true while an opinion can be true or false, as the Meno taught us in the discussion on true opinion (doxa alethes, Meno, 97b9-c2, c2, etc.; also called "right opinion", 97b5, etc.) and knowledge (epistèmè) that ends the dialogue. What characterizes opinion is therefore not that, unlike knowledge, it is always false, but precisely that it can be true as well as false. Transposed to the image / reproduction / similarity, this seeks to make us understand that an image is not necessarily false, even if it is not what it is only the image of, but can make us perceive something true about what it is only an image of. In other words, truth is not an all-or-nothing thing, in a binary way, but, like clarity, is a matter of gradation. And so the "images" that sight offers of what activates it (the pragmata), whether we take them for reality (the first sub-segment of the seen) or whether we are aware that they are only "images" (the second sub-segment of the seen), are not completely false and give us to see, "unveil" for us, something, but not everything, of the truth of what they are images of. This leads us to conceive of the progression through the segments, which will be illustrated by the allegory of the cave, as a progression towards truth, which is not reached all at once at the end of the journey, but which is unveiled (the primary meaning of aletheia) gradually from beginning to end.
There is nothing here that invites us to associate opinion with the segment of the seen (or with any segment whatsoever) and knowledge with the segment of what is perceived by intelligence (or with any other segment whatsoever) or that would suggest that the relation of the opined to the known is the relation which serves to divide the line the first time in order to deduce from it that the relation of the image to the original which is paralleled with it, and seems at this point of the analogy to be the one that was used to divide the segment of the seen, is the same as the one that was used to segment the line the first time. (<==)

(20) "The segmentation of the intelligible" translates the Greek tèn tou noètou tomèn. Two remarks on this expression: on the one hand, the word translated as "segmentation" is tomè, another noun derived from the verb temnein, just like tmèma, "segment", used until now. In the primary sense, tomè means "cut", either in the sense of the action of cutting, or in the sense of a trace left by the cutting tool, while tmèma refers rather to the pieces resulting from the sectioning. I translate it as "segmentation" to preserve in English the root kinship with "segment" as a noun and as a verb, which I have used so far to translate tmèma and temnein. It could also be translated as "section", which is one of the possible meanings of tomè in a mathematical context, since it is the term used to talk about "conic sections", for example, but then the relationship with "segment" and "segment" would be less apparent.
On the other hand, it should be noted that Socrates returns here to the term noèton ("intelligible"), used at 509d2, before the beginning of the analogy, to designate the second segment, whereas we would expect nooumenon ("perceived by intelligence"), as at 509d7, to be a counterpart to the horômenon used at 509d9 with regard to the divisions of the first segment. As I explained in note 11, the transition from the terminology noèton / horaton to the terminology horômenon / nooumenon reflected the transition from an "objective" to a "subjective" point of view by choosing verb forms that show more clearly that these terms refer to modes of apprehension by us of reality. And this change of perspective took place at the moment of talking about the visible, which is the one of the two orders that seems to us the most "objective", precisely to encourage us to relativize the confidence we can have in sight, which can sometimes have difficulty distinguishing a simple image, a reflection for example, from what it is the image of. But now that what is examined is the thought / thinkable, it is the "objective" character of what is at the origin of our thoughts that poses a problem, the reality of what leads our mind (noûs) to pose eidè associated with words designating "things" that have no visible schèma such as "beautiful", "just", "good", and this is probably why Socrates returns to the more objective vocabulary of noèton, as if to suggest that, if no one doubts that, if there is "seen (horômenon)", it is because there is something to see, "visible (horaton)", there is no reason to doubt that, if there is "thought (nooumenon)", it is because there is something at the origin of these thoughts, of the "thinkable (noèton)," which is no more a pure creation of our mind than the images that are formed in our eye are a pure creation of it. (<==)

(21) Where the division of the segment of the seen was done on the basis of objects of perception according to whether they were images (formed in the process of vision) or originals (of these images), the division of the segment of the perceived by intelligence is done on the basis of intellectual approaches, modes of reasoning according to whether or not they free us from the grip of visible images and words and give us or not access to coherence and to the understanding of the whole that constitutes the ordered Kosmos in which we live. "Approach", in the sense in which I use it here ("The method used in dealing with or accomplishing: a logical approach to the problem", American Heritage Dictionary, n. 4.), is moreover a possible translation of methodos, which is the penultimate word of the sentence, which I have translated there as "road map" to make its etymology perceptible: methodos is in fact formed by adding the prefix meta- to the noun hodos, which means "way, road, path" or "walk, journey". The prefix meta- adds an idea of succession in time, of the pursuit of something undertaken, of a change of place. Methodos means "following after, pursuit" in the primary sense, and has come to mean "research (of knowledge), investigation", or "method", which is the English transcription of it.
It is a mistake, however almost universal, to think that the difference between the two intellectual approaches described here by Socrates is made on the basis of the objects to which they relate, based on the fact that the example he then proposes to clarify his remarks is that of geometry, chosen most probably because geometric figures (schèmata) are the most elementary and easiest to understand category of eidè ("appearances"), as the Meno makes us understand where Socrates, in an attempt to make his interlocutor understand what he is looking for when he asks him what is the eidos common to everything to which he attributes the name of aretè ("excellence / virtue"), takes the example of the schemata ("(geometric) figures") (Meno, 73e3-76e7; on this section of the Meno, see my article (in French only) published in number 14, Varia, dated February 2010 of the online philosophical journal Klèsis, entitled « De la couleur avant toutes choses, les schèmas invisibles du Ménon », available on this site by clicking here). As I said in the introduction, the purpose of the division of the line is not to distinguish from each other categories of "objects" perceptible by human beings through their senses and intelligence (noûs), each category being in one and only one of the four identified sub-segments, but to distinguish the different modes of perception by them of the same objects by sight for those that are visible and by intelligence (noûs), therefore by reasoning expressed with the help of words uttered or thought, for all. It is therefore normal that, in the segment of the perceived by intelligence, the division should be made on the basis of modes of reasoning, whatever the "objects" on which these reasonings relate, whether they are visible (which does not prevent them from being also intelligible), such as a horse, a bed or a geometric figure drawn on some support, or only intelligible, like the just or the beautiful.
This reply of Socrates, which aims to explain by comparing them, the two approaches, each corresponding to one of the two sub-segments of the perceived by intelligence, is particularly dense and concise, to the point of arousing Glaucon's incomprehension, forcing Socrates to give additional explanations, and therefore deserves to be scrutinized in order to understand it properly, even if it implies anticipating Socrates' additional explanations. It speaks to us both by its structure and by the words used and poses both grammatical and semantic problems if we want to understand it properly. To highlight both its structure and the problems posed by this line composed with extreme rigour by Plato, I present it here in the form of a table in two columns, each column corresponding to one of the two approaches, clearly distinguished from each other by the to men..., to de... ("On the one hand..., on the other hand... / on one side..., on the other... ") which introduces them and splits the line into two parts of exactly the same size (111 letters on each side). In this table, I have used character and background colors to highlight the resonances of one to the other in the vocabulary (reuse of the same words or distinct words put in opposition) and the similarities and oppositions between clauses that contribute to understanding: colored words are the important words that are found on both sides; the parts of the sentence that are on a colored background are those that use different words but are to be compared to understand the differences between the two approaches; when two sentences containing colored words or words on a colored background are on the same line in each of the two columns, they must be placed next to another line where the same word or background colors are reversed (for instance, the line with a blue background on the left and green words on the right must be compared to the one with green words on the left and a blue background on the right) to better perceive the opposition between the two approaches, one aspect of which is explained in these phrases. In this table, each part of the sentence is presented on three lines: on the first line, the Greek text presented as it was in Plato's time, i.e. in capital letters, without diacritics (accents and breathings), without punctuation and without spaces between words, followed by the number of letters contained in this part of the sentence (the total of which for each column appears at the bottom of the column and allows us to check the perfect equality of the two parts in terms of number of characters), on the second line its transliteration into the Roman alphabet, and finally on the third line its translation into English.

If the cell in the fourth row of the second column containing the words ep' archèn is half on a white background and half on a green background with the text in duplicate, it is to show that these words are involved in two oppositions: they are in green on a white background on the first half of the cell to put them next to the words ouk ep' archèn which also appear in green on a white background below in the first column, and reproduced on a green background in the second half to show that, combined with the word anupotheton which follows them to form the expression ep' archèn anupotheton, they are to be placed next to the words all' epi teleutèn also on a green background at the bottom of the first column. The same goes for the word poioumenè in the penultimate line of the right-hand column: it is on a medium grey background to indicate that it should be placed opposite chrômenè higher up in the first column, also on a medium grey background, and reproduced in the second half of the cell on a pink background to indicate that it is the whole that it forms with the words di' auton tèn methodon which precede it, they too on a pink background, that is to say the expression di' auton tèn methodon poioumenè, which must be compared with zetein anagkazetai, also on a pink background in the first column. ICI

The sentence includes 6 verbs, only one of which is conjugated, anagkazetai, the third person singular of the present middle or passive indicative of the verb anagkazein, which means "to force, compel, constrain", from the root anagkè, "necessity, constraint, fate, destiny", which gives as translation either "is constrained (the soul, subject)", if it is read as a passive, or "constrains itself", if we read it as a middle, a choice that I will come back to when we have more elements to choose from. Its subject is psuchè ("a soul", any soul, since the word is used without an article) and its complement is the infinitive zetein ("to seek") inserted between the subject and the verb in a group of words, psuchè zetein anagkazetai ("a soul is constrained to seek" (passive) or "constrains itself to seek" (middle)), which seems to be the main clause common to both parts, which poses problems, precisely regarding the translation of anakazetai, to which I will return. The other four verbs are participles, all in the feminine for the agreement with psuchè, a feminine noun, which is their common subject since they all complement zetein ("to seek") by specifying the ways in which this search is conducted. There are two of them on each side, one of a verb of movement (poreuomenè, "advancing" or "driven", depending on whether it is read as a middle or as a passive, for the first approach; iousa, "going", for the second approach; on a light grey background in the array), the other of a verb evoking a means of action (chrômenè, "using"), for the first approach; poioumenè, "making", for the second approach; on a medium grey background in the array). The problems posed by anagkazetai are, on the one hand, that, as I said, it can be understood either as a passive ("is constrained") or as a middle ("constrains itself") and, on the other hand, that, if it is understood as a passive, nothing is said about the origin of this constraint and, moreover, it is not easy to determine what it relates to. Thus, with regard to the first approach, where the verb is explicit, we can understand that it relates to the use of images, to the fact of having recourse to hupotheseis, a word whose meaning, as the ensuing discussion will show, is not self-evident and that it would be premature to translate as "hypotheses", or both, or even on the whole first part of the sentence, including the fact of arriving at a result (it can be seen that not all translators agree on this point by consulting the various translations of this line that I have reproduced at the end of this note). And as far as the second approach is concerned, where the verb is only implicit if we admit that the words psuchè zetein anagkazetai are implied, the idea of external constraint does not fit well with the words di' autôn tèn methodon poioumenè ("through them the road map making", that is to say, in a more natural order in English "making the road map through them"), in which poioumenè can only be a means ("making for oneself") and which rather evoke an idea of freedom in the approach as opposed to an idea of constraint, which we do not see what else, in this second approach, it could apply to. But before going any further in solving these problems, it is necessary to take an overall look at this line using to a more functional division than the previous one and to provide some clarification on the meaning of the important words it uses.
One way of analysing the two approaches it opposes is based on the observation that each of them is described by means of four elements, a starting point (S), a goal (G), a means to achieve this goal (M) and finally something that this approach rejects (R), but that, as the table below shows, these four elements follow one another in a different order for each approach: MSRG for the first, GSRM for the second, i.e. the two extreme elements are reversed from one to the other: the first begins with the means M and ends on the goal G, while the second begins on the goal G and ends on the means M.

Before commenting further on this table, it is necessary to clarify the meaning of the words being used. The presentation I made of the structure of the sentence in the first table, using colours, shows that some words are found in both parts, either in close but different expressions (ex hupotheseôn, plural, on the one hand and ex hupotheseôs, singular, on the other), or in contrary expressions (hôs eikosin, "as images", on the one hand opposite to aneu tôn eikonôn, "without the images", on the other; ouk ep' archèn, "not toward a (leading) principle", on the one hand as opposed to ep' archèn, "toward a (leading) principle", on the other), and that these words and the expressions containing them succeed each other in the second part in the reverse order of their appearance in the first. Alongside these words common to both sides, there are words that appear only on one side or the other:
- teleutèn (accusative singular of the feminine noun teleutè), used to designate the term of the approach described by the first part, which is in some way opposed to anupotheton (accusative feminine singular of the verbal adjective anupothetos) archèn (accusative singular of the feminine noun archè), an expression that designates the term of the approach described in the second part, as shown by the ep(i) that introduces it, the counterpart of the epi that introduces teleutèn, but a term that becomes in a way a means of shedding light on the whole process, as its place at the beginning of the description of the second approach suggests;
- eidesi (plural dative of the neuter noun eidos), which appears only in the second part where it is presented as that which takes the place of images (eikonôn) and constitutes the "milieu" through which (di' autôn) the second approach progresses.
In a synthetic way, we can therefore say that the first process described by Socrates allows the soul to rely on multiple hupotheseôn (D) to progress to a teleutèn (F) using eikosin (M) and without looking for an archen in this search (R), while the second allows it, by relying on a single hupothesis (D), to reach an archen anupotheton (F) which will allow him to find his way through the eidesi (M) without having to resort to the eikonôn (R). To fully understand what Socrates means and what is at stake in these two opposing intellectual approaches, we must therefore take a closer look at the meaning of these words, which can be grouped into three pairs of opposites, hupothesis and anupotheton, teleutè and archè, eikôn and eidos, and see how they can be understood and translated into English without betraying Plato's Greek.
- hupothesis, which in both cases designates something that is located at the starting point of the process, is at the root of the English word "hypothesis", which is the pure and simple transcription of it. It is therefore tempting to translate it, or rather to transcribe it, by its English transcription "hypothesis". But, in so doing, we lose in English what the origin of the word could make perceptible to a Greek and all the multiplicity of meanings to which this origin had led. Hupothesis is indeed the substantive derived from the verb hupotithenai, formed by addition of the prefix hupo- ("under") to the verb tithenai, "to set, put, place", which therefore means in the primary sense "to put under". Hupothesis therefore literally means "the action of putting under", but comes to designate more generally "what is placed underneath": it can mean "foundation", "principle" (as a foundation, for example of a political organization), "basis" (of a reasoning, a speech, a writing, etc.), "subject, theme" (of a discussion, a debate, a deliberation submitted to a vote, etc.). etc.) or "guiding idea", "supposition" (which is the exact Latin equivalent of hupothesis, since this word comes from sub-ponere, "to put under" in Latin), or else "suggestion, proposal", or else "design, resolution". Hupothesis therefore has a much broader meaning in Greek than "hypothesis" in English, in which we most often lose sight of the relationship that its etymology implies with something else to which the hupothesis serves as a "foundation", a "basis", to retain only the somewhat "arbitrary" or uncertain character of this "foundation", about which we do not attempt to know what it itself "rests" upon. This leads to an emphasis on the undemonstrated, uncertain, "hypothetical" character of what is "put under", as shown by the usual meaning of the adjective "hypothetical" that derives from it. Of course, this meaning also exists in Greek, and this from the time of Plato (it is probably the one that the word has when used by Zeno, at Parmenides, 128d5, where he opposes the proponents of the hupothesis that "multiplicity is / it is multiple" (polla estin) to those of the one stating that "one is / it is one" (hen einai)), but hupothesis is a recent word at that time, of which there are no traces before the time of Socrates and Plato, unlike the verb hupotithenai which already appears in Homer in the primary sense of "to put under", and, according to the LSJ, the oldest meaning of hupothesis, contemporary with Socrates (the first example given comes from Xenophon), is "proposal", and hence "suggestion, advice", or even "pretext" (insofar as it serves as a "foundation" for a behavior), and it is quite possible that the evolution of the meaning of the word towards that of "hypothesis" in the modern sense is precisely due to Plato, relayed by Aristotle. Under these conditions, translating, or rather transcribing, the word by "hypothesis" presents two major drawbacks for the understanding of our text: on the one hand, it draws from it twenty-five centuries of semantic evolution of the word, in Greek, then in English, with in particular the emphasis placed on the uncertain, "hypothetical" character of what is "put under", and also the fact that modern language, at least in the mathematical register that most commentators assume to be that of this analogy, distinguishes more clearly than did the Greek definition, axiom and hypothesis, whereas, as will be shown below, the examples of hupotheseis that Socrates will give in the mathematical register, even and odd, figures, angles, have more to do with definitions, with the initial data of the problem, with what is "assumed" in its statement as given at the outset, than with hypotheses in the modern sense; on the other hand, to have the English reader lose what was perceptible to a Greek in Plato's time through the etymology of the word, the underlying image associated with hupothesis, that of something that is put under something else, literally or figuratively. This is why I preferred to translate hupothesis as "support" (and hupotithenai, used in the middle in Socrates' next reply, as "to set as supports"), in preference even to "proposal", which somewhat modifies the spatial image induced by the word hupothesis by evoking the idea of something simply "placed in front of (pro-)" the interlocutor rather than under something else to come, that is to say, a simple highlighting rather than a real support for a "construction". We will also see right away that the defects inherent in the transcription of hupothesis into "hypothesis" are the major cause of the incomprehension that most commentators show in the face of the pair's second word, anupotheton.
- anupotheton poses an even greater problem of comprehension, because it is a rare word that could well be a creation of Plato. In the entire corpus of Greek classics available on the Perseus website, there are only three occurrences, two in Plato, both in the section of the Republic here translated (here and at 511b6), and one in Aristotle's Metaphysics (Metaphysics, Gamma, 1005b14), in a passage where Aristotle explains that it is the same science, that of the philosopher, which deals with "what is called in mathematics axiômatôn (axiôma, derived from axios, "which has value, worthy, which is worthwhile", means in the primary sense "price, value, esteem", before coming to mean in mathematics "principle that is valid in itself", that is to say "axiom") and on the ousias (let us remember here that the usual, non-technical and metaphysical meaning of ousia is "that which is one's own, one's substance, property", which explains why Aristotle can relate it to another word that also evokes the idea of value)" (1005a20), that it is he, the philosopher, insofar as he studies "the things that are as being (peri tôn ontôn hèi onta)", who is the most apt to state "the firmest principle of all, about which it is impossible to be mistaken" (bebaiotatè d' archè pasôn peri hèn diapseusthènai adunaton), a principle that must be both "the most knowable / known" (gnôrimôtatèn) and something "anupotheton", since such a principle cannot be a "hupothesis" (here we see the modern difference between "axiom" and "hypothesis" in mathematics). And what he immediately afterwards states as this principle is the principle of non-contradiction: "it is impossible for the same [thing] at the same time to belong and not belong to the same [subject] according to the same [relationship]" (to auto hama huparchein te kai mè huparchein adunaton tôi autôi kai kata to auto; 1005b19-20). As a starting point for our search for the meaning of this word, let us look at the way it is translated in English and French by the translators I have consulted. In English, Jowett translates "which is above hypotheses"; Shorey: "that transcends assumption"; Cornford: "which is not hypothetical"; Lee: "which involves no assumption"; Bloom: that is free from hypotheses"; Grube: "that is not hypothetical"; Grube/Reeve: "that is not a hypothesis" Reeve: "unhypothetical"; in French, Chambry translates "absolu"; Robin: "anhypothétique"; Baccou: "anhypothétique"; Dixsaut: "anhypothétique"; Pachet: "non hypothétique"; Cazeaux: "qui n'admet pas de présupposés"; Karsenti / Prélorentzos : "absolu"; Leroux: "anhypothétique". As can be seen, Reeve in English, Robin, Baccou, Dixsaut et Leroux in French merely, but clumsily, transcribe the Greek word into an English or French word without trying to understand what it could evoke for Plato and his contemporaries and to verify that its transcription evokes the same thing for us (I say "clumsily transcribe" because "hypothetical" in English, or "hypothétique" in French are not the transcription of hupothetos, but of hupothetikos, which exists in Late Greek, and that the two suffixes, verbal adjective in -tos and substantive in -ikos, do not have the same range of meaning in Greek, so that this pseudo-transcription constitutes a betrayal, especially when we take into account the drift of meaning that has affected "hypothetical" in English and French). Expliciting the initial privative a(n) by translating as "that is not hypothetical" (Grube) or "non hypothétique" (Pachet), does not change the fact that we force the meaning of an-hupotheton on the side of the meaning that hypothetical has taken in everyday English and hypothétique in everyday French. To explain the supposed meaning of the word by "qui n'admet pas de présupposés", as Cazeaux does, is even worse since it completely reverses the idea underlying the Greek word, which does not say that the anupotheton principle is based on nothing, that there is nothing "supposed (put under it)" to sustain it, but that it serves as a sup(hupo-)port for nothing, that there is nothing "above" it. The problem is that hupotheton is one of those verbal adjectives ending in -ton as was the case with gnoston, doxaston, phuteuton, skeuaston, noèton and horaton used earlier (see notes 17 and 18), which, in principle, would mean "that can be put under" (expression of the possible). This form derived from the verb hupotithenai is theoretically possible, but extremely rare, and its meaning seems to have evolved in the medical register, towards that of "suppository" (its almost Latin analog), thus towards the "passive" meaning of "put under" rather than towards the expression of the possible. From this perspective, anupotheton, which derives from it by adding the privative alpha as a prefix and the nu added for euphony and is therefore the opposite, would rather mean "not put under", i.e. "put under nothing" or "support of nothing". And this is indeed how I propose to understand it, rather than in the sense of "which does not have a hypothetical character", that is to say, for us today, "which is certain, which does not need demonstration". In fact, as confirmed by the more precise analysis of the second approach proposed by Socrates, in which anupotheton is used to qualify an archè (a term whose meaning I will also clarify once we are finished with anupotheton) that must be at the end of the process, what Socrates is trying to say is that the archè that can be qualified as anupotheton, in an approach where each hupothesis is conceived as a "stool" on which one steps to rise a little higher after having placed it on the previous "stool" (which therefore serves as a hupothesis for it), it is the one that constitutes the top of the "pyramid" and beyond which there is nothing, that is to say the one which it is no longer necessary to place anything else upon and which is therefore not a hupothesis, "foundation, support" of something else. This way of understanding anupotheton takes us back to 505d5-9, where Socrates stated as obvious to everyone that if, when it comes to justice or beauty, people are ready to accommodate what has the appearance of it without really being that, for the good (agathon), no one will be satisfied with what only has the appearance of it and everyone wants what is really so: the reason for this evidence is that the just or the beautiful are not sought for their own sake, but as means to something else, pleasure, wealth, power, etc., and are therefore only hupotheseis, "foundations" for accessing something else, whereas the good, and more specifically the good in human life, that is, happiness, is sought for its own sake and not as a means toward something else, that is to say, it does not serve as a hupothesis for anything else and is therefore anupotheton, which does not prevent it from being obvious to all (in the sense that it is obvious that all want a good life, that is, true happiness), as Socrates said without being contradicted, of an evidence that everyone can experience in oneself and that does not require mathematical demonstrations (which is less obvious for all is what constitutes the good life for us).
As there is no English word that can convey this meaning of anupotheton, I translate it as "[which is] support of nothing", wich can be expanded into the longer expression "which is not itself posed to support something else", where "not" translates an-, "posed" translated -theton, redunded by the -port of "support", and the sup- of "support renders the idea implied by the prefix hupo-, the added words meant to make the periphrasis clearer. This more concise translation has the merit of using the same word I used to translate hupothesis, "support", thus recalling the common root of the two words. It puts us closer to the situation of the Greeks of Plato's time, to whom the word rare anupotheton was probably unknown, especially if, as I think, it was a neologism coined by Plato, but its etymology spoke (which does not mean that they understood the precise meaning that Plato gave it in this context). And it is quite possible that this was precisely the effect that Plato sought, to force his readers to think about what he meant with this word from the context, as we are still led to do today, where not everyone still agrees, more than two thousand years later, on how to understand it.
- archè is the noun derived from the verb archein, whose primary meaning is "to be first, begin, make a beginning", leading to the meaning of "to rule, govern, command". From the various meanings of this verb are derived the various meanings of archè, either "beginning, principle, origin," or "command, sovereignty, power." But the problem with this word, especially when it is, as is the case here, opposed to teleutè, which refers to the idea of termination, end, as we will see shortly, is that there is an inversion in our understanding, or at least in the images evoked by the word, which ends up completely distorting this understanding. Starting from the idea, implied by the primary meaning of the verb archein, of someone who is in front, who walks first and shows the way, and whom we follow, or from something in front of us that serves as an objective towards which we are advancing, we arrive, via the idea of beginning, then of principle, and finally of origin, to the image of something that is at the beginning and from which we move away, and therefore finally who is behind us. This inversion is particularly noticeable in the field of "physics" where we look for a "principle" of the world that we most often imagine at the beginning of time, and therefore far "behind" us, or in the field of mathematics, where the principles are the axioms posed at the beginning and from which the reasoning that leads us to conclusions follows (see the passage from Aristotle's Metaphysics quoted above about anupotheton, in which the terms axiôma and archè are compared). And since the analogy proposed by Socrates gives itself the air of a geometric analogy, the temptation is even greater here to associate archè with the idea of something that is at the beginning of reasoning. But this is not what Socrates does here, who places in both cases at the beginning of the journey one (in the second approach, where the word is in the singular) or several (in the first approach, where it is in the plural) hupotheseis, and who sees one, the first, lead epi teleutèn, and the other, the second, to lead ep' archèn. For him, therefore, archè is not at the origin and teleutè at the end of the intellectual paths he describes, but both are at the end and it is precisely the difference between these two outcomes that distinguishes the two approaches. This suggests that the meaning he gives here to archè, at least when he describes the second approach, is closer to that of the "guide" that we follow than to the "original principle" from which we start, that the mental image that this word must evoke for us has more to do with the star that the Three Wise Men followed to find their way than with the axioms that mathematicians initially posit or the principle of non-contradiction of which Aristotle makes his anupotheton archen to derive a logic from it and which is therefore "placed under" (hupotheton) all this logic whose validity it guarantees (from his point of view at least). But if it is a guide that we must "follow", it means that it is not enough to identify it, period, but that we seek to identify it in order to progress in its light, which means that, in a certain way, it is both at the beginning and at the end of the process, that it is the one who enlightens it and guides it from start to finish once it has been discovered, unlike a teleutè which is only a "term" after which there is nothing left: the problem posed is solved, we move on to something else (it is in this sense that it is "dead"). It is in an attempt to make this meaning perceptible that I have translated archè as "(leading) principle", an expression that allows us to preserve in the translation the two ranges of meaning of the word, that of "principle" and that of "leader"; But I have put "leading" in parentheses to make the reader perceive that this nuance of meaning is not self-evident, that archè can be understood in several senses and that, as the rest of the analogy will show, it is ultimately around the understanding of this word among all its possible meanings that the difference between the two approaches is played out.
We can now see, by comparing what I have just said about the meaning of archè with what I said before about the meaning of anupotheton, how to speak of an anupotheton archèn, is not to evoke an axiom that cannot be demonstrated on the basis of reasoning, but rather to suggest a "guide", a "guiding principle", which traces the path for us and towards which we move for its own sake and not as a means, as a springboard, towards something else, because there is nothing beyond it, a guide which is not itself guided by another guide further ahead. And we understand that the good itself (auto to agathon), perceived by our mind/intelligence (noûs) through the idea of the good (hè tou agathou idea) which Socrates told us in 505a2 is the most important object of study and in 508e2-3 that it is the one that is the source of truth for us, is this archen anupotheton towards which we are all progressing, seeking it for itself as our end, and that the question is not to decide whether this is really the goal, because we are all convinced of it, but to get closer to it in order to "see" it better, to understand it better and not to take the wrong path, because this archè that serves as our guide is not of in the realm of the visible, but can only be grasped by the "eyes" of the mind, by the noûs (intelligence).
- teleutè is a word from the family of telos, "performance, consummation, fulfilment, execution", but also "completion, end, finish", or else "supreme power", and, in another register, "what is due, tax, acquittal, payment". Teleutè has a more limited meaning than telos, centered on the idea of "end, accomplishment, outcome" and can be used to designate the "result" of a process and also the end of life, i.e. "death". And here again, it is important not to loose sight of any nuance of meaning of the word, including the one that refers to death, in order to understand what opposes the two approaches, the first, about which Socrates takes the trouble to tell us that it does not lead to an archen but only to a teleuten, and the second which leads not only to an archen, but to an anupotheton archen. The first type of process that Socrates mentions are reasonings whose archetype is a mathematical or geometric demonstration: we start from hupotheseôn that are initially posited as "bases" (in the plural, because there are several) of reasoning, definitions, axioms and simple hypotheses in the modern sense, and we unfold the implications of these pre-suppositions to arrive at a result, a teleutè, which is the logical consequence of the hupotheseôn posed at the beginning, as for example we can show that the square built on the diagonal of a given square has a surface area double that of the initial square, as Socrates makes the slave of Meno discover in the dialogue of the same name. But once we have arrived at the teleutè, at the end of the reasoning, at the result of the demonstration, at the end of the logical sequences implied by the initial presuppositions, we know no more about the only questions that should occupy us, those concerning the way in which we lead our lives in order to achieve happiness, for in fact, we have arbitrarily chosen a starting point and we have followed from there an approach that could be called "horizontal" towards conclusions that do not explain the reason for the starting point, but are content to enrich our knowledge by declining its properties, without "rising" us to "higher" considerations likely to make us better understand the universe that surrounds us and the place we should hold in it; and we will be able to start from other hupotheseôn, choosing just as arbitrarily another starting point, and, if we know how to reason, we will arrive with the same rigor at other teleutais just as useless to give meaning to our life, at teleutais who are not archai, leading principles for us men, and who therefore can only lead us to our "end" (teleutè), death for failing to tell us anything about what is good for us. It is to preserve all these resonances that I have chosen, to translate teleutè, the English word "end" which, as the Greek word it translates, can evoke death as the "end" of life.
- Eikôn and eidos have already been used in the first part of the analogy. I have examined in note 5 and at the end of note 17 the sense in which it seemed to me that Plato was using eidos and the close connection he made between eidos and name/word, showing that this way of understanding eidos implied that eidos in the sense in which Plato understands it (at least in some contexts) is not, contrary to the idea, an objective transcendent "reality", but something that each human being associates with each of the words one uses and which carries the meaning one gives them, according to one's intellectual evolution at the time one uses or thinks these words, which therefore varies from one person to another and over time for each person, that it is between eidè and not between words that are only placed one after the other that the logos suggests intertwinings that make it a bearer of meaning and that this eidos could, in each case, be analysed at the outset in terms of a visible schèma ("form, figure") common to everything to which one attributes the name/word that one associates with this eidos, if one sticks to the "visible" appearance, to the "how (is it for sight)?", as is the case for the prisoners of the cave caught in bonds, becoming more precise in the course of one's intellectual evolution until it points to a single idea ("idea") (objective, and represented in the allegory of the cave by one of the stars in heaven) when one seeks to discover the reasons, the logos, which make intelligible the "beings" that deserve the name one associates with this eidos, the "for what?", although Socrates did not use either of these two terms here (schèma and idea). I have also examined in note 13 the general meaning of eikôn, "image, likeness", before specifying in note 14 that, when he used this word in connection with the first sub-segment of the seen, Socrates was privileging a particular category of eikonôn, those that resulted from natural processes that had the particularity of producing something visible on a natural surface (e.g. the surface of a lake or a body of water) or man-made (e.g. the surface of a mirror) in which only the schèma remained, more or less faithfully reproduced from a certain angle of view and reduced to two dimensions (even if the supporting surface of the image, shadow or reflection, is not flat, the shadow or reflection has no thickness and is therefore limited to a "surface") elsewhere than where what it is the schèma of is located, leaving aside those that resulted from the creative activity of man, paintings or sculptures, from the fact that, even if they were inspired by models from nature, which is not always the case, they were above all skeuasta, manufactured objects, in which were reproduced only schèmata of their models figured according to the artist's own purposes and distinct from those of the models used (one does not sleep in (or on) a painting depicting a bed).
Here, as with hupothesis (which I have translated as "support") and anupotheton (which I have translated as "[which is] support of nothing"), the two words eikôn ("image / likeness / representation / semblance / similitude") and eidos (which I translate as "appearance", which must be understood bearing in mind that they are not only appearances for the eye, but can also be, in an analogical sense, "appearances" for the mind) must be understood in relation to each other. As I explained in note 5, the range of meanings of eidos is very broad, ranging from an "individual" sense, the "appearance" of a particular "thing", to an increasingly broad "collective" sense, the "appearance" common to several "things", which justifies they are associated with the same name or qualifier, and which may no longer have anything to do with sight. In the primary "individual" sense, in relation to sight ("visible appearance, external aspect") eidos is close to the meaning of eikôn, insofar as it is precisely this external aspect that is more or less precisely reproduced by the kind of "images" (eikones) that Socrates speaks of in connection with the first segment of the seen (shadows and reflections), and what shadows are supposed to "image" in the allegory, the "image" formed in the eyes in the process of sight of what it is the external appearance of. On the other hand, an exclusively intelligible eidos is reduced to logoi, thought or expressed, which locate it through its relations with other eidè, represented by the relations, the "intertwinings" (cf. Sophist, 259e4-6), that these logoi establish between the eidè underlying the words of which they are composed, logoi which translate and make communicable to others the mental representation that the one who produces them has of this eidos and makes it more or less comprehensible to others, since the word associated with this eidos by itself does not teaches anything of what the person who uses it associates it to someone who has not yet associated mental images (eidos) with this word (i.e. who does not yet know this word) and refers only to the common usage of those who speak the same language, without guaranteeing that the eidos that each one associates with it is the same, since it is for each one the product of one's individual intellectual history. And if Plato speaks only of eidè in the analogy of the line, and not of ideai, it is because the whole progression towards knowledge (epistèmè) through the different segments of the line and the different stages of the progression towards the outside of the cave in the allegory is made by means of the logos, accessible from the first stage to the prisoners caught in bonds, that is to say, by means of words (initially created by prisoners caught in bonds and attributed to shadows, that is to say, solely on the visual appearance of what they name, cf. 515b4-5) and therefore by relying on the eidè associated with them by the one who uses them or who hears them and seeks to understand them (cf. Republic X, 596c6-8), even if this association of an eidos with a word is, at least initially, unconscious. The ideai, represented in the allegory by the stars of heaven, are, for these eidè, only the objective targets that are beyond words and therefore cannot be adequately formulated by means of the logos. In other words, the eidè are what makes it possible to gradually move from images (eikones) to ideai ("ideas"), from the visible considered only as visible to the pure intelligible, without losing their status as eidè. But if, in the sentence we are dealing with, Socrates contrasts eikones ("images") and eidè ("appearances"), it is to invite us to understand eidè in the sense of "intelligible (noèta) appearances" as opposed to "seen (horômena) appearances", a distinction that he will precisely clarify in his explanation.
It is therefore not the fact of using the logos that distinguishes the segments of the perceived by intelligence from those of the seen, but the attitude of the one who produces logoi, mentally or vocally, in relation to the words and to what one supposes behind these words, the eidè which, consciously or unconsciously, one associates with them. We enter the perceived by intelligence from the moment we "attempt[] to see what cannot be seen otherwise than by thought" (510e3-511a1). But it is one thing for a geometer to become aware, for instance, of the fact that the properties demonstrated on the figures under examination (for instance, the fact that the square built on the diagonal of a given square has an area double that of the original square, as Socrates makes the slave of Meno realize, cf. Meno, 82a8-85b7) are not true of the drawing on which the demonstration is conducted, nor of any similar drawing that might be produced, no matter how careful one is in drawing it, but only of something which is accessible only to thought, it is quite another to have a clear apprehension of that "something" and, to use modern terminology, of its ontological status, of "what it is" (ho estin). And this is where the segmentation between the two sub-segments of the perceived by intelligence takes place.
The approach associated with the first sub-segment is that of people who do not try to know what is hidden behind the words, who are not aware of the fact that they implicitly associate eidè with them, which are not necessarily those that their interlocutors associate with them, even if they use the same words. Words serve as "supports" (hupotheseis) in their reasoning, but they are "unable to rise higher than the supports" (511a5-6), that is, to question the way in which the logos made of words points to something other than these words and the nature of what they point to. For them, to know something is to know its name, and they do not consider it necessary "to give a logos (logon didonai, word for word "to give reason" or "to give explanations") about them either to themselves or to others", considering that they are "completely evident" (510d6-8), so much so that they are reduced, in order to support their reasoning and to make clear the meaning they give to the words they use, to resort to "seen appearances" (horômenois eidesi, 510d5). Their only concern is to rely on these words which are supposed to be understood by all in the same way, to "use them as supports" (hupothemenoi, 510c3; poièsamenoi hupotheseis, 510c6), by considering them as "starting points" (archomenoi, verb derived from archè, 510d1) to solve the pratical or theoretical problems they pose without worrying about whether the answers they arrive at are consistent with each other, concerned only by the coherence of the reasoning (homologoumenôs, 510d2) which leads them in each case from the question posed to the solution. And, as we can see, this way of proceeding, thus described, is not peculiar to geometers and other "scientists," but is found in the vast majority of people in most of their reasoning, even in the problems of everyday life, where they use words much more problematic than words like "square" or "diagonal" used by geometers. If Plato's Socrates takes the example of geometers to make himself better understoot, it is not because this method is reserved for certain types of "objects", which could be described as "mathematical objects", but precisely because geometry reasons on "objects" simplified to the extreme, without personal, political, religious, ethical or other issues likely to be challenged and allows us to arrive at conclusions that are almost certain and can be understood and accepted by all, and that therefore the example is easier for everyone to understand.
The approach associated with the second sub-segment, on the other hand, supposes that before rushing headlong into logical reasoning based on words that we use without asking questions about them and take as starting points (one of the senses of archè) assuming the understanding resulting from common usage, from "received ideas" (nomima, cf. Republic, V, 479d3-5), we begin by trying to understand what the power of the logos, or rather of the practice of dialogue (hè tou dialegesthai dunamis, 511b4) is, which raises the question of mutual understanding and invites us not to take for granted that everyone understands the same word in the same way, by becoming aware that what gives meaning to a word for a person is the eidos that one associates with it, and that we look for a common principle (archè) that gives coherence to all the logoi that we can hold and the answers that we can give to the questions we are faced with. It is therefore one word at a time (hence the use of hupothesis in the singular in 510b7) that we question their meaning, that is to say the eidos that each one associates with them, before relying on them to move forward in a reasoning, which makes it possible to use them "really as supports" (511b5) which we can rely on to "rise" towards a global and coherent understanding of the world around us, of the place we occupy and the role that awaits us in it (which should be our main concern), which no longer depends upon the particular instances of use recorded in our memory and the images associated with them, not even upon specific words but depends only on the (intelligible) eidè, since it is their "intertwining" and not the simple juxtaposition of words that makes up the logos and allows it to carry meaning.
As we can see, it is not so much the way of reasoning (the logic dear to Aristotle) that makes the difference as the questions we ask and the goals we set to ourselves. And the importance we attach to testing the sturdiness of the supports (hupotheseis) on which we base our reasoning, that is to say the words we use in them, being aware that it is the eidè that each one associates with them that gives them meaning for that person. Socrates does not describe two alternative methods for dealing with the same problems, but two ways of questioning the world around us and of understanding what the logos is. He does not question the way in which a geometer conducts one's reasoning to solve the problem one is dealing with with the help of words that one assumes to be understood in the same way by all as soon as they understand them or claim to understand them, but the fact that such an approach can lead to knowledge in the sense he gives this word, not because he disputes the validity of the results, but because, for him, knowledge (epistèmè) is a whole that must allow each person to find the right path in his life, the one that will lead one to the excellence (aretè) that is accessible to that person when taking into account one's nature as a human being (anthrôpos), the specific limits of this nature in this particular case and the social context of this person's life and will therefore give one access to the true happiness to which this person can claim. But as Socrates says when he seeks to define dialektikè towards the end of Book VII, before reducing geometry and other similar disciplines to the rank of simple dianoia (the name given by Socrates to the pathèma ("affection") associated with the first sub-segment of the perceived by intelligence at 511d8 and again at 533e8), intermediate between opinion (doxa) and knowledge (epistèmè), reasoning, however rigorous, which is based on a "principle" (archè, 533c3, referring to the archomenoi of 510d1 and the hupothesesi of 533c1) that we do not know and of which we are "unable to give an account" (533c2-3; words accepted without asking questions about them and without going back, through eidè, to the ideai that gives them meaning) cannot lead to knowledge (epistèmè, cf. 533c3-5). To reach true knowledge (epistèmè), it is therefore not enough to use an ironclad logic, it is in the first place necessary to agree on what we are talking about, on the meaning of the words we use, that is to say on the eidè that we associate with them, and on the nature of what they refer to, which can only be done through dialegesthai (the practice of dialogue) and the sharing of experiences that it allows, and conducting one's research in the light of the guiding principle that gives meaning and coherence to everything (hè tou agathou idea, "the idea of the good"). Socrates is not saying that we have to take a detour through the idea of the good to prove any theorem of geometry, but that the research that geometers conduct to make these demonstrations does not lead to the true knowledge that could make us happy and is of no use to us in the search for this knowledge. Of no use, not quite, since Socrates uses a theorem of geometry in the Meno. But he does not do this for the theorem itself and the "knowledge" to which it leads (knowing how to calculate the length of the side of a square double in area of a given square), but as an example to show Menon that there is a difference between opinion and knowledge and that it is possible to learn what one did not know before, in a discussion that seeks to understand what constitutes excellence (aretè) for a human being in order to answer his initial question about the possibility of this excellence being the object of teaching (Socrates gives in this dialogue a good example of what is the method associated with the second segment: he refuses to answer Meno's question on the teachable or non-teachable nature of aretè ("excellence") until he has agreed with him on what he means by aretè, to the great displeasure of the latter who would simply like him to answer his question, no doubt to then develop some sophistic arguments made possible precisely by the fact that he did not bother to specify what he means by "excellence" (aretè) in the case of a human being). In short, the first method seeks to provide answers to specific questions on a case-by-case basis, while the second seeks to understand the world around us in a global and coherent way. The first uses words without asking questions about them, the second is interested in the naure of words before using them.
The example taken by Socrates when he mentions the tetragônon ("tetragon / square") at 510d7, which refers precisely to the Meno, is also interesting in relation to what he means by "giving an explanation " (logon didonai) of the words we use. Indeed, not to mention the fact that the geometer, who is aware of the fact that what he demonstrates does not relate to the drawn square on which he reasons, but to something that "cannot be seen otherwise than by thought (dianoia)" (510e3-511a1), does not seek to know what this "something" is, the word tetragônon that he uses to designate what he is reasoning about means etymologically in Greek "which has four angles", but unlike trigônon ("triangle"), which means "which has three angles" and has retained in Greek its general meaning of "triangle" without further clarification, tetragônon has taken on the special meaning of "square", that is to say, only designates what could be considered the most perfect of the figures having four angles, the one for which not only the four angles, but also the four sides are equal (which, in the case of the triangle, leads to the equilateral triangle, for which the Greek does not specialize the name trigônon, but uses the expression isopleuron trigônon). In other words, the word tetragonon is ambiguous (a Greek who hears it for the first time could understand it as refering to a tetragon other than a square) and a reflection on this ambiguity could lead to interesting considerations on the reasons that led the Greeks to give such pre-eminence to the notions of equality and symmetry, considerations that could quickly go beyond the framework of geometry and geometric figures to lead to aesthetic or even political considerations that would quickly bring us back to the question of the good (to agathon). In the same way, when Socrates says about the geometer that he posits as hupothesis ("support") "three kinds of angles" (gôniôn tritta eidè, 510c4-5), here too this deserves to be "explained" (logon didonai): why only three when there are an infinity of different angles? And why favor the "right" angle in relation to which the other two kinds are defined depending on whether they are larger (obtuse) or smaller (acute) than a right angle? Is it for "physical" reasons referring to the law of gravity in that it is the angle that the plumb line makes with the ground on which we want to raise a wall, and therefore the "right" angle to raise stable walls and build houses and palaces? Or aesthetic reasons referring to the notions of equality and symmetry in that it is the angle made by two straight lines that intersect each other forming four equal angles, and therefore the "right" angle to cut a round or square cake into four equal parts? or...?
And regarding the supposed "mathematical objects" (ta mathèmatika tôn pragmatôn) which Aristotle makes much of in his Metaphysics (cf. Metaphysics, A, 987b15-19) where he describes them as intermediate (metaxu) between the sensible (ta aisthèta) and the eidè (which for him are something different from what Plato means by this word, but not what he calls ideai either, but something akin to it), "differing on the one hand from the sensible by the fact of being eternal and without movement, on the other hand from the eidè by the fact of being in large number similar while each eidos itself is only unique", and of which many commentators who think that each sub-segment of the line corresponds to a specific category of "objects" want to make the specific "objects" of the first sub-segment of the perceived by intelligence, it becomes irrelevant, at least for a good understanding of the analogy of the line, once we have understood that there are no "objects" specific to each sub-segment, but only different ways of apprehending what acts on our senses and mind (noûs). But if we want to talk at all costs about "mathematical objects" intermediate between sensible and ideai (and not eidè as Aristotle suggests), here is where to find them, taking the example of the square. The idea of a square, unique, is that of a closed plane figure made up of four equal rectilinear sides forming two to two four equal angles, and this immaterial idea accessible only to thought does not imply any specific position or dimension, which allows it to be unique for all squares, whatever their size, which does not change their nature as squares. However, when Socrates is talking to Meno's slave, he offers him a two-foot square, and then goes on to highlight a different square whose side measures the length of the diagonal of the two-foot square, thus different in dimension from the two-foot square. And the theorem he proves is not true of the squares with approximate dimensions that he draws, but of squares that are only thought of, but thought with specific dimensions that are different for each one. And if we are now interested in all people reasoning on squares, there are a multitude of squares only thought of that are distinguished from each other by their dimensions, which we cannot even associate each one with a specific number since these dimensions depend on the units of measurement chosen (Socrates' two-foot square would have a different dimension, so would be associated with a different number, if we measured it in centimeters, or in inches, and yet it would be the same). And the reasoning would be the same for any other geometric figure, triangle, rectangle, circle, cube, etc. It is all these figures that are part of the "mathematical objects" that Plato must have had in mind when he spoke of "mathematical objects", if indeed he used this expression that is not found anywhere in the dialogues.
To consider all these squares endowed with specific dimensions as ideai in their own right distinct from one another would not make much sense and would only complicate an already complex problem. The fact remains that they are pure "intelligibles" distinct from the more or less approximate material representations that can be given of them. The more general question raised by this analysis is that of how far we should go in setting aside specific properties in the definition of an idea and whether the same problem is not found with regard to all ideai. The idea of anthrôpos ("human being") does not imply height, weight, eye or skin color, nose shape, or a multitude of other things. And yet, the "idea" of a blue-eyed or snub man, or a 120 kg man, is perfectly understandable without reference to a particular person, and is therefore purely intelligible. If Plato specifically evoked with his entourage the particular case of "mathematical objects", it is undoubtedly for the same reason that made him take the example of geometry to explain the division of the segment of the perceived by intelligence, their extreme simplicity, which makes it easier to understand what he wants to make understood. In the case of the square, the number of parameters not taken into account in the idea of square, as soon as we admit that the position in time and space is always ignored in an idea, is reduced to its simplest expression since only one remains, the dimension of the sides, the same for all four. But if, at the other end of the complexity, we return to the case of human beings, this number of characteristics not taken into account is considerable, especially if we go down to each individual and take into account behaviors and logoi, that is to say, we take into account not only the body, but the soul. And yet, one can conceive in a purely intelligible way of multitudes of sub-groups of human beings characterized by "values" specific to some of these characteristics (there is, for example, a purely abstract notion of "just man" or "beautiful man", referring to no particular individual, on which one can reason, as Plato's Socrates rightly does for the just man in the Republic, for instance at Republic, V, 472b7-d2), to the point of describing, always in a purely intelligible way, specific individuals. And this is how we arrive at the anthrôpoi ("human beings") present outside the cave, thus in the intelligible, who are neither ideai strictly speaking, nor sensible. And if Plato did not make much of "mathematical objects", to the point of not mentioning them in his dialogues so as not to complicate already complex things, leaving it to those of his readers who would have reached the end of the ascent out of the cave to find this for themselves (cf. Letter VII, 341b7-342a1, and more specifically 341e1-3), it is because, for him, they were only simple examples of an infinitely more complex reality, which he sought to make understood differently, by analogies and images, in particular through the allegory of the cave.
If, now that we have a better understanding of the division of the segment of the perceived by intelligence and of the two approaches associated with it by Socrates, we return to the question of whether anagkazetai should be understood as a middle ("constrains itself") or a passive ("is constrained"), it must be clear that, for the second approach, it must be understood as a middle, insofar as it is the person who uses this approach who chooses to question the meaning of the words and the nature of what they refer to and to seek a single guiding principle giving meaning to one's research, and therefore imposes this approach on oneself, is obliged to proceed according to the prescriptions of this "method", without anything forcing one to do so from the outside. On the other hand, in the case of the first approach, the doubt remains and the answer depends on the individual. For some, the very idea of questioning the meaning of words does not occur to them, so that they have no choice but to solve the problems they face by using the words that are available to them, taken in the sense given to them by usage and understood in the light of the "images" (eikones) that they associate with them; for others, it may be a methodological choice, a constraint that they impose on themselves in order to better concentrate on what seems important to them, to solve the problems they are faced with without drowning in philosophical and metaphysical considerations that would divert them (in their eyes) from this objective, as for instance for a geometer to question what is the nature of this "ideal" square, the only one for whom their demonstrations are valid, but which can only be "seen" by thought (it is this attitude that makes Callicles condemn philosophy for "serious" people who have reached adulthood in the Gorgias). But the fact remains that, for them too, their understanding of the words they use is the one provided by usage and the "images" (eikones) they associate with them, and that they are therefore "constrained" (anagkazetai, passive meaning) by the words they use without questioning them (if Socrates, in the Cratylus, uses the word nomothetes twenty times (etymologically "poser (thetes) of laws (nomoi)") in connection with those who "create" names, it is not to suggest that legislators, and they alone, are those who forge words and language, but to make us understand that words and language are the first "laws" that are imposed on all human beings from an early age and forge their understanding of the world). The first approach is the one that culminates in Aristotle's logic, the second that of dialektikè according to Plato, whose so-called "Socratic" dialogues give brilliant examples, provided that they are not read, under Aristotle's influence, as searches for "definitions" (horismoi) in the Aristotelian sense (cf. Metaphysics, A, 987b1-4, where Aristotle reduces the novelty introduced by Socrates to the search for definitions: "Socrates... seeking the universal (to katholou) and being the first to fix his thought (dianioan) on definitions (peri horismôn)"), but see them as attempts to clarify the contours of a notion (for instance, piety in the Euthyphro, friendship in the Lysis) through dialogue by seeking to better determine its boundaries with other neighboring notions through examples rather than seeking at all costs to summarize this quest by packing its result in the few words of a definition, words just as problematic as the one we are trying to define.
We may now attempt to see if there is a relationship of image to original between the approach associated with the first sub-segment of the perceived by intelligence, the one that relies on the words used without questioning their meaning(s) and the nature of what they claim to refer to, but accepting the common "opinion" on them and the sensible "images" associated with them by the person who uses them, and the approach of the second sub-segment, the one which, having become aware of the fact that what gives meaning to words for us are the eidè that each one associates with them, questions the meaning of words and the nature of what they refer to before relying on them to conduct reasoning and reasons on eidè and no longer on sensible images, and if so, in what sense, since we know that the two segments are segmented ana ton auton logon ("according to the same logos") and that this logos, in the case of the segment of the seen, seems to be a relationship between image and model. A clue toward an answer is given us, in pictorial form, by Socrates himself when, towards the end of Book VII, in the discussion on dialektikè, he refuses to call "knowledge" (epistèmè) geometry and other similar disiplines, saying: "we have seen that they (geometers and other "scientists") dream about what is, but that it is impossible for them to see [as if] in a waking state so long as, while making use of supports (hupothesesi), they let them unmoved, unable to give an account of them (logon didonai)", before adding these words, already mentioned earlier in this note: "for where a principle (archè) that one doesn’t know, an end [result] (teleutè) and the intermediate [steps] coming from what one didn’t know have been twined together, what contrivance could ever transform such an agreement (homologian) into knowledge (epistèmè)?" (533b8-c5).As we have seen, the supports that are left immutable and for which one does not want to account are the words whose meaning inherited from usage is accepted, that is to say, based for the one who uses them only on "opinions" that one does not question because one has not become aware of the fact that these words have meaning only through the eidè that we associate with them, unconsciously in their case since they are simply "inherited" from uses without questioning, and about which we do not try to understand the way in which they can give us access to knowledge about "beings". And Socrates tells us that to proceed in this way is to dream of a knowledge that is based only on opinions, and therefore to have only an "image" of knowledge, to develop only "images" of reasoning, because what distinguishes the two methods is not the rigour in the reasoning, but the solidity of the "supports" (hupotheseis) on which it relies, so that, as long as we have not put the solidity of these supports to the test, that is to say, as long as we have not ascertained the meaning of the words we use by bringing to light the eidè that each one associates with them and the way in which they are likely to give us access to something other than themselves, the most rigorous logical reasoning will not be able to transform what is only an assembly of opinions into knowledge, as the Parmenides brilliantly shows, which offers reasoning of unfailing logical rigour leading to opposite conclusions on the same issues because the one who leads the discussion has never bothered to specify what he means by "to be" (einai) and "one" (hen), two words whose meaning most people think is obvious when indeed it is not so as soon as one begins to scratch beneath the surface, and modifies the meaning he gives to these two words, and to the other words he uses, which he has not taken the trouble to define, from one reasoning to another without saying so. In other words, words play in the intelligible the role played by the shadows in the cave: if we do not become aware of the fact that they are only "images", "labels", referring through the eidè that we associate with them to "objective" originals, pragmata with or without a sensible dimension, which impose their "law" on them in that we cannot say anything about them if we want to say the "truth", and so to "know" what "is", we can only end up with opinions, which, if we are lucky, could be true, but which could also be false, and which, in any case, will only be "images" of knowledge. For the difference between knowledge (epistèmè) and opinion (doxa), which are the one or the other only in relation to a given person, is not in the fact that one is true and the other false, since an opinion can be true, nor even in the words used to formulate it, since someone can express what is for that person only an opinion (true, in this case) with exactly the same words as someone who has real "knowledge", for instance by having learned by rote the words heard or read from one "knowing", but precisely in the "foundations" that support these words in the mind of the person whose knowledge or opinion it is, in one's ability to "give an explanation of them" (logon didonai), to bind his words "by a reasoning on the cause" (aitias logismôi; Meno, 98a3-4), but a reasoning on the cause that ensures its foundations by ensuring the meaning of the words one's uses not merely by an appeal to usage, and above all that relies on the eidè to which one attaches these "labels", being able to change the label without changing the eidos, as Socrates does, for instance, in the analogy of the line and the allegory of the cave, speaking at times of phantasmata (510a1, 516b5), at times of eidôla (516a7), to refer to the "reflections" which he takes as an example of images (eikones) next to the shadows when he associates these images with the first sub-segment of the seen, but always taking care to immediately add the words "on the waters" (en tois hudasi) to make it clear that with different words, it is indeed the same thing that he is talking about, each of these two words with multiple meanings overlapping in part contributing to the clarification of the eidos he has in mind, the word phantasma insisting on the inconsistency of a reflection, the word eidôla on the resemblance to the original.
Before ending this note, I list hereafter the translation of this line by the various translators I have consulted:
- Jowett: "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."
- Shorey: "By the distinction that there is one section of it which the soul is compelled to investigate by treating as images the things imitated in the former division, 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, and in which it makes no use of the images employed by the other section, relying on ideas only and progressing systematically through ideas."
- Cornford: "There are two sections. In the first the mind uses as images those actual things which themselves had images in the visible world; and it is compelled to pursue its inquiry by starting from assumptions and travelling, not up to a principle, but down to a conclusion. In the second the mind moves in the other direction, from an assumption up towards a principle which is not hypothetical; and it makes no use of the images employed in the other section, but only of Forms, and conducts its inquiry solely by their means."
- Lee: "In one sub-section the mind uses the originals of the visible order in their turn as images, and has to base its inquiries on assumptions and proceed from them not to a first principle but to a conclusion; in the other it moves from assumption to a first principle which involves no assumption, without the images used in the other sub-section, but pursuing its inquiry solely by and through forms themselves"
- Bloom: "In one part of it a soul, using as images the things that were previously imitated, is compelled to investigate on the basis of hypotheses and makes its way not to a beginning but to an end; while in the other part it makes its way to a beginning that is free from hypotheses; starting out from hypothesis and without the images used in the other part, by means of forms themselves it makes its inquiry through them."
- Grube: "In such a way that in one section the soul, using as images what before were models, is compelled to investigate from hypotheses, proceeding from these not to a first principle but to a conclusion. In the second section which leads to a first principle that is not hypothetical, the soul proceeds from a hypothesis without using the images of the first section, by means of the Forms themselves and proceeding through these."
- Grube/Reeve: In one subsection, the soul, using as images the things that were imitated before, is forced to investigate from hypotheses, proceeding not to a first principle but to a conclusion. In the other subsection, however, it makes its way to a first principle that is not a hypothesis, proceeding from a hypothesis but without the images used in the previous subsection, using forms themselves and making its investigation through them."
- Reeve: "In one subsection, the soul, using as images the things that were imitated before, is compelled to base its inquiry on hypotheses, proceeding not to a flrst principle, but to a conclusion. In the other subsection, by contrast, it makes its way to an unhypothetical first principle, proceeding from a hypothesis, but without the images used in the previous subsection, using forms themselves and making its methodical inquiry through them." (<==)

(22) "I don’t fully grasp" translates the Greek ouch ikanôs emathon. The verb used here by Glaucon, which I translate as "to grasp," taken in the sense of "to understand," is manthanein, of which emathon is the aorist in the first person singular. The primary meaning of manthanein is "to learn", and by derivation, "to understand". It is from the aorist infinitive mathein that derive words such as mathèsis, "the act of learning" and also "education, instruction", mathèma, "that which is learnt, knowledge", and in the plural mathèmata, the "mathematical sciences", or mathètès, "learner, pupil, student". Unlike the katanoein used by Socrates at the end of the description of the visible (cf. note 15), translated as "to understand", which refers to comprehension by the noûs (root of the verb noein) outside of any master-student relationship, the verb used here by Glaucon emphasizes the learning process that leads to understanding and suggests that Glaucon implicitly places himself in the situation of "student" in relation to the "master" Socrates in the apprehension of the intelligible order where, as the allegory of the cave will soon show, there is a "guide" who shows the way. It is to mark this difference that I have not translated ouch emathon as "I have not understood", but as "I don't grasp" (in the present tense, which remains a possible translation of the Greek aorist which, as its name suggests, describes an action outside any consideration of time), having the same meaning, but which suggests the idea of someone "grasping" what someone else is "handing over" and is thus a little closer to the teacher-student relationship. To allow the English reader to identify the underlying Greek verb, I keep translating manthanein by "to grasp" in the other five occurrences of this verb in the analogy of the line (510c2, at the beginning of Socrates' reply to this remark of Glaucon, 511b1, in the form manthanô, "I grasp", as the first word of a reply of Glaucon, 511b3, at the beginning of Socrates' reply, 511c3 and 511e5, again in the form manthanô, "I grasp", as the first word of a reply by Glaucon). (<==)

(23) "You know" translates the Greek se eidenai (word for word, "you to know"), where eidenai is the perfect infinitive from the verb idein, "to see", which, at this tense, takes the meaning "to know" (for "to have seen": I have seen therefore I know), that is to say, to see with the eyes of the mind, and whose various forms are at the root of both eidos and idea, the words often translated as "ideas". Eidenai is phonetically close to eidesi, plural dative of eidos, which was used by Socrates in his first formulation to speak of the "appearances" through which one progresses in the second subsegment of the noèton. In his reformulation, Socrates thus emphasizes that he appeals to Glaucon's ability to reason about the eidesi. (<==)

(24) "Those who busy themselves in geometry and calculation": "calculation" translates the Greek logismous, accusative plural of logismos, a word derived from logos in its sense of "report, ratio, reason, proportion" via the verb logizesthai, "calculate", and which means in the primary sense "counting, calculation", and in the plural, "numbers", and hence "science of numbers, arithmetic". I retain the translation by "calculation" in the singular, since the English word has, like logismos, a dual meaning, depending on whether we are talking about dealing with numbers or planning action based on the various possible outcomes of the different courses of action avaiable.
It is indeed, as I have implied in note 7, and again in note 21, by taking his examples in the "propaedeutic" field of geometry and computation, which deal with the constructions that are easiest to "abstract", that Socrates will try to explain the approach characteristic of the first sub-segment of the perceived by intelligence he described in the extraordinarily dense sentence of his previous line. But, as I indicated in the note on this line, it would be a mistake to believe that this approach is limited to this register and that it can only relate to "mathematical objects", as do some commentators who want the four segments of the line to be distinguished by their "population" (on these supposed "mathematical objects", see what I say about it in note 21). (<==)

(25) "Setting as supports for themselves" translates the Greek hupothemenoi, a middle aorist participle in the masculine nominative plural of the verb hupotithenai from which hupothesis is derived, in a way that remains consistent with my translation of hupothesis as "support", justified in note 21. It will be seen that, as I implied in this note, the hupotheseis that Socrates presents here as examples are not hypotheses in the modern sense of the word, but rather elementary mathematical concepts designated by technical names that serve as a supposedly accepted starting point for the rest of the reasoning. In modern language, we would not speak of hypotheses, but of initial data. In fact, hupothemenoi here has exactly the meaning of the verb "to suppose" (which is the exact Latin equivalent of hupotithenai, as I indicated in note 21 when explaining the word hupothesis) in a sentence like "Let's suppose a circle with center O and a straight line D that intersects it..." uttered by a math teacher dictating a problem statement to his students. In this use of the verb "suppose", the emphasis is not on the uncertain nature of the elements listed, but simply means that, since these elements are not present in the students' immediate environment, it is necessary to "suppose" that they are in order to deal with the problem. The teacher could just as well have said "let be a circle with center O and a straight line D that intersects it..." without the meaning being changed. Unfortunately, this meaning of the verb "suppose" is not preserved in the noun "supposition" that derives from it, and which, like "hypothesis", emphasizes the uncertain nature of what is "supposed". This is the reason why I do not retain "supposition" to translate hupothesis, which prohibits me from using "suppose" to translate hupotithenai if I want to make the kinship that exists in Greek between the two words perceptible in English. (<==)

(26) "The figures" translates the Greek schèmata, plural of the word schèma. I have already mentioned this term, which has a meaning similar to eidos and idea, refering like them to the external appearance of something or someone, but which also has the more specialized meaning of "figure" in geometry, in the second part of note 17, where I tried to distinguish the nuances of meaning that Plato seemed to give to these three terms, suggesting that schèma insists on the strictly visual appearance of the thing or person under consideration, as it could be reproduced in a drawing, which is consistent with the geometric specialization of the word to speak of "figures" in the mathematical sense (a specialization that makes the word ambiguous since it can just as well designate the mental image we have of a visible reality by limiting itself to its more or less simplified contours as the material representation of it that we can give by drawing on sand, on paper or on any other material support). In the light of these remarks, we may think that it is probably not by chance that, in the Meno, Socrates chooses the word schèma to give Meno an example of what he is looking for when he asks him what, in his opinion, is the eidos common among all the aretai (a word often translated as "virtues", but which rather means "excellence, perfection" of a person or thing, the fact that it has to the highest possible degree the qualities proper to its species), which justifies calling them all by the same name of aretai, as I explain in note 7 to my translation of section 73c6-77a5 of the Meno (in French only): according to the explanation I propose in note 17 above, the schèma is in fact the most easily graspable variety of eidos, the one that is offered to the eye alone, and to which geometry familiarizes us precisely on examples that can be described as elementary (simple, regular shapes that are easy to reproduce even by someone who is not gifted in drawing).
It should also be noted that Socrates has just mentioned geometry and calculation and that the reference to schèmata makes the transition between two examples of arithmetic concepts, the odd and the even, and examples of geometric concepts, the three "appearances" of angles (see next note). And indeed, this notion of "figure" can be seen as meaningful in both domains. It should be remembered that, in the time of Socrates and Plato, it was customary to represent numbers by what are similar to figures: groups of dots organized into rows and columns that can form "rectangles" (for example, 21 represented by three lines of seven points each) or "squares" (for example, 16 represented by four lines of four points each), and that it is precisely this way of doing things that led to the name of "square" given in Greek as in English, to the product of a number by itself, which can therefore be represented by a number of lines equal to the number of points in each of the lines, and by extension that of "cube" when the third dimension is introduced (an example of this way of doing things can be found in the initial dialogue between Socrates and Theaetetus, at Theaetetus, 147d-148b, when Theaetetus explains to Socrates what he has just done with the young namesake of Socrates under the leadership of Theodore). Thus, a number is "even" if it can be represented by two lines containing the same number of points, and "odd" if one of the two lines contains one more point than the other. If it is important to keep this in mind, it is because it allows us to realize that our knowledge of numbers also originates in pictures: we know what three means by reference to the vision of three people, or three cows, or three chairs, or three dots on a sheet of paper or on the sand, and we know that two times three make six because we have visually experienced on our fingers or on dots that it was so. And even if we no longer use the graphic representation of numbers by dots and no longer count with abacus, the fact remains that it is still through visual experiments that children learn to count, starting with the first whole numbers, in particular by counting on their fingers before counting "in their head"; and it is by generalization of the confidence that our view has given us on the first numbers and on the tables of addition, subtraction and multiplication of the first ten whole numbers that, aided in this by the use of decimal notation (which the Greeks did not know, who represented numbers by letters and did not know zero), we extend this confidence to all numbers, whole or not, small or large, rational or irrational. And when modern mathematicians extend the concept of numbers to introduce what they call "complex numbers," they again resort to a geometric representation in a two-dimensional space to make what they are talking about apprehensible and to make us understand what they mean by the "real part" and "imaginary part" of such a number. In short, it is not only geometry that is dependent on sight and "images/figures", but, as Plato's Socrates rightly suggests here, it is also the case for calculation (logismos), even if we can more quickly free ourselves from the use of "figures" for calculation. (<==)

(27) "Three appearances of angles" translates the Greek gôniôn tritta eidè, which one might be tempted to translate simply as "three kinds / sorts of angles" by giving eidè a "neutral" meaning (all the translators I consulted except Lee and Bloom, who translate it as "forms", transalte it at "kinds".) But, as I said in note 5 when explaining why I had decided to translate all the occurrences of eidos in the analogy of the line by the same word "appearance", we would then lose sight of the fact that it is the same word, eidos, that is used by Socrates here and when he has just spoken of the eidesi in connection with the second sub-segment of the segment of the perceived by intelligence. This consistency in the translation of eidos will find its justification in the following reply of Socrates, when he speaks of horômena eidè ("seen appearances"), clearly showing that he does not limit the use of the term eidos to abstractions of the intelligible order (cf. note 34). The three eidè of angles he is refering to are the acute angle, the right angle and the obtuse angle, and these terms refer to concepts that have their origin in sight (I know at first glance an acute angle or an obtuse angle on a drawing), to different "appearances" of angles depending on whether they are more opened or more closed than the right angle which serves as a median reference and is the one formed by two straight lines perpendicular to one another, that is, which delimit at their intersection four equal adjacent angles. But if sight is at the origin of these concepts, it doesn't mean that they are "immediate" data of sight, which strictly speaking perceives only patches of color and not shapes, but concepts abstracted by our mind from the data provided by sight, which justifies the qualifier of eidè that is given to them, so that there is not much of a difference in nature between the eidos of acute angle which I abstracted from the vision of two segments of straight lines having a common end, by taking an interest, to follow up on the analysis of the second part of note 17, in the diagram which I contemplate, and the eidos of beauty which I abstracted at the sight of a master painting in a museum, by worrying more about the idea which presided over the creation of the painting: in both cases, we are dealing with something that is not given to me directly by the senses, but results from an elaboration of my mind. Simply, in the first case, what I am looking at is directly an image (eikôn) of acute angle, while in the second case it is not an image of the beautiful as such, but a beautiful image, and I can conceive a multitude of beautiful images which will have no purely visual characteristics in common with one another, not to mention that it is not only paintings that can be beautiful, so that beauty itself is not susceptible of direct visual representation.
The fact that Socrates uses here, in connection with the characteristic approach of the first sub-segment of the perceived by intelligence, the term eidè which he had reserved, in his first explanation (the one analysed at length in note 21) for the description of the characteristic approcah of the second sub-segment, is intended to make us understand that it is not the use of eidè that characterizes this second approach, since, as we have seen in note 5, there are eidè (visible or intelligible) at all stages of the progression from the bottom of the cave to the contemplation of the sun, since at all stages there is logos and that the words, all the words, presuppose underlying eidè, different from one person to another and for the same person from one moment of his life to another. As we have seen in note 21, what opposes the two approaches is the opposition between eikôn ("image") and eidos ("appearance"). What we are now invited to realize is that the perception we have of an "image" by sight is also made by means of an eidos, the "appearance" that this eikôn takes on for our mind from the data provided by sight. And the stake of all this reflection is ultimately to know whether we are ready to accept that there can be another category of realities, not visible, capable of nevertheless arousing eidè in our minds and whether we are capable of reasoning about these realities, or at least about the eidè under which they present themselves to our minds, In short, for a contemporary reader of Plato, the problem is not to "degrade" the highly abstract understanding that he might have of eidos in order to accept to apply it to the visible, but on the contrary to start from this common understanding of "appearance" that takes its source in the visible and extends it to the intelligible order by realizing that, even in the visible, eidos refers to a construction of the mind, not to an immediate datum of sight. And this is what the elementary examples of geometry and calculation can help with by highlighting the proximity between the schèma and the eidos for those who have less difficulty in accepting that a schèma such as the square or the circle does indeed refer to an abstract reality that is not the figure that we draw and that we can see than to accept that an eidos such as "beautiful" also refers to a transcendent reality of the same nature as the "square itself" (to tetragônon auton) or "the diagonal itself" (hè diametros autè). (<==) ICI

(28) I translate, here as in 510b8, the word methodon by "road map" so that the reader who has access only to the English translation will see that it is the same word that is used both times, which is not possible with most of the translations I have consulted, where, as is the case with eidos, the subject of the previous note, translators vary their translation of the same word from one occurrence to another, as can be seen by referring to the page of this site on "The vocabulary of the line", in which I present the way in which the important words in the analogy of the line are translated at each of their occurrences by the different translators I have consulted: It will be seen that in most of them, the eidesi of 510b8 and the eidè of 510c5 are translated by two different words, the first being always translated by a "noble" term referring to the vocabulary of the supposed "theory of ideas / forms", "ideas" or "forms", with or without a capital letter, and the second by a mere "kind" (the two exceptions are Lee and Bloom, who translate both by "form"), and that five out of eight translate the two occurrences of methodos by a different word (Bloom translates both by "inquiry", Grube/Reeve by "investigations" and Reeve by "methodical inquiries").
Socrates is here in the process of explaining for Glaucon a very dense sentence which, as we have seen, was constructed with the utmost rigour and where the words had been chosen with great care by Plato. In this synthetic description of the two approaches of the segment of the perceived by intelligence, which I have analysed at length in note 21, methodos, like eidos (cf. previous note), is used only in the description of the second approach. And as is the case with eidos, its reuse here in connection with the first approach suggests that it is not this word and what it suggests, the idea of a progression in reasoning organized according to a pre-established plan, following a "method", that is characteristic of the second approach. As we have seen in note 21, methodos means in the primary sense "path (hodos) through (meta)", and hence "following after, pursuit" and has come to mean "research (of knowledge), investigation", without this necessarily implying the "methodical" character that attaches to its transcription in English in the word "method". And moreover, the word is used here in the expression kath' hekastèn methodon ("according to each road map"), which clearly shows that Socrates does not have in mind a single "method" that would be specific to the first segment of the perceived by intelligence, but the way of proceeding used in each case, adapted to the specific problem posed and to the ways of doing things of the person conducting the research, which allows, from the "givens" of the problem, which he calls the hupotheseis ("supports") taken as a starting point (cf. archomenoi, "starting from these" at 510d1), to reach a conclusion (teleutè) proposing an answer to the question initially posed. In fact, any investigative intellectual approach to any subject and by any means can be described as methodos. What opposed the two approaches in the first description was not so much that one followed one methodos and not the other, but the more or less constrained character of the first as opposed to the greater freedom in which the second was exercised, due to a different attitude towards words: the opposition was between the poreuomenè ("driven" or "advancing" depending on whether it is read as a passive or a middle) on the one hand and the iousa ("going", active)... tèn methodon poioumenè ("making for oneself the road map") on the other hand. What differentiates the two sub-segments, more than the "method" used, are the questions we ask ourselves, solving problems on a case-by-case basis for the first or trying to understand the world in its own coherence in order to deduce rules of life that are "good" for the human beings that we are for the second, and the way in which we approach the "supports" (hupotheseis) on which we "build" our reasonings, which ultimately boil down to words, depending on whether we accept them as they are provided to us by usage without asking questions about them (which makes their understanding mere opinions) or whether we put them to the test before relying on them and manage to reason on eidè (intelligible) and no longer about words and visual images (eikones) associated with these words, which is manifested in particular by the ability to reason about the same "thing" by using different words to talk about it. In other words, with all due respect to Aristotle, the rigour of a reasoning, the logic on which it is based, is not enough to guarantee the validity of conclusions, it is first necessary to test the solidity of the "supports" on which it rests, the adequacy of the words with which it expresses itself to what it seeks to account for. In the first sub-segment, we are constrained by an interpretive framework that is imposed on us by usage through the words that we inherit without challenging their meaning(s) and that forge our understanding of the world and impose their "law" on it, whereas they only give an image of it and have moreover for the most part been created by prisoners who have never yet been freed from their bonds in the cave on the basis of the shadows they see there and take for reality (cf. 515b4-c2) whereas in the second sub-segment, it is admitted that words do not refer us directly to "beings", but do so through the mediation of eidè that we forge for ourselves in order to make them understandable to ourselves (cf. Republic X, 596a5-7), first from visual images that we must then get rid of in order to understand them only in the relationships they have with one another and that it is indeed by reasoning at the level of eidè and not words, even if these reasonings cannot do without words, that we have a chance of arriving at knowledge, even if this way of proceeding doesn't guarantee success. But since we are the ones who forge these eidè and make them evolve according to our experience, we have greater freedom in this approach than in the first, even if we cannot arrange these eidè as we please, since, in order to arrive at truth and knowledge, they must give an account of something that does not depend on us, of an objective reality in relation to which the greater or lesser truth of the result of these approaches is assessed. (<==)

(29) "As [people] knowing" translates the Greek hôs eidotes, in which we find the verb eidenai used at the beginning of Socrates' line, and which means, as I indicated in note 23, "to know" because "having seen". (<==)

(30) "To give a logos about them" translates the Greek logon... peri autôn didonai, without translating the word logos for reasons explained in the note on the prologue to the introduction. In the case that concerns us here, logon didonai is a ready-made expression meaning "to give an account, to give reasons, to give explanations", in which the meaning of logos, such as that of "account(s)" in English in such a context, straddles a purely linguistic meaning ("explanation" given by means of "speech", another meaning of logos) and a meaning having more to do with numbers, accounting and mathematical, which is the one that led to logismos in the sense of "calculation" (cf. note 24): this expression was used in particular with regard to elected leaders, who were asked, at the end of their mandate, to "give account" of it, which implied in particular to provide the accounting of the sums spent on the city's funds. It is therefore the context that makes it possible to know whether the "account" that must be rendered is limited to simple discourse or should take the more quantified form of "accounting", somehow like the expression ana ton auton logon ("according to the same logos") encountered at the beginning of the analogy and analysed in note 10, which, as we have seen, can also be understood in a general sense implying only a "rationale" expressed in words, or in a more technical and mathematical sense referring to a "ratio", a "proportion" between numbers. But if, as I pointed out at the time, in the case of ana ton auton logon at 509d7-8, doubt was allowed, here, the context shows unambiguously that what is at issue with the expression logon didonai is, for the geometer, to show by logoi that he has a real knowledge of what he takes as hupotheseis ("supports") of his reasonings and not mere opinions on it, that, for instance, in the case of the tetragônon ("tetragon / square") referred to below, he is on the one hand aware of the fact that its name is confusing, since it etymologically means "which has four angles", without implying that these four angles are equal and even less that the sides are equal, which leads to a very particular case of tetragon (it is to make this ambiguity apparent that I have translated tetragônon as " tetragon", adding "square" in parentheses after it), and that, on the other hand, he has a clear understanding of what "the tetragon itself" is (to tetragônon auto, cf. 510d7) accessible only to thought (noûs) and of which only the conclusions he demonstrates about the square are true, a question that precisely, most often, if he is only a geometer, he refuses to deal with, considering that it contributes nothing to the demonstrations he conducts. (<==)

(31) "Taking them as starting principles" translates the Greek ek toutôn archomenoi, in which we find the verb archein, from which archè is derived. Archè is another one of the words used in the previous reply, on the one hand to say that the approach of the first subsegment did not lead to an archèn, and on the other hand to say that, on the contrary, the approach of the second subsegment led to an archèn anupotheton, to an ultimate guiding principle that was valid for itself and not as a support (hupothesis) for something else. The association of this verb with the preposition ek ("out of") to introduce its complement shows that archein must be understood in a sense that refers to an archè "origin", which I have rendered as "to take as starting principles", using "principles" in the plural, rather than "point" in the singular ("to take as a starting point"), both to better mark the relationship of the verb with archè, which I have translated as "(leading) principle" and to make more perceptible the plurality that is at the beginning of this approach, indicated by the plural hupotheseôn, to which the plural toutôn of ek toutôn refers (one of the meanings of archein is "to begin" and we could therefore also simply translate it as "beginning from them", but we would then lose all these resonances with what has gone before). This confirms the fact that what Socrates refers to here as hupotheseis are indeed the data of the problem that were "supposed" at the outset, considered by the person who posits them as solid, as something which one can rely on, and not as simply "hypothetical", i.e. as uncertain.
"They end up" translates the Greek teleutôsin, a verb with the same root as teleutè, the word that was used in the previous line to designate what the characteristic approach of the first sub-segment led to, and which I have translated as "end". This choice of vocabulary confirms what I said in note 21 about the fact that, in the characteristic approach of the first sub-segment, the archè is not considered as a guiding principle towards which one advances, but only as an initial given from which one starts. In this approach, archè and hupothesis are practically synonymous and both designate what we start from.
The verb that I have translated as "going from beginning to end through" is diexiontes, the present participle of the verb diexienai, formed by adjunction of the prefixes dia- ("through") and ex- (which, in composition, implies an idea of completion) to the verb ienai, "to go", which also, like the prefix dia- ("through" or "by means of") was, in the first formulation, reserved for the second sub-segment. But, as I have already said in note 28 about methodos, it is not the fact of progressing from premises to conclusions any more than having a "plan of action (methodos)" that is characteristic of one or the other approach, but the attitude in relation to the visible / sensible and the acceptance or not of freeing oneself from it in order to "go up" to an ultimate principle that will serve as a guide for us to progress towards conclusions. (<==)

(32) "Consistently" translates the Greek homologoumenôs, an adverb derived from the verb homologein, constructed from the verb legein, "to say, to speak" (the verb from which logos is derived), and the prefix homo-, "the same". Homologein is either "saying the same thing" or "agreeing". The idea here is that the stages of "reasoning" are in agreement with one another and with the initial data, the hupotheseis ("supports") from which we started. It is this idea that I convey with the notion of "consistency".
This homologoumenôs can also be understood as meaning "in such a way that everyone can only agree", that is, in such a way that a present or supposed interlocutor will agree with you.
In a certain sense, behind the use of this adverb is the idea that any demonstration of this kind is in a certain way tautological. Everything that we discover in the course of reasoning was already contained in the premises and therefore only repeats the same thing: when I say "square", I immediately imply four equal sides, four right angles, a diagonal on which a square of double surface can be built, etc. (<==)

(33) "On what they had set their investigation in motion upon" translates the Greek an epi skepsin hormesôsi. Hormèsôsi is the subjunctive of the verb horman, derived from hormè, "assault, impulse, effort, zeal", and therefore implies the idea of a violent movement, the result of impulses or a zeal that is not always controlled.
Skepsin, translated as "investigation", is the accusative of skepsis, an action noun derived from the verb skeptesthai which means "to look about carefully, examine", and figuratively about the mind, "to view, examine, consider". The primary meaning of skepsis is "viewing, perception by the senses". Here again, a term whose primary meaning relates to sight is transposed to the realm of the noèton. (<==)

(34) "They also make use of the seen appearances" translates the Greek tois horômenois eidesi proschrôntai. The expression tois horômenois eidesi ("the seen appearances") confirms that Plato's Socrates does not lose sight of the etymology of the word eidos and its roots in "seeing", and that he does not limit the meaning of eidos to the sole case of intelligible "appearances / forms". When a geometer looks at the square he has drawn to carry out his reasoning, what he perceives is indeed an eidos, apprehended by sight and in which the noûs identifies a schema that he designates by the name of "square" (for Aristotle, De anima, II, 418a7-26, the schema is part of what he calls "common sensible", in the same way as movement (kinesis), rest (èremia), number (arithmos) and magnitude (megethos), that is to say, notions perceived by all or at least several of the senses, the "proper sensible" of sight, that which sight alone can perceive, being for him color (chrôma), which shows that he is well aware, after having frequented Plato for a long time, that it is not sight as such that recognizes "forms", but that he is reluctant to give up making of them data perceived by the senses). But this schema also arouses in his mind a purely intelligible eidos pointing to an idea, the idea of square, which is the way in which human intelligence supposed perfect can apprehend the "square itself" (to tetragônon auton), of which a visible figure gives only an approximate "image". And no more than the drawn square is in the mind of the geometer who looks at it, but only its eidos (visible) perceived by him, the "square itself", of which the drawn square is a defective image, is in his mind, but only the eidos (intelligible) that he forms of it according to his earlier experience; the drawn image of the square exists in time and space, as a material reality, on the sand or on the paper where the geometer has drawn it, and the "square itself", of which this material figure seeks to give the image of a particular instance as best it can, constitutes an objective "reality" with no relation to time or space, without size or specific position, which can only be represented for us by logoi.
In either case, whether regarding "visible" or "intelligible" eidos, we are therefore in the presence of an "appearance", a "representation" of a reality, visible (the drawn figure) or intelligible (the "square itself"), which is other than the "representation" that forms of it in our mind. And it is precisely because, in both cases, it is in our minds that these representations are formed, provoked in one case by the data provided by sight, in the other by the result of our reflections on these data, that we can use the same word for both, that of eidos. And it is because at this point it is more important to emphasize similarities than differences, that Socrates uses the same word here in both cases, rather than reserving schema for one and using eidos only for the other.
But let us also note that the difficulty experienced by translators influenced by twenty five hundred years of "Platonism" and whose native tongue is not ancient Greek faced with the words horômena eidè is most likely the opposite of that which readers of Plato's time must have experienced: for a contemporary of Plato, what must have seemed curious in these words is that Socrates feels the need to specify horômenon ("visible") when speaking of eidos ("appearance"), whereas for a modern "Platonist", who gives eidos, at least in the context of the analogy of the line, a "technical" meaning referring to a supposed "theory of forms / ideas", what seems unacceptable is that Socrates associates here two terms, horômenon ("visible") and eidos (supposed to mean "Form" in the "noble" sense, especially in an explanation that seeks to describe a specific approach of a sub-segment of the intelligilble) which are for them incompatible, which leads them to refuse to give eidos the same "noble" meaning as that of eidè in the line that Socrates is here explaining, as will be seen by examining the various translations of this line or by referring to the page on "The vocabulary of the line" already mentioned. Some translators (Cornford, Lee, Grube and Grube/Reeve) do not hesitate to use the same word, "figure", to translate in this line of Socrates both schèma and eidos, which prevents us from seeing that there are two different words in Greek, and to realize that the use here of one of them, eidos, could help us better understand the scope Plato gives it
It seems to me, however, that, with this choice of words, which, for his contemporaries, must have seemed redundant, Plato wanted to challenge them on the fact that, contrary to their usual way of thinking and using this term, not all eidè are "visible (horômena)" (since he feels the need to specify that some are, which implies that others are not), but that all of them have in common something that justifies using the same word, eidos, the fact of being "representations", "appearances", for our mind.
This impression is confirmed by the verb being used, proschrôntai. Proschrôntai is the third person plural of the present middle indicative of the verb proschresthai, in which the prefix pros added to the verb chresthai ("to use") introduces the idea of "addition", which therefore implies that what is mentioned after as "used" is not the only thing that is used. If the people of whom Socrates speaks "also make use of seen appearances", it is because they do not use only that. And if, as I have just said, he felt the need to specify that the eidè they also use are "visible" eidè while eidè naturally refers to the visible for his contemporaries, it is indeed to suggest that the things in addition to which they use eidè that he took care to qualify as "visible" are also eidè, but eidè that are not visible, but only intelligible. These eidè that are not "visible" are, for example, the "concepts" associated with what he has just given, in the previous line, as examples of hupotheseôn: the odd and the even, or the three eidè of angles (cf. note 27), which, insofar as he specifies "three", obviously refer to the notions of acute angle, right angle and obtuse angle, which, as eidè, are not visible. And it is in addition (pros-) to the fact that these concepts are mentioned as given (hupotheseis) in their statements, and because these words refer only to purely intelligible eidè, that geometers draw drawings that give them a visible "appearance" (eidos) to help them in their reasoning.
To be in the intelligible, as I said in note 21, is to be aware that, as Socrates will say at the end of this line, there are "things" that one cannot "see" other than by thought (dianoia), and the criterium that differentiates the two sub-segments of the perceived by intelligence is whether, to understand them, we turn downward toward the cave to look for visible images, horômena eidè ("seen appearances"), those "things" invisible to the eye (first sub-segment) or we turn upwards and try to get used to the light of the sun / idea of the good (the archè anupothetos) to try to understand them in this light without having recourse to visible images, relying only on noèta eidè ("intelligible appearances") that are associated with words in a flexible way (second sub-segment).
It is therefore regrettable here again, but consistent with their way of understanding eidè here, that most translators do not render this pros- in their translations: thus Jowett ("they make use of the visible forms", Cornford ("they make use of visible figures"), Lee ("they make use of... visible figures"), Grube, Grube/Reeve ("they use visible figures"), Reeve ("they use visible forms"). (<==)

(35) "They make for themselves their logoi" translates the Greek tous logous poiountai without translating logoi for reasons explained at the beginning of this page. According to dictionaries, logous poieisthai is often a simple periphrasis for legein, "to speak." But here this expression associates with logos, a word already encountered at 509d7-8 in the expression ana ton auton logon (cf. note 10) and at 510c7 in the expression logon didonai (cf. note 30), another of the words taken from the first formulation of Socrates at 510b4-9, the middle form of the verb poiein, which, in the form poioumenè, constituted the last word of this whole line and of which we have seen in note 21 that he played a important role, which invites us to compare this tous logous poiountai ("they make their own reasoning / discourse") with the tèn methodon poioumenè ("making for oneself the road map") read earlier, and at the same time, with the poièsamenoi hupotheseis ("making them supports for themselves") of the previous line, and also with the logon didonai ("give a logos / explain") of this same line, which concerns precisely the hupotheseis ("supports") made by the people Socrates took as examples (geometers and arithmeticians).
Let us notice first that the verb poiein has a very concrete meaning: "to make, create, produce", and that it is at the root of the word poiètès, the primary meaning of which is "maker" in the most general sense before coming to designate a particular "author", that of literary work, of logoi, written, initially in verse, that is to say a "poet" (the English word that is its transcription). The fact that the verb is used in the middle voice adds the idea that the creation, the making, is done in the personal interest of the subject to which it applies. Thus, we have seen in note 28 that Socrates seemed to oppose the approach of the second sub-segment of the perceived by intelligence where one is "making for oneself the road map (tèn methodon poioumenè)" through eidè ("appearances") in the light of an archèn anupotheton ("a (leading) principle [which is] support of nothing"), without having recourse to images (aneu tôn eikonôn), to the approach of the first sub-segment where the "road map", the "method" is rather imposed by the person's attitude towards the words designating the elements "set as supports for themselves (hupothemenoi)" at the outset and the rules of reasoning. And it is precisely with regard to hupotheseis and not to methodon that, in this case, Socrates had used the verb poeisthai in the expression poièsamenoi hupotheseis ("making supports for themselves"): the part of creativity that belongs to the geometer or the arithmetician is not in the path that leads to the result, but in the choice of the problem one seeks to solve and the data from which one starts. For even if one is the first to solve the problem one has chosen to address, one does not "create" the path to the solution, one only discovers it: the one who was the first to "find out" that the square double of a given square was the one built on the diagonal of the initial square (see Socrates' experiment with the slave in the Meno) did not "invent" this solution, was not the one who decided that the length of the side of the initial square should be multiplied by square root of two and that this was precisely the length of the diagonal, but this solution was imposed on this first finder as on everyone else, by the very nature of the square and one only had to "discover" it. This is precisely why once found, it imposes itself on everyone. One has only "marked" by logical reasoning a "path" that has always been there (or more precisely, that is outside time).
What Socrates adds here is that besides "making" hupotheseis, he also "makes" logoi, speeches, or even reasoning. Socrates thus reminds us here that thought can only be translated into logoi ("words, speeches"), whether they are spoken and audible logoi, or an inner "discourse", since, in the Theaetetus, Socrates defines thinking (to dianoeisthai) as "a discourse that the soul itself conducts from beginning to end with itself about what it is examining (logon hon autè pros autèn hè psuchè diexerchetai peri hôn an skopèi)" (Theaetetus, 189e6-7). Whether reasoning within oneself (logous poieisthai) or giving an explanation to others (logon didonai), it is always logoi that come into play, which introduces an additional problematic alongside those of seeing and thinking, that of legein ("to talk", the verb from which logos is derived), that of language and words. For who talks of discourses or speeches implies words, and, as we have seen, talking about words implies eidè, consciously or unconsciously associated with these words (cf. Republic X, 596a5-7). And this is indeed the question: to what extent the person is aware of the fact that, when one uses words, one is in fact assuming eidè, and to what extent one is able to reason about eidè beyond the words one associates with them by being able to recognize the same eidos behind different words or different eidè behind the same word used in different contexts with different meanings. To "make for oneself the road map (tèn methodon poioumenè)" through eidè, it is not enough to create words or to use those that are available in the language one speaks with the meaning given to them by usage, one must be able to "look" through (one of the meanings of the preposition dia) the words at what is beyond them, the "beings", in other words, not only legein ("to speak"), but dialegesthai, a verb whose usual meaning is "to dialogue" from the meaning "with one another" of the prefix dia-, but which can also be understood as indicating an achievement, a going beyond words, made possible precisely by the interpersonal exchange which, alone, can show us that words can have an "objective" meaning that does not depend on the fancy of the speaker but is regulated by " beings" (onta) distinct from the persons who speak. Human beings "make" (poiesthai) for their own use the words that allow them to "make logoi" (logous poiesthai), to "dialogue" (dialegesthai) with one another (cf. 515b4-5), but not most of the things that these words designate (everything that is not the product of human craftsmanship, starting with ourselves) and a word alone tells us nothing about what it designates.
In short, there is on the one hand those who, having words at their disposal as "supports" (hupotheseis), make for themselves speeches (tous logous poiountai) with these words without asking questions about them, and on the other hand, those who, starting from these same words, make for themselves a road map through the eidè that hide behind the words that serve them as simple "supports" (hupotheseis) to rise to the eidè to which they refer (autois eidesi di' autôn tèn methodon poioumenè). Or, to put it another way, with the words at their disposal, there are those who talk and those who think. (<==)

(36) "Thinking" translates the Greek dianooumenoi, a middle present participle in the nominative masculine plural of dianoeisthai. The verb dianoeisthai, formed by adjunction of the prefix dia- ("through", "to the end") to the verb noein, derived from noûs, "mind, intellect, thought", in the middle, means "to have in mind, intend, think, suppose". In the previous note I quoted Socrates' definition of dianoeisthai in the Theaetetus, according to which it is a logos ("discourse") of the soul with itself. With this verb, Socrates begins to introduce into the discussion terms built on the root noûs ("mind, intellect, thought"), which is also that of noèton ("intelligible"), the term used to designate the segment that we are trying to divide here. But this vocabulary is more problematic than that relating to the visible. Everyone knows, or thinks they know, what to "see (horan)" means, to the point that the spontaneous tendency is even to extend the terms relating to seeing to the register of thought and to use words derived from the various forms of the Greek verb meaning "to see", such as eidos and idea, to speak of what is grasped by the noûs ("mind / intelligence"). On the other hand, in the register of thought, the word noûs itself is quite open, since it can designate what, in the register of thought, could correspond to the organ of sight in the register of the visible, the mind that makes us capable of thinking, as well as the faculty that results from the existence of this "organ", the faculty of thinking or intelligence, or even thought itself, or even a particular thought, or the quality which results from this ability well used, intelligence as a quality and no longer as a faculty well or badly used. And from this multiplicity of meanings stems the multiplicity of meanings of the verb which derives from noûs, noein, which can mean "to think, reflect" as well as "to apprehend" or "to have in mind", or even "to show common sense".
Here, the verb used by Socrates is a compound of noein, dianoeisthai, which is derived from noein in the same way as dialegesthai, a verb that will soon be used by Socrates for the explanation of the second subsegment, is derived from legein ("to speak"). If we stick to dictionaries, the meaning of dianoeisthai is not very different from that of noein. It can be thought that here, Socrates uses this verb, in a way consistent with the definition he will give in the Theaetetus (cf. Theaetetus, 189e4-190a6), to designate the inner activity which is that of a person thinking to solve a problem, that is to say making his intelligence work to progress towards the solution of this problem "through (dia-)" a whole reasoning that will be translated into logoi with the help of words thought or pronounced.
Thus understood, this activity is no more specific to one of the sub-segments of the noèton than seeing is to one of the sub-segments of the seen: it is the same faculty of seeing that enables us to see shadows and reflections, or what these shadows and reflections are "images" of; in the same way, it is the same faculty of thinking that allows us to think about the horômena eidè ("seen appearances", see note 34) aroused in our mind by data from sight, and to think about the nooumena eidè, the "appearances" only accessible by thought.
But we will see in the rest of the discussion that there is perhaps another, more restrictive way of understanding dianoeisthai (and the associated name dianoia that will soon be used by Socrates) which, without contradicting what has just been said, restricts its scope to the first sub-segment of the perceived by intelligence. The fact remains that at this point in the discussion and because of the wide range of meanings of this term, there is still nothing to make dianoeisthai the operation specific to the first sub-segment of the perceived by intelligence. (<==)

(37) "Resemble" translates the Greek eoike, a form of the verb eoikenai, "to be like, look like, resemble", from which the word eikôn, "image", used by Socrates in his first formulation, is derived via the participle eikôs. (<==)

(38) Socrates uses here the expression tous logous poiesthai (the Greek text is tous logous poioumenoi, present participle in the nominative masculine plural) that he had already used two lines above (cf. note 35) to say that geometers "make their logoi on them (seen appearances)" (tous logous peri autôn poiountai, where autôn refers to tois horômenois eidesi which preceded it), seeming to say almost the opposite of what he said at the beginning of the line with the same expression as to the object of the logous that geometers make for themselves, since here he says that it is by "making their logoi for the sake of the tetragon (square) itself" (tou tetragônou autou heneka tous logous poioumenoi). But if we look carefully at the text, we notice that he has changed the preposition that introduces the complement, and it is this change that makes it not contradict himself: the first time, the logoi were peri autôn, "on them" (the horômenois eidesi, "seen appearances"), this time, they are tou tetragônou autou heneka, "in relation to the square itself" (which is not even an eidos, but what the intelligible eidos of the "square itself" is the "appearance" of in the mind of the geometer who reasons on it; about the ambiguity of the word tetragônon in Greek and the dual transation I give of it, to which are associated indeed two different eidè, see what I say about it in note 21). Peri is a preposition with spatial connotations, whose primary meaning is "around", while heneka, whose meaning is "on account of, for the sake of ", refers to an abstract relationship between two terms without any spatial or temporal connotation. Socrates therefore says the first time around that the geometer makes his reasoning (logoi) peri ("around") his figures traced on the sand or on a wax tablet, even in the most "physical" sense of the term (he turns around to trace and examine them, or else he and his listeners / students are placed in a circle around these figures), only to specify in a second step that his reasoning (logoi) is in fact "for the sake of" something other than these figures, "on account of" something else, "the tetragon / square itself" (to tetragônon auton), or "the diagonal itself" (hè diametros autè), unique and dimensionless, without spatial or temporal position, a pure abstraction of which alone the demonstrated properties are true, which are never strictly speaking the approximate figures he draws (we will recall here the expression that defines geometry as "the art of reasoning correctly on false figures"). The logos therefore has the power not only to speak of what is accessible through sight, but also to allow us to reason about what can only be apprehended by the mind and to allow us to express truths not only about what we see, but also about that very thing which, although not visible and "outside" time and space, is perceptible by the human mind through ideai that each person represents to oneself with the help of eidè that one associates with the names given to them.(<==)

(39) The choice as examples of the "square itself" (tou tetragônou autou) and the "diagonal itself" (diametrou autès) refers us back to the Meno and to the experiment with the slave (Meno, 80d-86d), to which I have already alluded in anticipation in the previous notes, where Socrates proposes to Meno's young slave to find the square double in area a given square, and ends up making him discover that it is the square built on the diagonal of the original square. On all the problems raised by this text and on Socrates' interpretation of this "experiment" with the help of the so-called "theory of reminiscence", which would be better called the "myth of reincarnations", see the notes to my translation of this part of the Meno (in French only).
That being said, it should be noted that, contrary to what most commentators think, for whom to auto ("the *** itself"), eidos ("appearance") and idea ("idea") are synonymous expressions, to tetragonon auton ("the square itself") should not be confused with to tou tetragonou eidos ("the appearance of the square"), nor with hè tou tetragonou idea ("the idea of square"), expressions that Socrates does not use here. The expression to *** auto or auto to *** refers to "the *** itself", as an objective reality not dependent on the way in which it can be perceived by the senses and intelligence (noûs) of a human being, or of any other creature capable of apprehending it by one means or another. The expression is therefore opposed to the notions of idea, the "objective" way in which this *** can appear to human intelligence (noûs) as such in its perfection, and of eidos, the way in which it "appears" to the senses and intelligence (noûs) of a particular human being at a given moment of his physical and intellectual development. What these auta ("(***) themselves") are, we cannot know because, by nature, we can only have access to the ideai which are what human nature can apprehend at its best (the stars of the allegory of the cave) and that each one, according to his own intellectual capacities and previous experience, apprehends through the eidè ("appearances") that they take for oneself at this point in one's life. And to say that they are "outside of time and space", or that they are "eternal", can only lead to confusion, because "outside" is a preposition with spatial connotations that still invites us to think of them "somewhere" and "eternal" implies a time without beginning or end, which again invites us to assume them in a certain way within time. What can be said about them is that they "are" (something) since they can be thought of and reasoned about, and that they are not a mere creation of the mind of the one who thinks or speaks about them, since that one can compare one's experience about them with that of other people by means of dialegesthai ("the practice of dialogue") and thus figure out that one cannot say anything about them (for instance, about the square, one cannot say, claiming that it is true, that the square built on the diagonal of a given square has an area three times that of the original square). In the myth of the Phaedrus, Plato's Socrates locates them on the other side of the vault of heaven, in a place that he describes as "above the heavens" (ton huperouranion topon, Phaedrus, 247c3) to which only the gods have access. If we want to qualify them at all costs with adjectives, we could say that they are "utopian" in the etymological sense of "without location, without place (or topos)" and "timeless" (without relation to time).
The whole question, to return to the example taken by Socrates, is then to figure out what each one of us, because of the limits of one's own mind, is capable of perceiving of this idea of the "square itself" (or of any other "abstraction") and of its "wealth" (ousia) beyond the purely visual appearance, the schema, which the visible images we make of it suggest to us, none of which being strictly speaking a square, and especially if we are able to realize that the square itself is neither the image we draw of it, nor even the mental representation (the intelligible eidos) that we have of it. And if we try to think of the square without giving it a dimension, without mentally representing to ourselves a more or less precise image of a drawn square, without "seeing" four equal sides and four right angles, is there not a risk that what we are thinking about will no longer be anything at all or no more than a simple word?... Plato illustrates this problem in the allegory of the cave by representing the ideai by the stars in heaven, which are all for us tiny spots of light almost identical with one another.
The fact that the geometer knows, when conducting a demonstration, that the square one draws is arbitrarily chosen, that it is only approximately a "square" and that the demonstration does not apply to it strictly speaking, and even less to it alone, but to any square, does not mean that one has a clear understanding of the nature of what that demonstration applies to, or even that one wants to address these kinds of questions. In fact, one can even think that it is precisely on these questions that, as Socrates said, one considers giving explanations (logon didonai, cf. note 30) as not needed. He makes his demonstration on "the square" and in fact, the word is enough for him, without him trying to know what is the ontological status of what is hidden behind it. What is "the square" which is no particular square but which is in a way all at the same time or, if one prefers, which is the model of them all, or rather that which makes them all intelligible, and yet which has neither precise position nor specific dimension? "Where" is it? And does this question even make sense about it? These are questions that do not interest geometers as such. The questions that interest them and to which they knows how to answer are of this kind: "I have built a square house and now I would like to build another, twice as big, but still square; what should be the length of the sides of my house?" To this, one will be able to answer: "Whatever the size of your first house, measure, for instance with a rope, the diagonal of the square it forms and take this measurement as the size of the sides of your new house".
But we can still explore this example in another direction. As I have already pointed out, the word tetragônon, is the substantive neuter of the adjective tetragônos meaning "which has four angles", as trigônos means "which has three angles" and leads to the substantive in the neuter form trigônon to designate a triangle. But where trigônon remains general and designates any triangle, tetragônon specialized in designating, not just any quadrilateral, but a very particular quadrilateral, the square, probably because it was considered by Greek geometers to be the most "perfect" of quadrilaterals since both its four angles and four sides are equal to one another. This name is therefore an example of an ambiguous name, which requires to be "explained" (logon didonai, cf. note 30) and it is to make this ambiguity perceptible in English that I have translated tetragônon by "tetragon", adding "square" in parentheses. But this ambiguity opens the door to a whole series of issues that can be extended to all kinds of things and concern the naming process and the issue of what there is an idea of. Towards the end of note 21, I briefly mentioned the problem of "mathematical objects", which Aristotle speaks of in connection with Plato, taking the example of squares, to say that alongside the idea of a square, without dimensions, one could also envision the abstract notion of a "two-foot square" or a square of any other dimension, which had in common with the idea of square that it was not the drawing that can be made of it but an abstraction regarding which all the properties that can be demonstrated about a square are rigorously true, but which was distinguished from the idea of square by the fact that there were an infinity of such notions, each with a dimension different from that of all the others. But now, if we take the word tetragônon in its general sense of "quadrilateral," is there an idea of tetragônon / quadrilateral distinct from that of tetragônon / square, although in Greek both are designated by the same word? Or is the tetragônon / square only a special case of quadrilateral in the same way as the two-foot square is a special case of a square, and is there only an idea of quadrilateral in the general sense, the different quadrilaterals having specific properties, such as the trapezoid (two parallel opposite sides), the rectangle (equal angles but not equal sides), the diamond (equal sides but not equal angles), the square (equal angles and equal sides), being only "mathematical objects" in the same way as the two-foot square or the two-by-three rectangle? And if we continue, quadrilaterals are special cases of polygons (polugônon), which are special cases of geometric figures (schema). The underlying question, which is not unique to geometric figures, is that of determining how far we must go in ignoring differences in order to arrive at ideai. Thus, for instance, is there is only one idea of dog, or also one idea of poodle, another of greyhound, etc. ? One idea of cat, or an idea of Siamese cat, an idea of angora cat, etc. ? To attempt to answer this question, we must start by getting rid of preconceivd notions about what ideai are and the links they have with what they allow us to understand, such as the idea that ideai are "eternal" and therefore pre-exist what they are the ideai of, which locates them in time while they are unrelated to time, timeless, and that therefore the very question of their possible "pre-existence" to what they are ideai of is meaningless; or that they are "causes" of what they are ideai of, which would only be "images" of them degraded by their materiality, for here again, the notion of causality implies a notion of time, even for the "final" cause (which owes its name precisely to the fact that it has a relation to what comes "at the end", that is to say, still in time), so that the very idea of a causal relationship between something that is in time and something that has no relation to time has no meaning. Suffice it to say, then, that the relevance of ideai is measured by their explanatory power, since they are principles of intelligibility. If we admit that there is an idea of square distinct from the idea of quadrilateral, it is because it is possible to demonstrate interesting properties about the square that are only true of squares, and if we reject the idea of square of two feet on each side, it is on the one hand because "two feet" is perfectly arbitrary and depends on the unit of measurement being used, itself perfectly arbitrary, so that this "two" is not an intrinsic property of the square with which it is associated, and on the other hand, because there is nothing interesting to discover about the two-foot square that is specific to it and is not a property of all squares apart from the fact that, measured in feet, it is two feet tall, nothing in any case that leads to useful generalizations. But this does not mean that the first to base a reasoning on an idea is the "creator" or the "inventor" of it. The first man who had the idea of building a bed did not "invent" the idea of bed, as is shown by the fact that it is perfectly possible, and even probable, that this idea came to different people in places far remote from one another and at different times without the productions of one being known to the other, and without them having assumed the same material appearance, but all responding to the idea of something facilitating rest and sleep for the people who use them. The idea of bed is simply something that relates certain aspects of human nature (the need to sleep) with certain possibilities of reflection and production of men to facilitate the satisfaction of these needs in a way that makes the purpose of these productions intelligible. (<==)

(40) "They mold" translates the Greek plattousin, and "they draw" the Greek graphousin. The verb plattein means in the primary sense "to form, mold" when speaking of a craftsman or artist working with soft substances such as wax, clay or any other malleable material (this is the verb that is at the root of the English word "plastic"), but it can also be used figuratively in the sense of "forming" a personality through education, or of "shaping" with art worked speeches, or in the sense of "fabricating" lies. Since Socrates refers to the "shadows" and "images" that the results of this "molding" produce, what he is refering to is probably the "molding" with wax, or with some other material, of three-dimensional models of geometric solids such as cubes or spheres. Graphein, on the other hand, means in the primitive sense "to scratch, graze", and, from there, "to draw" or "to write", and thus refers to the drawings made by the geometer of the figures on which he reasons.
By using these very concrete verbs, Socrates wants to insist on the material character of the images used by geometers or arithmeticians who draw figures to carry out their operations or use balls or pebbles to represent them, even assimilating them to the works of the human art of making images, since these two verbs refer to the two major arts of sculpture and painting.
And in a context where logous poieisthai ("to make speeches") has just been mentioned twice, we can also note that these verbs could not only concern the drawings of figures or numbers made by those of whom Socrates speaks, but also their speeches that they "make" by shaping them (plattein) before writing them (graphein), and that their use could therefore also be a way of discreetly reminding us that words too are human creations and "images". (<==)

(41) Even if we understand graphousin as refering only to "mathematical" drawings, the fact remains that these drawings, whether in wax or on sand, are visible only through the shadow-producing reliefs created in the soft material by the tool used for drawing, and that, if the resulting "scratches" are made on wax tablets, for instance, images of them can then be seen, along with those of the tablet, in any reflective surface. And this is all the more true of sculptures and paintings. These "images" are therefore not in the first sub-segment of the seen, but in the second, like any other man-made "image" that falls into the category of skeuaston (cf. note 17), and Socrates insists a little more by giving as proof the fact that they can themselves give rise to images of the first sub-segment of the seen by producing shadows or reflections of themselves. (<==)

(42) "See" translates the Greek idein, the closest verb form to idea ("idea"), which is not used here. Idein means in the first sense "to see with one's eyes" and can also have the figurative meaning of "to see with the eyes of the mind", so that its use here by Socrates, to speak of access to things that he will tell us are "visible" (second use of idein in the form idoi, optative, which, here, can only be understood in the figurative sense) only with the help of dianoia (translated as "thought" at the end of the sentence) is ambiguous because it can just as well be taken literally, as meaning that those people who need images to serve as "supports" for their reasoning thus prove that they would like to see with their eyes what can only be "seen" with the "eyes" of the mind, and therefore that their quest is doomed to failure and that they thus show the limits of their intelligence, as be taken in the figurative sense, as the idoi that follows, and understand that Socrates means that they thus seek to reach what can only be attained by thought, without ruling out the fact that they might succeed in doing so. (<==)

(43) I translate here dianoia, the noun of action derived from the verb dianoeisthai (cf. note 36), which is the last word of this line of Socrates and of which it is the first, but not the last, appearance in the analogy, by "thought", in a manner consistent with my translation of dianooumenoi at 510d6, earlier in this line, by "thinking", because this is the most natural meaning for this word and at this point, there is no reason to give it a more specialized meaning, which it will only take, as I have already implied at the end of note 36, later in the analogy by becoming the name of the "state of mind" (pathèma) associated with the first sub-segment of the perceived by intelligence.This translation has the advantage of assuming as little as possible about what is going on in the geometer's head, less at least than "understanding", another possible translation of dianoia, on the part of a geometer described by Socrates as not wanting to "give a logos" (logon didonai) for the "supports" (hupotheseis) of one's approach: a simple thought can remain cursive and not lead to undertanding on the object of this thought.
By declaring here that the realities of the intelligible order can only be "seen" with the help of dianoia, without taking the trouble to specify what he puts behind this word of rather open meaning as we have just seen, Socrates seems for the moment to make the latter the equivalent in the domain of the noèton ("intelligible") of sight in the domain of the horaton ("seen"). In doing so, he probably wants to lead us to wonder why we have no difficulty in granting existence (whatever that means, if it means anything) to what we see, but are reluctant to grant existence to what we think about as long as we cannot associate it with something "visible", even though, on the one hand, what we designate by names to describe what we see is not strictly speaking what we see, which is limited to patches of color, but abstractions derived from these sensible impressions by our mind, with the help of a process that has become in most cases automatic and unconscious with age (a so-called "common" name designates by nature a set of similar beings sharing the same eidos ("appearance", visible or intelligible, cf. Republic X, 596a6-7), and that, moreover, it is not sight alone that leads us to place such confidence in what it makes us able to see, but, if we think about it carefully, touch, which alone guarantees the "tangible" character (a word derived from the Latin verb tangere, which means "to touch") of what we see, Again, as a result of a process that took place in the early years of our lives, which taught us that most of the "things" we see can also be touched, either by extending our hands if they are close enough, or by moving towards them if they are further away, and that we have unconsciously generalized over the years by implicitly admitting that it is distance alone that can in some cases prevent us from touching some of the things we see, such as the sun and stars. And it is this same process that has taught us to distinguish things themselves from their shadows and reflections, that is to say, to differentiate between direct and indirect sight, that is to say, between the two visual processes by which Socrates distinguished the two sub-segments of the segment of the seen: if I try to touch the image of myself or of my mother who holds me in her arms in front of a mirror, it is not me or her that I am going to touch, but the surface of the mirror, which has neither the same texture, nor the same temperature, nor the same spatial extension; In short, my hand will not feel the same sensations if I caress my mother's arm or face and if I caress the image of it on the mirror, that is to say, by touching in fact the mirror and following as best I can the contours of what is only an image on its plane surface unable to guide my movements while touching it as the "real" three-dimensional face or the "real" three-dimensional arm of my mother would do.
One of the reasons why we can place more trust in sight than in thought is that sight imposes itself on us "from the outside" (at least that's the feeling we have of it), whereas thought seems to us to be a process taking place inside us and which we think we are causing to unfold: we do not choose what we see "around" us and we have learned, again with age, that, if other people are in the same place as us, they see the same thing as us, while we believe that we are masters of our thinking, and we know that others do not "read" our minds and cannot know, even when they are close to us, what we are thinking about, if we do not make it explicit through words. But in fact, it is only through language, and therefore through thought, that we can know that those who are next to us "see" the same things as we do, and what we point out to them to make sure of it with the help of the words we use are precisely eidè and not visual perceptions that we would transmit to their eyes so that they can compare them with their own perceptions. For sight, without the reflection it provokes and the logos that allows it to be accounted for, is no different from the sensations produced on a bull by the red rag waved before its eyes by a toreador and which makes it move in all directions without understanding what is happening and fall under the power of the one who leads it by the nose and ends up killing it. And it is in exactly the same way that, by means of the dia-logos, we can compare our thoughts, our understanding, and even develop them out in common, as Plato's Socrates never ceases to do, so that there is no more objective reason to believe that, because we agree on what we see (which only means that we put the same words on what we suppose to be the same visual perceptions), it proves that it exists, than to doubt the existence of what we think about together but which is not apprehended by our sight (and above all by our touch).
For Plato, there is a very close link between logos, and more specifically dia-logos, and dia-noia, as will be seen by refering to Sophist, 263e3-8, where the Elean Stranger defines dianoia as an inner dialogos of the soul with itself, almost taking up the formulation of Socrates at Theaetetus, 189e4-190a6, to which I referred in note 36, except that Socrates defines dianoeisthai (the verb from which dianoia is derived) by describing it as an inner logos of the soul, while the foreigner defines dianoia (the action name derived from the verb dianoeisthai) by making it an inner dialogos of the soul. Beyond these differences in detailed wording, what is important in these two series of terms, dia-logos / dia-legesthai and dia-noia / dia-noeisthai, is the prefix dia- which they have in common and which must be taken in all its overtones: it evokes both an idea of means (it is "by means" of the logos, "by means" of the noûs, that we can account for our sensible and intelligible perceptions and precisely make them intelligible), an idea of difference in directions (it is by knowing how to distinguish our perceptions from one another that we can conceive them as relating to distinct entities, identify them individually and name them appropriately), an idea of reciprocity (it is in the exchange with others, the dia-logos, or with oneself, the dia-noia, that we give meaning to what we perceive) and that of "going (all the way) through" (dia in the sense of "through"), a "range" of meaning that refers us to a beyond (beyond words as well as beyond thoughts) that words and thoughts seek to apprehend (the "beings" themselves (auta) that we see or think about) without our having the certainty of grasping them as they are, or rather, if we have understood the limits of the logos as well as of our sensible perceptions, by having the certainty that we only grasp "appearances", eidè, either visible / perceptible, or intelligible. And dia- also evokes an idea of dispersion ("in different directions") that should not be neglected when the time will come to give the word a more specialized meaning, because it suggests that thought, reflection, and even reasoning, can sometimes wander, fool around, scatter and get lost (not so much by reasoning falsely as by not taking the right target and not knowing how to turn towards the light that gives meaning to everything).
To return to our text, what Socrates wants us to perceive here is that geometers, thanks to the simple and "schematic" character of the concepts they deal with, and independently of the complexity of the problems they seek to solve on these simple concepts, have in hand everything they need to understand the mechanisms of thought and discourse and would have only one more step to take to achieve it. In fact, they stand at a point where, depending on whether they turn to one side or the other, depending on whether they are content to seek to solve the problems that have been posed to them or that they have posed to themselves by "descending" to a death-smelling teleutè or whether they seek to understand what they are working on and what makes this work possible for them, that is to say, taking a further step "upwards", towards abstraction and understanding of themselves as animals endowed with these capabilities, they remain geometers hardly more advanced than the surveyors from whom they took their name (geometrès in Greek means "measurer of the earth (gè)"), even if their elucubrations may impress the layman, or they take a big step towards the state of philosophoi, even if it means losing their admirers. This is exactly what was meant by the motto that is said to have been engraved at the entrance of the Academy, ageômetrètos medeis eisito ("let no one unfit for geometry enter" (see on this motto and its sources, the page dedicated to it in the frequently asked questions of this site): if one is not able to perceive that the geometric constructs on which one reasons are not the figures we use as "supports (hupotheseis)" for our reasoning and that they are accessible only through dainoia, not through sight, which gives us only approximations, there is no need to go further down the path of abstraction and philosophia, for we will be even less able to admit the existence of the beautiful of which a beautiful girl gives us an example!
And it is a safe bet that Plato gave his analogy geometric colorations by speaking of a segmented line and used expressions that could have a technical meaning in geometry, such as ana ton auton logon (cf. note 10), to test his readers and separate those who would remain at the level of geometry by focusing for example on the question of whether the two segments are isa ("equals") or anisa ("unequals") (cf. note 8), and on the meaning that this (in)equality may have, from those who would be able to see that it was only a "dressing" by images and analogies which had to be overcome in order to understand what he was trying to make us discover, through and beyond geometry, and to see that the logos in question was indeed not numerical, was not a ratio but a rationale. (<==)

(44) "So I indeed said intelligible this appearance, but the soul constrained / constraining itself to make use of supports in its investigation about it... " translates the Greek touto toinun noèton men to eidos elegon, hupothesesi d' anagkazomenèn psuchèn chrèsthai peri tèn zètèsin autou. If all the translators don't care to preserve in English the indeterminacy found here, already encountered on the anagkazetai form of the same verb anagkazein at 510b5, on the way in which the participle anagkazomenèn, which can be either a middle or a passive (cf. note 21), should be understood, they do not agree among themselves on the way in which this part of the sentence should be understood and therefore on the general meaning of this reply of Socrates (see at the end of this note the various translations I have consulted), and the source of their difficulties is not grammatical, but has more to do with the meaning that must be given here to eidos. Grammatically, in fact, the sentence can be analysed quite simply: a main verb with an implicit subject, elegon (1st person singular of the imperfect of the active indicative of the verb legein), "I said", commands two series of double complements at the accusative opposed by men... de... (the de being elided in d' before the initial vowel of angkazomenèn), the first complement indicating in each case about what Socrates "said" something and the second what he said about it; the first group of two complements is made up of touto to eidos ("this appearance") as that about which he says something and of noèton ("intelligible") as a qualifier applied to what he is talking about; the second group is made up of psuchèn ("the soul") as that about which he says something, and of all the rest of the line, made up of successive participles (anagkazomenèn, "constrained to / constraining itself to"; ouk iousan, "not going"; chrômenèn, "using", three verbs taken from the "short" version of his explanations, analysed in note 21) each completed by more or less complex phrases (in the case of the first participle, the only one I kept here, anagkazomenèn ("constrained to / constraining itself to"), it is completed by the infinitive clause hupothesesi chrèsthai peri tèn zètèsin autou ("to make use of supports in its investigation about it"). The problem is that this natural translation does not satisfy translators (and commentators) who would like each sub-segment to host a "population" of its own and that Socrates is here in the process of distinguishing a "kind / class / type (their understanding of the eidos used here by Socrates)" of noèta ("intelligible") "populating" the first sub-segment of the perceived by intelligence (what Aristotle will call "mathematical objects") which would be opposed to another "kind / class / type " populating the second sub-segment of the perceived by intelligence (the "forms / ideas" in the noble sense). And this temptation goes back a long way since it was probably already that of Aristotle with his "mathematical objects" and that it has left traces in the manuscript tradition, some manuscripts giving noètou (genitive) instead of noèton (accusative), which makes it a complement to the name of touto to eidos and forces the translation by "this kind of intelligible", but, simultaneously, unbalances the grammatical construction of the sentence by making it impossible to consider elegon as applying to the two groups opposed by men... de..., which leaves the whole second part of the reply, from hupothesesi d' without a main verb.
But even for those who respect the grammatical structure of the sentence, the idea that Socrates describes a distinct category of intelligibles is not lost, because of the anagkazomenèn ("constrained"), which, read as a passive rather than a middle, gives them the possibility of assuming two distinct categories of intelligibles by locating "necessity" (anagkè, root of the verb anagkazein) in the very nature of the intelligibles being described here, imposing the method of reasoning that would then be specific to these intelligibles and exclusive of any other to apprehend them. For them, in view of Socrates' two previous lines, what is at issue here is the hypothetico-deductive method of reasoning of geometers and other mathematicians, implied by the geometrical examples previously used by Socrates and by expressions such as hupothesesi chrèsthai understood as meaning "to make use of hypotheses", giving the word "hypotheses" its modern and specifically mathematical meaning, and, in this perspective, they come to consider the "mathematical objects" (of which Plato never speaks in his dialogues, but of which Aristotle attributes the paternity to him) as a particular category of noèta of which they make the "population" of the first sub-segment of the perceived by intelligence, reserving concepts that are less easy to define but more "noble" such as beauty, the just, the good, for the second sub-segment.
But we saw in the introduction that the allegory of the cave makes us understand that it is the same "objects" that are found in the four sub-segments of the line (or in two for those that are only intelligible), but apprehended in a different way, both by sight and by intelligence (noûs). And besides, is there anything in Socrates' earlier words that suggests that the geometric examples he uses are more than examples, that the mode of reasoning he describes concerns only geometers and mathematicians, and above all that it is the only one that can be used to reflect on concepts such as the square and the diagonal? Certainly not! I would even say that, on the contrary, all of what he said earlier suggests that it is perfectly possible to reason about mathematical concepts as a philosopher and not as a mathematician, and that he precisely gives an example of this in his discourse, when he speaks of the "square itself" (tou tetragônou autou) and of the "diagonal itself" (diametrou autès) and insists on the fact that these are concepts that "cannot be seen otherwise than by thought". And several elements of his two previous lines lend credence to the idea that, at least in the case of the geometers taken as an example, anagkazomenèn, and therefore the anagkazetai of 510b5 of which it is the repetition, should be read as a middle and not as a passive: he in fact presents geometers to us as taking as obvious the concepts they take as the starting point of their reasoning (the hupotheseis understood as "supports", as what is "assumed" in the statement of the problem, cf. note 21) and as refusing to give a logos about them (on the expression logon didonai, see note 30). In short, they are the ones who forbid themselves certain types of questioning and who constrain themselves to make only deductive reasoning "descending" from the data "supposed" at the outset, and who forbid themselves to go back higher than these hupotheseis towards principles; But nothing suggests that it is not possible for others to logon didonai ("to hold a (explanatory) discourse") on concepts such as the square, the diagonal, etc. by agreeing to question their supposed "evidence" and by privileging another type of approach, as Socrates does by distinguishing the square itself from the sensible images of squares that geometers need in order to carry out their deductive reasoning.
In fact, the difficulty experienced by the translators here is directly tied to the difficulty they experienced with regard to horômenois eidesi, who opened his previous line at 510d5 because, as we have seen in note 34, the two expressions, noèton eidos here and horômena eidè (in the dative horômenois eidesi) in the previous line, answer each other, so that, if one has not understood the first, or has not accepted the natural meaning that must be given to it, one cannot see that the other refers to it, nor understand why the one is in the plural, that concerning the visible, and the other in the singular, that concerning the intelligible. This is not to mean that there would be only one eidos that would be noèton in all that Socrates spoke of before, where he mentioned both the "square itself" (tou tetragônou autou) and the "diagonal itself" (diametrou autès), but to suggest that, in relation to each term mentioned by the geometer, square, diagonal, or any other geometrical object that one uses in one's reasoning, there is a multiplicity of images that can be given of it and that will produce in us horômena eidè distinct in size, position, orientation, the matter in which they are traced, etc., but which all refer to a single noèton eidos which is the perception by our mind of the idea underlying this term, the idea of square, or of diagonal, or of whatever. Thus, in order to make Meno's slave find the solution to the doubling of the square, Socrates is led to draw on the ground at least four squares which together form a fifth square four times larger in area than each of the first four and four diagonals, which together form a sixth square, twice as large in area as each of the first four; But all of them are "images" (eikones) in the visible of the "square itself", unique and unrelated to time and space, of which the "idea of square" is what the human mind (noûs) can perceive of it and which each of us apprehends in a more or less precise way by means of a single noèton eidos ("intelligible appearance"). And this unique noèton eidos is the one "in addition" of which (see note 34 on the pros- of the verb form proschrôntai ("they also make use of") of which tois horômenois eidesi ("the seen appearances") is the complement) geometers "make use of" the multiple horômena eidè ("visible appearances") around which they build their reasonings. But how can we understand that Socrates feels the need to specify here what he has in mind when talking about noèton eidos ("intelligible appearance (singular)") when we refuse to see that he was previously talking about horômena eidesi ("visible appearances (plural)"?... ICI
And as I have already mentioned in previous notes, if we want to find "mathematical objects" in Plato at all costs, we must look for them in things like the notions of a two-foot square, a four-foot square, a square whose side is the length of the diagonal of the two-foot square, which Socrates needs in his discussion with Meno's slave, which are only special cases of the idea of square, of which it would be useless to make distinct ideai, but which, however, like the idea of square, are not Socrates' drawings of them, but something that can only be apprehended by dianoia ("thought") and has no relation to time and space (having a dimension does not mean to be located in space), something of which the propositions that can be demonstrated about them, such as the one that Socrates highlights, are rigorously true. But these objects are no more the "proper objects" of one or other of the subsegments of the intelligible than are the idea of square or the idea of diagonal.
Following are the translations of this part of the sentence by the various translators I have consulted, with, in parentheses afterwards, the reminder of their translation of the words toishorômenois eidesi proschrôntai at 510d5, and, in red in each translation, the word used in each case to translate eidos.
Jowett : "And of this kind I spoke as the intelligible, although in the search after it the soul is compelled to use hypotheses..." (510d5: "they make use of the visible forms");
Shorey: "This then is the class that I described as intelligible, it is true, but with the reservation first that the soul is compelled to employ assumptions in the investigation of it..." (510d5: "they further make use of the visible forms");
Cornford: "This, then, is the class of things that I spoke of as intelligible, but with. two qualifications: first, that the mind, in studying them, is compelled to employ assumptions..." (510d5: "they make use of visible figures");
Lee: "This type of things I called intelligiblle, but said that the mind was forced to use assumptions in investigating it" (510d5: "they make use of... visible figures...", with a note on "figures" stating that the Greek word so translated is eidos, but "non-technical again", the "again" refering to an earlier note on eidos at 509d4, where he translates it as "orders of things" and describes this use of eidos as "a good example of Plato's non-technical use of the term, to mean 'kind', 'sort', 'type' (as also at 511a, 'type of things)'"; see note 6, toward the end);
Bloom: "Well, then, this is the form I said was intelligible. However, a soul in investigating it is compelled to use hypotheses..." (510d5: "they use visible forms besides");
Grube: "This is what I called the intelligible class, and said that the soul is forced to use hypotheses in its search for it..." (510d5: "they use visible figures");
Grube/Reeve: "This, then, is the kind of things that, on the one hand, I said is intelligible, and, on the other, is such that the soul is forced to use hypotheses in the investigation of it..." (510d5: "they use visible figures");
Reeve: "This, then, is the kind of things that I said was intelligible. The soul is compelled to use hypotheses in the investigation of it..." (510d5: "they use viisible forms").
To sum up, Socrates is not talking here of classes (Cornford) / types (Lee) / kinds (Reeve) of (intelligible) things, but of different appearances (eidè) under which things, any thing, may appear to our eyes (homômena, "visible") or mind (noûs) (noèta, "intelligible") depending on how we "look" (with the eyes or the mind) at them, these different looks / appearances (eidè) depending both of the thing being "looked" at ("objective" dimension) and of the state of mind (pathèma) which is ours when "looking" at it ("subjective" dimension). (<==)

(45) This entire line uses a spatial vocabulary with words such as anôterô ("higher") and hupo ("below"), which suggests that the line of the image must be seen as vertical, with the visible at the bottom and the intelligible at the top (which is consistent with the imagery that we will find in the allegory of the cave that follows, which involves a lot of ascent and descent): here, in fact, the soul is said to be "unable to rise higher than the supports" (tôn hupotheseôn anôterô ekbainein) "toward a (leading) principle" (ep' archèn) (the primary meaning of epi, of which ep' is the elided form, is "upon"), and it "us[es] as images the very [things] that are copied by those below" (autois tois hupo tôn katô apeikastheisin) (tôn katô, "the things below" refers to the first segment of the seen, that of images, and the preposition hupo, used here in the sense of "by the fact of", means in the primary sense "under").
Moreover, the end of the sentence uses pronouns whose antecedents are not easy to determine: « eikosi de chrômenèn autois tois hupo tôn katô apeikastheisin kai ekeinois pros ekeina hôs enargesi dedoxasmenois te kai tetimèmenois » (mot à mot « as_images but making_use of_themselves the by the below being_copied and those in_comparison_with those as visible held_in_esteem and also honored »). The problem is posed by the "ekeinois pros ekeina", in which the same demonstrative pronoun, ekeinos, "this one", is used twice, once in the neuter plural dative ekeinois and once in the neuter plural accusative ekeina. The difficulty arises from the fact that ekeinos is generally understood in relation to another demonstrative, houtos, "this one", the first, ekeinos, referring to the most distant person or thing, and the other, houtos, to the closest. Now, here, the two things that are related are designated by the same ekeinos, and the sentence refers to three "kinds" of things: the "appearances" (eidè) that are called "intelligible", which this entire line is about and which are mentioned first, the visible realities apprehended according to the approach specific to the second sub-segment of the seen, which serve as images for the apprehension of those mentioned first, and, mentioned last, the visible images of these in the sense of the mode of apprehension specific to the first sub-segment of the seen, since they can produce shadows and reflections. All the words in red are in the dative plural (for verbs, in the aorist participle or perfect passive), and we can therefore think that they all refer to the same thing, that is to say, as the beginning of the sentence clearly shows, to the visible realities that serve as images to facilitate access to the intelligibles according to the approach specific to the first sub-segment of the perceived by intelligence and are themselves copied by the "images" mentioned in connection with the first segment of the seen. This settles the case of the ekeinois. It remains to be seen, then, whether the ekeina refers to the tôn katô, to "those below" which are images of the preceding ones (the sub-segment of images in the visible), or to the intelligibles whose visible realities give us an image represented by the touto... to eidos at the very beginning of the sentence, which itself refers to an auta ekeina at the end of Socrates' previous line. Most translators opt for the first solution and see in it an opposition between objects and images of the visible order. But, beyond the grammatical clues that preach for the other option, the fact, on the one hand, that, if this were the case, the two ekeinos would ultimately refer to the two things that are closest to the reference in the sentence, and therefore in the memory of the listeners (in this kind of abstract sentences, the "proximity" or "distance" referred to by a "this" or "that" are more "linguistic" than spatial, and rather concern what was spoken of last compared to what was spoken of before), and not to the two things that are most remote, and on the other hand, the presence of another ekeina at the end of Socrates' previous line, it is above all the meaning of what is said next that makes the first solution unlikely: the qualifications that are attributed to the ekeinois (the fact of being "reputed and esteemed"), and to which I will return in a moment, are "because they are visible" (hôs enargesi). But enargesi, the neuter plural dative of the adjective enargès, which means "visible" (for instance, when speaking of a god who makes himself visible to men, but also of images seen in dreams), etymologically "made bright /shining (argès)", and also "clear, distinct", is not a comparative, as one might expect if what was intended was comparing two kinds of visible things, that of the originals and that of the images. Not to mention that some "images" are just as "clearly visible" as some "originals", which raises the question of how our sight alone allows us to distinguish visible "images" from "originals". Whereas, if the comparison is between visible objects (those that are not images) and intelligibles (the two kinds of things mentioned the farther from where we stand in the discussion), the simple qualifier "visible" is sufficient to explain the value given to them.
Let us now see how this "value" is qualified. The two verbs used, dedoxasmenois and tetimèmenois, are the perfect passive participles of doxazein and timan respectively, verbs that refer, the first to doxa, "opinion", and the second to timè, the "esteem", the "value", the "worth". These two verbs are ambiguous to say the least in the mouth of Socrates, for whom opinion, doxa, is not what he holds in the highest esteem (see, at the end of Book V, the section on knowledge and opinion). In short, what Socrates means is that what we esteem and what we have a good opinion of is what is enargès, "brilliant", and that our appreciation of the respective value of the original and the image is reversed when we pass from the relation between visible images and visible originals, where more value is given to originals, to the relationship between visible and intelligible "images": here, more value is given to "images", because they are visible, without realizing that they are only images, precisely because they are conceived as original in relation to the visible images of the first category. (<==)

(46) Glaucon is here only stating the obvious, namely, that Socrates has just spoken of geometry, and, as we shall see in the next line, Socrates does not feel the need to point out this observation and goes on directly to explain what concerns the second sub-segment of the perceived by intelligence. Glaucon also notes here that Socrates did not limit his considerations to geometry, but that they also apply to tais tautès adelphais technais ("the arts akin to it", or, word for word, "the sister arts of this one"), a rather vague and open wording, which echoes an even vaguer wording of Socrates, who began by referring Glaucon to hoi peri tas geômetrias te kai logismous kai ta toiauta pragmateuomenoi ("those who busy themselves in geometry and calculation and the like" 510c2-3). Technais, plural dative of technè (feminine in Greek, hence Glaucon's reference to his "sisters"), the word used by Glaucon, from which the English "technique" derives, refers in the primary sense to the "manual arts / crafts", to the "skill", to the "cunning of hand" of the craftsman, and is often opposed to epistèmè, "science", the knowledge that allows one to dominate one's subject. To try to better understand what Glaucon refers to with the word technè in this context, we will have to wait for his next line where, after hearing Socrates explain the second of the two approaches characteristic of the two sub-segments of the perceived by intelligence, he will reformulate his understanding of these explanations. Let us simply note for the moment that, as I have already said in note 43, geomètria derives from gè-metrein, that is to say, from "measuring (metrein) the earth (gè)", and originally designates the entirely practical art of the surveyor who measures the fields before designating a nobler "art" reasoning on "abstract" figures, but whose purposes had perhaps not been completely dissociated from the practical applications that could be made of them, not only by surveyors, but also by architects and other trades who have to work on regular shapes, surfaces and volumes.
We cannot, therefore, use this reply of Glaucon to argue that Socrates has in mind for the first segment of the perceived by intelligence only the abstractions and methods of mathematics in general. I have already said that he did not seek to isolate distinct noèta in each of the two sub-segments, but only to describe modes of thought specific to each of them, but, even at this level, there is no obligation to suppose that he limits what he has described to mathematical reasoning. The expression tais tautès adelphais technais, in the absence of clarifications or corrections on the part of Socrates, is sufficiently open to be able to encompass all reasoning of a "scientific" type. And, as I have already said, geometry is put forward by Socrates only as the most elementary example of such an approach and within the reach of the greatest number. (<==)

(47) "What logos itself reaches through the power of the dialegesthai" translates the Greek touto hou autos ho logos aptetai tèi tou dialegesthai dunamei (word for word "that which itself the logos reaches by_the of_the dialegesthai power"), without translating the two most important words of this part of the sentence, logos and dialegesthai, for which any translation would be reductive and misleading. This part of the sentence, which the rest of Socrates' reply only makes explicit, wants to highlight in a synthetic way what constitutes the essence of the approach associated with the second sub-segment of the perceived by intelligence, the highest level of understanding accessible to human beings. However, it does so by using two terms, logos and dialegesthai, which at first glance seem to have nothing to suggest a particular approach that would be reserved for a small elite: "speech" (the primary meaning of logos) and "converse / discourse" (the usual meaning of dialegesthai) are banal activities practiced every day by everyone! It is therefore by means of a formulation that could not be more open and which seems like nothing that Plato's Socrates describes what seems to him to be the culmination of the entire process of education (paideia) of the human being (Socrates introduces the allegory of the cave by saying that it describes "our nature regarding education (paideia) and the fact of not being educated (apaideusia)", 514a1-2). However, he is in no way making logos, or at least logos in the usual sense, or even dialegesthai, the specific activity of this second sub-segment since, in the allegory of the cave, as I have already said in the introduction, Socrates describes prisoners caught in bonds since childhood as "able to dialogue (dialegesthai)", and thus to make use of the logos, for which they even give names to the shadows they see (515b4-5).
Before going any further, let's start by seeing how the English translators of the Republic that I consulted render these few words (in red their translation of logos, in blue their translation of dialegesthai):
Jowett : " that other sort of knowledge which reason herself attains by the power of dialectic";
Shorey: "that which the reason itself lays hold of by the power of dialectics";
Cornford: "all that unaided reasoning apprehends by the power of dialectic";
Lee: "that which the very process of argument grasps by the power of dialectic";
Bloom: "that which argument itself grasps with the power of dialectic";
Grube: "that which reason itself grasps by the power of dialectic";
Grube/Reeve: "that which reason itself grasps by the power of dialectic";
Reeve: "what reason itself grasps by the power of dialectic".
Logos is translated as "reason" (Jowett, Shorey, Grube, Grube/Reeve, Reeve), as "reasoning" (Cornford), as "process of argument" (Lee) or as "argument" (Bloom), and dialegesthai, which in Greek is the infinitive of a verb, as "dialectic", which is a noun (all except Shorey), or as "dialectics", also a noun (Shorey), but never by a verb. And to "translate" dialegesthai (middle present infinitive) by "dialectic(s)" is to use as a translation of a verb a noun that is only the transcription in English of a Greek word derived from this verb, dialektikè, and which, over the centuries, has taken on meanings that no longer have anything to do with Plato, which does not help us to understand what he was trying to tell us here. But, if Plato's Socrates speaks here of to dialegesthai, by substantizing an infinitive in a formulation that means "the [fact of] dialegesthai", it is to make us understand that what he is talking about is not an abstraction or a particular technique (hè dialektikè (technè)), but an activity that implies a practice, and a practice in the interest of the person who practices it (this is the meaning of the middle in Greek). What he is talking about here is a certain way of making use of the logos. But then, of what logos? Speech? Discourse? Reasoning? Reason? Precisely, by speaking of its dunamis, of its "power", of his "capacity", he exempts us from choosing: what he has in mind is logos at his best, what it is the only one to allow us to achieve when it is used to the best of its capabilities, and this therefore implies "discourses", but discourses implementing "reason" at its best, therefore "reasonable" discourses, but this implies above all to understand logos in the sense given to this word by the Elean Stranger at Sophist, 259e4-6, when he affirms, at the beginning of a discussion that will allow him to specify how the logos works, that [it is] by means of the intertwining of eidè with one another [that] logos happens for us" (dia tèn allèlôn tôn eidôn sumplokèn ho logos gegonen hèmin): words alone have no meaning; they only take on meaning through the eidè that each one of us associates with them, and it is therefore the intertwining of eidè and not the juxtaposition in sequence of words that gives meaning to the logoi. From this perspective, it can be said that there is logos properly speaking, that is to say, meaning-bearing discourse, only when the person who produces these logoi is aware of the fact that one associates eidè with the words one uses in one's thought, and that it is these eidè that are meaning-bearers when one intertwines them in one's logoi.
The key underlying question, which Socrates invites us to investigate, is therefore that of the power (dunamis) and limits, of the logos, of the way in which, when properly used, it is capable of giving us access to something that is beyond the words it uses and the mere images, and even the eidè, that we associate with these words. And what he's saying here is that this power results from a specific way of making use of the logos, the practice of to dialegesthai. So here we are, back at the starting point: how to understand this verb? What meaning should be given to dialegesthai, the middle present infinitive of the verb dialegein, formed by adjunction of the prefix dia- (which means in composition "through", "in different directions", "one with another", "to the end", "thoroughly") to the verb legein, the verb from which the word logos derives? In the middle dialegesthai, the verb dialegein means, in the time of Socrates and Plato, as I have already suggested, "to converse, to discuss", or, in Herodotus for instance, "to speak such and such a language or dialect" (the English word "dialect" comes from this use), but it is a relatively rare verb before Plato: while the Perseus site lists, in all the Greek classics available there, 19 400 occurrences of the verb legein, distributed among all the authors, it lists only 663 occurrences of dialegein / dialegesthai, only 38 of which are in authors prior to or contemporary with Socrates (Homer: 5; Herodotus: 10; Thucydides: 3; Aristophanes: 7; Antiphon: 6; Lysias: 7), while there are 216 in Plato, 100 in Xenophon, and 76 in Isocrates). As for other words built on the same root:
- dialogos, which is at the root of the English word "dialogue", appears only 13 times in all, 9 times in Plato, 1 time in Isocrates, and the rest in late authors (Plutarch and Flavius Josephus);
- dialexis, an action name derived from dialegein, never appears in Plato, and only once, in Aristophanes (in the Clouds, at line 317, and in the mouth of Socrates, glorifying "the Clouds, the great goddesses of idle men, through whom knowledge (gnômèn), dialexis, intelligence (noûn), and also fabulation (terateian), verbosity (perilexin), chicanery (krousin) and forceful talk / understanding (katalèpsin)"), while there are 120 occurrences of lexis ("action of speaking, speech (as an act)", or "manner of speaking, elocution, style", and hence "word, expression"), the equivalent derivative from legein, including 28 in Plato;
- dialektikos, from which the English "dialectic" derives, and which exists in Greek only in the form of an adjective, not a noun (although the adjective can be substaized in hè dialektikè, implied technè), is listed 52 times (including the adverbial form dialektikôs), 21 times in Plato, 29 times in Aristotle (of whom Perseus provides only a small part of the works) and 2 times in Xenophon.
It therefore seems that dialegesthai and the related words took on a new importance in Socrates' time, and perhaps under his influence, although we do not know the meaning (or meanings) given to them by Socrates. As for Plato's understanding of them, we have his "dialogues", precisely, but it is not the fact that we have transposed the words identically into English that sheds light on their meaning, and it is not from the meaning that "dialogue" and "dialectic" have taken on in English that we should try to understand Plato, but on the contrary, on the basis of Plato and the uses he makes of these words, we must try to determine the meaning he gave them, without being influenced by the meaning that their transcription into English may have taken. This is why it seems to me that the best way to reduce the influence of English in this matter and to force the reader to question the meaning of the word and try to understand it based on the context rather than importing a preexisting understanding of the word taken in isolation is not to translate it.
The first thing to remember in order to try to understand dialegesthai is both the plurality of meanings of the prefix dia-, recalled in note 43 in connection with dianoia ("thought"), and the multiplicity of meanings of logos / legein. What is at stakes is to determine whether the dialegesthai is limited to the practice of verbal exchange between several people, that is to say to the art of "dialogue" in the sense that this word has taken in English, if necessary according to a particular technique that would have to be specified, in which case it would not be easy to determine what would differentiate Socrates from the rhetoricians he opposed, aside from a different oratorical technique, or if it has a broader meaning that is not limited to interpersonal exchanges by means of spoken words, which would open the way to an understanding of dialegesthai that is not limited to questions of oratorical techniques. We must then recall the words of the Elean Stranger at Sophist, 263e3-8, mentioned in note 43, where he defines dianoia as an inner dialogos of the soul with itself, echoing the words of Socrates at Theaetetus, 189e6-7, where he defines thinking (to dianoeisthai) as "a discourse that the soul itself runs through from beginning to end with itself about what it is examining (logon hon autè pros autèn hè psuchè diexerchetai peri hôn an skopèi)". In short, it seems that, for Plato, in some contexts at least, logos and dialegesthai are not limited to vocal sound expression but can include inner dialogue. And let us note that this idea of a dialegesthai that would remain interior might well be a mere return to the sources, because the 5 uses of this verb listed in Homer all refer to the same verse used 5 times with the exact same wording (Iliad, XI, 407; XVII, 97; XXI, 562; XXII, 122 and XXII, 385), an example of ready-made expressions found in his work, the text of which is as follows:
      Alla ti è moi tauta philos dielexato thumos;
which can be translated as:
      But why does my soul argue with me in this way?
These words are uttered by different characters (Ulysses, Menelaus, Agenor, Hector, and Achilles, respectively), but always, except for the last one, in the context of an inner "monologue" introduced by another ready-made expression:
      Ochthêsas d' ara eipe pros hon megalètora thumon :
      « Ô moi egô(n)...
that is:
      Afflicted, he then said to his great-hearted soul:
      "Poor me...
and in which the hero balances between two options. And if, when the words are used by Achilles (last occurrence), it is in the context of a speech to the Achaeans in front of the corpse of Hector whom he has just killed, the fact remains that it introduces in the middle of this speech an aside in which Achilles returns to himself to remember the sad fate of his friend Patroclus. In short, for Homer, dialegesthai is exclusively "interior", and implies only the thumos of the hero (a word that I have translated, in Homer's verses, as "soul", and which serves, in Plato, to designate the middle part of the psuchè, of the soul, the one that is precisely torn between the voice of the logos / reason and that of the "passions (epithumiai)"). Now, any Athenian in the time of Socrates or Plato, who had spent one's childhood memorizing Homer's verses while learning to read, could not have failed to have this all-purpose verse in mind when one heard of dialegesthai, even if the word had since found other uses. Let us keep in mind from all this that, when Plato's Socrates speaks here of the "power of dialegesthai", he probably does not limit this activity to the exchange of words between people.
However, we should not underestimate the decisive importance of this practice (the "dialogue" in the usual sense) for Plato, who made it the privileged form of all his written work, because it is it, and it alone, that makes it possible for us to ensure the "objectivity" of what we put behind at least some of the words we use, that is, that they are not pure creations of the mind of the one who uses these words without external reality. And so, even if the activity of the dialegesthai requires an internalization of the "dialogue" that it implies, it cannot do without verbal confrontation with others. If we think about it carefully, the logos exists, and could only have come into being, solely in view of the dialegesthai understood as the practice of interpersonal dialogue allowing cooperation in action between individuals living in society, and each of us can only learn to speak through dialogue with those close to us, even if, at the beginning, this dialogue is reduced to simple words accompanying gestures with a child who, precisely, does not speak yet, or only in a fragmentary way. But learning to speak does not imply understanding the power of the dialegesthai and the role of words in our understanding of the world around us, nor above all the fact that it is through the mediation of the eidè, and not of the words that we associate with them, most of them unconsciously, that the logos takes on meaning. And it is indeed of all this that both the analogy of the line (static vison) and the allegory of the cave (dynamic vision) deal with. The final stage, which corresponds to the second sub-segment of the perceived by intelligence, presupposes an understanding of how the logos works, what decisive role the eidè play in it, and what its practice in dialegesthai gives us access to beyond the images that, when learning to speak, we have associated with the words we use to unconsciously make eidè of them, yet very rudimentary eidè.
But to understand all this, and what Plato means more precisely by dialegesthai, which he makes the ultimate knowledge to which must have access the philosophos according to his own understanding of this word, who is certainly not the one he portrays in the digression of the Theaetetus, who flees from exchange with his fellow citizens, that is to say precisely the dialegesthai, by retiring to his ivory tower, we will have to wait until he has his Socrates describe a little better what he puts behind this word at the end of Book VII, because, in order to be able to understand it, we must first have swept away what it is not, everything that haunts the first segment of the perceived by intelligence by pretending to be "knowledge" (epistèmè) and is in fact only "wandering" in the meanders of the logos (dianoia) leading to nothing beneficial for those who practice them and for their fellow citizens (arithmetic, geometry, etc.) as long as they use them only to solve specific problems and without yet being aware of the fact that words have meaning only through the eidè that we associate with them and not with a view to a global and coherent apprehension of our universe aimed at understanding what is good for us as human beings (to agathon) in this life, individually and collectively, by seeking to grasp the ideai of which these eidè are only approximations more or less similar depending on the person who poses them for oneself (cf. tithesthai in Republic X, 596a6-7). And since what specifies the human being is the logos, the "excellence" (aretè) for a human being is to know how to make the most of the logos. The rest of the line will tell us what is the right way to make the most of the logos by using the dialegesthai, of which Plato's dialogues give us brilliant examples.
When we understand this, we see that it is practically impossible to translate logos and dialegesthai here without closing doors that Plato took care to leave open. But if one wants to translate at all costs, one must do so with words that are as unspecialized and "technical" as possible, and therefore in particular avoid any reference to "dialectic(s)" and not restrict logos to the realm of reason alone. (<==)

(48) "Making" literally translates the Greek poioumenos, the middle present participle of the verb poiein, "to make, produce", which has already been mentioned in notes 21 (in relation to the expression tèn methodon poioumenè which closes the elliptical sentence in which Socrates introduces the two approaches specific to the segment of the perceived by intelligence), 35 and 38 (in relation to the expression tous logous poiesthai). This meaning of the middle poieisthai here echoes the middle in the words "making for oneself with the appearances themselves the road map through them" (autois eidesi di' autôn tèn methodon poioumenè, 510b8-9) and anticipates what Socrates says at Republic X, 596a6-7, that "we are in the habit of positing for our own use some eidos, unique in each case, for each of the pluralities upon which we impose the same name" ( eidos ti hen hekaston eiôthamen tithesthai (middle) peri hekasta ta polla hois tauton onoma epipheromen). In each case, the middle, either of poiein ("make, produce") or of tithenai ("posit, make"), stresses the leading role the person talking plays in choosing the words one uses in the logoi one produces and giving them meaning through eidè. One doesn't simply make a road map (methodos), but one makes / builds a road through (the etymological meaning of methodos), paving it with words that play the role of "supports" (hupotheseis), of "stepping stones / footholds" (epibaseis, 511b6), when properly associated with eidè that give them meaning. This personnal implication implied by the middle is what makes the difference between opinion (doxa) in the segment of the seen, which is accepted after being received from others, and knowledge (epistèmè), at the top of the segment of the perceived by intelligence, which is challenged before being accepted, a challenge which begins with the meaning of words, which are the stepping stones (a possible meaning of epibasis at 511b6) of any logos.
But here, where the subject of poioumenos is the logos itself (autos ho logos), it is harder, and it would be misleading, to make the use of the middle felt in the translation. The personal involvement is not that of the logos itself, but that of the one producing it. (<==)

(49) First part of the explanation of the approach constituting the proper use of the logos: the role played by the hupotheseis ("supports"). We have seen that, in the first approach, the one for which the geometers serve as an example, these hupotheseis are what is supposed to be known at the outset and for which we do not seek to give a logos (explanation) (logon didonai) because we consider them obvious. But in the end, to the extent that what we are talking about manifests itself in logoi, what is given are the words with which these logoi are made, words like "square", "angle", "diagonal" in the example taken by Socrates, and the fact that we do not want to give explanations (a possible translation of logon didonai) about them is tantamount to saying that we do not distinguish between a word and what we are talking about that it designates and that we are content to use the words in their usual sense in the context at hand without really trying to investigate what they cover. In other words, the association of an eidos with a word is unconscious, and one does not seek to question this eidos when one uses the word. In the second approach, on the contrary, these hupotheseis are not principles / starting points (archai) considered as intangible principles but really simple supports, since it is necessary to rely on something to progress, but supports that nothing prevents from being called into question if the reasoning requires it, that is to say if they are not stable enough to serve as firm points of support for progress or, in other words, if the meaning of the words on which one relies is not sufficiently established and shared to allow constructive progress in dialegesthai. Socrates speaks of them as "stepping stones" (epibaseis) and "springboards" (hormas). Epibasis, formed from the prefix epi- ("on, towards") and basis, "step / base / pedestal", a noun derived from the verb bainein, "to walk", means in the primary sense "the action of stepping upon, or towards", either "approach", or "attack", or "means of approach, access". The primary meaning of hormè is "rapid motion forward, attack, invasion". The word can also mean, in the moral sense, "impulse, effort, eager desire." These terms are explicitly opposed to archè ("principle") used in connection with the first approach. They evoke the idea of the beginning of a movement by emphasizing the resulting movement, the energy involved, but without giving any particular value to the starting point, as archè does by making it a "principle". You have to start from somewhere and rely on something to move forward, but what counts is not the starting point but the energy you deploy and the ability to rise from the supports that serve as simple initial points of support. This is what my translation of hormè as "springboard" seeks to convey, which illustrates the notion of "impulse", one of the meanings of this word, in coherence with my translation of hupothesis as "support" and echoes the words of Socrates in his description of the first approach, when he says that those who practice it are "unable to rise higher than the supports" (511a5-6), that is to say, if we return to the idea that these hupotheseis ("supports") are the "supports" of the logoi, i.e. the words, to free oneself from the grip of words. In this perspective, what is described here is an approach that relies on words to rise toward what words are only "representations" (not even resemblance) of and which is capable of giving them meaning and regulating their use, the "beings" themselves, or rather the ideai that are the intelligible perception accessible to human beings through "the power of the dialegesthai" (hè tou dialegesthai dunamis), who, because of the limitations of human nature, which is not the divine nature and therefore cannot do everything, is not able to give them access to the auta ("the *** themselves"). (<==)

(50) Second part of the explanation of the approach constituting the proper use of the logos: the principle (archè) considered as its target. The Greek text translated as "going all the way to that [which is] support of nothing, toward the (leading) principle of the whole" is mechri tou anupothetou epi tèn tou pantos archèn iôn. Here we have, with mechri tou anupothetou ("to that [which is] support of nothing") and epi tèn tou pantos archèn ("toward the (leading) principle of the whole"), two distinct complements of the verb iôn (going) introduced by two different prepositions with similar meanings:
- mechri with the genitive means "as far as" and introduces tou anupothetou, a singular neuter genitive, which is indeed a substantized adjective (such as "the beautiful", to kalon, or "the good", to agathon);
- epi with an accusative, whose meaning is much broader, starting from the primary meaning of "(up)on", and which can mean "up to" or "in the direction of, towards", and also "until", which introduces tèn archèn, accusative feminine singular.
This reformulation thus disconnects what was joined in the first formulation, which spoke of going ep' archèn anupotheton, which could be translated literally as "up to the principle put under nothing" and which I had explained in the form "up to a (leading) principle [which is] support of nothing", making to anupotheton (here in the genitive tou anupothetou called by the preposition mechri, "all the way to") another name for the intended target ("the put under nothing") next to hè archè ("the principle", here in the accusative tèn archèn, called by the preposition epi, "toward"), this time qualified as the principle "of the whole" (tou pantos). This way of dividing the statement makes it possible to better highlight each of its elements and to better understand what Socrates means by archè ("(leading) principle") in the context of this approach. Starting from supports (hupotheseis), that is to say from something that is under something else and on which one can rely to rise to something higher, the approach described here progresses to something that is under nothing (anupotheton, where a-n-(h)upo, a combination of the privative alpha and the preposition hupo ("under"), means etymologically "not under"), that is to say, constitutes the unsurpassable summit at the top of the progression. And this principle (archè) / summit is now also qualified as the principle of the whole (tou pantos), it being understood that, since we are in the intelligible, it is a principle of intelligibility, and not of existence, whatever that means. It is therefore the principle (in the singular) that gives meaning to all (pan) the rest, taken individually, and not as a whole, as implied by the use of to pan ("the whole" in the multiplicity of its parts) as opposed to to holon ("the whole" thought as one set).
Socrates thus opposes an approach, the one associated with the first sub-segment of the perceived by intelligence, which progresses in time from a beginning (archomenoi, 510d1, which resonates with archè and explains its meaning in this context, that of a simple beginning, as I explain in note 31) to an end (teleutôsin, 510d2, which resonates with the teleutè of 510b6 and explains its meaning in this context, that of a simple term of the approach undertaken) on a path that can be imagined to be horizontal (along the axis of time) and "paved" with hupotheseis placed one after the other to support the "walkers", who advance without rising (" unable to rise higher than the supports", 511a5-6), to an approach, that associated with the second sub-segment of the perceived by intelligence, which can be imagined as the ascent from the bottom to the top, therefore vertically, of a pyramid in which each floor "supports" the floor above it up to a topmost floor which supports nothing since it is not placed under anything (an-upo-theton in the etymological sense), an ascent in which it is not the axis of time that is important (properly linking the steps of a reasoning), but the progression in comprehension, a word that must be understood here keeping in mind its meaning of "inclusion", and in which it is therefore not so much the order in which one progresses in the reasoning (temporal dimension) that is important as the result one arrives at, ideally the "principle of the whole", which is not a point of arrival where one stops as if one had reached the end of the process, as do those of whom Socrates speaks a little later when "deciphering" the allegory of the cave, who, having reached the end of the ascent out of the cave and the contemplation of the sun, "think[] they have already been carried alive into the islands of the blessed" (519c5-6) and "[are] not willing to go down again among those prisonners" (519d4-5), but kind of a scenic viewpoint (in the sense of tourist guides) from which one can encompass the whole landscape and understand everything, giving those who reach it "a single aim in life aiming at which they must do everything they do privately and publicly" (519c2-4), and in particular to solve the problem they started with, which must incidentally be solved. This vertical progression is illustrated by the allegory of the cave, which does not limit it to the second sub-segment of the perceived by intelligence, but widens it to the progression through the four sub-segments, which invites us to draw the line as vertical and not horizontal as we most often see (the fact that the progression between the successive segments is in the direction of an ascent, that is to say, vertically, does not preclude the progression in one of these segments, for instance in the first segment of the perceived by intelligence the "wandering" of the dianoia, from being made horizontally as long as one has not understood that one has not reached the top).
But this "principle of the whole" which makes everything intelligible and serves as a support for nothing above it, what is it? As I have already said, it is the good (to agathon) or at the very least, for us human beings, the idea of the good (hè tou agathou idea), which is the way in which the good itself (auto to agathon) can be grasped by us because of our human nature. Plato's Socrates does not say this explicitly and leaves it to us to find out (it is part of the ascent!), but given the position of this analogy in the Republic, in the course of a discussion on "the idea of the good" (hè tou agathou idea) (cf. 505a2), between the parallel between the good and the sun, which this section purports to complete, and the allegory of the cave that Socrates, immediately after completing it, explains to us as an image of the soul's march towards this "idea of the good" (cf. 517b9-c1), and the way in which the analogy of the line itself is introduced, which presents the first segment, that of the seen, as the image of the "kingdom" of the sun and the second, that of the perceived by intelligence, as an image of the "kingdom" of the good (cf. 509d), there is no doubt about this answer. And moreover, its anupotheton ("support of nothing") character is highlighted by Socrates at the beginning of the discussion, at 505d5-9: if human beings can be satisfied with what only looks (dokein) beautiful or just without really being so, no one will be satisfied with what only looks good (agathon), but is not so in truth. In short, the good itself is not sought for anything else superior to it, as when justice, or its semblance, is sought for honors or power or riches, but it is for its sake that all do what they do. What is open to discussion is not whether it is true that every human being is always looking for what is (or seems to onesemf) good for oneself, which, for Socrates, is self-evident (a statement that some, misunderstanding it because they think exclusively of the moral "good" and forget the "for oneself", view as a "paradox"), but what is really good for oneself. To understand anything, since each one of us always acts with a view to what one thinks, rightly or wrongly, is the best for oneself among the different options between which one has a choice, is to understand how it can be more or less good for oneself. And insofar as we are by nature made to live in society (which is precisely what made the logos possible through dialegesthai), this search for the good for me cannot do without dialegesthai since others can thwart my plans to achieve what I think is good for me if they do not think it is good for them and have the means to prevent me from achieving it or to make it more difficult for me and reduce or cancel out the beneficial nature of it for me.
Having said this, it may be interesting to return now to Aristotle's version of the anupotheton principle, to which I alluded in note 21. In the passage quoted (Metaphysics, Gamma, 1005a19-b34), we see the good student making intense and yet fruitless efforts to try to rise to a "principle" that meets the criteria suggested by Plato: he has understood that this "principle" is not specific to this or that field of investigation and that therefore it is not one of those geometers or arithmeticians ("oute geômetrès out' arithmètikos...", 1005a31) are looking for; he has well understood that it transcended the realm of the "physical", and that it could not therefore be reached by the "physicists" ("tôn phusikôn enioi", 1005a32) who think that phusis is the whole of being; he has well understood that it was in the realm of logos and that it was up to the philosophos to seek it; but alas!, for him, dialegesthai has become sullogizesthai in the most technical sense of the term ("reasoning by way of syllogisms"), the philosopher must set out in search of the "principles of the art of the syllogism" (peri tôn sullogistikôn archôn), anupotheton is for him synonymous with "certain" (bebaios) and implies that this principle "must not be a hypothesis (touto ouch hupothesis)" (1005b15), and what, in this perspective, is for him the first and most certain of principles is the principle of non-contradiction! In short, he exhibits a principle of reasoning where Plato expects a principle of intelligibility. The principle of non-contradiction does not allow us to understand anything, it only allows us to reason rightly, and what Aristotle did not understand is that it is not enough to reason correctly to be in the right, because it is the "beings" that impose their truth on reasoning and logos and not the reasoning that, when they are just, impose their conclusions on the facts. Plato does not dispute that it is necessary to reason well in order to progress in knowledge, and moreover, he himself speaks of sullogizesthai in the allegory of the cave, at 516b9, but this is in the context of the efforts to understand the role of the sun / idea of the good as precisely the light of intelligence and the guiding principle of the whole, a necessary prerequisite for reasoning correctly about the rest and understanding it (the meta of the expression kai meta tauta ("and in the midst of these [reflexions]") at 516b9, which opens Socrates' line about sullogizesthai, must be understood in the "spatial" sense of "in the midst of / among", and not in the temporal sense of "after", as do all translators and commentators, who think that the prisoner who comes out of the cave can indeed "see clearly and contemplate as it is the sun itself by itself in its own place" in this life, and that it is only then that he comes to reason about the role of the sun / idea of the good; cf. note 55 to my translation of the allegory of the cave). Short of being able to see the sun / idea of the good in this life, the freed prisoner out of the cave can nevertheless understand the preeminent role he plays as the light of intelligence, as the Socrates of the Republic seeks to make his interlocutors understand in these pages, and this is what must have been understood in order to claim to be in the second sub-segment of the perceived by intelligence. Besides, Plato uses the verb sullogizesthai in a sense that is probably less "technical" and more general than Aristotle, that of "reasoning"; he disputes the fact that it is enough to reason correctly to arrive at the truth. And this is precisely what he tried to make Aristotle understand (and which the latter seems not to have understood) through the lesson of the Parmenides, in which he took the trouble to choose as Parmenides' pale interlocutor in his "tedious game" a namesake of his pupil, a young Aristotle, for an exercise in pure logic in which Parmenides (and behind him Plato) demonstrates with equal rigour everything and its opposite, in the absence of agreement in each case with his interlocutors on the meaning to be given to the words he uses, and in particular to words as abstract as "to be / being" (on), "one" (en, neuter of eis)", or ousia (which does not seem to have been part of the vocabulary of the historical Parmenides, but made a fortune with Aristotle), in contrast to the "dialectic" that the Elean Stranger will practice in the Sophist (I only have to open my eyes to see that "Theaetetus is sitting" is true and "Theaetetus is flying" is false, cf. Sophist, 263a-b, and this is enough for me to prove the existence of false speech).The principle of non-contradiction cannot be the "principle of the whole" in the sense in which Plato's Socrates understands it here, for the simple reason that it has to do with means, and more precisely with means for human beings to make good use of the logos that specifies them, and not with the ends to which the logos is meant to give them access, which presuppose an understanding of the whole of which they are a part. It is therefore at most only a part of the user's guide for the use of our intelligence (noûs), of our reason (logos), and it can therefore at most only contribute to allowing us, human beings and us alone, to make good use of what specifies us as human beings in order to understand and explain what surrounds us and pre-exists us. It cannot, therefore, in any way be the principle of the whole, and cannot even by itself enable us to comprehend the whole without recourse to experience, for, as I have just said, it is not the logos, language, or even logic, which imposes its rules on the whole, but, on the contrary, the whole which imposes its results, observable and verifiable through the data of experience, to reasoning, to logos, which only account for it. This is exactly what Plato sought to make Aristotle, and all his readers, understand by example, with Parmenides: at first sight, a succession of reasonings conducted on the same concepts ("one" and "being") with the same logical rigor, seems to arrive at conclusions contrary to one another, and it is only when we realize that these same words are not taken in the same sense from one reasoning to another, as Mitchell H. Miller, Jr. (in Plato's Parmenide, The Conversion of the Soul, Princeton University Press, Princeton, 1986) clearly shows, that it is possible to make contradictions disappear and find an overall coherence in their sequence in a "second-degree" reading, which clearly shows that it is beyond words (dia logos) in what they seek to "image" and not in words and logic that we must look for the truth. Aristotle is more or less aware of the problem posed by his choice of the "principle of the whole", since he begins his presentation by speaking not of "principle (archè)", but of "what is called axioms (axiômatôn) in mathematics", and by questioning whether they are of the same order, and in the field of investigation of the same epistèmè as ousia (cf. 1005a20): by seeking to bring axiôma, whose primary meaning is "price, value, esteem", closer to ousia, whose primary meaning is "property, fortune, wealth", he seems to want to cajole Plato by rising into abstraction about what makes the "wealth" of beings, by passing from a term, ousia, which evokes a "wealth" that is entirely material, landed or monetary, to a term, axiôma, which evokes a much more abstract "value", of a qualitative rather than quantitative order, which refers to the ideas of "dignity", "fitness", "quality", "worth" (the axiômata of mathematicians are propositions that are valid in themselves, without the need to try to demonstrate them, and therefore have a "value" of their own), but at the same time, he seems to lean towards the Pythagoreans, looking for the ultimate principles of the whole in mathematics, while for Plato, mathematics, as well as the logic dear to Aristotle (which ended up being absorbed by mathematics, since the set theory proposes a reformulation of the rules of Aristotle's logic through the rules of inclusion of sets, and that Boolean algebra allows logical operations to be transformed into arithmetic operations), remain of the order of means and can at most only help us to understand "hows?", but never "whys?", while the only questions that should interest us are those concerning "whys?": why is it better for us to do this than that? It is the "value" implied by questions of this kind that alone should matter to us, and this value is what we all call "good," without agreeing on what is "good" for us. As to whether this principle, which Aristotle would like to make the principle of the whole, is certain and indisputable, he is the first to make us doubt it, since, in order to affirm it irrefutable, he is already obliged to deny that some, even before him, wanted to doubt it (at 1005b25, he doubts the word of those who claim that Heraclitus doubted it). And we know that since then, others have tried, with more or less success, to develop logics that free themselves from this principle... (<==)

(51) "Having taken hold of it" translates the Greek hapsamenos autès. The verb haptesthai, of which hapsamenos is the aorist participle, means in the primary sense "to grasp, take hold of", and is the root of words such as aptos, "tangible", and apsis, "touch" (as one of the five senses). It can be translated as "to come up to, reach, attain", but it seemed important to me to keep in the translation the "tangible" character that Plato seeks to give to his "principle of the whole" by the choice of this verb. This is one more way of suggesting that he doesn't refer to a pure "invention" of our mind, but to something that is, if not "external" to it, at least "transcendent" in that its existence does not depend on us. (<==)

(52) "Reaching further in turn to what can be reached from it" translates the Greek palin au echomenos tôn ekeinès echomenôn. These words pose a problem of translation, in the sense that it uses twice the present participle middle of the verb echein, whose primary meaning is "to hold, possess", and hence "to have", in two different roles: on the one hand, as a verb with autos ho logos at the beginning of the sentence as its subject, in a series of clauses using verbs in the present or aoristic participle to describe the successive stages of the process followed by the logos (iôn, "going"; apsamenos, "having taken hold of"; echomenos, "reaching"; and a little later, proschrômenos, "making also use of"); on the other hand, as the object complement (in the genitive plural tôn echomenôn) of this present participle, in the neuter, and substantized by the article. It is not uncommon to find in Greek a turn of phrase where the noun of the same root as the verb is used as the direct object complement of a verb, but the problem here comes from the multiplicity of possible meanings of echein, and from the difficulty of finding one that suits both uses, active (to translate the middle echomenos) from the point of view of the logos and passive from the point of view of what it exerts its action upon. The meaning that seems to be appropriate here among those who command a complement to the genitive is a meaning that derives from "holding" through the idea of proximity, of contiguity, which is that of "leading toward, reaching to" and hence "come next to, follow closely". Thus, ta echomena means "what follows," or "what is connected with" something that has come before, or "what concerns" something or someone. I have chosen among these possible translations one that allowed me to keep the same verb for both uses of echomenos.
What is important to note is that the emphasis of this verb is not on the idea of a rigorous deductive process that leads from the principle to its "consequences", but rather on the idea of proximity: once the principle has been found, one looks for what is in its vicinity (one of the possible meanings of hoi echomenoi is "the neighbors"). If we want to translate as "drawing the consequences of it", or something similar, it is by remembering the etymology of "consequence", formed on the Latin verb sequi, "to follow". (<==)

(53) "It follows a downward path toward an end" translates the Greek epi teleutèn katabainèi. In the first formulation of the description of the two segments of the noèton, Socrates characterized the method implemented in the first sub-segment by the fact that it progressed, not ep' archèn, but epi teleutèn. Now, here, by explaining the characteristic method of the second sub-segment, he also makes it reach a teleutèn. Once again, as in the reformulation of the characteristic approach of the first sub-segment, where he used the terms eidos (cf. note 27) and methodos (cf. note 28) that he had in his synthetic formulation reserved for the approach specific to the second sub-segment, Socrates inverts certain words between the two approaches, to show us that it is not these words that are characteristic of one or the other of the two approaches, if only because the same word can have different meanings in different contexts and it is the relationships established between the words used that are more important than the words themselves and that it is these relationships that make it possible to specify the meaning of each of them. Thus, we have seen in note 31 how the context allowed us, from the use of the verbs archein (from which archè is derived) and teleutan (derived from teleutè) in Socrates' description of the approach of the first sub-segment of the perceived by intelligence, to understand the words archè and teleutè in this context, namely, in the purely temporal sense of "beginning" and "end", without this preventing that, in the first description of this same approach, when Socrates says that it does not lead ep' archèn ("to an archè") by contrasting this with the fact that the second leads ep' archèn, archè must be understood in the same sense in both cases and, since this word is used with the preposition epi (towards), this meaning cannot be that of a temporal beginning, but rather that of "principle." But going back to a principle, especially if this principle is a principle "of the whole", does not prevent this ascent from being made within the framework of logoi unfolding in time and therefore having a beginning and an end, and that they have been undertaken for a certain purpose, with a view to a certain end (teleutè) and that the ascent towards the principle must not make us forget the descent towards the answer or answers to the questions that were at the origin of this particular use of the approach.
And it is deliberately that Plato seems to be playing with our nerves by breaking in his repetition the "codes" that he seemed to have set in his initial explanation by letting us think that certain words would be specific to one or the other of the approaches and by using these words about the other approach, precisely to make us understand by example that it is not the words that are important but what they refer to deduced from the context and relationships they have with one another.
Having understood this, if we try to use the vocabulary to determine what is specific to each approach, we see that the only word in the synthetic description of 510b4-9 that is never used in connection with the first approach, neither in its synthetic description nor in its more detailed presentation, is the adjective anupotheton, which is most probably a neologism formed by Plato for the occasion, whose meaning can therefore only be deduced from its etymology ("not placed under") and the context, which explains why, using it for the first time, he does not make things harder for the reader / listener and reserves it for a single approach, of which it is then the key word. In the synthetic description, this adjective qualifies the word archè (which I have translated as "(leading) principle"). Now, if this word also appears in the synthetic description of the first approach, the one that characterizes the first sub-segment of the perceived by intelligence, and, in this case, as we have just seen, in the same sense, it is in a negative clause, to say that this approach does not progress "toward a (leading) principle (ouk ep' archèn)" (and for good reason, since we have seen in note 51 that, for it, archè means "principle" as a starting point). And in the same way, in the synthetic description of the characteristic method of the second sub-segment of the perceived by intelligence, we find a negative formulation taking up a term used affirmatively for the first method, the word eikôn ("image"), in the words aneu tôn peri ekeino eikonôn ("without the images [revolving] around that"). We can conclude that what best characterizes each approach is not something that only one of the two does (or uses) and the other does not, but something that only one of the two does not do (or does not use) while the other does (or uses). And this something is, for the characteristic approach of the first sub-segment of the perceived by intelligence, not to try to ascend toward an archè that is anupotheton, and for the approach proper to the second sub-segment of the perceived by intelligence, not to have recourse to the eikones that provide to our senses, and not to our intelligence, visible realities, thus pulling us down and making us return to the cave before having seen all that there was to see outside, starting with the intelligible dimension of what was merely seen inside the cave, to ascend toward this principle and draw the consequences stemming from it.
To put it positively, then, what characterizes the first approach is that it cannot do without "images," and what characterizes the second approach is that it begins by trying to ascend toward the guiding principle of the whole, the good, in order to form an idea of it (that is to say, to be more precise, to try to perceive something of the idea that it presents to our minds as human beings endowed with logos) that will allow it to give to the teleutè towards which it will then progress, in addition to a character of certainty that will depend on the rigour of reasoning (one of the foundations of which is indeed that principle of non-contradiction which Aristotle wanted to make the anupotheton archè, cf. note 51), and which is associated nowadays with "truth" as we understand this word, a criterion of "value" which presupposes placing oneself under the light of the good, the principle giving value to the conclusions, and not only to reasoning, and the only one capable of unveiling the truth (a-lètheia, which etymologically means "unveiling") of "beings" as understood by Socrates (cf. 508e1-6). In fact, in his first formulation, Socrates had somewhat obscured this "descent" of which he speaks here in order to better oppose a strictly "horizontal" approach (which progresses over time while always remaining on the same plane, that of visible realities), which starts from any premises provided by circumstances and hastens towards concrete conclusions without the means of appreciating their value, whatever the rigour of the reasoning that leads to it, in short, who is only interested in the "how?" and never in the "why?", to a process that begins the "descending" phase towards conclusions only after having sought how what it is interested in is tied to the principle of the whole which makes it intelligible by allowing us to determine its greater or lesser value and can therefore guide our choices by allowing us, even more than knowing how to do this or that, to be able to determine whether it is better for us in relation to the ultimate end that is ours, what is good for us, our greatest good (megiston agathon), to do this rather than that. For the value of a conclusion, or of an end proposed to our action, does not result from the mere certainty resulting from the rigour of the reasoning that leads to it. For Plato's Socrates, "truth" is not the same thing as this certainty: it is not because I have found the "right" result of the problem posed to me and that I am "certain" of my reasoning and the conclusions to which it leads that the result is "good" in the sense that Socrates gives this word. It is a certainty that, if I build a square house whose length of the sides is that of the diagonal of my present house, this house will be twice as large in area, but this does not tell me whether it is good for me to incur the expenses of building for myself a house twice as large; it is also a certainty that, if I bring together two half-spheres of enriched uranium of appropriate size, I will trigger an atomic explosion, but that does not tell me whether or not it is good for me to cause this explosion.
In all this reflection, Socrates does not deny the practical interest of the conclusions to which the first type of approach can lead by the mere fact that he hierarchises the four approaches he presents to us, any more than he condemns, in the realm of the visible, the observation of shadows and reflections, which can sometimes even be useful to us if we do not fall into the excesses of a Narcissus (this is, for example, the only way we have to know what our face looks like, or our back). And in fact, the allegory of the cave that follows the analogy of the line will try to make us understand that it is impossible to reach this level of apprehension without having first lingered at the level corresponding to the first segment of the perceived by intelligence in order to habituate (cf. sunètheia, 516a5), to familiarize our mind with what it can "see" in the intelligible. He only wants us to understand that this approach in the intelligible register is not the only one possible, or even the most important for us, and that we cannot stop there, that the one who does not go further can perhaps become the most "learned" man in the world, a Hippias capable of making himself everything he carries on him, to answer with confidence all the questions that can be asked of him on any subject whatsoever, to know how to recognize a pretty girl or a beautiful life according to the criteria of the greatest number of his fellow citizens, but that he will not deserve the name of sophos ("wise") because his knowledge will only be an "image" of knowledge, a "know-how" and not a "knowledge", since he will lack the knowledge that would allow him to figure out, in each situation, which is best for him, that is, for his soul, to enable him to attain true happiness, because he has not taken the time to look for the principle of the whole before hastening to practical conclusions and doing things because he can do them without knowing whether he should do them (Hippias, however "learned" he is, is at the same time the most "unjust" of men according to the criteria of the Socrates of the Republic, since he has not been able to understand that the good of man, an animal endowed with reason (logikos) and destined to live in society (politikos) implies the reasoned sharing of tasks in social life). (<==)

(54) I translate the Greek aisthètos (used here in the form aisthètôi, singular neuter dative), a verbal adjective derived from the verb aisthanesthai ("to perceive by the senses") as noètos ("intelligible") is derived from the verb noeisthai ("to perceive by the mind") or horatos ("visible") from horan ("to see"), by "perceptible by the senses" rather than by "sensible", which would be the most natural translation to oppose it precisely to "intelligible" or "visible", to counter the natural tendency in a "classically" understood Platonic context to understand this word as designating a category of "beings", the "sensible" "objects / creatures" that would be opposed to another category of "beings", the "intelligible", in the idea that a "being" is either one or the other, but never both, whereas, as I said in the introduction in the "decoding" of the allegory of the cave, for Plato, properly understood, everything that is visible in the cave is also intelligible outside the cave, and that therefore for him, these adjectives, visible (horaton), sensible (aistheton), intelligible (noèton) characterize ways of apprehending what is offered to our attention, without excluding, but quite the contrary, that the same "being" may be perceived in different ways, which does not pose any problems when it comes to the five senses, and therefore makes it possible to group these perceptions under a single term, "sensible" (aisthetos, used here by Socrates), which nevertheless covers a plurality of modes of perception that are completely different from one another, but seems to pose problems as soon as intelligibility, assumed in Plato's "classical" (and erroneous) understanding incompatible with apprehension by the senses, comes into play. Plato is therefore not saying here that the method associated with the second sub-segment of the perceived by intelligence must not be interested in everything that is "sensible", that is to say, in the creatures of the material world, but that it must consider what it envisages only in its intelligible dimension, abstracting from all that is sensible in it when what is examined are "beings" also perceptible by the senses.
It should be noted that this is the first appearance of this word in the analogy and that here, and here only, it takes the place of horaton ("visible"), which Socrates used until now, precisely so that we do not have too quickly the desire to oppose "sensible" in the general sense to "intelligible", and to give us time to understand that he is not talking about categories of "objects", but of modes of perception. (<==)

(55) "without making also use in any way of anything perceptible by the senses, but with appearances themselves, through them, into them, it ends also into appearances" translates the Greek aisthètôi pantapasin oudeni proschrômenos, all' eidesin autois di' autôn eis auta kai teleutai eis eidè. This part of the sentence echoes the "without the images [revolving] around that, making for oneself with the appearances themselves the road map through them" (aneu tôn peri ekeino eikonôn, autois eidesi di' autôn tèn methodon poioumenè) of the first concise formulation of 510b4-9, by describing, in similar terms, successively what not to do and what to do, and is opposed to the "they also make use of the seen appearances and make for themselves their logoi on them" (tois horômenois eidesi proschrôntai kai tous logous peri autôn poiountai, 510d5-6) of the detailed description of the approach of the first segment, an opposition clearly marked by the reuse of the same verb proschresthai ("to also make use", of which proschrômenos is the present participle and proschrôntai the third person plural of the present indicative), here in a negative clause (on this verb and its use at 510d5, see note 34).
Let us start by noticing that the more detailed description of the process associated with the second segment of the perceived by intelligence provided here by Socrates distinguishes two successive phases in this process that were not explicitly distinguished by the synthetic formulation of 510b4-9: an ascending phase towards "that [which is] support of nothing, toward the (leading) principle of the whole" (ton anupotheton, tèn tou pantos archèn), followed by a descending phase (katabainèi, cf. previous note) towards an "end" (teleutè), and that, in this new context, the part of the sentence that interests us, by virtue of its position in the sentence, seems to concern only the descent, and not the whole process. It is only from what was said in the concise wording that we can assume that it concerns the whole process. But in fact, this problem is a false problem because the approach in question here, unlike the first, the one of which geometers give an example, is not an approach that imposes a precise unfolding in time. Once the unifying guiding principle (the idea of the good) has been found, it is not necessary to start the ascent to it each time, but to see in each particular case how it sheds light on it and gives it meaning. It is, as is suggested by the poioumenè ("doing") of the first formulation, the one who reflects or leads a discussion, who traces one's own path of investigation without being constrained by the rules of logic and the data of the problem to be solved, but by taking advantage of the light that the unifying principle casts on the problem under discussion. This does not mean that one must ignore logic and reason in any way one likes, but that research is not limited to constrained reasoning. Plato's dialogues, beginning with the so-called "Socratic" dialogues, which confuse logicians who see them as failures because they do not lead to the definition they think is the object of the research, give examples of this way of proceeding: in order to better understand a notion (for instance, piety in the Euthyphro, or courage in the Lachès, or, to take examples outside the "Socratic" dialogues, justice in the Republic, or the sophist in the dialogue of that name) and not simply to come up with a "definition" of a few words just as problematic as the one they help to define, we give a first try at it that sheds a first light on what we are trying to understand, in exchanges that can be the occasion for perfectly rigorous logical reasoning, and, once a first understanding has been reached, a counter-example is brought out or the discussion is taken up again from a different point of view which obliges new investigations, which do not render the previous investigations obsolete, but relativize their scope, and these new investigations enrich the understanding of what is the object of the discussion, and so on, each new try adding a new light on what one is interested in and allowing one to gradually understand that it is illusory to want to compress the fruit of this search into the few words of an Aristotelian-type "definition" and that what matters is what is formed in the mind of each participant beyond the words used, which is precisely of the order of eidè, increasingly rich mental representations that one associates with the word that was the support (hupothesis) of the whole discussion, and which will probably be different in the mind of each one, according to one's own understanding of the discussion (the eidos that Euthyphro associates with the word "piety" at the end of the dialogue with Socrates is undoubtedly very different from the one associated with it by Socrates, and from the one associated with it by each reader of the dialogue, itself different from one reader to another).
This being said, in order to fully understand what Plato's Socrates is saying here, it is important not to lose sight of what I have already emphasized several times before, the fact that what he is talking about in the analogy is not "things" that would be objects of investigation specific to one and only one subsegment of the line (images, material objects, mathematical objects, forms/ideas), but different modes of apprehension of the same "things" from one sub-segment to another (apprehension by visible appearance, apprehension by sensible reality, apprehension by names, apprehension by means of a "cartography" of ideai). And so, when he says that we must not "mak[e] also use in any way of anything perceptible by the senses (aisthètôi)", he does not mean that we must ignore all the material world subject to change and no longer reason about everything that acts on our senses, but that we must no longer use, in order to reason about this too, data provided by our senses about specific instances of what is in question, changing "images" concerning each time only one "object" perceived at a given moment in a given place. He does not say that the approach of the second sub-segment of the perceived by intelligence does not apply to man as a material creature, but that this approach cannot be based on the data perceptible by the senses concerning them. To do this, it may be interested in the specific case of Socrates taken as an example (which almost all dialogues do), provided that it is not by taking an interest in the shape of his nose, the color of his eyes or hair (which changes with age), the sound of his voice, the smell of his breath, etc. etc., or in a purely incidental way, for instance to evoke the links between physical ugliness and moral beauty, or the difference between physically resembling (him) (like Theaetetus) and resembling (him) by name (like the young Socrates, a character who is in any case imaginary), in order to shed light on what constitutes excellence (aretè) for a human being from the case of the one whom Plato considered to be "the man, of those of [his] time of which [he] had experience, [who was] the best (aristos), in other words, the most sensible (phronimôtatos) and the most just (dikaiotatos)" (Phaedo, 118a16-17, the last words of the dialogue that recounts the death of Socrates).
The fact that, in this new formulation, the word aisthèton ("perceptible by the senses") replaced the word eikones ("images") of the first formulation, but that both are set in opposition to eidè ("appearances"), does not call into question what I said in note 21 about the fact that the two words thus set in opposition to each other were to be understood in relation to each other and that this opposition should make it possible to specify the meaning to be given to eidos in this context by limiting it precisely to noèta eidè ("eidè intelligible"), quite the contrary: the two terms successively contrasted with eidos help specify these restrictions: with eikôn ("image"), Socrates, choosing a word that share some meanings with eidos, as I said in the aforementioned note, showed that what he had in mind with eidos was the collective meaning of this word, the one in which it is precisely opposed to eikôn ("image"), the one referring to what is common to the different elements of a plurality to which, as a result, the same name or qualifier is attributed to designate this community (cf. Republic X, 596a6-7). By now replacing eikôn ("image", which is spontaneously understood as being in connection with sight) by aistheton ("perceptible by the senses"), Socrates wants us to understand that the eidè he has in mind are indeed noèta eidè ("intelligible eidè") which are not based on any criterion of resemblance of a sensible order, but only on criteria of intelligibility, which should go without saying since we are in the intelligible, but which it is better to say in plain words, and especially by specifying by contrast what "intelligible" implies, that is to say "not dependent on the data of the senses". But I must be more specific on what I mean by "not dependent on", insofar as our thoughts always find their origine, immediate or more remote, in the data of the senses: the discarding of the data of the senses invites, as I have just said, to eliminate the individual meaning of eidos as "the visual appearance of a single object of attention", where it is almost synonymous with eikôn ("image") and focus only on its collective meaning and in so doing eliminate all considerations related to the situation in time and space, that is to say, to pass from the particular to the general, which alone is susceptible of generalizable reasoning.What Socrates is saying here, when he speaks of using "the eidè themselves" and "through them, into them, end also into eidè", is that, in order to reason and not simply to account for observed facts, it is necessary to abstract from everything that is specific to a particular instance, which can be located in time and space and is provided by the senses, in order to concentrate on the similarities and differences between the multiple instances (of the same "object" at different times or of distinct "objects" in different places and/or times), which are at the source of our ability to name (present in the prisoners of the cave from the beginning, when they are still caught in bonds), itself a precondition for being able to reason by means of logoi. And the last stage of this progression is the one where we free ourselves from the last hindrance, the one that we have made for ourselves, that of words, by becoming able to reason in full knowledge of what we are doing about the supposed eidè behind these words by being able to recognize the same eidos behind different words and different eidè behind the same word taken in different senses, a practice for which Plato included practical work into his dialogues without explicitly saying so, and in particular in the analogy of the line and the allegory of the cave read as forming a whole: thus, in the analogy, when he describes the first sub-segment of the seen, he associates it with "images" (eikones, 509e1) and gives two examples of what he means by "images", shadows (skiai) and what he designates by the word phantasmata, complemented by the words en tois hudasi ("in the waters", 510a1) to make us understand that what he has in mind are reflections; but when, in the allegory of the cave, he again mentions shadows, at 515a7 inside the cave, and at 516a6 outside the cave, and when the parallel between the two images (line and cave) makes us expect phantasmata afterwards, we are up for a deception in both cases! Inside the cave, what is mentioned as another example of "images" is èchô (the "echo" of the words of the bearers sent back by the wall of the cave, 515b7), and outside the cave, eidôla (516a7), but eidôla about which he specifies en tois hudasi ("in the waters") (he returns to phantasmata at 516b5 about the sun, about which he cannot mention shadows, but only reflections, and always specifying en hudasin ("in the waters"))! It must therefore be clear to any reader that what is designated here by the word eidôla is the same thing as what was designated shortly before, and will be designated a few lines later, by the word phantasmata and that it is in all cases what we call "reflections" in English. Therefore, we are in the presence of two different words referring to the same eidos, the one that we associate with the word "reflection" to designate what we see on the undisturbed surface of a body of water or in a mirror (in the analogy, next to phantasmata "in the waters", Socrates also evokes those that occur on surfaces "at the same time compact, smooth and bright"" (510a2), i.e. in particular mirrors). And if we want to pursue the parallel to the end, we can notice that an echo is a kind of "reflection", but a sound reflection and no longer a visual one, so much so that we can conceive of an eidos that is associated with three words, phantasma, eilôlon and èchô, at least in certain contexts, evoking the idea of the "reflection" of a wave, of light, of sound or of any other kind, on a surface havind a reflective power. And finally, to confirm that phantasma and eidôlon refer to the same thing in the analogy and the allegory, in the summary of Socrates' allegory of the cave towards the end of Book VII, he returns to the word phantasmata (and still further qualified as en hudasi, 532c1) when he recalls the stage of observation of what was designated by eidôla in the allegory. And for an example of the opposite, the same word used in several different senses, we can cite the case of noèsis, which is used in the analogy, at 511d8, to designate the "affection" (pathèma; cf. note 74 later in this page) that Socrates proposes to associate with the second sub-segment of the perceived by intelligence, the one we are interested in here, and which, in the recalling of the analogy towards the end of Book VII, will serve to designate the grouping of the "affections" (pathèmata) associated with the two sub-segments of the perceived by intelligence (534a2), what was designated by noèsis in the analogy being then designated by epistèmè ("knowledge", 533e8).
In short, to use "the eidè themselves" and "through them, into them, end also into eidè", is to be able to reason by freeing ourselves not only from the sensory perceptions associated with what we reason about, insofar as they relate to unique "facts" (pragmata) located in time and space and which we witness at that time or retain "images" in our memory, but also of the influence of the words we use (which does not prevent us from associating the color "white" with snow, as long as we think of white as a color in general and of snow as a meteorological phenomenon and not as what I see at that very moment on the summit of Mount Olympus, that is to say as intelligible eidè). And it is precisely this liberation from the grip of words that makes the difference between the first and second sub-segments of the perceived by intelligence: the geometer who does not want to "give a logos" (logon didonai) about the words he uses shows that he is a prisoner of them and does not know how to think of what he is reasoning about other than by means of the unique word he associates with it, to rise from the words that serve as "supports" (hupotheseis) to the eidè that he unconsciously associates with them: he has understood that what he is talking about and about which he is reasoning can only be "seen" through dianoia ("thought"), but because he does not take the time to reflect on what it is, he does not manage to dissociate this both from the words he uses and from the sensible "images" (eikones) that he associates with these words and is forced to return to these visible "images" in order to understand it.
And in the end, what is the power of dialegesthai for Plato's Socrates, which he places at the heart of this second approach? It is not so much the fact of being able to cooperate in action or in the random prediction of the future (cf. 516d1-2, in the allegory of the cave, about the prisoner freed and having climbed out of the cave, remembering, after having understood the role of the sun / idea of the good as the light of intelligence and the guiding principle of the whole, how behave in the cave the prisoners who have never been freed from their bonds), nor even the fact of being able to solve practical or even theoretical problems by reasoning, in order to then do what can be done merely because one can (the dianoia ("thought") without a compass of the first sub-segment of the perceived by intelligence, where dia can be understood in the sense of "in different directions"), than the fact of being able, thanks to the properly used logos, to access what is true beyond words (dia logon, this time taking dia both in the sense of "by means of", in that of "all the way through" and in that of "one with another", but not in the sense of "in different directions") in the light of the good (hè tou agathou idea), which allows us to know not how to do this or that, but whether it is good for us, individually and collectively, to do this rather than that, that is to say, not to try to understand what is (a question that means nothing), but what is good, with a look toward the future and capable of serving as a guiding principle (the archè anupoteton of Plato's Socrates) for our action, and not toward the past to try to know where we come from, what is our "origin" (the archè as understood by geometers and other "scientists"), a question that we can never answer concerning facts that no one has witnessed and that we cannot change, while we can influence the future to make it evolve towards the better, in short, we can be interested in the value (ousia) of things with regard to the good (to agathon) which is "beyond ousia" (509b8-9) precisely because it is what the ousia of each "being" is measured by. And we cannot do this work alone since, destined by nature to live in society (which has allowed us to develop the logos through dialegesthai), we cannot avoid taking others into account, if only to minimize their power to harm our individual lives, and must therefore look for a common "good" in the sharing of experience that interpersonal dialegesthai allows, using it in search of the best for the "city", that is, for the community of life that we form with those who live around us and are able to influence our lives, for better or for worse. In short, as I have already said, the difference between the two approaches is not in the way we reason, but in the way we ask ourselves questions, in the target we set for our investigations and in the greater or lesser confidence we have in the words on which we base them (their "supports" (hupotheseis)), words that have no meaning in themselves, and only take on meaning through the eidè that each one associates with them and between which he establishes relationships by "intertwining" them (cf. Sophist, 259e4-6) in logoi in which the words are only put one after the other in order to try to account for one's experience of the "beings" that activate our senses and our mind / intelligence (noûs) and to confront it with that of the others to progress together towards the truth about the good. (<==)

(56) "A long-term work" translates the Greek suchnon ergon. Ergon refers to action as opposed to inaction, but also with regard to logos, when, for example, we say that we must translate our beliefs "into words and deeds". Glaucon uses it here to talk about an activity that Socrates has just described as that of the logos. Glaucon's qualifier for this activity, suchnon, means "long" in a temporal sense, or "many, great, difficult." (<==)

(57) "Declare" translates the verb diorizein, formed by addition of the prefix dia- (like is the case with dianoein, or dialegesthai, already encountered in the words of Socrates) to the verb horizein, derived from horos, "limit" (from which the English word "horizon" comes), which means "to limit, bound", and also "to delimit", and from there, "to define" (i.e., "to bound the meaning"). Diorozein may mean "to draw a limit through", or "to distinguish" (and therefore, in a way, still "to define", by separating words with similar meanings), and also "to determine, declare", and, when constructed with an infinitive as complement, as is the case here, where the complement is saphesteron einai (to be clearer), "declare [one] to be so and so".
Glaucon seems to have understood that Socrates was trying to "distinguish" (one possible translation of diorizein) from one another related concepts such as epistèmè ("science, knowledge") and technè ("art, technical knowledge"), or even noûs ("intelligence"), dianoia ("reflection") and doxa ("opinion"), but what his use of this verb betrays is that he stays at the level of words and would have liked Socrates to give more structured "definitions" of them, whereas, as usual, the later tries to make the concepts, and not the words, understood in the interplay of relationships they have with one another. And this is what Glaucon's reformulation reflects, who cannot extract "definitions" of these words from the words of Socrates, and can only try to reproduce these interplay of relations by summarizing them. (<==)

(58) "Clearer" translates the Greek saphesteron, a comparative of the adjective saphès, which means "clear, plain, distinct, manifest, sure", from which the noun saphèneia ("clearness, distinctness") is derived, which Socrates had used at the beginning of the analogy, at 509d9, to make it the criterion for distinguishing the different sub-segments into which he asked to divide the two segments of the seen and the perceived by intelligence (cf. note 12). Glaucon is therefore a good student in reformulating the distinction made by Socrates between the two sub-segments of the perceived by intelligence in terms of saphenia and thus assuming that Socrates considers the apprehension linked to the approach characteristic of the second sub-segment clearer than that linked to the approach characteristic of the first sub-segment. But by retaining the concept of "clarity" which refers in its primary meaning to sight, rather than that of "truth" (aletheia) introduced by Socrates at 508a8-10 (cf. note 19), which is more appropriate to the register of the intelligible, and even if one of the derived meanings of saphes is "accurate, true", he betrays his still important anchoring in the sensible. (<==)

(59) "Among what is and also [is] intelligible" translates the Greek tou ontos te kai noètou. Glaucon does not say "of what is intelligible", which would be in Greek tou noètou, or tou noètou ontos (word for word "of_the intelligible being"), but does distinguish two "attributes" of what he is talking about, the fact of "being" (ontos) without further precision, and the fact of being "intelligible" (noètou) separated by te kai (and also"), thus leaving open the possibility of one being, without being intelligible. He thus introduces a problematic of "existence" that is not in Socrates' words in the analogy: he speaks of "domains" (topos), "families" (genos), "appearances" (eidos), and "seen" (horômenon) or "visible" (horaton), "perceived by intelligence" (nooumenon) or "intelligible" (noèton), but without associating the verb einai ("to be"), and without speaking of "beings" (onta) and even less of "being" (on) by substantizing the participle (to on, which leads to the genitive tou ontos) and using it without explicit attributes.To find such a use in Socrates' words, we must go back to 508d4-9 in the parallel between the good and the sun, when he says that "when what truth lights from above and that which is, it relies upon this, it conceives and gets to know this and appears to have intelligence; but when it’s upon what is diluted in darkness, what becomes and perishes, it produces opinions and sees dimly, turning those opinions up and down and then it seems not to have intelligence" (Hotan men hou katalampei aletheia te kai to on eis touto apereisètai enoèsen te kai egnô auto kai noun echein phainetai hotan de eis to tôi skotôi kekramenon to gignomenon te kai apollumenon doxazei te kai ambluôttei anô kai katô tas doxas metaballon kai eoiken au noun ouk echonti; on this text, see note 86 to my translation of the parallel between the good and the sun). And in this line, by "to on", associated with truth (aletheia), Socrates designates that which serves as a regulator of the logos by determining its truth (if it accounts for "what is", in the sense of "what is the case") or falsity (if it states "what is not (the case)"), and is opposed to "what becomes and perishes" (to gignomenon te kai apollumenon), which designates what the senses allow us to apprehend of what solicits them, which, as a pragma ("activity" (on our senses), fact, thing") "is" indeed something, a "being" (on). It may be that Glaucon is trying to shine in Socrates' eyes here, but it is questionable whether he understood his words correctly, because, for Socrates, everything that "is" is intelligible, but not everything that is is always perceptible by the senses, and therefore in particular "visible". Glaucon's way of rephrasing suggests, or at least does not exclude, that, in his mind, some of the "beings" are intelligible and others are visible, but that they are not the same, which, as we have seen, is totally contrary to what Socrates is trying to make us understand. (<==)

(60) "What is observed" translates the Greek to theôroumenon, present passive neuter participle substantized of the verb theôrein, derived from theôros, whose primary meaning is "spectator" and from there, "official deputies" sent to represent a city at public games, and by extension, to consult an oracle or make an offering to the god, then finally, to serve as an ambassador (a meaning quite similar to that of the word "observer" used nowadays to refer, for instance, to UN envoys in this or that country faced with internal difficulties). Theôrein is therefore in the primary sense "to attend games, or a religious festival". In the rest of this line, Glaucon uses twice a verb of similar form and meaning, but of distinct origin, theasthai, which I have translated as "to consider," and which derives from thea, "sight, spectacle, contemplation," from which the word theatron, "theater," also comes. Hoi theômenoi, which is found a little further on, and which I have translated as "those who consider," generally means "the spectators" (word for word, "the contemplating"). The discussion on the difference between science and opinion that closes Book V begins with an attempt to distinguish the philosophos from the philotheamôn, that is, from the lover of all kinds of spectacles (who will become a philodoxos, "friend of opinion") at the end of this discussion. The word theamôn is also derived from thea and designates one who contemplates, who observes. "To consider" is also one of the possible meanings of theôrein, but, to distinguish it from theasthai used in the same sentence, I have preferred the translation here as "to observe". It is from this idea of "contemplation" by the eyes of the mind that stems the meaning of the word theôria which led to "theory" in English (alongside other meanings of this word more in line with the original meaning of theôros). In a different register than with idein, we are again faced with words refering to "seeing" used analogically to talk about the activities of the mind. (<==)

(61) "under the guidance of the science of dialegesthai" translates the Greek hupo tès tou dialegesthai epistèmes (for the reasons that lead me not to translate the word dialegesthai, see note 48). If I translate epistèmè here as "science" and not as "knowledge", it is because "knowledge" is appropriate to translate epistèmè when this word is used in the absolute, to speak of "knowledge" without further precision, whereas "science" is appropriate when what is being considered is a particular epistèmè or the Greek word is used in the plural since the English word "knowledge" doesn't have a plural, as is precisely the case here where Glaucon speaks of a particular epistèmè that he qualifies as hè tou dialegesthai epistèmè ("the science of dialegesthai"). And this is precisely the key problem, because Socrates spoke of hè tou dialegesthai dunamis (in the dative tèi tou dialegesthai dunamei), that is to say "the power of dialegesthai", not of an epistèmè, of a "science" of dialegesthai. And this shift in vocabulary on Glaucon's part is not neutral, because it is also one of the characteristics of the approach linked to the second sub-segment of the perceived by intelligence as conceived by Socrates, to aim at a global and coherent knowledge (epistèmè) in the light of the unifying principle that the good is and not at "items of knowledge / sciences" (epistèmai) fragmented by domain and not necessarily coherent with one another. The fact that such "knowledge" is extremely difficult to acquire, or even inaccessible to human beings, should not lead us to play with words to give us the satisfaction of saying that we have gained access to "knowledge" by degrading this notion, rather than admitting, like Socrates, that we "know" nothing in the sense given to the word "knowledge" (epistèmè), even if, like him, we know how to prove theorems of geometry and possess other particular and therefore fragmentary "knowledge" (epistèmai), of which he will tell us shortly (in the recalling of the analogy of the line towards the end of Book VII) that it is by abuse of language and habit that they are called epistèmai and that dianoiai would suit them better, when he speaks of "the arts (technai) we have been reviewing (arithmetic, geometry, stereometry, astronomy, harmony), which we often called “knowledge / sciences” (epistèmai) out of habit, but would require another name, connoting more clearness than “opinion” (doxa), but more obscurity than “knowledge” (epistèmè) and adds "(discursive / wandering) thought” (dianoia), I think, [is how] we defined it earlier" (533d4-7). By this shift, Glaucon betrays in a way the state of mind of a good part of the Athenian youth in Socrates' time, seduced by the sophists and other professors of rhetoric: what they are looking for is indeed a "power", a dunamis, and this, in the most political sense of the term, and, in order to conquer it, they want to be taught a technè, or rather an epistemè, a more "noble" term (two words that we find, in our section, in the mouth of Glaucon, but not in that of Socrates), in short "recipes" to impose on the people and rise to the highest places. But this is not at all what Socrates proposes to them, at least not in the way they imagine, and the "power of dialegesthai" of which he speaks has nothing to do with any "oratorical technique", an "art of dialogue", which would ensure the domination of one interlocutor over the others. And this is another reason not to translate dialegesthai as "dialectic", which, nowadays, suggests too much of a specific technique (for instance, in the case of Plato, the method by successive divisions implemented by the Elean Stranger in the Sophist, which some would like to make the nec plus ultra of Platonic "dialectics"), of which we do not always know what it consists of (at least when we attribute it to Plato)... It may well be that, more than the confrontation of the interlocutors, it is the clash of words with each other, and the web of relationships that is thus woven through the words, whether they are pronounced or simply "thought", by the same person or by several interlocutors, and, beyond the words, between what they are trying to designate, that makes those who are able to see clearly in this "web" progress. But learning to see clearly and to "unveil" the true (alèthès) in the labyrinth of the logos has little to do with learning a "recipe" to better convince, as Socrates suggests in the Gorgias, when he compares the rhetoric that Gorgias claims to teach to cooking, the former being to the soul what the latter is to the body (Gorgias, 463a-465e), and reproaches him for not caring about the truth (Gorgias, 454c-455a... (<==)

(62) "Arts" translates the Greek technôn (plural genitive of technè), a term already used by Glaucon in his previous answer (cf. note 46), but which, again, Socrates did not use. It seems that Glaucon is trying to play the good student by suggesting an opposition between technè and epistemè that Socrates is careful not to endorse, at least for the time being. (<==)

(63) "Are constrained / constrain themselves": here we find the form anagkazontai which echoes Socrates' anagkazetai at 510b5, the main verb of the synthetic description of the division of the segment of the perceived by intelligence, commented at length in note 21, which had the soul as its subject and about which I had then pointed out that it could be read either as a middle or as a passive. This is again the case here, and that is why I propose the two options in my translation, but it is likely that, for Glaucon, in the context of his sentence, it is a passive that he has in mind: it is the fact that those he is talking about are interested in abstract realities such as the square itself that compels them to examine them through reflection rather than through the senses. Socrates' wording was more ambiguous, since the verb anagkazetai appeared in the set of words psuchè zètein anagkazetai ("a soul is constrained / constrains itself to investigate"), which left open the possibility that it was the soul itself that set methodological constraints in the way it conducted its research and not the nature of the realities studied that imposed them on it. (<==)

(64) The word used by Glaucon, which I translate as "thought," is dianoia, a word that Socrates used at 511a1 to talk about mathematical concepts such as the square itself, "which cannot be seen otherwise than by thought." If the translation of dianoia by "thought" did not pose any problems then, any more than that of dianoeisthai, the verb of the same root, by "to think" at 510d6 (cf. note 36), it becomes more problematic here insofar as it makes us lose sight of the root community between noèton (translated as "intelligible"), dianoia (translated as "thought") and noûs (translated as "intelligence") in a sentence in which it seems that Glaucon is playing on this community of roots to reformulate what he understood of Socrates' thought about the noèton by establishing a gradation between dianoia and noûs and by associating the first with the first sub-segment of the noèton, which Socrates had not done, and the second with the second sub-segment, taking care therefore to use for both a term derived from the root of noûs, which is also that of noèton. In any case, it is not certain that Socrates and Glaucon give the same meaning to this word among the multiplicity of meanings it can have and it is therefore prudent here to translate it by a word that is as open as possible in English, but the same in both cases, so that the English reader sees that it is the same word that is used by both interlocutors. (<==)

(65) "They investigate" translates the Greek skopein, yet another verb whose primary meaning relates to vision (it is the verb from which the English suffix "-scope", that is found in all words describing devices intended to see or "observe", is derived, as opposed to "-graphe", derived from graphein, which suggests writing or recording). (<==)

(66) "Intelligence" which is opposed here to "thought" (dianoia) is noûn, accusative of nous, which is found, as I pointed out in note 64, at the root of dianoia. It should be noted that this word, although it is at the root of several words used by Socrates in the analogy of the line (noèton, nooumenon, dianoeisthai, dianoia), is never used by Socrates, probably because of its ambiguity, pointed at in note 36.
Noûn ischein, "to possess intelligence" suggests an idea of stability and of a goal reached, whereas dianoia evokes rather a process being followed, a "progression" (dia in the sense of "through"), and therefore a search that has not yet reached its goal. (<==)

(67) It is deliberately that I have not split Glaucon's long convoluted sentence when translating it into English, any more than I have split Socrates' long sentences in the previous lines (one sentence per line, or almost, since he began to describe the noèton), because I think that the oratorical style that Plato lends to each of his characters is one of the "clues" he offers us to help us perceive what is going on in the dialogues. Socrates' sentences, in fact, were long, but quite easy to analyze and grammatically correct. Glaucon's, on the other hand, poses more problems, and he seems to have lost the thread of his initial construction before reaching its completion. What is well conceived is clearly stated, it is said. Comparing the greater or lesser grammatical rigour of each of the interlocutors also provides indications about the clarity of their ideas. It is all the more regrettable that some translators, in order to show their mastery of style, and under the guise of "facilitating" our understanding, "torpedo" what Plato so masterfully orchestrated. (<==)

(68) "The state of mind" translates the Greek hexin. The word hexis is the substantive derived from the verb echein, "to possess, to have", of which ischein, from the same root, which we have just found in the set of words noun ischein (cf. note 66), is a variant. Hexis is therefore in the primary sense possession, from which the meaning of "state of being" derives, which is the result of the "habits" that one "possesses", and therefore also the meaning of "habit, state of mind" (it can also be noted that "habit" derives from the Latin habitus, itself derived from the verb habere ("to have"), the Latin equivalent of the Greek verb echein, via the supin habitum, which makes hexis the exact Greek equivalent of the Latin habitus, with a very similar range of meanings). It is a word that Socrates did not use in the analogy of the line proper, but shortly before, in 509a5, in a expression, tèn tou agathou hexin, which I had translated as "the possession of the good", that is, in a somewhat different sense (on this expression and its translation, see note 97 to my translation in the previous section, entitled "The Good and the Sun"). (<==)

(69) "Of those dealing with geometry" translates the Greek tôn geômetrikôn. Geômetrikos is not a noun, but an adjective meaning "which concerns geometry". Hoi geômetrikoi is therefore word for word "the geometrics", that is to say something like those whom Pascal would describe as having "the mathematical / geometrical mind". There exists in Greek a name from the same root, geômetrès, but it is likely that, for Glaucon, it evokes more the "surveyors", i.e. those who make practical use of "geometry", the science of measuring portions of the earth (see note 46), than the practitioners of the more theoretical geometry to whom he thinks Socrates refers. (<==)

(70) "Opinion" translates the word doxa, a word that Socrates did not use in the analogy of the line proper, but shortly before, again, as for ontos at the beginning of this sentence, in his line at 508d4-9 (cf. note 59), where he uses both the verb doxazein and the noun doxa to characterize the result of the efforts of the soul interested only in to gignomenon te kai apollumenon ("what becomes and perishes"). It was in this same line that Socrates contrasted the soul incapable of arriving at anything other than opinions with the soul which, being interested in to on ("what is") in the light of the good (to agathon), ended up appearing noun echein ("to have intelligence"). Glaucon thus tries to make coherent the general words of Socrates in this reply and what he has just said here about the two modes of apprehension associated with the segment of the noèton ("intelligible"), in order to specialize (and thus to delimit, cf. the initial diorizein and note 57) some of the words used by Socrates to designate one or the other of the approaches he described, which Socrates has been careful not to do until now, preferring to make concepts understood rather than to rule upon words by forcing technical meanings on them. (<==)

(71) We have seen, as Glaucon's reply unfolded, which words he had taken from Socrates and which he had introduced on his own initiative. Having reached the end of his reformulation, it is interesting to take an overview of it and to investigate the more or less faithful way in which Glaucon reformulates Socrates' words and seeks to specialize specific words to designate the different approaches he has just described.
Let us first note that Glaucon has tried to characterize the two specific operations of each one of the two subsegments of the noèton, by means of words derived from the same root as noèton, dianoia for the first sub-segment and noûs for the second, but that in this he betrays Socrates doubly, on the one hand because the latter did not make dianoia a specific operation of the first subsegment, noting only that mathematical concepts such as the square itself were visible only through dianoia, unless we suppose that dianoia, a word that, in the context of his line, I have simply translated as "thought", refers only to such concepts (but which ones exactly? by what would they be characterized? and what would be outside the scope of dianoia within the noèton?) and that these concepts are the exclusive "population" of the first sub-segment, which I have abundantly implied was not the intention of Socrates, who did not seek to characterize each sub-segment by different "populations", but by specific modes of apprehension, and on the other hand because he never used in the analogy the word noûs, which he probably considered too ambiguous, since it can just as easily designate a "faculty" of man, almost an "organ" giving him this faculty, than the good use that can be made of it, or, according to the meaning that Glaucon undoubtedly gives it here, the result of this use, "intelligence" in the sense of "understanding", of "the fact of understanding" a given situation, fact or reality. In fact, at this point, Socrates had not yet sought to give a name to each of the two approaches that he described as specific to each of the two sub-segments of the perceived by intelligence, any more than he had given a name to the operations specific to the two sub-segments of the seen.
But if Glaucon has tried to remain in the register of noûs ("intelligence, mind ") in order to reformulate Socrates' remarks on the specific operations of this segment, he has just as conscientiously avoided the words of the family of horan ("to see"), idein, which serves as his aorist, and eidenai, another related verb meaning "to see", which probably referred to the segment of the horaton for him. Even in the choice of verbs he used to describe the activities of those he was talking about, he preferred the verbs theasthai ("to consider"), theorein ("to observe"), and skopein ("to investigate"). And above all, he did not use the word eidos, even though it was used several times by Socrates and plays a major role in his explanations, probably because he found it too ambiguous (while he did not find noûs ambiguous to speak of the noèton segment), or too connoted by what he associates with the first segment, or simply because he had not understood what Socrates meant by speaking of both horômena eidè ("visible appearances ") and noèta eidè ("intelligible appearances"), or because he had deduced that, since eidè can be both "visible" and "intelligible", they do not play a discriminating role in all these explanations and distinctions, which is both true and false: true if we take eidos in its most general sense, including both horômena eidè ("visible appearances") and noèta eidè ("intelligible appearances"); false if we admit that Socrates, in his explanations, changes the meaning of this word to limit it, in the intelligible, to noèta eidè ("intelligible appearances") and makes the ability to understand this and the role that these eidè play in relation to words the condition for accessing the second sub-segment.
In the continuity of this refusal to mix any reference to the visible with its reformulation, Glaucon has completely obscured Socrates' reference to eikones ("images"), which the latter nevertheless made one of the discriminating elements between the two approaches, the one associated with the first sub-segment could not do without it while the one specific to the second sub-segment did without it. As a result, his reformulation almost reverses Socrates' words because, by having placed too much emphasis on the fact that Socrates admitted that geometers and other proponents of the approach specific to the first sub-segment realized that realities such as the square on which they reasoned were only perceptible through dianoia, to the point of making it the proper operation of the first sub-segment, starting from the idea that dianoia was opposed to grasping by the senses (aisthesesin, 511c8), he went so far as to say that these people consider these through thought rather than through senses", whereas Socrates said on the contrary that, even if they realized that what the conclusions they arrived at were true about were not the material "images" they contemplated of them, they could not do without visible "images" to carry out their reasoning.
On the other hand, there is one aspect of Socrates' explanations that Glaucon seems to have perceived well, and that is the discriminating role between the two approaches played by the fact of going back or not to an archè ("(leading) principle"), even if nothing in his words allows us to figure out whether he understood to which "principle" Socrates was implicitly referring when he spoke of the anupotheton ("support of nothing") "principle", a term (anupotheton) that Glaucon does not use, probably precisely because he did not understand what it meant and what it was aiming at. He has retained that the full and complete intelligibility of the noèta presupposes a "principle" of intelligibility, without it being known whether, for him, this "principle" is the same in all cases, for lack of knowing what meaning he gives to the word archè and whether he gives it the same meaning as Socrates.
But he makes no reference to the logos, either to clarify how the division of the segment of the perceived by intelligence is made ana ton auton logon ("according to the same logos") as that of the segment of the seen or to make it what the proper use of conditions the approach associated with the second sub-segment of the perceived by intelligence, as Socrates does at the beginning of the previous line, and on the other hand he transforms the "power (dunamis) of dialegesthai" to which Socrates referred into a "science (epistèmè) of dialegesthai", which completely distorts his words and may lead to a different understanding of dialegesthai, giving it a more "technical" character and suggesting that Socrates would have in mind a more efficient kind of "rhetoric" (science of discourse, that is to say, of the logos), capable of giving us access to a clearer (saphesteron) understanding of the world around us. For Glaucon has well remembered that at the beginning of his analogy, Socrates had announced that the division into subsegments would be done in a perspective of greater and greater clarity (sapheneia), and his reformulation aims to show how, according to Socrates as he understood it, one of the two approaches leads to a clearer apprehension (saphesteron, 511c4) of the intelligibles than the other. Perhaps he even sees in this greater or lesser clarity the logon that presides over the division into sub-segments, which would explain why he does not try to look for it elsewhere. Unfortunately for him, "clarity" is something that varies continuously and is therefore not suitable to serve as a discriminating criterion for a clear-cut division.
In short, Glaucon probably did not understand all the subtleties of Socrates' explanations. As a zealous "student", he does not hesitate to correct some of Socrates' remarks (as when he speaks of the "science" (epistèmè) rather than of the "power" (dunamis) of dialegesthai, and seems more concerned with giving appropriate names to the notions of which Socrates speaks, in particular by trying to give a name to each of the two approaches of the intelligible (noèton) that he evokes, and this, by using only terms that have a kinship with noûs ("intelligence"), the word at the root of noèton ("intelligible"), even if it means introducing words that Socrates had not used, such as noûs itself, and eliminating all the words used by him that refer directly or indirectly to sight, such as eikones ("images") and especially eidè ("appearances"). And even when he uses terms used by Socrates, such as archè ("(leading) principle, beginning"), it is not certain that he understands them in the same sense as Socrates. Certainly, he anticipates something that Socrates will take up again towards the end of Book VII where he will specify what he means by dialektikè and make it the crowning achievement of the studies of the future philosophos, when, speaking of the subjects of study reviewed previously (arithmetic, geometry, etc.), he will refer to them as "the arts (technai) we have been reviewing, which we often called “knowledge / sciences” (epistèmai) out of habit, but would require another name, connoting more clearness than “opinion” (doxa), but more obscurity than “knowledge" (epistèmè) before adding, referring both to Glaucon's words here and to his words that immediately follow, "dianoia, I think, [is how] we defined it earlier" (one can also wonder if Glaucon's words here did not play a role in the fact that Socrates chooses to call, as Glaucon does here, dianoia the pathèma ("affection", cf. note 74) which he associates with the first sub-segment of the perceived by intelligence). But the fact remains that, as shown by his use of epistèmè to speak of dialegesthai, it is clear that, as I pointed out in note 61, Glaucon does not give the same meaning as Socrates to the word epistèmè: where Socrates refers through this word to "knowledge" conceived as a coherent whole, Glaucon understands it as a specialized "science" of which there can be several instances depending on the field concerned, and of which dialektikè would be only one instance among others. In short, one may wonder if his way of correcting Socrates' vocabulary is the mark of a great freedom with regard to words, showing that he is more attached to eidè than to words, which is doubtful when one sees that he did not mention eidè to reformulate remarks in which they played a major role, or if, on the contrary, because he does not really understand what Socrates is trying to say, he reduces it to words that he thinks Socrates uses in the sense in which he, Glaucon, understands them. (<==)

(72) I translate the Greek apedexô as "you have followed", in the sens in which we say that a student "follows" a class, because I think that the choice of the verb used by Socrates is somewhat ironic, which would not be perceived by a more classical "you have understood", another possible translation, especially when the translation of the adverb in the superlative ikanôtata by "most adequately" or something equivalent gives Socrates' reply the air of a most favorable compliment, while this ikanôtata in Socrates' mouth could well be nothing more than a backlash in response to what could only be a fit of false modesty of a Glaucon beginning his pedantic summary by an ikanôs men ou, "definitely not sufficiently", to qualify his understanding of Socrates' explanations.
Indeed, Socrates is not using here one of the verbs he had used before, such as katanoein (see note 14) or manthanein (see note 21), the first which implies apprehension by the noûs (of which Glaucon had his mouth full), the second which implies the result of a learning process, but the verb apodechesthai, whose primary meaning is "to receive". Figuratively, the verb also means "to receive favorably in one's mind, admit, approve" and finally "to understand", but it is clear that the emphasis is not on understanding, but on reception by an attentive student who "follows" what the teacher says, with the implicit idea that if one has "followed" the words of the teacher and "received" (heard) them, one has properly understood them. It would remain to be demonstrated that, because Glaucon did indeed "follow", "received", what Socrates said, he also properly "understood" it. But we have just seen in the previous note that this was probably not the case. The only point that could justify Socrates' sincere compliments is that Glaucon has properly identified what, in the end, as I have explained through the previous notes, constitutes the major difference between the two approaches, even more than the use or absence of use of eikones ("images"), which is nonetheless the common logon of the two division in sub-segment, or the reference to eidè, the fact of ascending or not ascending toward an archè. But even this point must be modulated by the fact that there is nothing allowing us to know whether Glaucon understands the word archè in the same way as Socrates, whether he understood what anupotheton (which he doesn't reuse) means, or, above all, whether he realized that this principle to which we had to go back was the good (to agathon) of which Socrates had made at the beginning of the analogy the "king" of the intelligible.
In short, contrary to the translators who see in these words of Socrates a sincere compliment to Glaucon (probably because they themselves have not understood Socrates' explanations), I see in them at best a half-compliment not devoid of a good dose of irony on the part of a Socrates who thinks that what Glaucon has understood is sufficient for the moment and that it is premature, given the young age of his interlocutor and most of his listeners, to want to go further in the explanations. (<==)

(73) "Take with me" translates the Greek moi... labe, in which moi is the dative of the personal pronoun ego ("me"), which has here sort of an expletive role, as "moi" in the French expression "regarde-moi ça", and labe is the active aorist imperative of the verb lambanein ("to take, to receive" in the literal and figurative sense), taken here in the same sense as at the beginning of the analogy, when Socrates says: "taking (labôn) a segmented line..." ». This is a somewhat specialized meaning, close to « assume » or "suppose", used when stating a problem. Through the use of this verb, Socrates reconnects with the geometric dressing of his analogy, which he had somewhat neglected in his last explanations. (<==)

(74) "Affections / states of mind produced in the soul" translates the Greek pathèmata en tèi psuchèi gignomena. The word pathèma (of which pathèmata is the plural) is derived from the verb paschein which means "to suffer" in the sense of " to have something happen to one, to be or come to be in a state or case, to be affected by", as opposed to poiein ("to do, to act") or prattein ("to accomplish, to act"), taking "to be affected by" in the most neutral sense possible, implying no idea of sadness, that is, not as in "he was affected by the death of his wife", but as in "the rate of the euro is affected by the economic situation in Greece", according to definition 1° of "affect" of the American Heritage Dictionnary online: "To have an influence on or effect a change in." Pathèma can refer to any kind of event that affects the body or soul, often, but not exclusively, in a bad way, illness, accident, affliction, etc., or else the state that results from it, the dispositions that it induces, hence my translation by "affection", which must be understood in the sense of "fact (for the soul) of being affected by something" or "result of the fact (for the soul) of being affected by something", taking "affect" in the sense I have just specified. This is meaning II.1. of the LSJ. The verb gignesthai, of which gignomena is the present participle passive in the neuter plural, adds the idea of "begetting", of "becoming". Another translation would be "the state of mind produced (or occurring) in the soul". What is certain is that Socrates is trying to suggest here a certain "passivity" (an English word having a certain root kinship with the Greek paschein, which gives pathein at the aorist, from which pathèma is derived, via the Latin patior, which makes passus in the past participle) of the soul in everything he is going to talk about. It is important to note this, because, if we have no difficulty in admitting that, in the order of the visible, our sight, and our senses in general, are from a certain point of view "passive" in relation to affections that come to them from the outside and that they "suffer", "are subject to", without soliciting them, the whole problem is precisely to figure out whether the same is true in the order of the noèton: are we "producers" of our thoughts, which would then be nothing more than creations of our mind without autonomous reality, or are they also induced by "impressions" coming from the "outside"? By putting the four pathèmata he is going to talk about on the same footing, Socrates takes a clear position on this point: our "mind", as much as our senses, is, if not totally "passive", at least stimulated, activated by "solicitations" from the "outside", and not only by the senses: just as the "visible" eidè are "appearances" for the sight of something external, the "intelligible" eidè are also "appearances" for the intelligence (noûs) of something "external", whether this something is also visible or not. Thus, to translate pathèmata as "faculties", as Jowett does, or by "processes", as Grube does, is quite simply to betray Plato by obscuring the part of passivity implied by the Greek word (Shorey translates by "affections", Cornford and Lee by "states (of mind)", which is much better, and Grube/Reeve and Reeve by "conditions", which is halfway in between, but rather on the wrong side since it doesn't imply either activity or passivity).
Moreover, Socrates' use of the word pathèmata to introduce the conclusion of his analogy confirms that what he had in mind throughout this analogy was indeed the modes of apprehension by us of the realities that surround us, both by our senses (for the realities capable of activating sight, as horata) and by our mind (for the realities capable of being perceived by the intelligence, as noèta), and not a classification of these realities supposed to affect only one or the other (sight or intelligence), and this, in two different ways in each case, again exclusive of each other, leading to four sub-segments partitioning the whole of reality into four disjointed subsets. In the preceding notes, with regard to the two sub-segments of the perceived by intelligence, I spoke of "approaches" rather than "affections," thus using a term which implies an activity of the individual and not a passive attitude, because what Socrates described in it were in fact modes of reasoning. By speaking here of pathèmata, he changes register and is no longer directly interested in intellectual processes, in the different ways of reasoning, but in the "states of mind" (one of the possible meanings of pathèma) from which they proceed, induced by external solicitations acting on sight (and more generally the senses) and intelligence (noûs) according to the greater or lesser attention we pay to each one and the degree of "reality" that we grant them. We must therefore not lose sight of the part of passivity that Socrates suggests by the choice of the word pathèma and remember that these "states of mind" are produced in us by the combination of "solicitations" (visual, tactile, auditory, intellectual, etc.) coming from external realities, of the constraints and limits of our organs of sensible perception (the senses) and intelligible perception (the noûs) and of the influence of the "environment" through which they reach us (such as the laws of light reflection when we see reflections, which allow us to see things where they are not, or the rules of language through which our thoughts are expressed).
Whatever the precise meaning to be given to pathèma here, to which we can return when we have examined in more detail the specific pathèmata to which Socrates refers, what is certain is that, if, as far as sight (and more generally the senses) are concerned, we have no difficulty in admitting that, as I have just said, we are passive in relation to the apprehension that it allows (that they allow), in the sense that we see as soon as our eyes are open, that we do not doubt the reality of what we see and that we assume that others see the same things as we do when they are next to us, as far as the "sight" of the mind is concerned, it is quite different, and it is only through the logos (inner or vocally expressed) and reasoning that we can make perceptible to ourselves and to others the fact that we are "thinking" about this or that reality, concrete or abstract. And it is the way in which we articulate discourse and reasoning that allows us to understand the degree of "knowledge" we have of the realities we talk about and the thoughts we develop and the way in which we situate ourselves in relation to them. Thus, when the geometers taken as an example by Socrates in his explanations of the first sub-segment of the perceived by intelligence reason on the square or diagonal, without seeking to "give a logos about them" (logon didonai, 510c7; cf. note 30) and contenting themselves with positing them as "supports" (hupotheseis), as "data", and hastening to carry out reasoning on them leading them to conclusions (teleutai) about them, even if they know that what they demonstrate is not, strictly speaking, true of the figures they draw but only of a reality that is accessible only to the dianoia, they implicitly show that they do not care about the actual nature of these realities, the square itself, the diagonal itself, and still less about the relationship that these realities may have with the good (to agathon), with what is good for them, but only about the arguments that they can carry out about them with the words they use through the images they produce of them. They are in a way like Narcissus, fascinated by his visible reflection (the first sub-segment of the seen) on the surface of a pool of water, who will never succeed in knowing himself by the simple blissful contemplation of his reflection and can only fall in love with his own visual appearance, his schèma, without being able to draw the slightest principle of action from it, without learning anything about his soul (psuchè), which is nevertheless his intimate being in the fullest sense, since it does not produce a visible reflection, and which will end up falling into the water and drowning, attracted by this purely visual appearance. But, you may ask, what other approach is available to the geometer interested in squares and diagonals, which could help him to know himself better rather than "drowning" in more or less sophisticated demonstrations? The answer is given by Socrates in the rest of the discussion, when, after the allegory of the cave, he undertakes a review of the disciplines suitable for the training of philosophers, explaining how they contribute to their formation: after having explained at 523a-524d how arithmetic can be "solicitor and awakener of intelligence" (noèseôs paracklètikon kai egertikon; 523d8-e1), he attributes as merit to geometry the fact that it deals with the "knowledge of what always is" (tou aei ontos gnôseôs; 527b5), an assertion which he does not elaborate, but which must be read in the light of what has been said in the analogy of the line about the square itself and the diagonal itself, which are the real subjects of the geometers' demonstrations.It is this understanding that Meno lacked in order to grasp what Socrates wanted him to understand through the properly understood experience with the slave, which focused precisely on the square and the diagonal, but took place in the context of a discussion on aretè ("excellence, perfection") of man. The avowed objective of this "experiment" by Plato's Socrates was to show Meno that it is possible to learn something that one does not already know, in order to ruin the paradox he had just stated according to which one does not need to seek what one already knows and it is useless to seek what one does not know since, not knowing it, one will be unable to recognize it if one comes across it. But the most important thing is elsewhere, in the reflections that this experience should have provoked in him, and that it should provoke in us: what it makes every reader grasp who can relive within oneself the memories of one's beginnings in geometry, without it being necessary to suppose that it really took place in the historical life of Socrates, which is obviously not the case, is that the truth of the theorem that Socrates makes the slave discover (the square double in area of a given square is the square built on the diagonal of the original square) is a truth that imposes itself upon any sane person capable of understanding the terms used to state it and of following the reasoning that leads to it; in other words, this truth is not the product of a human brain that would have been the first to state it, but a transcendent truth independent of the human mind, and therefore the square itself, the diagonal itself, realities that only the dianoia allows us to "see", about which it states a truth, are not mere products of the human mind, but have an immaterial transcendent reality "outside" time and space and possess properties that can be discovered (and not invented). And so their perception by the human mind is indeed a pathèma undergone by it in the same way as the sight of Socrates by his interlocutors, or the sight of his reflection in water by Narcissus, is a pathèma produced by a reality external to the subject affected by this pathèma. And the reflection should not stop there! For the next question is: if this is true of the square and the diagonal, is it also true of the other noemata ("intelligible") that my mind, my noûs, discerns? And so is it true of "just", "beautiful", "good", etc.? And is the aretè, the excellence of man, then what each individual wants to put behind these words or what the greatest number in a given place and time judge as such according to the uses and customs of one's "city", or is it the conformation to an idea/ideal of man that does not depend more on what Meno, I or most of them think than the answer to the question of the doubling of the square put to Meno's slave by Socrates depend on what he thought of it before the exchange with Socrates, on the opinion he had on the subject at the beginning of the experiment? This is the kind of approach that constitutes for Socrates an ascent back to the archè, the principle, and which allows us to have a correct idea of intelligible realities. In short, the problem is not to evaluate the respective value of two approaches that a geometer could use to solve problems of geometry, but to evaluate the respective value of two approaches in their way of asking questions about the understanding of the world around us and in what they reveal about the state of mind of the one who implements them under the effect of the solicitations one's mind / intelligence (noûs) is subjected to, one that wants to solve problems by aiming for multiple teleutai ("results") and the other that wants to understand this world as a whole in a coherent way by aiming at a single archè ("leading principle"). (<==)

(75) The four pathèmata mentioned by Socrates are, in the order in which he names them, "descending" from the second sub-segment of the perceived by intelligence to the first sub-segment of the seen, that is to say, in the reverse order of the one in which he described them in the analogy, noèsis, dianoia, pistis and eikasia. The diagram on the left relates each segment of the line to the pathèma that corresponds to it. Although this representation of the line does not correspond to the usual representation, the diagram respects the instructions given by Socrates in the division of the line since the segment of the nooumenon ("perceived by intelligence") is larger than that of the horômenon ("seen"), which means that they are indeed anisa ("unequal", 509d6), and the two segments are indeed further segmented "according to the same ratio" (ana ton auton logon, 509d7-8), in this case, equality (the two subsegments of each segment are equal to each other), which is therefore different from the ratio between the two original segments, which are unequal to each other. Moreover, I represent the line by a vertical line, and not a horizontal line as is most often done, to take into account the fact that the vocabulary used suggests a "vertical" hierarchy rather than a horizontal line, as I pointed out in note 45: Socrates says that noèsis must be brought epi tôi anôtatô, "upon the highest" segment, and not "on the first", even if he then speaks of the second (epi tôi deuterôi), the third (tôi tritôi) and the last (tôi teleutaiôi) without spatial indications. If he took the trouble to introduce this asymmetry into the formulation, by imposing a spatial position at the outset on the first of the sub-segments to which he successively refers at the very moment when he reverses the order, it is probably not without reason. Of course, one might think that Socrates uses anôtatô ("the highest") in the "figurative" sense to mark the primacy of noèsis over the other segments, but since his analogy is precisely intended to "figure", to make visible and representable, what he is talking about, why not use the "figurative" indications that he gives by the choice of his vocabulary to "illustrate" his words?! For everyone, the analogy of top and bottom is telling to "illustrate" a hierarchy of "nobility", while horizontality evokes rather an idea of equality. So, why not take it into account, as I do in this diagram? Finally, the part in red, which shows doxaston ("opined") and gnôston ("known"), seeks to illustrate the words of Socrates at 510a8-10, where he draws a parallel between the relationship between the opined and the known and that between the image and the original, which served to distinguish the segment he now associates with eikasia from the one he associates with pistis (cf. note 19). This representation does not mean that pisits is on the side of the known and eikasia on the side of the opined, but that the relationship between eikasia and pistis is analogous to the relationship between the opined and the known. This is the reason why the two groups of words are on lines orthogonal to each other, the dotted lines only inviting them to be related.
As far as the names that Socrates gives to each of the four pathèmata ("affections") are concerned, it must be borne in mind that he is proposing something new here for which he therefore has no pre-existing names. Creating new words to designate these new concepts would not have enlightened its listeners and the readers of the Republic. Plato therefore chose to have Socrates use pre-existing words, even if it meant forcing their meaning, so we must consider the names he gives to these pathèmata as partial insights into notions that have no pre-existing names and not absolutize them. He will give us an indication in this sense, as I have already pointed out, by changing one of these names, and not the least, since it is that of the pathèma associated with the upper segment, when, towards the end of Book VII, he recalls the analogy, not only by replacing noèsis by epistèmè ("knowledge"), but also by using noèsis to associate it with the set of the two sub-segments of the perceived by intelligence, therefore to epistèmè and dianoia taken together cf. 533e7-534a5). Moreover, it is in the absence of perfectly adequate specific names that he prefers to let the readers make up their own mind about what he is talking about in their own words before proposing to them, in one go at the end, four names that must be understood by reference to what has been said before and to what will follow in the allegory of the cave which illustrates the dynamic of progression between these different pathèmata ("affections"), on which they shed a complementary light, and in relation to one another, and not be used thinking that their usual meaning, which is not unequivocal for each of them, adequately defines each of the pathèmata that Socrates had previously described. Rather, we must seek, from the possible meanings of each of these names, what may be the main idea that led Plato to choose this name rather than another to designate what he associates with each sub-segment of the line. So we can see how difficult it is for a translator to translate these terms. To help the reader to do this work, I propose in the rest of this note some light on the possible meanings of each of these words, taking them up again, not in the order in which Socrates introduces them here, but in the order in which he described the four segments to which he associates them, insofar as this is the order in which the progression in comprehension takes place and that it is therefore in relation to what has gone before that what follows is understood and not the other way around.
- Eikasia ("imagery" in my translation; or "perception of shadows" (Jowett); "picture-thinking or conjecture" (Shorey); "imagining" (Cornford); "illusion" (Lee); "imagination" (Bloom, Grube, Reeve); "imaging" (Grube/Reeve)), is a new term in the analogy, of which it is the only occurrence in all the dialogues (here and taken up twice in the reminder, at 533e7-534a8, of the pathèmata introduced here), and the multiplicity of translations that are given to it alone shows that its understanding is not self-evident. It is formed from the verb eikazein, "represent by an image or likeness, liken, compare, infer from comparison, form a conjecture, guess", derived, like eikôn ("image"), which has been mentioned in the analogy both in the register of the seen and in that of the perceived by intelligence, from the verb eoikenai which means "to be like, look like", and also "to seem" or "to beseem, befit", and it means "likeness, representation", "comparison" or "conjecture". My translation by "imagery" is not given by dictionaries, but it reproduces in English the derivation of eikasia from the verb eikazein ("to represent by an image"), itself related to eikôn ("image"). No English word can, in my opinion, convey the meaning that I think Plato gave here to eikasia, and, before justifying more precisely in what sense I take "imagery" by which I translate it, a little tour of its other uses in Plato and other authors can help us to clarify its meanings. It is a rare word, of which there are only two occurrences in the entire Greek corpus available on Perseus apart from the three occurrences in Plato's Republic, one in Xenophon (Memorabilia, III, 10, 1, where the word is put in the mouth of Socrates discussing with the painter Parrhasios the possibility of representing the feelings of the soul in a painting, considered as eikasia tôn horômenôn, "representation of [things] seen"), and one in Plutarch (an author several centuries later than Plato, who uses it in his life of Themistocles to refer to a "comparison" made by the latter before the king of the Persians Artaxerxes between the discourse of men and a tapestry). The LSJ gives a few other examples for the meaning "conjecture", one taken from the Hippocratic corpus, one from Plato's Spuria, one from Philo of Alexandria and one from Hierocles, a stoic philosopher of the IInd century AD. All this does not facilitate the precise understanding of the meaning that Plato wants to give it.
Since the word eikasia itself is rarely used, we can turn to the uses of the more frequent verb eikazein, of which eikasia is the quality noun (the abstract notion to which the verb refers). There are 23 occurrences of this verb in the 28 dialogues of my tetralogies:
- Alcibiades, 105c7: Socrates describes Alcibiades' ambitions by declaring that he would only agree to live if his name and power were known to all the earth and that he cares little for anyone except Cyrus and Xerxes, and he adds: "That therefore you have this hope I know well and I do not conjecture it" (hoti men oun echeis tautèn tèn elpida, eu oida kai ouk eikazô)
- Meno, 80c1, 4: (with anteikazein at c3, 6): Meno has just compared Socrates to the torpedo-fish that numbs its prey, to which Socrates replies: "I know why you have given an image of me! (gignôskô hou heneka me èikasas.) ", before specifying in response to Meno, "So that in return, I give an image of you! For I know this of all those who are beautiful, that they take pleasure in being portrayed in pictures. Indeed, it gives them an advantage: beautiful indeed, I suppose, are also the images of what is beautiful. But I won't make an image of you in return." (Hina se anteikasô. egô de touto oida peri pantôn tôn kalôn, hoti chairousin eikazomenoi--lusitelei gar autois: kalai gar oimai tôn kalôn kai hai eikones--all' ouk anteikasomai se.)
- Meno, 89e2, 2: Socrates doubts that aretè ("excellence" in man) is an epistèmè ("science / knowledge") because, he says, what is epistemè, is "teachable" (didakton) and there are teachers (didaskaloi) and students (mathètai) of that, the question then being to figure out whether there are teachers and students of aretè, because "if it is the opposite, if there are no teachers, nor students, should we conjecture well in conjecturing that it is not teachable?" (Oukoun tounantion au, hou mète didaskaloi mète mathètai eien, kalôs an auto eikazontes eikazoimen mè didakton einai;)
- Meno, 98b1, 3: In the discussion on the difference between right opinion (orthè doxa) and knowledge (epistèmè), Socrates declares: "And for sure too, I speak, not as knowing, but as conjecturing. But that they are two different things, right opinion and knowledge, I think I do not conjecture at all, but if there were anything left that I would say I know--and I would say that of very few things--, one for sure, and it's that one, I would admit among what I know" (Kai mèn kai egô hôs ouk eidôs legô, alla eikazôn: hoti de estin ti alloion orthè doxa kai epistèmè, ou panu moi dokô touto eikazein, all' eiper ti allo phaièn an eidenai--oliga d' an phaièn--hen d' oun kai touto ekeinôn theièn an hôn oida.)
- Symposium, 190a4: Aristophanes describes the original men of his myth, spherical, with four legs, four arms and two similar faces (prosôpa), and specifies: "the head for the two faces arranged in opposite directions [was] unique, with four ears, and [they had] two copies of the shameful parts, and all the rest as one can imagine from that" (kephalèn d' ep' amphoterois tois prosôpois enantiois keimenois mian, kai ôta tettara, kai aidoia duo, kai talla panta hôs apo toutôn an tis eikaseien.)
- Symposium, 216c6: Alcibiades praises Socrates using the image of the statues of Silenus, or the satyr Marsias; After a first development of these comparisons, he adds: "But listen to me again [telling you] how similar he is to those by which I have given an image of him and how astonishing is the power he has" (alla de emou akousate hôs homoios t' estin hois egô èikasa auton kai tèn dunamin hôs thaumasian echei)
- Phaedrus, 248a2: Socrates describes, in his second discourse, in the context of the comparison of the soul to a chariot carrying a coachman driving two horses, the procession of the gods that souls follow to the other side of the vault of heaven, where they can contemplate "justice itself (autèn dikaiosunèn), moderation (sôphrosunèn), knowledge (epistèmèn)" before speaking of the human souls who try to follow them, to say that "of the other souls, the one who best follows a god and has best imitated him raises the head of the coachman towards the outer place and is drawn with [the god] in the circular movement, disturbed by the tumult of the horses and observing with difficulty the [realities] that are [there]" (hai de allai psuchai, hè men arista theôi hepomenè kai eikasmenè huperèren eis ton exô topon tèn tou hèniochou kephalèn, kai sumperiènechthè tèn periphoran, thoruboumenè hupo tôn hippôn kai mogis kathorôsa ta onta)
- Phaedrus, 250b5: Later in the same discourse, Socrates describes the situation of the souls who have returned to earth and says: "Of justice indeed and moderation and other [goods] precious to the soul as many as they are, there is no brilliance in the similitudes of this world, but by means of organs lacking in precision, with difficulty, some of them in small number, by looking at their images, see the gender of what is reproduced in images" (dikaiosunès men oun kai sôphrosunès kai hosa alla timia psuchais ouk enesti pheggos ouden en tois tèide homoiômasin, alla di' amudrôn organôn mogis autôn kai oligoi epi tas eikonas iontes theôntai to tou eikasthentos genos)
- Republic, II, 377e1: Socrates blames poets like Homer and Hesiod for the lies found in their works, because one commits an ugly lie "when one gives a bad image by discourse, about gods and heroes, of what they are, like a painter painting [things] that bear no resemblance to those to which he would like to paint them similar" (hotan eikazèi tis kakôs tôi logôi peri theôn te kai hèrôôn hoion eisin, hôsper grapheus mèden eoikota graphôn hois an homoia boulèthèi grapsai)
- Republic, VI, 488a2, 5: Socrates announces that, in order to explain why philosophers should govern cities to put an end to their evils, he will have recourse to an "image" (eikôn), and he adds, in introduction to the analogy of the unmanned ship: "Come on! I say, you are laughing at me after having thrown me into a discussion where demonstration is so difficult. But listen to the image, so that you can see even better how relentlessly I can make images! For so painful is the fate of the most suitable at the hands of the cities, that such a fate is not reserved for anything else, but must be composed from several things in order to give an image of them and speak in their defence, just as painters paint goat-stags and that kind of things by making mixtures." (Eien, eipon: skôpteis embeblèkôs me eis logon houtô dusapodeikton; akoue d' oun tès eikonos, hin' eti mallon idèis hôs glischrôs eikazô. houtô gar chalepon to pathos tôn epieikestatôn, ho pros tas poleis peponthasin, hôste oud' estin hen ouden allo toiouton peponthos, alla dei ek pollôn auto sunagagein eikazonta kai apologoumenon huper autôn, hoion hoi graphès tragelaphous kai ta toiauta meignuntes graphousin.)
- Phaedo, 99e6: in the course of his intellectual autobiography, Socrates, in introduction to the presentation of his "second navigation", evokes people who burn their eyes by looking directly at the sun during an eclipse, rather than looking at an image of it by reflection and explains that he decided, rather than looking at things with his senses, to seek the truth by taking refuge in reasoning (eis tous logous kataphugonta), but he immediately adds: "perhaps, in fact, what I am giving an image of, in a certain sense, is not suitable, for I do not agree at all that he who examines beings in logoi examines them more in images than he [who examines them] in acts / facts" (isôs men oun hôi eikazô tropon tina ouk eoiken: ou gar panu sugchôrô ton en logois skopoumenon ta onta en eikosi mallon skopein è ton en ergois)
- Cratylus, 425c3: Socrates, after having proposed to Hermogenes to define names as an "imitation" (mimema) by the sound of what one seeks to name, then invites him to examine in what way the elementary constituents of words, letters and syllables, can be a correct imitation of what they serve to name by imitation and, as Herrmogenes declares himself incapable of doing so, Socrates nevertheless suggested that he embark on this attempt "by declaring beforehand, as a short time ago, to the gods, [we declared] that, knowing nothing of the truth, we imagine the opinions of men about them..." (proeipontes, hôsper oligon proteron tois theois, hoti ouden eidotes tès alètheias ta tôn anthrôpôn dogmata peri autôn eikazomen)
- Cratylus, 432b3: Socrates explains to Cratylus that the rectitude of an image (eikôn) does not consist in reproducing its original in all respects, "but on the contrary, it must absolutely not render everything as it is in what it pictures, if it is to be an image (alla to enantion oude to parapan deèi panta apodounai hoion estin hôi eikazei, ei mellei eikôn einai)", and he immediately gives the example of an image of Cratylus made by a god who would reproduce not only his appearance according to the figure (schèma) and color (chrôma), as a painter would do, but also the interior, the movement, the soul and the thought (phronèsin) down to the very last detail, which would lead to two Cratyles, and not to an original and an image
- Cratylus, 439a8: Socrates makes Cratylus admit that, if it is by their composition from letters and syllables that names give a correct image of what they designate, it was necessary that those who instituted names had some means of knowing the things designated by these names other than with the help of names, to which he adds: "If, then, it is possible to study things as completely as possible by means of names, but it is also possible by means of [things] themselves, which of the two will be the fairest and clearest study? From the image, to study it, [to see] if it is itself fairly similar, and the truth of what it was the image of, or from the truth, [to study] it and whether the image of it has been properly realized?" (ei oun esti men hoti malista di' onomatôn ta pragmata manthanein, esti de kai di' autôn, potera an eiè kalliôn kai saphestera hè mathèsis; ek tès eikonos manthanein autèn te autèn ei kalôs eikastai, kai tèn alètheian hès èn eikôn, è ek tès alètheias autèn te autèn kai tèn eikona autès ei prepontôs eirgastai;), to conclude that it is from the things themselves and not from the names that designate them that we must learn.
- Ion, 532c5: to Ion who wonders why he falls asleep when the conversation is not about Homer, but wakes up and has much to say as soon as Homer is mentioned, Socrates replies: "Not very difficult to conjecture, comrade, but it is clear to all that you are incapable of talking about Homer with any technical skill or knowledge" (ou chalepon touto ge eikasai, ô hetaire, alla panti dèlon hoti technèi kai epistèmèi peri Homèrou legein adunatos ei)
- Parmenides, 132d4, 6: in the discussion between Parmenides and Socrates, Socrates offers an explanation of the "participation" in eidè / ideai in these terms: "this is what most of all, to me at least, appears clearly as [the position] to have: that on the one hand these eidè stand as models in nature, that on the other hand the other [things] resemble them and are copies of them, and this participation by other [things] in the eidôn comes to be nothing other than being made in their likeness (malista emoige kataphainetai hôde echein: ta men eidè tauta hôsper paradeigmata hestanai en tèi phusei, ta de alla toutois eoikenai kai einai homoiômata, kai hè methexis hautè tois allois gignesthai tôn eidôn ouk allè tis è eikasthènai autois)", to which Parmenides replies: "If, then, anything resembles the eidos, is it possible that this eidos is not similar to that which is made in the likeness of it, according to what it is made similar with it?" (ei oun ti eoiken tôi eidei, hoion te ekeino to eidos mè homoion einai tôi eikasthenti, kath' hoson autôi aphômoiôthè;)
- Stateman, 260e3 (with pareikazein): in the context of his attempt to define the art of politics by the method of divisions, the Elean Stranger has come to place it in the category of those giving orders (epitaktikè) and then proposes to divide it by analogy with the way in which, in the Sophist, he had divided the art of commerce between merchants selling their own production (autopôlikè) and those who buy the production of others to resell it, which leads to a distinction between those who give orders of which they are the authors and those who, like the herald, are content to transmit orders given by others, and since there is no name for these categories, the Stranger proposes: "Or do you want us, insofar as we proceed by comparison at this very moment, to proceed in parallel by comparison also for the name" (è boulei, kathaper èikazomen nundè, kai tounoma pareikasômen) and suggests that the art of giving orders of which one is the author oneself should be called autepitaktikè)
- Philebus, 55e5: Socrates wonders what would remain of the manual arts and techniques (cheirotechnikai) if we took away from them the knowledge (epistèmai) that plays a leading role (hègemonikon) such as the art of calculating (arithmètikè), measuring (metrètikè) and weighing (statikè) and replies that there would not be much of value left, adding: "we would indeed be left [as the only possibilities] to make use of imagination with regard to them and to exercise our senses by experience and long practice using in addition the powers of the art of conjecture that most call "arts / techniques", [these] having built their strength through application and laborious effort" (to goun meta taut' eikazein leipoit' an kai tas aisthèseis katameletan empeiriai kai tini tribèi, tais tès stochastikès proschrômenous dunamesin has polloi technas eponomazousi, meletèi kai ponôi tèn rhômèn apeirgasmenas) (there are two words in this passage that evoke the idea of "conjecture", eikazein and stochastikè, the first of which can be translated in this context as "to make use of conjecture" and the second as "the art of conjecture", but they do so from two different starting points: eikazein starts from the idea of "image" (eikôn), of "resembling to" (eoikenai), while stokastikè starts from the notion of stochos, whose primary meaning is "target, goal" and the derived meaning "conjecture", the idea being that to conjecture is to try to find a path that leads to the goal one has set for oneself, without the certainty of reaching it; this is the reason why I have not translated eikazein here as "to conjecture", but as "to make use of imagination", reserving the vocabulary of conjecture for the translation of stochastikè).
These uses of eikazein can be classified into three main registers of meaning:
1. the problem of the relationship between an image and its model (7 occurrences): Phaedrus, 248a2 (the soul as imitating a god) and 250b5 (deficiency of the reflection of justice itself, moderation, etc. in human behavior); Republic, II, 377e1 (the false image of gods and heroes in poets); Cratyle, 432b3 (to be an image, the image cannot be a reproduction rigorously identical in all respects to its model) and 439a8 (is it better to start from the image or the original to determine how faithful the image is to its original?); Parmenides, 132d4, 6 (the participation of other things in the eidè would be a relation of image to model);
2. the notion of image used in an analogical way through comparisons, analogies, parallels, metaphors, allegories, etc. (7 occurrences): Meno, 80c1, 4 (Meno's comparison of Socrates with a torpedofish); Symposium, 216c6 (the comparison made by Alcibiades of Socrates with a Silenus or the satyr Marsias); Republic, VI, 488a2, 5 (the analogy of the unmanned ship); Phaedo, 99e6 (Socrates' comparison of various intellectual approaches with the different ways of observing an eclipse of the sun); Stateman, 260e3 (same dichotomy "activity concerning what is produced by oneself / activity concerning what is produced by others" in the context of commerce and in the context of the activity of giving orders)
3. the register of conjecture, which implies more or less a weakened form of analogy presupposing an implicit parallel between the present situation and the experience acquired in similar cases, and which is opposed to the register of knowledge, explicitly in the references followed by an asterisk (9 occurrences): Alcibiades, 105c7 (*); Meno, 89e2, 2 and 98b1, 3 (*); Symposium, 190a4 ; Cratylus, 425c3 (*) ; Ion, 532c5 ; Philebus, 55e5.
We could also add to this 55 occurrences of apeikazein (more frequent in dialogues than eikazein), a verb formed on eikazein by the addition of the prefix ap(o), which induces an idea of completion that hardly changes the general meaning of the verb, and 3 occurrences of apeikasia, the quality substantive derived from this verb, but this would not substantially change the results. I will content myself with mentioning two occurrences of apeikazein that are close to the text that concerns us, one in the analogy of the line itself, at 511a7, in the explanation of the first sub-segment of the perceived by intelligence, where Socrates speaks of people "using as images the very [things] that are copied by the ones below (eikosi de chrômenèn autois tois hupo tôn katô apeikastheisin)", and the other at the beginning of the allegory of the cave which immediately follows the analogy of the line, at 514a1, which Socrates introduces with these words: "liken to such an experience our nature regarding education and the fact of not being educated (apeikason toioutôi pathei tèn hèmeteran phusin paideias te peri kai apaideusias)". The first is in the register of the relationship between image and model, the second in the register of analogy.
What can be learned from this inventory is that eikazein, when used to speak of an intellectual activity (registers 2 and 3), and not of a relationship between realities (register 1), refers to an alternative to true knowledge or to the ability to reproduce this knowledge directly through words, but that, if this alternative is indeed further from the truth than knowledge properly so called (epistèmè), its use is not to be proscribed in all cases, at least for Plato, as evidenced by the intensive use he makes of comparisons, analogies, allegories and other myths in the dialogues. The important thing in such cases is to be aware that images are just that, images, and not to take the image, whatever it may be, for the reality that it only illustrates or tries to reproduce in the best possible way.
In the context of the analogy of the line, where eikasia is associated with the first sub-segment of the seen, the one that Socrates associates with the "images (eikones)", to designate the affection (pathèma) corresponding to the initial situation of the anthrôpoi ("human beings") in the early stages of their lives (which some may never overcome), that is to say to the situation of those who will be represented in the allegory of the cave by prisoners caught in bonds from their birth at the bottom of the cave without them being able to look at anything other than the shadows on the wall of the cave in front of them, that is to say, the images provided by sight, the main idea that this word seeks to impose is that of eikôn ("image"): whoever is in this situation lives and thinks in a world of images (the shadows in the cave, which represent all that sight offers us) of which he is not aware that they are only images, not reality itself, thinking that things are as sight shows them to us and that therefore, once we have seen something, we know it. It was in order to transpose into English this reference transmitted by the root of the word retained by Plato that I finally chose, after much hesitation, the word "imagery" to translate it, aware that none of the possible meanings of this word are exactly appropriate in this context. "Imagination", the translation used by Bloom, Grube and Reeve, "imagining", retained by Cornford, or "imaging", the word used by Grube/Reeve, have the merit of preserving the root kinship with "image", but reverses what Socrates has in mind since people who are in this state of mind have everything but imagination: they only believe what they see and therefore do not show any "imagination", "imaging" or "imagining"! And, no more than they show imagination, they do not "conjecture" (Shorey, who complement this word with the words "picture-thinking", which is ambiguous) since, for them, it goes without saying that things are as they see them; this is not a conjecture for them, but a certainty. Jowett's "translation" by "perception of shadows" is not a translation, but an interpretation based on the allegory of the cave. Lee's translation by "illusion" is the best of all these, since what characterizes this state of mind is indeed the "illusion" that what is seen is the real thing and not a mere image of it provided by sight, but has the defect of dropping the link with eikôn ("image"), a word central to the analogy and used several times in it. Besides, it puts too much emphasis on the judgment that can be made from the outside on this attitude of mind: those who are in it do not think they are victims of an illusion, for them, what they see is the only reality. This is also partly true of my translation as "imagery", since I emphasize that what characterizes these people is that they are not aware that what they see is only images. But I can at least allow myself with this translation of the example of Plato himself, who chose a word with a common root with eikôn, the Greek word meaning "image", as is the case in English between "image" and "imagery". In short, "imagery" suggests an environment made up of images, whether those who see them are aware of it or not, and this is what Socrates has in mind when he talks about eikasia.
- Pistis ("trust" in my translation and Bloom's; or "faith (or conviction)" (Jowett at 511e1); "belief" (Jowett and Grube at 534a1 and a5, Shorey, Cornford, Lee, Grube/Reeve, Reeve at 511e1 and 534a5); "opinion" (Grube at 511e1, Reeve at 534a1)) is also a new term in the analogy, which means "trust, faith, confidence inspired in others or inspired by others", hence "guarantee, assurance, pledge". This is the second occurrence of this word in the Republic, the first being in 505e2, in the section where Socrates states the anupotheton character (without, however, using this qualifier) of the "good" (to agathon): after declaring that if, when it comes to seeking "just or beautiful [things] (dikaia kai kala)", most people are satisfied with those that only look like them (ta dokounta), when it comes to good [things] (agatha), they look for those that are really good (ta onta) and not only those that look good, and in this case hold the mere opinion (doxa) in low esteem, he adds that these same people, concerning that in view of what they do all they do (the "good"), are "unable to grasp appropriately what in the world it is nor possess a stable confidence (pistei) about it". Apart from its reuse in the reformulation of the conclusions of the analogy of the line at 533e7-534a8, the word reappears only once in the Republic, in Book X, at 601e7, in a discussion, through the example of the flute, on the difference between the "knowledge (epistèmè)" that the user, in this case the flute player, has of it, and the "right belief" (pistis orthè) that the craftsman who makes it (poiètès) has about it through one's frequentation of flute players.
The main idea that this word conveys is that of "trust". But what "trust"? It is not so much confidence in the data of sight to, if not always present to us adequately what surrounds us, at least give us enough apprehension of it so that, with a few precautions to avoid falling into its traps, we can find our way around in most situations of daily life, but trust as the foundation of opinion. Indeed, as long as we remain in the register of the visible, and more generally of the sensible / material, and that we refuse to submit our opinions to the examination of intelligence (noûs) by leaving the cave, there are only two grounds for the apprehension of the world around us: direct sight or trust in the opinion of other people. All the "knowledge" we can have if we stay in the cave can only come from direct sight or from inherited opinions from those around us, whose choice is based on the trust we have in those whose opinion we adopt. In fact, since what we "know" from personal experience, that is, from having seen it (literally) ourselves, is very limited, trust is present practically from the start. But when we are still at the level of eikasia ("imagery"), not only are we not yet aware of the fact that sight only offers us "images" of what we see, but we have not yet become aware of the fact that most of the things we think we "know" or admit, we have never seen in person but "know" them by hearsay and that therefore the "knowledge" we have of them is based on the trust we have in our sources of information.
It should also be noted that the use of the word pistis to designate the pathèma associated with the second sub-segment of the seen can help us understand in what sense Socrates is speaking here of pathèmata: the pistis he is talking about here in connection with the seen is of course not the vision itself, in which the one who sees is passive, but the "state of mind" of those who admit that, in their apprehension of the world around them, sight alone allows them to apprehend only a very small part of this world, what enters into their field of vision in the place where they are at any given moment and that they must therefore trust others to broaden their field of "knowledge", from their words or their writings, that is to say, ultimately from their logoi.
- Dianoia ("(discursive / wandering) thought" in my translation; or "understanding" (Jowett, Shorey); "thinking" (Cornford); "reason" (Lee); "reasoning" (Grube); "thought" (Bloom, Grube/Reeve, Reeve)) is the only one of the four terms chosen by Socrates to designate the four pathèmata that has already been used in the analogy, both by Socrates (at 511a1) and by Glaucon (at 511c7, d2 and d5). But, even if Glaucon takes up a word previously used by Socrates and makes a big deal of it, it is not certain that he understands it in the sense in which Socrates used it. It is likely that, when Socrates says that geometers are aware of the fact that they reason about "what cannot be seen otherwise than by thought (dianoia)", he is taking dianoia in a very general sense, which is more or less "thought" in English: dianoia / thought is simply opposed to sight without prejudging in which way geometers arrive at this conclusion or the exact meaning they give this word (cf. note 43). the point is only to oppose what is seen to what is simply thought, whatever the modalities and content of this thought. In the primary sense, dianoia is a noun with the ending in -ia, which designates a simple quality in the abstract, a faculty as such, in our case, the simple faculty of thinking, without prejudging the use that is made of this faculty. But in a second step, the word comes to designate the use of this faculty, "reflection", thought expressing itself in discourse (logos), in particular inner discourse, as attested by the Elean Stranger defining, at Sophist, 263e3-8 dianoia as an inner dialogos of the soul with itself (more details on the different meanings of noûs and the words that derive from it, including dianoia, can be found in the Vocabulary section of this site (in French only), which reproduces for some of them the corresponding entry in the Bailly Greek-French dictionary and, in the entry devoted to noûs, the corresponding entry in Chantraine's Dictionnaire étymologique de la langue grecque, which examines noûs and all its derivatives). And we may think that when Glaucon seems to want to make dianoia the name of the specific approach of geometers and other "scientists", as opposed to noûs becoming for him the specific name of the approach that Socrates associates with the second sub-segment of the perceived by intelligence, it is this second meaning of dianoia, the one that refers to the use made of thought through reasoning that he has in mind and that he opposes it to noûs understood as a form of intuitive "intelligence" resulting from illuminations inexpressible through words and therefore beyond any form of logical reasoning. But Socrates corrects Glaucon by replacing noûs by noèsis for the pathèma ("affection") associated with the second subsegment of the perceived by intelligence, but retains dianoia for the pathèma associated with the first. And even if these names are chosen without conviction, since, as I have already pointed out, noèsis will be replaced at 533e7-534a8 by epistèmè ("knowledge") and reused in a more general sense covering the two segments of the perceived by intelligence, that is to say both what he calls here noèsis and what he calls dianoia, and that, moreover, at 533d4-7, he refers to his use of dianoia as the name of the pathèma associated with the first sub-segment of the perceived by intelligence in a sentence reminiscent of Glaucon's words making dianoia "something intermediate between opinion and intelligence", he does it using words, "dianoia, I think, [is how] we defined it earlier", which suggest that he might as well have used another word if Glaucon had not whispered this one to him, it remains interesting to understand why he can accept, even provisionally, these two terms as the name of the two pathèmata ("affections") associated with the two sub-segments of the perceived by intelligence. The replacement of noûs by noèsis indicates, as I have already hinted earlier in this note about this word, that what he has in mind is not a kind of enlightenment beyond words, or at least not only that, but an activity, as implied by the suffix -sis of noèsis. And his preservation of dianoia for the other pathèma, now accepting this word in the sense he clearly sees Glaucon has given it, referring also to the activity of thought, means that we now have two terms implying an activity of the mind (noûs). We must therefore look for what can distinguish these two activities. To do this, we can note that what differentiates these two words, and therefore undoubtedly provides the key we are looking for, is the dia- of dianoia, a prefix that we have already seen occurs in words loaded with meaning in Plato, such as dialegesthai, which I have given up translating (see note 36 on dianoeisthai, note 43 on the first occurrence in the analogy of dianoia, and note 48 on dialegesthai).To try to see more clearly what is at stakes, let's start once again from the various possible meanings of this prefix: when used in composition, the prefix dia-, whose primary meaning as a preposition is "through", can introduce an idea (1) of separation, in the sense, depending on the case, of division (as in di-airein, which means "to divide, separate", from the verb airein meaning "to grasp, to take in one's hands"), of distribution (here and there, as in diaphorein, which means "to disperse, to spread", from the verb phorein meaning "to carry"), of diversity (in a different way, variously, as in certain senses of daipherein, which can mean "to differ, to be different", from pherein meaning, as the previously mentioned phorein which derives from it, "to carry") or distributivity (one with the other or against each other, as precisely in dailegesthai, whose etymological meaning is "to speak with one another, to dialogue"), (2) penetration (through, as in diabainein, which means "to cross", from bainein meaning "to walk"), (3) superiority or (4) completion (through from one end to the other, that is to say, to the end, as in diaprattein, "to complete", from prattein meaning "to act"), it being understood that several of these nuances of meaning can be found in the same compound, leading to different meanings of the same word. The idea of completion, which is the one that is generally attributed to the dia- of the verb dianoeisthai and the noun dianoia derived from it, ends up weakening in common usage to give the compound meanings very close to those of the original verb (noein in this case): originally, dianoia refers to a thought brought to its conclusion, that is to say, a firm intention, a well-fixed resolution, but in practice, the word ends up designating a thought like any other, or simply intelligence, like noûs. In the end, we can think that what interests Plato here in the dia- of dianoia is the idea of the movement of thought through the successive stages of reasoning, but of a movement which, in the absence of a compass (the "guiding principle" that the other approach seeks), carries itself according to the problems being delt with in different directions ("from here, from there", one of the possible meanings of dia-) and therefore wanders and ultimately arrives at nothing coherent. And the absence of dia- in noèsis then rightly underlines that the activity of thought associated with the second sub-segment of the perceived by intelligence is coherent thanks to the light of the guiding principle previously identified (the idea of the good) and leads to a single apprehension of the real constituting a true knowledge (epistèmè, which will become the new name of this pathèma) and no longer to a Brownian movement of autonomous reasoning that leads to something that is not a mere opinion, but not yet "knowledge". The dia- that remains relevant for the approach associated with the second sub-segment of the perceived by intelligence is the dia- of dialegesthai, as Socrates sought to make us understand by suggesting that, in order to move from dianoia to noèsis / epistèmè, it is necessary to make use of the "power of dialegesthai", which Glaucon wants to make a "science (epistèmè)" among others, whereas what Socrates simply means is that, in order to progress toward knowledge, we must accept to "dialogue (dialegesthai)" with others, to confront our noetic perceptions with those of other people, just as it is by confronting our visual perceptions with others that we have been able to learn to speak: just as we end up realizing, over the years of childhood, that we see the same thing as others and learn to distinguish the things we see, to understand that they are external to us and to name them with the help of those around us, it is a fortiori by confronting our purely intelligible perceptions with those of those around us by means of dialogue (here in the sense of "one with the other") that we will have a chance to realize that our thoughts are not a pure creation of our mind and to better approach the reality of what arouses these perceptions in us through the ideai that are what is accessible to the human noûs ("intelligence"), the starry heaven of the allegory of the cave and the sun / good that enlightens it.
- Noèsis ("apprehension by intelligence" in my translation; or "reason" (Jowett at 511d8); "intellect" (Jowett and Grube/Reeve at 534a); "intellection or reason" (Shorey at 511d8); "intellection" (Shorey at 534a, Bloom); "intelligence" (Cornford, Lee at 511d8, Grube at 534a); "understanding" (Grube and Grube/Reeve at 511d8, Reeve)), is yet another new term in the analogy, and this is its first use in the Republic. It is formed on the verb noein using the suffix -sis, which refers to the action corresponding to the root verb. It is a rarer word than noûs, used by Glaucon, which has the advantage over it of not being ambiguous between a "faculty", the use we make of it and the result for us of a good use of it. As I have already said in the previous notes, noûs can in fact be used in the sense in which we say that "man is an intelligent animal", that is to say "endowed with a faculty called 'intelligence'" more or less developed according to the individual, or in the sense in which we say that "Einstein was very intelligent" to show that, in him, intelligence was greater than in others, or in the sense in which we say that "this girl made an intelligent decision in acting in such and such a way," to indicate that she has made good use of her faculty in specific circumstances. Noèsis is more specifically the apprehension by the noûs, considered as the "faculty" within us that gives us access to the noèta, themselves seen as external to our noûs, which "suffers / is affected by" the more or less distinct impressions of it (see previous note on pathèma), hence my translation by "apprehension by intelligence" rather than by "intelligence" itself, intended on the one hand to show that the word used by Socrates is not the one just used by Glaucon, and on the other hand, to remove the ambiguity that would remain with a more classical translation by "intelligence" when precisely noèsis used instead of noûs is supposed to remove it by emphasizing the activity that the noûs ("mind / intelligence") allows. By choosing this word to replace the one used by Glaucon, Socrates corrects him discreetly, without insisting, but, as I pointed out above, when he takes up this enumeration of the four pathèmata when recalling at 533e7-534a8 what was said here, noèsis will become the common name of the two pathèmata of the noèton ("intelligible") (and doxa ("opinion") the common name of the two pathèmata of the horaton), and it is epistèmè ("knowledge") that will take its place to designate what is here called noûs ("intelligence") by Glaucon and noèsis by Socrates, and this two-step correction is one more indication that Socrates' compliment to Glaucon was not devoid of irony, that he tried here not to push it too deep and that he corrects his approximations only gradually. In view of all that is at stake in these subtle shifts in vocabulary, it is therefore regrettable that most of the translators I have consulted translate both Glaucon's noûs and Socrates' noèsis at 511d8 by the same word (Jowett tranlates both by "reason", Cornford and Lee by "intelligence" (but Cornford translates noûs at 511d1 as "true understanding"), Grube, Grube/Reeve and Reeve by "understanding"; the exceptions are Shorey, who translates Glaucon's noûs at 511d1 as "true intelligence" and both occurrences at 511d4 as "reason", and Socrates' noèsis at 511d8 as "intellection or reason", and Bloom, who translates noûs as "intelligence" and noèsis as "intellection").
To conclude this note, we can note that the instruction given by Socrates at the beginning of the analogy to divide the two initial segments, that of the seen and that of the perceived by intelligence, ana ton auton logon ("according to the same logos") finds here a "graphic" representation through the spelling, and the morphology that underlies it, of the names given to the four pathèmata ("affections / states of mind"). Indeed, in each of the two segments, the first affection is assigned a name ending with the suffix -ia (eikasia for the first sub-segment of the seen, dianoia for the first sub-segment of the perceived by intelligence), the second affection a noun ending with the suffix -(s)is (pistis for the second subsegment of the seen (where the first sigma of the suffix -sis fades before the tau of pisteuô), noèsis for the second sub-segment of the perceived by intelligence). And we have seen earlier in this note that the suffix -ia designates a quality in the abstract while the suffix -sis designates the activity associated with the root verb, which can be interpreted here by saying that nouns ending in -ia refer to a form of "passivity", in front of the images provided by sight for eikasia, in front of the words used as supports in the dianoia, which, in each case, are accepted without questioning, whereas the nouns in -(s)is refer to an activity, that is to say to an active involvement of the person in what the verb implies: pistis ("trust") is an inner state of mind that presupposes prior reflection on the part of the person who trusts somebody else, and noèsis is the activity of the noûs ("intelligence / mind"). And this is in line with and confirms what I have already implied about the identical logos that presides over the two divisions: faced with "images", whether they are those produced by the biological process of vision or words, one can either accept them without asking questions and without looking for what is hidden behind them, what they are images of (first sub-segment), or scratch to see what is hidden behind them, understanding that they are just that, "images", and seek to access, as far as is possible for human beings, what they are only "images" of (second sub-segment). Of course, one may think that this is mere chance, but, in my opinion, it would be to misunderstand Plato: it fits too well to be the result of chance and it is entirely in Plato's way to choose his words with the utmost care and not to hesitate to use all the means at his disposal to get his message across. And the fact that, as I have announced, he breaks this beautiful symmetry in the reminder of the analogy towards the end of Book VII by replacing noèsis with epistèmè is not a counter-argument if we admit, as I announced at the beginning of this note, that none of these words adequately and exactly describes what Plato has in mind, for which he has no pre-existing names at his disposal. Under these conditions, varying one's vocabulary goes in the direction of helping readers not to become slaves of the names he chooses and to avoid freezing a "technical" vocabulary, so that first choosing a series of names that can "illustrate" an aspect of the image he proposes in the analogy, even if it means questioning some of these choices later, has only advantages from his standpoint by illustrating in many ways the freedom he advocates in the face of words. On the contrary, we can even see in this "fluctuation" on the name of one of the pathèmata ("affections / states of mind") a proof that this symmetry is not the result of chance: if the name of noèsis was not the one he wanted to give to this affection, but he nevertheless used it at first, thus creating the correspondence of the endings between the two groups of names, it was because he had seen this correspondence and that he had grasped its significance, and wanted to place it at least once, in the analogy itself, even if it implied getting rid of it later.
But how do you transcribe this into another language?! ... It's mission impossible... (<==)

(76) "On this logos" partially translates the Greek ana logon, an expression similar to the ana ton auton logon used at the beginning of our section, at 509d7-8, which I had translated, again only partially and leaving the word logos untranslated, by "according to the same logos", specifying in a note (cf. note 10) that the word logos could in this case be understood either in the mathematical sense of "proportion" or (numerical) "ratio", or in the non-mathematical sense of "rationale" (behind the segmentation) referring to the idea of a criterion of segmentation that has nothing numerical about it and not to the supposed size of the segments. I have specified in the course of the notes that a proper understanding of the analogy leads to choosing this second meaning, the "rationale" for dividing each segment into two sub-segments being the fact of having or not having become aware of the fact that what sight is working on in one case, the mind / intelligence (noûs) in the other, the visual images and the words, are only "representations" adapted to the organ involved, eye in one case, mind (noûs) in the other, of something of which it gives only a partial apprehension. Here, what is at stakes is no longer the "rationale" that presides over the division into sub-segments, but the reason that presides over the hierarchy of the four pathèmata that have just been listed by Socrates, which is not the same thing. And as a result, there is hardly any ambiguity left about the meaning to be given to logon, which cannot here refer to a numerical relationship. But this does not seem to have prompted the translators to ask themselves questions about the meaning of logon in the initial ana ton auton logon, as it is obvious to them that, since Socrates is talking about segments, he can only envision numerical relationships! And most of them do not hesitate, despite the resemblance between the two expressions, to translate logon in a different way in the two cases, as can be seen by referring to the page on the vocabulary of the line.
By translating the Greek taxon auta ana logon... hegesamenos... as "order them being led by this logos that...", in my translation, I understand this ana logon as applying to the hegesamenos which comes later in the sentence (in the Greek, it is the penultimate word of Socrates' line), understandind the verb hegeisthai, of which hegesamenos is the aorist participle, in its etymological meaning of "to walk ahead, to lead, to guide", but which, after Homer, also comes to mean "to believe, to think". More literally, we could probably translate it as "order them according to a logos, thinking that...", but this would introduce into the English a verb "to think" with no means to see that it renders a Greek original that has no connection with the noûs, but which, on the other hand, highlights the "guiding" role of the logos in us.
It should also be noted that Socrates here invites Glaucon, through this imperative taxon ("put in order"), to order the four pathèmata on the basis of a criterion that he provides immediately after, whereas in naming them, he himself has introduced an order ("the highest", "the second", "the third", "the last") which is not the one in which he presented the segments, but the exact opposite order. He thus opens the door to a possible challenge by Glaucon to this order, now that he gives the key of this ordering. (<==)

(77) "And order them being led by this logos that as the ***s upon which it is partake of truth, so these partake of clearness." translates the Greek kai taxon auta ana logon, hôsper eph' hois estin aletheias metechei, houtô tauta saphèneias hègèsamenos metechein.
To conclude the analogy, Socrates, after having, so to speak, pulled out of the hat at the last minute (but not just any old way, as we have seen at the end of note 75) in the first part of this line a name for each of the four pathèmata ("affections / states of mind" engendered in the soul) that he proposes to associate with each of the four segments of the line, giving them names which, except for one (dianoia), are new terms in the analogy, seems to propose to Glaucon to classify them "(being led) by a logos" (ana logon, cf. previous note) which would relate each pathèma to something that is designated by the words eph' hois estin (which I translate as "the ***s upon which it is" rather than by "that upon which it is" in order to preserve its unexplicit character and stay at the same time closer to the Greek, where hois is a plural, rendered by the "s" after ***), and which would make it possible to determine for these affections an order of greater or lesser "participation" in "clearness (saphèneia)" (notion of "participation" expressed by the verb metechein, formed from the prefix meta-, which, from the primary meaning of "in the middle of, among", implies this idea of participation, and from the verb echein, "to have, to possess", and which therefore means something like "to have something to do with, to share in") based on the greater or lesser "participation" (same verb as before) of these eph' hois estin in the "truth" (aletheia), which brings us back to the two criteria mentioned by Socrates at the beginning of the analogy, saphenia ("clearness") at 509d9 (see note 12) and aletheia ("truth") at 510a9 (see note 19), which are here related to each other.
From the point of view of vocabulary and grammar, the expression eph' hois estin takes up the preposition epi ("on", which here becomes eph' because the word that follows, hois, begins with a diphthong with a rough breathing) which has just been used by Socrates several times at the beginning of this line to ask Glaucon to bring the affections (pathèmata) that he is going to list epi tois tettarsi tmèmasi, "upon the four segments", and then, apprehension by intelligence "upon the highest" (epi tôi anôtatô), (discursive / wandering) thought "upon the second" (epi tôi deuterôi), associated with a relative, hois, plural neuter dative of hos, without explicit antecedent, the two together beginning a relative clause reduced to the verb estin ("is"), without an explicit subject, but of which it is clear that the implicit subject is the auta which preceded and which itself refers to the four pathèmata that have to be classified (it is usual in Greek to put in the singular a verb whose subject is a neuter plural). On the other hand, it is more difficult to decide whether the hois ("which" in the dative commanded by the preposition epi ("(up)on")), is plural simply because there are four affections, or because the ***s upon which it is are multiple for each affection.
In any case, these words reminds us of the discussion on knowledge and opinion at the end of Book V (cf. note 19), where this same kind of formulations using the preposition epi associated with various verbs is used, and in particular Republic, V, 477c1-d6, where Socrates explains what he meant by dunamis ("power") and explained that he distinguished the dunameis from one another by two criteria associated with each of them: "what it is upon and what it accomplishes (eph' hôi te esti kai ho apergazetai)" (V, 477d1). Of course, what we are talking about here are pathèmata ("affections") and not dunameis ("powers"), but on the one hand, these "affections" are not purely passive, since we have seen that Socrates distinguishes them according to the way in which the affected soul interprets them (this is indeed what justifies the second division of the line, the one who segments in two ana ton auton logon ("according to the same logos") each of the two initial segments of the line, that of the seen and that of the perceived by intelligence), and on the other hand, the example of the dunameis ("powers") can help us understand what Socrates means by the expression eph' hoi esti ("upon what it is") in a way that is transposable to the case of the pathèmata ("affections") in the sense in which he seems to use the term here. By introducing these general considerations on the notion of dunamis ("power") as a parenthesis within a discussion of "knowledge" (epistèmè) and "opinion" (doxa) that he has just appropriately described as dunameis ("powers"), Socrates takes the trouble of giving us two other examples of what he calls dunamis ("power"), sight and hearing. Of course, he does not elaborate on these examples and it is up to us to do so, but these examples, if we take the trouble to dwell on them for a moment, are particularly enlightening. What sight "accomplishes" is to see, and what hearing "accomplishes" is to hear, it is not rocket science to know that. On the other hand, when it comes to the "upon what it is" (eph' hoi esti), that's when it becomes interesting and instructive. Because the first intuitive answer is that the "upon what it is" of sight are the objects seen. But, in Socrates' mind, each "power" has both a specific "upon what it is" and a specific "what it does": at 477d2-5, in fact, when the problem is giving names to these powers, he only considers the cases of powers that accomplish the same thing on the same "upon what it is", to which he assigns the same name, and that of powers that perform different actions on different "upon what it is", to which he assigns different names, but not the case of powers that would perform different actions on the same "upon what it is" or identical actions on different "upon what it is". This being the case, the objects seen cannot be the "upon what it is" of sight, since the same objects can also be tangible, therefore acting on touch, odorous, therefore acting on smell, sapient, therefore acting on taste, and finally producing sounds, thus acting on hearing, and even be intelligible, therefore acting on intelligence. What Socrates has in mind when he speaks of the "upon what it is" (eph' hoi esti) of sight, or hearing, is what Aristotle will call the "proper object" of the sense being considered, namely, color for sight and sound for hearing. In this same perspective, the "upon what it is" of knowledge cannot be the known "objects," even if we limit these "objects" to pure "ideas" inaccessible to the senses, and the "upon what it is" of opinion cannot be the "objects" about which we have an opinion, even if we limit these "objects" to sensible objects, if only because these limitations are untenable, since everyone knows, and Socrates first and foremost, that a person can have an opinion about an "object" and another person, or the same person at another time in one's life, can have a "knowledge" about this same "object". Socrates gives us two examples of this in the Meno: in the experiment with Meno's slave, he first shows us that the slave has a (false) opinion on the question posed (the length of the side of a square having an area twice that of a square having sides of two feet) when he answers that this length is four feet, then tells us at the end of the experiment that, if the slave, to whom he has finally shown the right answer (the length of the diagonal of the square of two feet), continues to confirm his understanding of the demonstration in which he participated with Socrates, he will end up having a "knowledge" on this question, on which he can no longer be made to change his mind; and in the final discussion on right / true opinion and knowledge, he gives the example of two people, one with knowledge about the road to Larissa and the other with mere opinion, to show Meno that, from the standpoint of the end result of the action, they will both be able to lead travelers to Larissa with equal success. And in the end, what Socrates will identify as the "upon what it is" of knowledge is to on ("the being / what is"), and as the "upon what it is" of opinion, the nomima ("received ideas", 479d4) of the majority. What he means by this is that knowledge aims to know "the being / what is as it is" (hôs esti to on, 477b10-11) and therefore seeks to give an account in a coherent and validated way through the sharing of experiences of what acts on the senses and intelligence, whereas opinion is content to reproduce logoi heard from others without knowing the justifications, logoi which, even if they are true and may have their origin in someone who "knows", are no longer knowledge as soon as they are taken up by others, if the latter have not mastered the justification that makes them knowledge (this is what explains why Socrates does not consider that Meno's slave knows the right answer to the question he had asked him immediately after he (the slave) had given it to him, with his help, at the end of the experiment). This discussion shows us that the notion of "knowledge" (epistèmè), which Socrates ends up making here one of the four pathèmata ("affections"), has meaning only in relation to a given person and is therefore not in the objectivity of the words used to state it, but in the mind of the person who states them, depending on whether or not one knows the "reasons" why it is true, whether or not one is able to logon didonai ("to give the rationale").
And it is in this perspective that we must understand the expression eph' hois estin ("the *** upon which it is") used here in connection with pathèmata ("affections"). Not as suggesting that each affection relates to a different category of objects, since, as we saw in the introduction to this page, the allegory of the cave makes us understand that there are four distinct ways of apprehending the same "objects" when they are "visible" (i.e., capable of activating sight), two in the cave, through shadows and (sound) reflections, or in direct view, and two outside the cave, through shadows and reflections or in direct view, and two when they are only intelligible, outside the cave, through reflections or in direct view, corresponding indeed to the four segments of the line, and thus to the four pathèmata listed here.
But before attempting to better understand this expression and what it covers in each case, it should be noted that Socrates himself, and therefore Plato who makes him speak, are perfectly aware of the fact that it is not easy to understand in the case of the pathèmata ("affections") they are talking about here. Indeed, when Socrates returns to the analogy of the line in the discussion on dialektikè towards the end of Book VII, at 533e7-534a8, he recalls the list of the four pathèmata that he has just introduced here, in a context where he is no longer speaking of a line, but of intellectual development, and where he then speaks, no longer of pathèmata ("affections"), but of moirai ("parts / lots / shares", with the underlying idea of "fate / destiny" suggesting that not all have access to the four because of their own nature), taking up for them the names given here, except for one of them, noèsis, which becomes, as I have already said, epistèmè ("knowledge"), before specifying that the two "affections of the soul" that were here associated with the segments of the seen together constitute doxa ("opinion") and the two associated here with the segments of the perceived by intelligence noèsis; and he adds that opinion (doxa) is in the realm of becoming (peri genesin: peri followed by the accusative means "around, in the region of", or "in respect of, concerning") and noesis ("intellection") in the realm of ousia ("beingness", with a notion of "value" with regard to the good) (see diagram opposite, which can be compared to the diagram proposed in note 75), that ousia ("beingness") is to becoming (genesis) what intellection (noèsis) is to opinion (doxa), and that what intellection (noèsis) is to opinion (doxa), knowledge (epistèmè) is to trust (pistis), and (discursive / wandering) thought (dianoia) to imagery (eikasia), to conclude by inviting Glaucon, concerning "the relation of analogy between what those [are] upon" (tèn eph' hois tauta analogian, 534a5-6), and the division into two parts of each one of these two, the doxaston (that which is in the realm of opinion) and the noèton (that which is in the realm of intelligence), to be "drop[ped], so that it doesn’t fill us full with discussions many times longer than those having preceded" (534a7-8), Glaucon to whom he had a few moments before, when the latter asked him to explain to him "what [is] the manner [of working] of the power of to dialegesthai (which Socrates had made in the analogy the "marker" of the second segment of the perceived by intelligence) and also in how many kinds it is divided and, further, what paths [it follows]" (532d8-e1), refused these explanations on the grounds that he "will no longer be able to follow [him] (akolouthein)" (533a1-2). And indeed, in order to be able to follow the explanations that Socrates might have given, one must have reached the last stage of the progression through the four sub-segments of the line, described in the allegory of the cave, that is, not think that one has "see[n] clearly and contemplate[d] as it is the sun (idea of the good) itself by itself in its own place," but at the very least to have understood, without having reached that ineffable vision inaccessible to the majority in this life, that it is the light of intelligence and the guiding principle of the whole. And, as Plato says in the philosophical digression of the 7th Letter, only a small number of people are capable of this, and only after much work.
That being said, if we nevertheless want to get an idea of what could constitute the "upon what it is" (eph' hoi esti) of each of the four pathèmata / moirai ("affections / lots") that he has listed, we can suggest that:
- for eikasia, it is images formed by sight considered without critical analysis as adequate and sufficient representations of what they are only images of;
- for pistis, it is the opinions on what one admits regarding things that are visible without having seen it oneself on the basis of oral or written statements by other people deemed trustworthy, even if they are not themselves direct witnesses;
- for dianoia, it is what is accessible only by thought about "beings" that are also accessible by the senses or "beings" that are accessible only by thought, but understood outside the light of the guiding principle that the idea of the good constitutes and without really distinguishing between words and what they claim to designate, understood by means of the eidè that are associated, most often unconsciously, with these words;
- for noèsis / epistèmè (noèsis emphasizing the process, the activity of intelligence (noûs), epistèmè the result of this process), it is these eidè, understood as both distinct from the words with which they are associated and from the auta ("the ***s themselves") which arouse our thoughts which can only grasp the ideai of which these eidè are only the approximations depending for each one on his past experience and evolving over time during one's life.
We see that indeed, the thought that attaches itself to the eidè is closer to the truth, reveals more about what it is thinking about, and therefore allows it to hold a truer, and therefore clearer, logos about what it is interested in, than the thought that attaches itself to words, which itself allows it to hold a logos closer to the truth, and therefore clearer about what it is talking about than the opinions accepted on the basis of the trust one has in those one receive them from, which are themselves clearer, because they are expressed in words, thought or uttered, than simple visual images in perpetual evolution that give one access only to the external material envelope of what they allow us to apprehend. Truth (aletheia) is a property of the different logoi that each of the four pathèmata allows us to hold about the same "object" (material or immaterial) with regard to this "object" itself, and clarity (saphèneia) is a property of these logoi with regard to their ability to be understood by others and therefore to make known what they are talking about. (<==)