Education and the Mind’s Eye

We turn next to the descriptions of the actual education of the philosopher who is destined to return to the city as ruler. The exposition of two images in the two previous chapters, The Divided Line and the Cave, suggested just how our interpretation and construction of images both structures and responds to our world. The Cave story depicts how collectively and individually we lose and find our way. The Line story gives the structure of a discursive image-world. This is a world of increasing clarity and justifiable certainty that we make in our moving from opinion through to knowledge. In the present chapter, the description of education (as contrasted to the earlier training) will give the details of how individual souls can actually make their way along that journey in a series of practices that are an intellectual training of the soul in the habit of seeing more clearly. At the same time, it illustrates for the reader what there is to see. It is in this sense an education in the reading of images. As revelatory as this is, it is still an argument by illustration. Albeit pointing to what we actually do, the description of each stage and of each kind of study is set out in a series of images that will be seen as subtly different from their traditional counterparts. The differences imbedded in the descriptive images are to lead the potential philosopher/ruler forward, and are in support of the basis of the knowledge/opinion distinction. At the same time, it is an argument for what it is about the constituents of experience that poses an obstacle to this, and how the world can be read differently. The key, once again, is in the seeing of images for what they are, in reading correctly the image as image. Certain key moments of the interlocutor’s misunderstanding—for example in the discussion of astronomy—reveal the difficulty of this and its ontological origin, and the correction in course is instructive.

The higher education of the guardians, that is, the preparation of the philosopher/ruler (521c-535a) is not often given detailed attention—certainly not as much as the similes of Sun, Line and Cave. But this is unfortunate, since the details of the studies, their description and their sequence, can tell us much about those similes and eventually about dialectic and its goal. The studies are, after all, described in such a way as to make them more than just the conventional list of a Greek intellectual’s education. Though the description is not the education itself, and not that vision itself, its description, and particularly the conversation about it, does shed considerable light. Furthermore, the path “upward to the light” is not a preparation for a sudden illumination at the end, but rather a gradual upward path along which more and more light is shed through the sequence of studies. They thus deserve more attention. Socrates says that the better educated thus is “better able to share in both kinds of life [the public and the private, politics and philosophy] . . . (they) must each in turn go down to live with the other men and grow accustomed to seeing in the dark. When you are used to it, you will see infinitely better than the dwellers below; you will know what each image is, and of what it is an image, because you have seen the truth of things beautiful and just and good” (520b-c). So upon his return from his journey out of the Cave and up the Line, the philosopher now sees the truth of each image. He sees it as an image—sees “what each is,” that it is an image, and in what sense in the particular case, and also “of what it is an image.” That is, he sees the original and the relation of image to original for the particular kind of original, which depends on at what level the relation and the objects are being considered.

The education of the philosopher is reviewed as an answer to just how the relatively disinterested ruler can be got for the city. The Line and the Cave sketch what that view of reality entails, i.e., in Socrates’ “opinion”—thus, the Cave gives the “effect of education and the lack of it,” and the Line image before it gives the world as knowable. The educational program for the philosopher then gives the particular kind of “studies” that will result in an ontological reorientation that forms the basis for that higher vision. This vision is, in turn, the basis for the rule of the just city, both the city in the visible world and that of the soul itself. It is important to note at the outset that the use of the studies is not necessarily the normal one, the normal way or reason that they are taken up. Just as what we might call the earlier virtues of a visible world, those which require and result from “training and habit,” were redefined and reoriented to provide the initial training of the guardians, so here, the disciplines of the future philosopher-ruler are redefined and reoriented, after the “excrescencies [have been] knocked off which belong to the world of becoming [which] have been fastened on the soul by feasting, gluttony and similar pleasures, which like leaden weights draw the soul to look downward” (519c-b). The knocking off of the “excrescencies” is an external habituation to virtue and corresponds to 1) the earlier education/training of the guardians, 2) the first struggles at the bottom of the Cave, and 3) the passage in the Line from Imagination to Trust (eikasia to pistis). This is the pre-struggle, that is, the struggle for a reformation and reorientation of values at the level of habit or doxa or that of uncritical philosophy, and is pre-ontological—as contrasted to the education of the philosopher, where the studies are also a sort of habituation but of the active intellect. Rather than a habit of action (ethical) it is a habit of perception. After this initial turning away, the primary function of the studies then, is to “draw the soul from the world of that which becomes to the world of that which is” (521d). But this is unintelligible without an awareness of their relation, and a detailed look at just how these studies function, to which we now turn.

Socrates begins the discussion of the education of the philosopher-ruler, which studies are also to be a test of who is to be fit to rule, by going back to the beginning, that is, by recalling the training studies of the guardians in order to stress their limits—“these men, in youth, must be athletes in war” and “we educated them before this in the arts and physical culture (gymnastic).” Now this education was described initially as a good preparation for the defenders of the city, and it remains so, since those who are to be the leaders of the city are chosen from amongst those who have already been formed. The shortcoming or limitation of these early forms of education is that they are a result of habit, conditioning of the soul, the hammering or tempering of it.

Physical training is disqualified as a higher study because it deals with, or in, “what comes to be and dies” and the turning of the soul is from the world of becoming to that of being. But the training in the arts are disqualified for the higher studies not because they deal with becoming but because they “educated the guardians through habits; [and] its melodies gave them a certain inner harmony, not knowledge . . . its stories . . . cultivated certain habits akin to those [harmony and grace] but there was no knowledge in it which would lead to what you are looking for now” (522a). The link between these two aspects of disqualification is important to note: the physicality and exteriority of the objects, the body, music, the arts, corresponds to an exteriority of method—training or habit. This is a learning imposed from without, through imitation on the whole; and especially initially through in some cases quite literally a copying of images (art school, gymnastic, rhetorical forms, and the poets, etc.). It does not deal with active intellect except at its lowest level, that of memory, as opposed to recollection which requires comparison and judgment. The link between the two disqualifications of object and method then is in the nature of our hold on what we have—and not the changeableness of the world of generation and decay per se from which we are said to turn away.^[1] For in a sense, nothing is more fixed than habit (the training of the gymnast, the skill of the musician or artist) nor for that matter anything more reliable, perforce, than the everyday world in which we live—a stability on which we rely.^[2] But we operate within this reliability without understanding it, without knowing it. Nor is there a need to understand it, for the most part—though occasionally something goes wrong, something is out of place, or there is a seeming violation of the “laws of nature,” and we are puzzled, disoriented (or sometimes while the occurrence is a part of the everyday world, it is one from which we are usually hidden, or hide ourselves. Such is the personal experience of death in the death of a loved one). At such a point we either rest with our puzzlement, or it drives us to seek to understand what we see, drives us to a higher level (a fuller discussion of this dynamic is to follow). Apart from these shocks, our trust (pistis) in these worlds does change as interpretations either of nature or art change, but the changes of craft or habit change passively, so to speak—that is, in response to an external force—the teacher, the culture, etc. and not as a result of the active rethinking of the agent. The flux of nature itself is best taken, as we noted earlier, as a metaphor for the whole realm of doxa, which on the one hand is intractable, on the other, changes, when it does, arbitrarily. One might in fact go so far as to assign the physical training mentioned to the lowest segment of the Line—since its basis is in a way pure imitation, and then assign the conditioning of the soul via the arts to the second section, where at least the making of images is involved, and where there is the use of intellectual elements (though not their understanding).

Again, these new studies are to be distinguished from the earlier training because they develop and depend upon the noetic capacities inherent in the non-visible soul, as distinguished from the skills and habits which are given to the subject, imposed from without. To cast this in terms of our image of the just city—we have been presented with its outer form, the visible city of imagination with its mirror image in its inner form, the soul, and the larger world as cast or corresponding to that inner form. We are now to look at the activities which contain an image of the noetic soul at work on its natural objects, just as before the soul was given activities that formed in a sense its external nature. The virtues of the latter that were called into being—that is, courage, moderation, etc.—are practices that depend for their very existence, if not their meaning, on the embodied soul, on the soul at the level of the desiring/feeling soul that presents us with the battlefield of the passions. Even justice, conceived of in its particular application to the human situation (as opposed to the bare principle of harmony of parts, unity of a many) might be said to exist as human justice only in the embodied and embattled soul (the plan/design laid up in heaven of the just city is not a form—it is an image drawn by the philosopher/artist, an ideal not an Idea). Thus though the turning around is of necessity of the whole soul it is “not the art of putting the capacity of sight into the soul: the soul possesses that already but it has not turned the right way or looking where it should. This is what the higher education has to deal with” (518d). The virtue of intelligence is, it seems, not a moral quality acquired with the experience of life, since it may be turned either to good or evil ends, and yet in the latter case “its capacity for sight is not inferior” (519a); the capacity to know (diverse in its general sense—all the pathēmata of the Line) is itself an excellence or virtue belonging to something “more divine” simply by virtue of its alignment with what is knowable, what truly is as opposed to what appears to be—or, appears to one to be, in the form of doxa, as opposed to what one knows to be in the form of epistēmē.

Having distinguished in general what these studies are to be, we may now look at them in particular. Glaucon, not too surprisingly from my point of view, asks, “Yet what study is there left apart from the arts and physical training and the crafts” (522b). The answer, Socrates suggests, is something “which bears upon them all … used by all crafts, all modes of thought and all sciences . . . that trifling thing [to phaulon], namely to distinguish [diagignōskein] one, two and three, or to give it one name, I call it number and calculation [arithmon, logismon]” (522c).^[3] The seriousness and weight attached to this beginning is indicated by the playfulness of to phaulon: (pl. hoi phauloi ) meaning 1) slight, light 2) trivial, mean, low or common (of persons), ugly (of persons), mean looking (of outward appearance) 3) careless, thoughtless—hoi phauloi means the vulgar, the common sort, also weak or uneducated, as opposed to hoi sophoi. (Liddell and Scott). It is possibly significant that Plato says legō de auto en kephalō translated by Grube as “or to give it one name, I call it. . .” and by Shorey simply as “in sum.” For kephalō is a pun of sorts, meaning, in the dative as here, a summary, the sum of the matter, but also literally and generally “of or belonging to the head physically—and by obvious extension metaphorically the chief or main point” (Liddell and Scott). If number and calculation are truly the beginning, broadly conceived, of all knowing and the base of all knowability in the things known, then other forms of knowing must all partake of these and these be always “in the head” and “at the head.” This starting point is, I believe, so significant that I propose to discuss it at length.

One is reminded of the Meno and its mathematical example. One might suppose that the use of a geometrical example there would have predicted a different order of studies in the Republic, with geometry preceding arithmetic or calculation. The metaphor of sight for knowing and the immediacy of shape as the embodiment of eidos (which is translatable as shape or look) would suggest this, and so with the other examples of idea, such as bed. In the slave boy episode, the first question to Meno about the slave boy is, does he speak Greek, while the first question to the boy himself is, “Do you know that a square figure is like this?” (the drawn figure in the sand). “Yes.” “Now a square figure has these lines, four in number, all equal?” “Certainly” (Meno 82c). And they then go on to compare lines and figures drawn in different ways and the sizes of the spaces delimited, etc. The exercise is indeed one in geometry, or geometrical reasoning, and does have its beginning with recognition of figures (what is a square, etc.). But the actual beginning of the analysis, the activity of knowing, is in counting, however primitively, the sides of the square, in recognizing, however instantaneously, that the figure before one (in the sand) is indeed a four-sided equilateral one. The examination then goes on to logismos or calculation of the divisions of the figure constructed.^[4] There is also the act of comparing, and this might be said to be more basic and more general than calculation. Or, one might say that counting and calculation are themselves the images or instances of comparison as the more general dialectical essence. In the Meno too, after all, the slave boy episode is meant to illustrate the general notion of recollection. If we consider the figure as known only as it is identified as a square, and that, only through the first implicit and then explicit awareness of its equal four-sidedness, then recollection is only a kind of knowledge when it is active analysis, however primitive in awareness, as opposed to an image as precognitive or aesthetic.^[5] The adult mind may identify the object, recognize it as a square, though put to aesthetic uses; the squares of say, the paintings of Joseph Albers, and even more so those of a Renaissance painter, are focused on as such, though in a different way, and as means to an end. The very young child lumps pretty much all figure together, and only later begins to distinguish the famous round from square pegs and holes. But this is at that point primitive and instant calculation and counting, even in the geometrical modes, as opposed to the more obvious counting and calculation of discrete units.

The key point is that counting and calculation here are offered as fundamental, and that they themselves are, in their literal form (the mathematical), images of comparison and judgment—they are taken up, after all, as what is in common in all the arts, crafts and sciences and “all modes of thought.” Even the shoemaker and the pastry cook must count and calculate, and more generally, compare and make judgments. In their literal form as simple counting—“to distinguish one and two, and three”—they correspond to the simple distinguishing of this and that, hence the distinguishing of the visible in the lower half of the Line. Recall that those in the Cave in chains pass the time of day noting and counting the shadows as they pass before them. It is in distinguishing what each one is that they err. And there is some confusion even in their counting as to whether x shadow is different or new, or simply passing again—much like children who cannot remember if they have counted this one already before or not. Number and calculation in the higher, limited, more technical sense, as mathematical per se, is of course to be found in the third highest section, as dianoia. And it is in this form that it is recommended as a study that prepares the ruler, consciously turns his soul. But its introduction into the conversation here is as helpful in the second level, that of pistis—where the training of athletes in war, the soldier-guardians, is at issue.

Again, it is the level at which Glaucon lives, and from which Socrates tries to pry him loose, which is relevant. And so the beginning discussion is of number and calculation in its practical utility. Glaucon docilely listens to Socrates’s joking story about Palamedes, which is supposedly to illustrate the usefulness of calculation and numbering for the art of war. Socrates says that the figure of Palamedes in the tragedies “shows up Agamemnon every time as a quite ridiculous general” because “[Palamedes] says that he invented numbers, which men did not know before, then arranged the companies in the camp in Ilium and counted the ships and everything else.” To which Socrates adds, “Agamemnon apparently did not even know how many feet he had, since he did not know how to count. What sort of general do you think that made him?” Glaucon answers solemnly, “A very strange one, in my opinion, if that was true” (522d). Glaucon may or may not get the joke, depending on the tone of his response—I would tend to think not—but he apparently misses the implication. Agamemnon may not have been the greatest general but he surely knew how to count—so what is the point of the story? It is, in fact, an indication of where Glaucon’s mind is, and an implicit comment to the reader about the nature of calculation and counting. The point is that Palamedes precisely did not “invent numbers” (another calling to account of the poets, and tradition). Counting is a natural capacity of everyone, not a craft or art to be “invented”—hence what is ridiculous is not Agamemnon’s generalship but the idea that “Agamemnon did not even know how many feet he had.” Of course he did, and if counting is thus natural then Palamedes did not “invent” it. Glaucon’s probable cultural acceptance of the contrary is of unreasonable doxa—significantly, embedded in a story of the poets—“Palamedes in the tragedies. . .” He takes the story to grant the authority of the poets as to the usefulness of calculation in war, but not the significance of the quip about Palamedes. The truth of the matter comes out of Glaucon’s mouth in spite of himself. When asked whether then this study is “necessary for a warrior,” he replies, “More necessary than anything else, if he is to understand anything about marshalling troops, indeed if he is to be a man at all” (522e).

But what then precisely is this study, calculation, which “leads to intelligent thought” but which “no one uses correctly”? Socrates begins Glaucon on a search for a criterion not just for calculation, but to “distinguish . . . the studies which lead in the direction mentioned” and asks Glaucon to “observe them along with me and agree or disagree.” Socrates considers the relation of sense impressions to intelligence. Sometimes the “decision of perception is sufficient, while others certainly summon the help of intelligence to examine them because the sensation does not achieve a sensible result” (523b). Glaucon takes the distinction to be between the visually clear and distinct and what is fuzzy to sense perception: “You are, he said, obviously referring to things appearing in the distance and to shadow painting.” To which Socrates replies, “You are not quite getting my meaning.” As usual it is Glaucon’s mistakes that are instructive. There is a test for when we are in need, when a capacity and a kind of knowledge or study is called into play, and in such a way that it is seen to “lead us in the direction we mentioned”—and this test applies not only in the limited case of calculation but will identify or characterize the other studies as well. The visually fuzzy, any more than a sharp image, does not actively “call for help,” parakalounta, Plato repeats the word several times; their vagueness is not provocative, at least not in the way that is useful here, not thought provoking. “They do not call for help, I said, if they do not at the same time give a contrary impression; I describe those that do as calling for help whenever the sense impression does not point to one thing rather than its opposite, whether its object be near or far” (523c. Cf. Lesher, Saphēneia, 173). The essence, then, of those experiences that will drive us further along is the tension of contrary impressions—the experience not just of ambiguity or vagueness but of possible opposite judgments about the same perception—eis enantian aisthēsin hama. Only these are worthy to be called provocative, or calling forth. A visual example of the merely contradictory or ambiguous would be the drawings of Escher. Here the visual flip-flops, staircases which can be read either as going up or going down etc., are not provocative in the right way; they are not dialectically opposed such that there is ever a resolution possible, a higher point of view from which intelligence may view the opposites and contain them. The example or image that Socrates now introduces, of the three fingers at different heights is, by contrast, of this latter sort. It is a basic, non-technical calculation. In the finger example, the one finger is both taller and shorter, each in a different relation, but in Escher’s staircase, the staircase goes both up and down in alternating perceptions of the same view.

The first step in Socrates’s account is the comment that each of the fingers presents itself clearly enough, unambiguously, as what it is, as a finger, regardless of variations in thickness, color, whether in the middle or at the end, etc. “In all this, the soul of the many is not compelled to ask intelligence (noēsis) what a finger is, for the sense of sight does not indicate that the finger is the opposite of a finger” (523d). This is true only at the level of everyday sight, hence the reference to the “soul of the many.” To a biologist, by contrast, the identity of the finger—just when something is not a finger, when it is, and what makes it so—is precisely what is “provoked” by the observation (and framework of questions). For the evolutionary biologist, the articulation of the fin of the fossil fish Panderichthys, which it used to lean on in shallow water, counts as a rudimentary finger. Sight and seeing thus provides initially here an unthinking example. Sight, the very thing that had been used to represent the noblest and highest perceptions in the Sun/Good analogy, is here used for the simplest and most unambiguous, the presentation of the kind of a thing—at least until further notice—hence our trust in the visible world. (Cf. 532c where Socrates repeats the assertion of sight as the “clearest sense in the body.”) Each instance of sight is clear, clearly of what it is—taller, shorter—intellect is provoked only when we take the two judgments together. In any event, here we are dealing with the very beginnings of the process of intellect’s probing, again non-mathematical, responses to the world—the bare physical identity of kinds is left alone as evident. We recall that all in the world of pistis, in which a finger is reliably a finger, receives eventually a reinterpretation when seen and judged from the standpoint of dianoia—our biologist is an example.

The same object may be judged big or small depending on what it is compared to. The identity of the kind, the finger, is not at stake, but the identity of relational qualities in the object may be, e.g., relative bigness. One might be tempted to say that some relational qualities are subjective as contrasted, for example, to specifying density or length i.e., sense impressions and comparisons of the perceiver. But whatever truth there might be to that is not Plato’s concern here. Rather, each sense, including sight, is capable of presenting the tension of contradiction that will provoke intelligence. Because each sense is capable of perceiving the opposite quality in a particular instance, for the same thing perceived, unawares it places the perceived object in an ambiguous position. Every staircase, for example, does in fact go both up and down; the relatively minor tension between them can be resolved and unified by giving each direction its context (e.g., social rules, signs, etc.) and understanding that the same staircase can serve through both contexts. But in the case of Escher’s staircase the fascination of it is in our awareness that the tension is permanent—the two contexts are irreconcilable, are in fact the same context—the person walking up is at the same time walking down (or sideways).

For Plato in our present context, the identity or unity of the sense object is not in fact what is at stake here, but rather the intelligibility of the quality applied to it. “Does not each sense behave in the following way: in the first place the sense concerned with the hard is of necessity also concerned with the soft and it declares to the soul that it perceives the same object to be both hard and soft” (524a). The sense personified is not all that troubled, while thus confronted with conflicting decisions of perception the soul becomes “puzzled,” Socrates says, not about the thing perceived, but about the “meaning of the light and heavy, if sense perception indicates that what is light is also heavy, and what is heavy is also light. . . . Then, in those cases, the soul is puzzled as to what this perception means by hard, if it says the same thing is also soft.” At issue then is the intelligibility of heavy if the same thing can be considered both heavy and light, both hard and soft. (The same motion, that of seeking the resolution of contradiction, is contained in dialectic, hence the danger of its perversion in misologos and sophistry.) The point of Plato’s argument is in the direction that it takes here. Obviously there are many ways to respond to perceptions. Plato is considering just which experiences of perception will provide the ground for the turning of the soul from becoming towards being, and this because he sees these as parallel to the distinction between opinion and knowledge. But the sense of becoming is not limited to the sense world—it is the term for the realm of all doxa, of all contradiction (actual or potential). It is rather a natural example for this argument to start with a case of supposed opposing sense impressions, hence conflicting doxai (or rather contradictory judgments provoked in one sense impression). The direction of the argument is to resolve this conflict by focusing on the problem of identity, and opting for the preservation of the self-identity of the quality, rather than the identity of the object qualified by opposing predicates. This solution is to provide the “springboard to the Ideas.” The principle that only the self-identical is ultimately intelligible is here presented in a sense as a choice, that is, amounting in effect to the denial that the sensible can form an intelligible unity precisely because it is subject to contrary predicates. For there is of course the rejoinder that the contradiction “x is taller than y but shorter than z,” is only apparent, since each is in a separate relation. But since Plato actually carefully states the principle of identity with this same qualification earlier, at 436e^[6] we must ask why he would not employ it here—since the answer cannot be that he was not aware of it.^[7] The answer is that the relational aspect of sensibles holds satisfactory for the self-identity of each relation, even while it remains problematic for the unity of the subject that enters into these relations. Qualities themselves, and other Ideas, do have relations as part of their own identity, and are thus in a sense a multiplicity to be dealt with. But the direction here is to single out Ideas at all, to distinguish them from sensibles that are their temporary and fluctuating referents. The Republic does not deal, at least explicitly, with the behavior and complex identity of the Ideas—its topic being the soul in relation to them.

We return to the case at hand, the consideration of the function of calculation and the judgment of qualities:

Here, from the last sentence, it is clear that the argument is focused on the distinction and intelligibility of the two qualities, e.g., tall/short, and that it is the act of thinking of them as separate that demonstrates, at this stage, their independent being or identity—for if they were not indeed separate we would not be able to conceive of them as two—“for if they were not separate, it would not be thinking of them as two, but as one.” The ability to use a name or term at all, and to intend different qualities by different terms (and the same by the same) is a reflection of the intellect’s response to at least the appearance of sameness and difference in the world (primarily the visible world at this point). It is more an argument about intellect and intentionality (though clearly not entirely so) at this point. What is important is the one or two of the qualities and not of the subject, which quickly drops out of the picture. The function of opposites here is to focus the intellect, through their paired opposition in the sensible world in one thing, on precisely the radical separation of them with regard to intelligence, and, it is argued, hence being. The argument, such as it is, is that such tensions of contradiction, of opposites, call forth through the puzzlement they provoke the only resolution that intellect is capable of—namely, leaving the sensibles behind for the moment and rising to the self-identical qualities themselves; that only so can the terms we use in describing have any meaning.^[8] Whether we ever return to these sensibles for a resolution at their level (or rather, in their world) is doubtful. But since the whole enterprise is eventually directed at resolving the conflicts of doxa, specifically about justice and happiness and the soul, by replacing doxa with knowledge, there must be a return from the Ideas that does deal with these sorts of objects of opinion—the sensibles of doxa in the realm of human action.^[9]

At any rate, the distinctions here are in terms of the visible as representing the realm of opposition and doxa, as contrasted to the stable world of intelligence, here in isolation or at least in retreat from the sensible. “But we say that the sense of sight saw big and small not as separate but as commingled. Is that not so?” “Yes.” “So in order to clarify this, intelligence is compelled to see big and small not as commingled but as separate, the opposite way from sight” (524c).

And from this we are to move from a consideration of the original perception, of an object, to a consideration of the nature of the meaning, and source, of the concepts that perception contradictorily applies: “And it is from some such circumstances that it first occurs to us to ask ‘What is the nature of bigness, and again of smallness?’ And so we called the one intelligible and the other visible.”

Socrates then shifts to a consideration of something a further step removed from the visible (as indicated by Glaucon’s hesitation), that is, to a consideration of “number and unit,” which are, as contrasted to say tall and soft, non-visible images. He turns, that is, to our use of these in eidetic image-making. The earlier question, whether the qualities are “one or two” implicates number and counting, but not yet, as here, in an overt consideration of number itself and its order in itself, i.e., arithmetic. In the previous operation we were distinguishing the qualities as either one or two, as presented first to perception and, as a result of conflicting judgments, considered in themselves “not as commingled but as separate, the opposite way from sight.” In the consideration of number and unit Socrates urges Glaucon to “reason it out from what was said before.” Unit and number are to be put to the same tests as in the case of the comparison of fingers example of bigness and smallness, i.e., by the provocation of intellect in conflicting judgments of perception. But unit and number are already somewhat more removed from immediate sensation. The fingers appear immediately fat or thin, taller or shorter; though comparison is necessary, it is more direct, more natural from the point of view of sense experience than counting, which employs a device that is natural in a different sense—natural to intellect. Perhaps this accounts for Glaucon’s difficulty in extending the previous example of the comparison of fingers. The question asked is, to which of the two groups do number and the unit belong, those that provoke thought by affecting the senses in contrary ways at the same time or those which do not. In this elenctic invitation to a dialectical sorting, Glaucon is “unable to decide.”

Glaucon’s reply is odd, especially considering that he did not know which group to place unit in: “Certainly, he said, the sight of the unit possesses this characteristic to a remarkable degree, for we see the same object to be both one and an infinite number [hōs hen . . . hōs apeira to plethōs]” (525a). Probably what Glaucon has in mind is that any physical unit is also infinitely divisible into parts. This is not true of natural parts, like the limbs of a man or the segments of an orange, but perhaps Glaucon is thinking of unit as the unit of spatial extension in general, hence not as natural points of division. The framework just given by Socrates was, by extension from the example of the fingers, whether “the unit, in and by itself, is adequately seen or perceived by any other sense . . . (or) something contrary to it is always seen at the same time so that it does not appear to be more than the opposite.” It would seem then that the perception of the unit itself is a kind of aisthēsis—at least analogously. Glaucon quite literally says the sight of it. The infinite divisibility of sensible units could “provoke” if one thought about it.

Consideration of the tensions of number, the unit per se as concerns arithmetic and not the countable of sense, goes on at the level of dianoia, the third level. But these contraries at the level of dianoia, where one is “at a loss,” result in a movement to “search for an answer, stir up intelligence within itself, and ask what the nature of the unit in itself is” at a higher level, that of noēsis, where the nature of unity itself is considered, of which the unit of arithmetic is one, rather pure, example. But these considerations can result, it would seem, either directly from a consideration of arithmetic or of the one/many composites of the sense and social world. In either case the question of the nature of the unit is beyond the level of dianoia, of arithmetic and political science for example, within which the unit is assumed. The higher question can be conceived either as restricted to the foundations of mathematics, the unit of arithmetic, or, the larger question in a sense, of the nature of unity such that the unities of the world of appearances can be accounted for, those of the social world, the soul and the city, wherein action is the sensible that results in a unity out of many.^[11] Here, Glaucon considers with Socrates only the unity of arithmetic and hence the usefulness of it as a study to lead one to the truth—but the application is still concrete enough to support Glaucon in his understanding of the ongoing analogy of the inner (personal) and outer cities. “They [calculation and arithmetic] would then belong it seems to the studies we seek. The warrior must learn to marshal his troops, and also the philosopher because he must emerge from the world of becoming and grasp reality, or never be a rational thinker. … And our guardian must be both a warrior and a philosopher” (525b). The philosopher must also learn to “marshal his troops” if he is to be able to turn first his own soul and then those of others; and the study of arithmetic, that is, of counting and calculation, is recommended on two grounds.^[12] First, it is an activity which in itself turns one from preoccupation and pleasure with the sensible and doxa, to pleasure and preoccupation with the non-visible and knowledge. But secondly, and as important if not more so, is the occasion that it gives for intelligence to be forced by the contraries of both arithmetic itself and calculation in application to the visible world to consider the nature of number and the unit—and it is in this service that it is of particular value. The unit itself is indivisible, but the other numbers are both a unity and each made up of, and divisible into, a sum of ones/units and a sum of parts—a one as the fifth numeral yet a many as composed of units. Five is both five ones and three plus two, etc. (the Greeks did not use fractions, rather ratios). Just as the virtues were considered in a different light to the end of reforming or retraining the soul to a new definition of justice, so here, the study of arithmetic is appropriated to different ends and considered in a different light: its ability to train the soul, to turn it, and as a beginning point for dialectic in the driving of the soul upward to consider “by pure thought the contemplation of the nature of numbers.”

The dynamic of the Line is here illustrated again as a forcing (as in the Cave) of the activity of intellect from one level, that of dianoia, to a higher one, noēsis, when confronted by a contradiction, whose resolution is sought perforce at a higher level, in this case, just how the one can be a kind of many. The nature of unity allows for it to be both many and one if the particular kind of manyness that is part-whole is considered. This is not the case with tall-short, which are never a unity though they are a pair. The one finger is subject to both tall and short, though always in different relations; otherwise the intelligibility of each contrary would not be maintained in its self-identity, which was to be the case, hence to lead to the study of the nature of each. But in the case of unit and unity and certainly the other numbers (“every number has it too”) in counting, the quality of divisibility is a kind of unity that is co-extensive with the self-identity, and they thus contain the tension within them—and in this way like the finger—but do so in a way that is essential to the nature of number, and that tallness is not to finger. A finger must be of a certain height, while whether it is tall or short depends on the external relation. In one crucial sense the unit must be indivisible in itself, and only that to which it is applied, whether to sensibles or to a number as one specified aggregate of units, is a one out of many. But one and many, the number two and its divisibility, its composition of units, though co-extensive, are logically distinct, and it is the questioning intellect that is forced to go beyond this to consider the nature of number and of unit/unity to see how this can be so. An easing away from the literal level of the discussion of the advantages of the studies is apparent in the irony in Socrates’s tone and the coupling in one sentence: this study will not be pursued “for the sake of buying and selling like merchants, but both for the sake of war and to attain ease in turning the soul itself from the world of becoming to truth and reality” (525c). Yet war is clearly as practical a use as that of “buying and selling” unless we remind ourselves, as we are meant to here, that the “war” of the guardian who is “both warrior and philosopher” (525b) is only an image, by now well inverted, of the inner war of the soul with itself, hence the real necessity of “turning the soul itself from the world of becoming to truth and reality.”^[13]

In sum, the soul in this study “does not allow one to discuss them [numbers themselves] by presenting numbers with visible and touchable bodies.” The characteristic selected to illustrate this prohibition against the introduction of “visible and touchable bodies” in the study of calculation, as would be the case in illustrations drawn from the “buying and selling like merchants and retailers,” is the indivisibility of the arithmetical unit. Socrates mentions “experts in these matters” as thwarting any attempt to “divide the unit up in argument”—the experts in logismoi are, then, models in their behavior, which is to preserve the wholeness of the unit. “If someone tries to divide the unit in argument [they] laugh at him and do not allow it; if you divide it, they multiply, taking care that the unit should never appear to be many parts and not one” (525e). The ironic use of temnō for “divide” stresses the intended contrast with those physical units that can be physically cut up in pieces, like a butcher cuts up an animal, though through this literal meaning one also arrives at the figurative one, to cut or draw a line. In the Phaedrus the cutting up of an argument properly as precisely compared to the butcher’s art in speech is cited as the sign of the proper use of dialectic in the pursuit of ontological reality (Phaedr.265e, cf. Parmen. 129ff.). There is a kind of one in many amongst the forms in the identity comprised or composed in their complex relations, and in the operations of dialectic in the pursuit of analysis, and at the lowest level in the physical divisibility of bodies. Thus the unit per se (as contrasted to the divisibility of numbers), that of the expert or arithmetician, is an important special case—the unit of numbers, illustrates, as does the zeal of the arithmetician, the indivisibility of a whole; in the case of numbers this means a unit without parts, but in the case of other unities the whole is a complex one, in which what is important is the indivisibility or unity of an Idea in its identity or justifiable wholeness. In a system like that of Greek mathematics where there are no fractions, the role being played by whole number ratios, there is a ready-made contrast available between the physically divisible and the indivisibility of the numerical unit. When a geometrical figure is constructed and, say, divided or cut up—as in the case of Plato’s Line image—the comparison of lengths is always carried out in terms of the commensurability of the lengths described as ratios. Hence the incommensurability of the hypotenuse of a (right) triangle can be described by appeal to the ratio of the (commensurable) opposite sides.

The singleness of unity extends beyond the arithmetical form to that of unity itself (the study of which is indicated by the introduction of arithmetic as the first study). The concept of unity itself, which has broader application than the unit of arithmetic, is of something whole and indivisible, though not in the same sense as arithmetically indivisible—the Idea of unity has no parts itself but is capable of applying to things that do. What it says, or is employed to say, about such things is that they are a whole in some sense, and thus though the whole that partakes of this particular Idea may have other separable parts (even as another Idea) the Idea of wholeness itself does not (though it has relations to other Ideas). Its whole content is filled up with the idea of unity. Of the application or participation of unity in the sensible this is clear, and we may include other unities that are sensible in the larger sense that we have been arguing for—those in the realm of space, time and human action and passion—hence the city and the soul. How and whether this applies to Ideas themselves and the relations among them is a more difficult question. But an organic or natural unity is divisible because of its unity while a merely physical unity is divisible arbitrarily—it can be cut up into arbitrary pieces and is an image of all arbitrary divisions. It is thus the task of dialectic to distinguish true from false unities, and hence precisely to see just where a unity can be divisible naturally.

But we also ought not to confuse arithmeticians with dialecticians—the former take number, and the unit, and claim or maintain it to be as here described, hence as hypotheses; while these are to be examined and here expanded by the dialectician, who examines the nature of unity, and of the one-many relation, prompted by the tensions of one-many in numbers themselves and in the unit in its application. As Grube points out in a note to his translation (178, note 8) the skill of those at issue here, calculation, or logistikos is the same term used “for the highest of the three parts of the soul, the reasoning part,” to logistikon at 439ff. Hence Socrates’s statement that “those who are good at calculation are, one might say, naturally sharp in every other study.” But Grube concludes, wrongly I think, that “because of this double meaning the statement that the man who is good at arithmetic is good at every other study is less startling in Greek, and this fits in with Plato’s mathematical approach.” Rather it seems clear (on the basis of the text here at least), that an arithmetician is not yet asking the right questions about the enterprise. He falls short, to use the quasi-mathematical term that Socrates employs ironically elsewhere. Arithmetic in itself only serves as a training and turning of the soul, and then only when properly directed. Secondly, the mathematical is used here as the clearest image of the more general sense of calculation—of the soul’s ability to take the right measure of something, hence an example of the perception by intellect of the stable non-visible. This is not say that the contents of the other non-visibles, or the method of perceiving them, are themselves mathematical in any conventional sense, even as objects of dianoia. It is an example, but one with a unique ontological and propaedeutic status. If anything, just the reverse is suggested, since the images employed by the arithmetician/expert and his studies are offered here by Socrates precisely as images, examples, and are seen as such by the dialectician, in contrast to the expert—who continues to be satisfied with life at the level of hypothesis and the limited realm of dianoia and its objects (the Ideas as hypotheses is another story). One such expert is Theodorus in the Theaetetus. Though his hold on his expertise is of course much more like knowledge than his acquaintance with Heracleitean views of nature, more systematic and explanatory, it is still doxa as regards its starting points; and Theodorus is initially both unaware and unperturbed by any contradiction or tension between the practice of mathematics and his views of nature.

The complexities of the relation of the dianoetic technē of arithmetic to its function as a model for noēsis, and the special significance for Plato of numbers, the numerable, and of that singularly human capacity for counting, hence for separating and collecting—are all dealt with in careful detail by Jacob Klein in Greek Mathematical Thought and the Origin of Algebra (cited above, n.13). My own discussion is in part stimulated by his analysis. The argument of the book goes beyond the central chapters on Plato, since the overall subject is rather the difference between ancients and moderns, as seen through the ontological shift in the perception of number. The following comment will be seen as more relevant after the issue of separation (chōrismos) is dealt with in the next chapter, on the Phaedo. By arithmetic Plato does not mean the skills of addition, subtraction, etc. These would hardly be a newly acquired knowledge for the philosopher. But neither is the Greek study of arithmetic the theory of numbers in the modern sense. As Klein points out, the conventional Greek understanding of arithmetic was the study of numbers themselves—the attempt to say what they are—which means in effect to define them, limit them, as to their kinds. Plato’s advance on this is to examine, as the Line suggests, what are the presuppositions or hypotheses of both counting (calculation of) things, either in the sense world or units as monads, and the presupposition of number itself and its divisions (e.g., into odd and even). That is, by examining how they are to be defined we are forced to look at what it is about them that makes them knowable—the ontological preconditions that are behind their use by the knower, their behavior in their pure relations and in the wholes that are formed. The nature and function of the unit in particular is fundamental and points the direction for a resolution, or at least the context, of the general problem of the one and the many.

Concerning the intimate relation between mathematics and Greek ontology in general, Klein suggests that “mathematical truths as mathēmata, i.e., as things ‘to be learned’ serves as a model for all teachable and learnable knowledge, and it is knowledge so understood which determines the sphere of Greek ontological inquiry” (61). His study is “an attempt to discover the basic presuppositions of Greek arithmetic and the ontological arguments connected with it within the structure of the arithmos concept itself” (62). He stresses that “Greek scientific arithmetic and logistic are founded on a ‘natural’ attitude to everything countable as we meet it in life. . . . The peculiarity of the Greek concept of ‘number’ lies therefore less in an ‘archaic’ [Becker] or ‘intuitive’ [Stenzel] character (which is not at all its primary property) than in the kind of relation it has to the ‘thing’ it intends” (63). “For Plato generic identity is the ultimate foundation of all possible unity,” and the basis of Aristotle’s attack on the chōrismos thesis as not explaining the relation of eidos to sensibles. (105 ff.) Calculation (logismos) can rely on the natural articulation and original source of delimitation, but the pure monads of arithmetic are by definition indivisible, thus absolute. “In this sense the ‘one’ (or the one thing subjected to counting) makes counting and thus ‘counting number’ possible: In this sense it takes precedence over number and may be called its archē. The priority of one over number does not follow from a relationship of superiority of genus over species but rather from the character of one as ‘measure.’. . . We comprehend a number as one because we do our counting over one and the same thing, because our eyes remain fixed on one and the same thing” (108). But dianoia never considers “single being as such” rather always a “relational series of beings such that the members of this series are carefully distinguished from, and thus simultaneously related to, each other” (76). Dianoia cannot come to face with the one itself, since it is always dealing with it through a multitude, “because, although it is directed to noēta it nevertheless always remains related to that aisthēsis which first ‘called upon it’ to clarify an obscure state of affairs,” (78) that is, of contrariety (enantion). What dianoia can do is “replace the more and less, which always attaches to the realm of aisthēta by the exact relations of numbers, thereby accomplishing the most important step toward gaining that true epistēmē which no longer has any use for aisthēsis and whose object is the realm of those other noēta which ascend to something ‘unsupposed.’ But the dianoia itself is not able to appreciate the full range of significance of its accomplishment, because its own noēta, which it ‘supposes’ to underlie the aisthēta, appear altogether lucid and without further need of foundation” (79). There is an opposition in appearances that is provisionally reconciled by the implicit hypothesis of the eidē. But the how of this is beyond the capacity of dianoia. Number, numbering, counting provide the pre-eminent example and model that this is possible. It lies between the sensible—to which it can be applied, from which it can be separated and from which it arises (but which is not its cause)—and that which is above it, to which reflection on its relations points.^[14]

Turning from the Visible: The Philosophical Use of the Studies

I have spent so much time on what is just the first of the studies because I think it has far-reaching importance, in spite of the limited and literal context in which calculation is here presented in the text. The function of arithmetic and calculation in the more limited sense is of course itself a preparation—since it reveals itself as presupposing nonsensual monadic units, which turn us, when properly studied or reflected on (an issue with all of the preparatory studies), away from the sensible numbers of aggregates to the eidē. But to fully explore the meaning of unity, of calculation as taking the right measure, and the issues here raised, would take us further into the nature of the Ideas than we can go here. I have at this point only wished to stress that we are dealing with an activity as an image. Let us recall that the idea of right measure, the right metron, was to be key to our understanding of the application of the ideal city analogy to the healthy soul—and so the capacity for arithmos and logismos, is crucial.

The selection of the studies, and to a certain extent the ordering, follows the conventions.^[15] It is interesting however that although calculation is first in the studies, it is described as already compelling “the soul to use intelligence alone to discover the truth” (526b) while geometry and astronomy, which follow it, seem more to be linked to visible images, both in their initial objects and to the constructions of mathematical imagination in the actual practice of their respective calculations, as contrasted to the non-visual field of arithmetic. The link is misleading of course, as Glaucon’s comments regarding both the study and utility of astronomy in particular show. In a sense, as we shall see, the six studies before dialectic—arithmetic and calculation (worth keeping separate as number and counting), geometry, stereometry, astronomy, and harmonics—are arranged in inverse order with regard to their physicality: from number, which would be by tradition ostensibly the least overtly physical, to harmonics, which is the ratio of the sensed, that is, observed, vibration of strings. But when seen in their proper perspective, so to speak, we begin with the most concretely linked, the most elemental and constitutive of our image relation to the world of experience, numbering, and rise to the most abstract study, divine harmonics. The turning of the soul from the visible/sensible to the non-visible/intelligible is the reversing or inverting of the conventional perception of each study, if not of their sequence. Thus, as with the civic virtues earlier, which were taken over from the conventional list, both cases are given a different interpretation and a different purpose. This may be said to parallel the dual nature of the guardian-philosopher, evident throughout, and represented here by the dual task of the studies, that they be useful in both war and in themselves, i.e., for philosophy. The warrior-philosopher distinction, as already noted, must also be read as an image, and so we get war as a metaphor for the prevention, or the quelling, of civil strife in both the ideal public polis and the community of the soul itself. The warrior/guardian-philosopher image also serves as an image for the visible/non-visible distinction, and the division of doxa and epistēmē; the laws as laid down for the polis are doxa, as contrasted to the inner comprehension of their ground and necessity. But doxa and the war is left behind, as Glaucon is with some difficulty progressively weaned, at least temporarily, from the practical-external, i.e., doxastic level of his understanding of the utility of these studies. Since the turning of the soul is arduous and the result of the actual practice of the right study of these subjects, and Glaucon gets the point with difficulty, if at all, in theory alone, it is thus to be expected that his turning is a halting and fleeting one. The studies are of course practical, but not in the sense he attributes. Their practicality lies in their ability to turn the soul, and in the superiority of the warrior equipped with knowledge of what is (and what is not) on his return to the world of practical action.

Again, the order of the studies, the ascending order of apparent physicality (from number, through figure, to the vibrations of strings, etc.) of each study must be seen in its true nature as inversely and increasingly not this at all but rather intelligible, and serve in this way to increasingly wean the soul of its doxalogical dependence on the sensible, both by the contents of the studies and the way in which they are taken up, i.e., seen as not what they at first appear to be. Counting, however, is still first as primary, essential in the way we have suggested, and also as the most primitive, and in one sense the most physical of intellectual operations—separating and uniting the physical thises and thats. Socrates also says of calculation that he does not “think that you will find studies which give more trouble to the practicing pupil, or not many of them” (526c). It is hard to know what to make of this unless he means: a) the labors of the beginning young student; b) the larger sense of calculation, when it becomes in effect dialectic, the “right measure” of much more than the nature and objects of number, since “those who are by nature good at calculation are, as one might say, naturally sharp in every other study. . .” (526b, c) that considered rightly in its contrariety (regarding the nature of unit, etc.) it forces the soul upward by necessity (a term repeated often) to use the intellect alone to reach the truth.

Preserving the image of calculating and ordering, Socrates then says, “let that then be one of our studies [literally, ‘Let this one be so placed or situated for us’]. In the second place let us look at that which is next to it” (526c). Glaucon, in response, suggests geometry, adding that “In so far as it pertains to war, it is obviously suitable. As regards encampments, occupations of positions . . . etc., being a geometer or not makes all the difference to a man.” It is clear what level Glaucon is still operating on. Socrates’s rejoinder, with perhaps a slight suggestion of impatience, is, “Yes, but for those purposes even a small amount of geometry and calculation would suffice. We must examine whether the greater portion of it which is more advanced tends to make it easier to see the Form of the Good.” The use of “small amount” and “greater portion” is perhaps a poke at Glaucon’s literal mindedness for the reader. But it is not only the quantity or degree of study that is missing from Glaucon’s vision but rather the purposes of these higher studies. “All things tend in that direction which compel the soul [anankazei] to turn itself towards the place in which the most blessed part [to eudaimonestaton] of reality exists, which the soul must see at any cost. . . . Therefore if it compels the soul to contemplate reality it is suitable, if to see only what comes to be, it is not” (526e). To which Glaucon replies, “So we have said.”

Since Glaucon is still not clear on what exactly is going on, Socrates continues by, as in the case of arithmetic, pointing to a contradiction or tension which, when noted and properly considered, will hopefully point Glaucon (and the reader more likely) in the right direction. “No one who has even a little experience of geometry will dispute that the science itself contradicts the expressions used by those who practice it” (527a). This time the contradiction is not in the matter itself (though it can also be found there) as was the case with number and the unit, but rather, Socrates says, in the way geometers talk about the activity of geometry, that is, the nature of the discourse they must use in the activity itself. “They use ridiculous terms, and indeed it must be so. They speak like men of action and all their words point to actions. They talk of ‘squaring’ and ‘applying’ and ‘adding’ and do so throughout, whereas the whole study is pursued for the sake of knowledge” (527a). Though a feature of their discourse, contradiction is inherent here too since, as Socrates says, it is in the very nature of geometry, in the activity, that they use such terms: “they must do so.” Not that the geometer does not see the difference between digging a square hole and constructing a square on the side of a mathematical triangle, he knows that his actions do not occur in physical space and time—and that the side contains the square upon it, determines its area absolutely quite independently of the activity of the geometer. But the construction is a necessary activity of the geometer in order for him to see or release this knowledge^[16] —and it is combined with the drawing out of the implications of the relations of the figure (original and manipulated), which as rational relations are neither visible nor constructed (with the possible exception of congruence, though even here we can distinguish between the fact of congruence, the construction of it, and the judgment of it and the consequences (Cf. Theat. 186d-187a). Thus, though the figures themselves do not possess this contradiction, being themselves fixed, there is a necessary contradiction between the activity of the geometrical imagination and the fixity of the figures.^[17] The terminology of action is a verbal image and is “ridiculous” in this context, if necessary. But the reason it is ridiculous yet necessary, the cause and significance of this necessity, is hidden from the geometer, who knows perfectly well that his objects are not subject to “birth and destruction.” What he does not know is the ultimate significance of the difference, and how the activity of the knowing mind, its dianoetic eikasia, its capacity to reconstruct knowledge, in this case geometrical, is itself only an image in activity of the truth of objects themselves.

Of course another reason for Socrates’s saying that they “speak like men of action” is that Glaucon has just confounded the true nature of the activity (and its purpose in this context) precisely with the practical uses of geometry “as it pertains to war.” Hence we are meant to contrast such “men of action” and those who “speak like men of action,” the geometers. The geometer, like the poet/philosopher, creates in words, in images, but more real ones than those actions of the geometer-generals (both more real actions and more real figures created), whose figures on the battlefield “come to birth and [are] destroyed.” Not that the square formation on the battlefield is any less a square than that on the geometer’s chalkboard (or the converse)! But the one in the general’s mind is had as doxa (Alexander etc. excepted), put to work in the field, while the square in the geometer’s mind is possessed as epistēmē put to work in the theorem. Glaucon mouths the conventional wisdom that the study of geometry is “pursued for the sake of knowledge,” and that this is, as he argues, “knowledge that exists forever,” as contrasted to “knowledge of that which at some time comes to birth and is destroyed.” But Plato would claim that the latter is not, in fact, even susceptible to knowledge—and we know from the next discussion that Glaucon does not really understand what is meant by the permanent versus the changeable.

Astronomy is praised first by Glaucon as useful to the “farmer and navigator, but no less the general.” To which Socrates replies, “You amuse me, you are so like a man who is afraid of appearing to the many to prescribe useless studies” (527d). But clearly this is generous; it is not just that he is “afraid of appearing to the many” etc. but he really does not understand himself the difference, as the following discussion shows him still struggling. For astronomy is next praised by him as physically turning us towards the “eternals” in the heavens (see earlier discussion). It is only the distinction between the practical and the theoretical as rightly understood, between doxa and epistēmē, epistēmē as the image of being and the measure of justice and happiness, that eventually provides the answer to his question regarding the just and unjust life. Hence this study properly conducted will “draw the soul toward truth and produce philosophic thought”; again, conspicuously it is the awareness of necessary contradiction which forces the intellect to a higher level and, so considered, will “make us direct upward those parts of ourselves which we now direct downward when we should not” (527b). Shorey stresses the importance of a “sharp distinction between mathematics and dialectic,” and that Plato “affirm(s) the general inferiority of the mathematical method (i.e., that it deals with hypotheses) and the intermediate position for education of mathematics as a propaedeutic to dialectics.”^[18] But just how is it “propaedeutic”? Here it would seem rather important to see how mathematics when seen correctly is itself dialectical in its application. If it is to be truly propaedeutic it must have some essential features in common with dialectic and further, as we have shown, be seen as revealing certain tensions or contradictions or contraries which drive one to a higher kind of thinking, hence the dynamic of the Line is not only of levels but dynamic interrelation. These tensions or contradictions are present but unconsciously so, dianoia lives with them; they only serve as the conscious stimulus for upward motion when “studied rightly” with this purpose.

Concerning astronomy then, we should not confuse its higher purpose with what are its practical side benefits (from the present standpoint) and even its benefit “with regard to all other studies,” though this helps to recommend it generally. Astronomy properly studied, the science of the laws of heavenly motion, turns one’s gaze upward intellectually, inward, and hence the opposite of the “stargazing” (in Socrates’s mocking words) that Glaucon proposes. Here the tension or contradiction is between the visible upward gaze at the immediate object of study, the heavenly bodies, and the downward and inward gaze of the mathematical astronomer whereby “the embroidery of the sky [serves] as models in the study of those other things,” the “motions by which true speed and true number and true shapes, both in relation to each other and carrying along what these true shapes contain. These things can be perceived by reason and thought, but not by sight” (529d). The study thus results in a cleansing of the organ of sight, preparatory to dialectic, hence not (immediately) for the sake of others, as would be the case with astronomy for the farmer and navigator, who would see the present study as useless in itself; but rather here it is carried out for “one’s own sake.”Thus the “true astronomer” will admire the “beautiful workmanship” of the “movements of the stars,” but even these “fall far short of true existences” since their motions, “the relation of night to day, of these to month, and month to year, and of the other stars to these and each other,” are such that “he will consider it strange to believe that they are ever the same and do not deviate at all anywhere, since they are material and visible, and he will undertake, in every way he can, to understand the real truth of these things” (530a). Thus the “material and visible” nature of these, means that even their motions and relations are not “ever the same” and hence provide once again in their differences, the contradictions that stir intellect to search for the “real truth of these things,” that is to say, of the concepts brought to bear on them in the study of their relations and the laws of their motions. Hence the true way to conceive of the study of astronomy is analogous to that of geometry, through the study of “problems” with regard to the mathematical relations and the study of their elements, to which the objects themselves only roughly correspond.

The study divides into the exercise of the study itself, of the now mathematized objects, and that of the study that arises out of this, the study of the nature of its elements and laws. “Let us then study by means of problems as we do in geometry and let the things in the sky go, if the study of astronomy is to make the naturally intelligent part of the soul useful instead of useless” (530b). The study of “problems” may be taken to refer to the mathematical-astronomical studies themselves or to the higher studies. It is not clear if this must be within the mathematical study, (not the variation of the physical events, since we are to “let the things in the sky go,”) or if it is the study of the variation in the mathematical figures (reflecting the physical variations of motion), since Socrates says that the philosophical astronomer will undertake to “understand the real truth of these things,” where “things” refers to the mathematical models of the relations of night to day, etc., that is, of the physical of which they are more intelligible images or models.

One might also be led to ask what the difference between geometry and astronomy is, once the latter has seemingly been made into the study of problems and having let the things in the sky go. However, astronomy even in its mathematical form, is still the study of relations of relative position and of motion. Motion, it is true, that is formalized and absolutely regular, but motion nonetheless. And it is as the study of motion that it perhaps has the advantage over geometry to drive intellect further to ask about motion and rest per se, as well as about “the equal, the double, or any other ratio,” and again, about same, other, unit, etc. If the motions of the heavens are explained with the use of these, even when turning away from the physical representation of them to the representation of them in thought, i.e., on paper and in imagination, they themselves can become the object of study precisely in the observation of their paradoxical use, as with the actions of the calculating arithmetician and geometer—which can drive one in each case away from the actual field of their use, as hypotheses, to the study of the elements.^[19]

It is worth noting as a result that action and motion has been introduced at each stage, for each study, in a different form. And that the progress from one study to another, while all within the provenance of dianoia (the studies, not the progression and the study of it) is from apparent stasis (numbers) to increasing degrees of motion—but motions which are each time revealed as intelligible, inner images as opposed to their outer shell. So, arithmetic is dividing up, counting, geometers use action words, astronomers study motions of the heavenly bodies. In each case this outward motion is a reflection of an inner motion, rationally perceived, ending in dialectic, motion completely freed of its external context, the motion of thought itself. That is, it is motion in the study of dialectic itself, and of its application in the understanding of what might be called its pure objects, the elements themselves. By contrast, in the application of pre-ontological dialectic in the everyday world, the results of its divisions and collections can be physical, still in only a relative sense: the mule standing over there with the horses can be moved to stand with the other mules, though need not be in order for thought to make the distinction, and so with the unjust man among the just.

So it is that Socrates continues, after finishing with astronomy, that “motion is not of one but of several kinds,” and as astronomy is the image that the eyes are “fixed upon” so the sense of hearing is “fixed upon the movements of harmony” (530d). An external physical motion or movement, of the vibrating string, is opposed to the actual object of study, which is the non-sensible, non-visible image—that of the laws of the relations of tones—the silent mathematical ratios between the sounding tones, which give the isolated physical pitch events their identity as tones.^[20] The kinship of these sciences is specifically mentioned. In astronomy it is natural to look at the physical images in the sky and it is intellect that must turn downward (the inverted physical motion again the image of its true meaning). Harmony has its own physical images which are attractive for the natural imagination as explanation; those who literally listen closely place the ear before the mind, and imagine that they are measuring—imagination, eikasia, is used to generate doxa about the relations of the tones.

But there is a second group of people who study harmony wrongly, or in a limited fashion, though implicitly further along than the first. These are those who find number in the tones and relations, whose images are thus non-visible as contrasted to the perceptible images or vibrations sought by the former group. They are introduced as representative of dianoia, like the mathematical astronomers, and thus study the laws of the motions of the tones—but only to observe them and produce what might rightly be called dianoetic doxa. For though dianoia is systematic by nature, there is the danger of mythologizing the system, of turning it into a cause in itself rather than seeking its cause higher. They do not, as does the first group of “harmonists,” study “consonances and sounds which they . . . measure against each other and so labor in vain,” rather, the second group measure or calculate the laws—the ratios of the string lengths, etc. But they turn these, the ratios or numbers, into causes: “for they seek for numbers in the consonances they hear but do not make their way up to problems to examine which numbers are consonant and which are not, and what is the reason in each case” (531c). Only when these “consonances” and the laws of their motion are studied as images rather than themselves, are they “useful,” “in respect to the search for the Beautiful and the Good, while pursued in a different way they are useless” (531c).

Images of Dialectic and the Provocations of Intellect

It is the “kinship” of all these studies (and any others) that they are capable of turning the soul from the visible images to the non-visible images that reflect intelligibility.^[21] In each case the point is to see the model, law, or term not as a cause in itself but as a non-visible image, having its self-identity, meaning or intelligibility at some other level where it is truly self-identical and not an image of something like itself. The studies, ending with the “movements of harmony” are but the “prelude to the song that must be learned,” that is, to the exercise of dialectic proper. The use of nomos, as Grube points out, puns in Greek on the double meaning of both song and law. But the pun also has the significance of establishing a continuity, the series of studies which end with harmony and with the contrast between a study of the physics of music, the esotericism of the Pythagoreans, and the platonic study of the laws or nomos of harmony, that is, its intelligibility and necessity. The same matter, the regularity seen as laws, is seen by the dialectician as images of necessity, so “those who cannot give an exact and reasoned account of what is said [do not] know anything at all of the things we say they must know” (531e). Dialectic, then, produces the image of discourse as the song of reason, of which the relations of Ideas as a harmony of the whole is the ultimate objective content. It is the true noetic science that produces the image in speech of the laws that are the song of, for example, musical harmony. The identity of the tones of the literal, musical harmony is inseparable from the relations of the tones, and is expressed in the laws formulated by the dianoetic science of harmonics. Analogously, the identity of Ideas, the eidetic elements reflected in the discursive elements that noetic imagination images, is inseparable from the relations of or among the Ideas. These eidetic relations are expressed in the relations revealed by dialectic and formulated as the laws of their motion both in their inter-relation, and as the motion of the Ideas downward. These two motions, the horizontal and the vertical, as reflected in the Line, are seen in both subject and object. That is to say, it is seen in the medium of visible and non-visible existence, and in the corresponding and co-temporal motions of intellect in the reinterpretation of existence as mirroring knowledge. Seen in the light of truth, what before was merely existence and doxa, in both visible things and in human action, becomes knowledge (though even as doxa it is possible only as a melee of images that express and rely, unperceived, on the Ideas). The study of musical harmony participates in this motion, and is the last image of it before dialectic, but does so unknowingly. It is the last science, paradigmatically and mythically, that lacks noetic self-consciousness. Dialectic, on the other hand, has as its law precisely the self-awareness of reason—whereas other sciences are images of reason and take their objects as embodying reason objectively. Hence the Line’s forth level, noēsis, is the home of fully self-conscious dialectic, arriving at its conclusions “by means of Ideas and through Ideas,” and as the highest form of the hierarchy of capacities. These capacities of the intellective soul, or ways of knowing, are distinctive for each kind of object knowable, but also linked vertically as intelligence is driven from a lower to a higher level of object and corresponding activity. We shall return to a further consideration of dialectic momentarily.^[22]

How literally are we to take Plato’s prescription for just these studies as those which prepare the philosopher for the higher studies of dialectic proper? The particular studies chosen can in fact serve to turn the philosophical soul from considering the visible as the real to the non-visible present in the everyday. Similarly, it is important that the image of the city constructed is really possible—though I have argued, only when that possibility is considered as having its locus in the individual soul, the personal city within. These particular studies are chosen partly because, like the virtues of the city and individual, they are both the conventional wisdom of what training in the latter case and learning or education in the former consist in. With the arrival of the sophists on the scene there is of course much controversy over what is to constitute a proper curriculum, but at least one tradition, represented by Hippias, is contrasted to that of Protagoras (cf. Prot. 318d-e) and is closer to the one proposed by Socrates. That conventional wisdom has merit in both cases. But in both cases the description of them in the Republic is seriously altered from what it would be conventionally, to enable us to see them in the light of the larger purpose they are made to serve as “useful and beneficial” in the turning or re-education of the soul. Thus, for the reader they serve to illustrate, are images of, teaching the soul to see rightly. And the arrangement of them in the particular order given is, I have suggested, also meant to illustrate the criteria and methods by which what is real is increasingly revealed. For those who practice the studies, again stressing seen in the right light and continued far enough, the experience of the practice of them is a training of the intellect, a habituation of judgment via the experience of the reality of the non-visible in such studies. But in the Republic and in life they also serve as images, though actually working ones, of both the Ideas and human intellect at work—and other activities may serve both as this practical experience, and as images to be reflected on for their ability to lead us further. (This is not to deny that these particular ‘sciences’ are perhaps essential paradigms for insight into the intelligible structure of the natural world, and in particular calculation.)

Furthermore, the ones chosen are described as activities in such a way as to bring out, even in their non-technical form, basic truths about knowing, being and human being—such as counting “one, two, three” and the divisions of like and unlike, unity, etc. For the perception and study of these activities and ideas at work, clearly other studies might serve—biology, for example—though some better than others. On the other hand, arithmetic, or more basically, counting, is clearly a special case, serving as both fundamental image and primary activity of human calculation or “measuring rightly.” That the studies are not meant to be strictly prescriptive is particularly clear when one considers the last study, that of harmony, even in its Pythagorean incarnation.^[23] The choice of harmony ought to instruct those who would limit insight into knowledge and being to insight into the special, particularly mathematical, sciences. True, the study of music proposed is with regard to its mathematical properties, etc. yet it is still what is the basis of sensual sound, and of the presence of the Beautiful. And the placing of music as higher than astronomy and geometry, which could have served as well for the strictly mathematical aspect to illustrate these, is instructive. (Though there is also the motive, as described above, of arranging the studies to illustrate increasingly intellectual kinds of motion and the inversion of the physical world.) The choice of the structuring elements of an art as the last study, one designed to bring pleasure, albeit here an intellectual one, is surely significant. For it occurs in a discussion whose overall purpose after all is to demonstrate the beauty and ultimately the desirability of a unity that is beyond the limited, if pure, one of mathematics and the sciences—the unity and being of the just soul and its beauty, both within itself and in the world that it knows (astronomy, as well, had a movement from the beautiful of the starry heavens (530a) to the study of “true speed and true slowness”). Geometry and astronomy could have served if only the perception of the mathematical intelligibility of Nature were at issue. Harmony, though here as purified, is significantly aesthetic and erotic, as befits a study that prepares the soul for dialectic’s approach to the Good. Thus finally my answer to the question as to whether these studies in particular are to be taken seriously as the preparation of the philosopher is no, not if taken literally as the only ones capable. And that it is their description in the Republic that is ultimately important to understand—for it is in understanding this that one would be able to decide “which studies are beneficial and useful and which are not,” and taken up in which way, to what final end.

Mitchell Miller, in “Beginning the ‘Longer Way’”^[24] provides a detailed and careful analysis of the five studies and their specific and apt role in preparing for a final insight into the Good. Miller distinguishes, as do I, between the argument addressed to the interlocutors and that addressed to the careful reader/listener. The studies are both a progression away from attachment or immersion in the visible, a “purgation” of the soul, but also a return via the studies to a reinterpretation or “reconstitution” of the visible, of the world now seen in terms of its intelligibility. Miller argues that the last of the sequence, “harmony,” takes us to the doorstep of the Good, by introducing us to the notion of ratio contained therein, which unlike the others is “neither visible nor spatial” (321). “With the turn from the geometrical disciplines to harmonics, then, we recover in its pure intelligibility the innermost structure of all that becomes, both the corporeal and the incorporeal” (322). Further, “the study of harmonics brings to the fore the complex unity that makes for wholeness and harmony” and provides the philosopher-in-training the basis for appreciating the “normative status of complex unity” and “a keen appreciation for what is whole and harmonious” (333). Miller develops on the basis of this a notion of intermediates that are “perfect exemplars” shaped by the “forms they instantiate,” and in particular the notion of “perfection as such” (326-8), which, he argues is “like the forms in having the character of a function, not an object [my emphasis throughout] . . . providing the context in which the purely intelligible perfect figure can come to mind. “Perfection as such,” according to Miller, is the very thing that is the “goodness of the Good,” (328) and which is the ground of the ontological functionality of the Good as cause. Miller thus attributes, rightly I think, a kind of indeterminancy to the Good, as a kind of richness. Each Form is “an itself determinate way of determining perfection as such,” for example, in triangularity, as prior to its instantiation in a particular intelligible instantiation in a perfect triangle. “Perfection as such, on the other hand, is in and of itself indeterminate, hence it both transcends and lends itself to the host of ways of determining it that the Forms just are and that their intelligible instantiations exemplify” (327). The Forms themselves are the expression in their “basic nature” of “these different ways of determining” perfection as such. That is to say, the Good is an openness out to the world that becomes concrete in the multiplicity of its instantiations. Miller’s understanding of the whole apparatus is much like my own. It is as a dynamic set of functions set in, and setting, a context that structures the world, and our soul’s capacity when equipped with the right philosophical education, for our reading of it in its intelligibility. We have then our rule in this light of both others and ourselves—mirroring the rule of the Good over their Forms and the world in its intelligibility. This is both ontological for the productions of being and epistemological for the production of epistēmē. This is very close, I would say, to what I argue. The difference in our interpretations is that I would not take the educational program as prescriptive of the sole route of access but rather more themselves as images, as well as models, albeit singularly apt, actually functioning and productive ones. Otherwise, it would seem that one is limited to the reading of the intelligibility of the world as an object of the natural sciences, and the practice in the years spent back down in the Cave if literally only so equipped as somewhat reduced to a preliminary. Miller is led to acknowledge at the end that “the breadth of the longer way” in including the ethical, political and other dimensions of human life and virtue, is yet to discover, in the “turning of our soul to all of being.” “The more deeply our path is illuminated by mathematics, the more urgently we want to understand whether we can extend our insight to, for example, the spheres of the ethical, the political, and the religious” (341). I would add, to the ontological ground of these.

We turn now to a further consideration of dialectic, but it must be said that with regard to dialectic as a precise method not much is explicit in the Republic itself. We should note, however, that it is the whole process of the movement from “the deliverance from the bonds and the turning from the shadows” up to the final vision of the Good, that is described as dialectic (532a). Thus, “the power of sight would be imitating it [be an image of it] when we described it as attempting to look at actual living creatures, then at the stars, and finally at the actual sun.” Dialectic is clearly distinguished from the operations of the intellect at each of the first three levels of the Line, since dialectic is to operate “without any help from the senses but by means of reason,” and only then and because of this condition, if one does not give up, “does one reach the final goal of the intelligible as the prisoner escaping the cave reached the final goal of the visible” (532b). The other three levels are in one way or another, as already discussed, tied to the visible—always understood here as comprising not just the physical but all the objects of doxa, and of dianoia insofar as it operates from hypotheses.

And yet dialectic may also be said to be both present from the beginning in all knowing, as well as being the special technique of the forth section. It is present in the other levels in two ways. First, as the inner tensions of the objects themselves which, in the contradictions of their appearances or showing of themselves, have the potential to call forth reason to resolve them—as in the examples of the fingers of different lengths, and received doxa about matters of piety and justice (shadow identities). Secondly, as discourse: each level of discourse is a reflection or image of the truer discourse of dialectic in its pure self-conscious form. Thus even the blind arguments of the cave dwellers are a dim image of real dialectic; their arguments lead nowhere, but at least they reflect the ability of human beings to distinguish, if not rightly, this from that, to argue about the identity and sequence or relations of the shadows. The turning of the soul away from the shadow wall and towards the cause of the shadows, in the puppets, the parading puppeteers and the fire, is not a complete negating of their earlier discourse but rather a resolution of it (primarily aporetic at this level, hence already the danger of misology), a resolution in the light of which their former discourse is seen as mock dialectic, yet still having the dialectical features of all discourse and hence prototypic to the keen observer.

But the ability of objects of study to call forth intellect in such a way as to drive it upward is dependent on the openness of intelligence to ontological and epistemological tensions at each stage, and thus the necessity for the reformation or turning of the knowing soul towards those tensions which otherwise remain implicit, hidden in ordinary discourse. Once again, it is important to see the several senses in which the journey of dialectic is the whole journey, and hence, in the recapitulated image of the Cave journey, at 532b-c, the images of “living creatures and plants and the light of the sun” are called “divine.” But not the shadows seen in the cave, which are the received images of the doxalogists. The images which the latter carry (though including ‘objects’ found in nature) are made images constructed or based on elements that for the makers are still doxa—they do not ask for, and cannot give, an account of their source. In ascent, we are said to be looking “no longer at shadows of images thrown by another source of light [the creations of the doxalogist] which is itself an image as compared to the sun.” Only the last set of images, those that the sciences make or use as dianoia and which correspond to the images and reflections outside the cave, approach knowledge closely enough to merit being called divine. “The practice of every study we have described has this power to move the best part of the soul and lead it upward. . .” (532c).

So far, the prelude and the surrounding discussion has given but an image of dialectic, but when Glaucon asks for the song itself, “the way in which dialectic works, what its parts are and what path it follows,” Socrates replies, “Not yet, my dear Glaucon, will you be able to follow—it is not that keenness is lacking on my part,—for you would no longer be seeing an image of what we are discussing but the truth itself, or so it seems to me” (533a). One cannot, then, describe what dialectic does to one who does not already have the understanding of its realm of operation and some experience of its method. And Glaucon has just demonstrated in his consistent misunderstanding of the nature and purpose of the higher studies, that he is in fact either incapable or not ready for more than an external image, in the description of the studies, poorly understood at that. That is, as a friend of the Ideas, he operates still primarily at the level of pistis. “It is not worthwhile insisting that it is so in fact, but we must maintain [ichuristeon] that one would say something of this kind. Is that not so?” “It surely is.” “And that the power of dialectic will only appear to one who is experienced in the studies we have described, and cannot ever appear otherwise” (533a). And then too, one may add, when they are taken up “rightly” and not “as now presently practiced.” For the sciences, while they “grasp at reality to some extent,” (533b) as long as they continue as sciences in making use of hypotheses without examining them, are cast as “dreaming about reality, unable to have a waking view of it.” The task of dialectic is to be in a position to do what the sciences themselves cannot do for themselves, which is to “give a reasoned account of them.” We are not, however, in the same position as Glaucon, to the extent that we grasp the nature and purpose or use of the studies, and see in their description, as well as that of the Line etc. the hypothesis of the Ideas that they are built on and point to. In this, we have an initial giving of an account of knowing and being.

Though we are not given an immediate example of dialectic, we are given an external description or image of the behavior of dialectic. This is given in the descriptions of the philosopher’s preparatory studies, and in those of the Line and Cave. In other words, we are given images of the world as necessitating and supporting dialectical questioning if knowledge is to be possible. Hence the appropriateness of a recapitulation of the Line in the text at just this point, since it is the Line which is an implicit image of dialectic, not just as a human endeavor but as a dynamic mapping of knowledge and being in its appearances and images. The Line is a schematic representation of the ways in which the non-visible is available in experience, and the basis of it. It is therefore at this point that the divisions of the object-world presented in the schemata of the Line get matched more concretely to the pathēmata as corresponding ways of knowing (cf. 534a, 511d-e). The Line discussion per se, gave the divisions of the world into objects of knowledge, and a schematic description of the behavior of the knowing subject, but only gave names at that point to the capacities of the soul that specifically deal with each segment. What was formerly noēsis becomes here epistēmē, and the former term, noēsis, is reserved for both upper sections taken together. There is, further, a significant difference in the characterization of the seeming repetition of terms of the two lists. The first time, at 511d, these terms applied to their respective divisions of the object-world of the Line are called pathēmata, and we are “to consider that each has as much clarity as the content of its particular section shares in truth.” Glaucon characterizes them just above this as an “attitude of mind.” “You seem to me to call the attitude of mind of geometers and such reasoning but not understanding.” On the other hand, at 534a, after the elaboration in detail of the studies and culminating with the description of dialectic, these same activities are now seen differently. First, they have now become visible as the active image-making powers of the soul (dunameis as opposed to pathēmata, though the former term or name does not explicitly occur in the passage, but neither does pathē). Secondly, they are themselves a prime subject matter of dialectic, the new science productive of true epistēmē.

Each of these (eikasia, pistis, dianoia, epistēmē ) are in their own way and degree dialectical—but only the final practice of dialectic as a self-conscious study of being and knowing attempts to “apprehend methodically, with regard to each thing, what each really is.” Note that dialectic is itself not listed as a kind of knowing or knowledge. In the original description of the Line, it is the “other [highest] section of the intelligible [as a whole] . . . which reason itself grasps by the power of dialectic” and then “having reached this [what is beyond hypothesis, the first principle of all that is] . . . it does come down to a conclusion.” The mathematized objects of reasoning of dianoia are the conclusions via hypothesis; while for the reproduction of the Ideas themselves noēsis is the product, and dialectic the method. But another reason for not including dialectic in the list of ways of knowing is perhaps that, as I have tried to suggest, dialectic is implicit in all discourse about experience and only in its explicit form does it (gradually) take leave of the sensible, i.e., the individual. Further, it might be said that the dynamic of discourse has broadly the same implicit features at every stage. This is so even at the lowest level, where it is seen in its satiric form in the shadow play and dynamic of the Cave, and in the analysis of what calls forth intelligence, beginning as it does with Agamemnon’s feet. Finally, the tension of dialectic is implicit in both Being and its appearances—the relation of Ideas as the inner tension in the former and the flux of identity in the latter. Thus the dialectical essence of discourse is an image in its motion of the dialectical essence of what is in itself and of what appears. In its motion it points to reconciliation in the principles of knowing as the principles of any intelligible being. Whether the forms and principles of this knowing are as Plato thought, and hence are the forms of the being that they reveal, is a question for reflection, but even another analysis, say Hegel’s, would be looking for the mapping of intelligible discourse and intelligible being, and discourse would remain in principle still the non-visible and dialectical image of being.

At the end of the brief discussion of dialectic, Socrates suggests, after listing the powers or capacities that correspond to the levels of knowledge and objects of knowledge, that their relative order, the original order in which they stand in the description of the Line, can be alternated, hence their relations: “As being is to generation, so intelligence is to opinion, and as intelligence is to opinion, so knowledge is to belief as reasoning to imagination” (534a). The main divisions of the Line, and hence the world that it images, is into Being and Generation, corresponding here to Knowledge (noēsis) and Opinion (doxa). Generation must obviously, we have argued, be understood as referring not only to the literal world of physical nature and our opinions about it, but to all the realm of the opinable, hence the sphere of human action and judgment, which is the external image of our opinions about justice and the like. We “generate” collectively and individually, our changing opinions about the world—how we take it to be, how we think it ought to be, and how we think we ought to live in it. Dialectic attempts to sort this out, to replace opinion with the broad category of knowledge. But the alternation suggests as well that the same relationship holds between knowledge (epistēmē) and belief (pistis) and reasoning (dianoia) and imagination (eikasia): the second term is less stable, less reliable than the first in each case—but we knew that already from their place in the Line, the hierarchical relationship is not changed by the alternation. We are instead asked to focus on the specific relations of these smaller pairs. As Being is to Generation so is epistēmē to doxa. The difference between the two halves of the Line, between noēsis and doxa is one of relative reliability or firmness, relative absence of change or motion. This same difference holds between the terms of the divisions of each half, and, it is suggested, between the terms in alternation. Dianoia is a kind of Intelligence as contrasted to Opinion, but relies on assumptions (hypotheses) while epistēmē characteristically does not. Epistēmē is firmer than dianoia. A similar comparison is elicited by the alternation Socrates suggests: between the kind and source of the firmness held by the dialectician and the characteristic firmness of pistis, which is firm belief, tenacious but ungrounded (at least with regard to consciousness, that is, knowledge). The second half of the alternation, “as reasoning (dianoia) is to imagination (eikasia)” presents, analogous to the first pair, the contrast between two kinds of system making: the logical, if hypothetically based, ordering of dianoia, contrasted with the random, baseless (or intuitive?), the ordering of imagination. In each alternation one member is an image of the other, not of their respective objects (as is the case before alternation) but as to their respective processes or methods—the one being a weaker or defective image of the firmer method of the other.

But these relations of contrast or tension, negative relations, are matched by a more positive relation of the pairs. For the first pair, it may be said that the firmness of belief, while itself ungrounded is not altogether unfounded. Fortunately, much of what we believe is true, or true enough to get along in the world, in both the physical and even the ethical world. The ideas and appearances that we trust get us through the world precisely because they are trustworthy: the physical world is reliably the same, lawlike, and the social world is more often than not, reliably ethical, though wholly conventional. Lest I be misunderstood, I mean only to say that though there is much vice, false virtue, and virtue misconstrued, still, conventional virtue is the cement that held Plato’s world together as much as ours—we do not live for the most part in ethical chaos. This is not, of course, to deny Plato’s challenge, examination, redefinition and transformation of the virtues as already discussed above and elsewhere, for example as we see even in the earlier education or training of the guardians, and to be further examined in the Phaedo chapter to follow (67b-69e)—let alone in the other Socratic dialogues. The function of philosophy, in Plato of dialectic, and hence of recourse to the idea of a necessary relation between true virtue and knowledge (noēsis, epistēmē in the alternation) is not necessarily always to replace this behavior with something else but to ground it. It is to show why it works, when it does, to justify it, and to provide the grounds for its reinterpretation when it doesn’t. (Then of course there are those elenctic dialogues where the final outcome at least appears to be aporetic. Furthermore, Socrates’ redefinitions in the Republic are, I have argued, in the light of special ends.) Hence, Cephalus makes the right choice, not to return the weapon held in trust, but instead of re-examining his ground, the maxim concerning telling the truth and paying one’s debts that he had offered, he retreats to doxa, the sacrifices and the poets, rather than seeking higher ground. The opinions of firm belief, at least certainly their elements, are reliable, to the extent that they are, because they are a workable arrangement of the elements, albeit unreflectively and in unrelated contexts—in short, they are images of noēsis in a much more direct way than dianoia, in that they are reflections in the everyday world of the highest elements of thinking and being. To take Polemarchus’s line, or rather Simonides’s, justice is indeed “giving each his due,”—there is something intuitively right there, and it is what we conventionally expect, strive for, etc.—the trick is to figure out what it is that is due—and what the who is to whom it is owed. Dianoia’s relationship to this world, as sketched earlier in the account of the Line, on the other hand, is in a sense more temporary—as system or theory it is a bridge, or as Socrates says a “springboard” to the Ideas of noēsis. Dianoia is epistemologically more stable of course than anything in the bottom half. Yet perhaps the ontological link between the Ideas of noēsis and their images in everyday discourse and the experienced world is stronger.

The positive link of the second pair, between dianoia and eikasia, Reasoning and Imagination, is in a sense easier to see. The circle we construct, physically or in imagination, is an image that feeds and supports the reasoning and is recognized as an image of what dianoia is really talking about, really looking at. The former are the visible circles, whether drawn with our finger in the sand or with a compass (or again, imagined)—the circle as an object of pistis on the other hand, is what common sense believes or unreflectively understands as a circle. The court judge in considering the case before him can only judge it as it conforms or not to the law. Dianoia works on the object of common understanding (pistis) but works with the image-making that represents the concrete particular. But the individual case is much more than a visual aid. The roughly drawn circle is in fact roughly circular, the man before the judge is in fact roughly either acting justly or unjustly, and in both cases the individual is a particular roughly similar to, the same as, other cases—the circle in the sand, the moon, the orbit of the earth. The ability to see these resemblances and also that they are individual and different, is the beginning of philosophy, the philosophical capacity par excellence; and the possibility of this activity rests on the general ability or capacity to see sameness and difference, and the provocative reality of the approximate resemblances and differences before us. The reason that this faculty is called image-making is that we bring these resemblances together (and exclude the different) all the time—sometimes wrongly sometimes rightly, sometimes well sometimes badly—but without a critical awareness of what we do, or of how it is possible, which are the questions of philosophy. When eikasia performs this noetic function for dianoia it is taken up into dianoia as dianoetic eikasia. This is contrasted to its normal or everyday natural functioning. The verb eikazō can mean either to represent by a likeness, or to be like or resemble, or to liken, compare: to infer from comparison, to conjecture, guess (Liddell and Scott). Dianoia is directly related to this image-making because it is reason’s first attempt to get it right, with the aid of hypotheses from above and the use of images, and examples of and for comparison, from below. It is thus in the middle as both temporary and on the way, between. Its relationship to pistis is that it is a better sorting out of the appearances in image-making. (Pistis as doxa has more to say than image-making, but it tends to babble rather than explain.) Both of these pairs, then, help to explicate the relation of the non-visible world to its manifestation, appearance, and use in the world of appearing, of action and activity. Both relations are at once aporetic or unsettling and purifying and confirming; and each higher term is in a peculiar way not only an image but a measure of the method or activity of the other term.

There is one more topic to take up with regard to both the preparatory studies and dialectic itself. Neither leads inevitably to its natural end. Even when studied rightly it is possible to be derailed and misdirected, and this is particularly true, Plato urges, of that early stage of the practice of dialectic which involves one in the technique of questioning people and things. It is for this reason that Socrates concludes the brief discussion of dialectic in the Republic with a discussion of the dangers of dialectic, or what turns out to be a false image of it. In effect, “great evil . . . has now come upon dialectic. . . . Those who practice it are filled with lawlessness” (537e) which is the very opposite of what one would expect. The problem is, that while we cannot assume that with age comes wisdom, as the episode with Cephalus shows, nor can we assume that with youth comes foresight and prudence, in fact the opposite is more likely. Socrates suggests that we consider the image of the vigorous, enthusiastic student—a good candidate for further studies. What, he asks, should happen, if the child of rich parents, who had sent their child away to school with “beliefs about just and beautiful things,” suddenly discovers that “he was not the child of his professed parents … he would honor and care for them less, while his relations with the flatterers would become more intense, he would be persuaded by them far more, would now live in the way they did.” These “flatterers” are in fact his teachers, or at least offer themselves as such along the way. “And then a questioner comes along, and asks a man in those circumstances what is the beautiful. and, when he answers what he has heard from the lawgiver, the argument refutes” both him, his parents, and his spiritual parents, the laws of the lawgiver of the community, from whom he had heard about “beautiful and just things.” This reduces him to the “belief that this thing is no more beautiful than it is ugly, and the same with what is just and good and things he honored most” (538d-e). True dialectic would have led to “what each thing is,” while false or misused dialectic leads to believing that nothing is; that neither of the opposing predicates can be applied with justice. The result is that, when he no longer believes these original values, and has not yet discovered true ones, he will lead the life and adopt the principles of the flattering kind. Such youths are attracted by the “pleasure” of argument which “flatters our soul”—those who resist this absorption in dialectic as an end in itself are those “who are at all moderate.” “These honor the principles of their fathers which we mentioned and obey them;” instead, he “becomes lawless.” This use of dialectic, in place of uncovering and confirming order or law, is rather a misuse—resulting in lawlessness in both the individual and the realm of activity, whether literally the ethical dimension or as an image representing the object-world.

The difference between an “older man” ready for dialectic and “those young puppies” who discredit philosophy, both in the eyes of others and for themselves as a method of arriving at the truth, revolves around a difference of character. The young have been seduced by the flattery of soul, that of the sheer power of reason: “When youths get their first taste of reasoned discourse they take it as a game and always use it to contradict. They imitate those who cross-examined them and themselves cross-examine others, rejoicing like puppies to drag along and tear to bits in argument whoever is near them” (539b) The attitude, if not the skill, of some of the interlocutors who periodically interrupt, trying to catch Socrates out, and elsewhere in the dialogues, e.g., Meno—in contrast to Socrates’s gentle treatment of Cephalus. Note that Adeimantus arrives on the scene with the opposite party, with Polemarchus. The use of the puppy image is interesting since it had just been recalled in a positive light to describe part of the early formation of the guardians: “You remember that we stated [at 467] the boys were to be led into war, as observers on horseback, and, wherever it is safe to do so, they should be brought close and taste blood, like puppies” (537a). But even there, if we read the present image, which is the metaphor revealed, back on to the earlier discussion, where the image is still veiled heavily, we see the distinction at work. For although the “future warriors” are allowed, it is said, to “taste blood,” they do so actually only visually, from a distance—they are to be taught to ride early and observe the battles “mounted on horseback” to allow for “easy flight.” And so, “We must make opportunities for the children to observe war while contriving to keep them safe.” As future warriors it is worth exposing them somewhat to risk. But precautions are taken to minimize it. Fathers will not choose expeditions that are too dangerous, and more importantly, “they will put in charge of them officers whose age and experience qualifies them to be leaders and tutors.” And if the unexpected happens, they are mounted on docile horses, “not fond of fighting or high spirited … and if the need arises, they will find safety in following their older commanders.”

Rereading this in the context of the later passage about dialectic as education, the earlier image reveals more of itself as extended metaphor for education in the later sense, the war between ignorance and knowledge in which dialectic assists. Thus the children are put under tutors to “observe warfare”—which one could see as perhaps the reading and discussion of the schools before entry into actual participation, whether in the actual debates of philosophy or the debates of the city. And the “retreat by horseback” is suggestive of the refuge of conventional beliefs, as in the later context—so the horses are docile and not fond of fighting and the retreat is to “their older commanders,” the guides of right opinion. The earlier image is a fainter image of the later—i.e., more physical and more disguised an anticipation. They are related, as images, as discussion of training (the formation of character by habit, etc.) is to true education, physical war to the wars of discourse, and as degrees of disclosedness about what truly “is and is not to be feared.” The older man exercises dialectic with restraint—“in order to discover the truth rather than one who is merely playing and contradicting for play, he will himself be more measured [kai auto metriōteros] and will bring honor rather than discredit to the pursuit of philosophy” (539c). The use of metriōteros here is significant since while it also means moderate, its use in place of sōphrosunē stresses the context of the discussion of dialectic, hence the right measure of truth which reflects the internally ordered soul, itself both rightly measured by philosophy, and brought into right measure by sōphrosunē, and thus in turn capable of the dialectical right use of its capacities to rightly measure.

Moral harmonia is an existential precondition for accessing the harmony of subject and object noetically, the correspondence of knower and known. But the ontological justification of the harmonia of the soul (justice) and of the construction of its pre-philosophical image in Books II-V, is the philosophical insight into the whole which is the firmest grounding of this harmonia of soul in the moral sense:

But this vision of the whole is the fruit of hard labor, and the true dialectician is to be distinguished from the individual who merely plays at dialectic for amusement and the pleasure that is flattery of the soul and a destructive mere technique, negative in its results. The ordering of the soul permits its ascent, but is carried out in the light of what is seen, only fully justified at the end of that same ascent.^[25] This is the Platonic circle, the image of which is the Republic itself. The final end of the journey for the whole soul, as opposed to the rational part alone, is its return to the place where the insight of reason is put to use. It is here where the unity of the whole seen as a reflection of the Idea of the Good is “useful and beneficial,” in the return back down the Line to know what each thing is and “give a reasoned account” i.e., engage in discourse, and in the world of action to “live well,” to “put in order the city and its citizens as well as themselves” (this is not to say, however, that such a return, such a comprehensive vision of the unity of the whole is ever actually fully completed).^[26] Still, one might have a lingering suspicion that the rational part, albeit not a person, is being treated unjustly. But even here, one might reply, the rational part is deprived only of continuous uninterrupted contemplation, while its work in the city is a deepening insight into the way the forms have an ordering presence. Dialectic fully formed and put to right use thus produces not a deprivation but a deepening fulfillment, in which the rule of the rational part has the opportunity to increasingly mirror image the rule of the Good.^[27] To rightly understand the ordering, both of the individual soul/city and that of the reform of the community represented in the Cave, we must understand the soul’s capacities (dunameis), its powers, and what it would mean for these to function rightly. Understanding this ordering is in effect, speaking generally, also to understand the relation of justice and virtue. ^[28]

In the next chapter, we shall take a closer look at the notion of soul, and its capacities or powers (dunameis),^[29] through examining the notion of the separateness of Ideas and the separating and image-making abilities of the soul, as well as the soul’s separateness itself. These are the activities that were introduced by Plato in the Republic as the unseen necessary hypotheses that precede the construction of the Line and are embedded in it. Coupled with this will be a further examination of the notion of image. But for a discussion of these topics we turn to the Phaedo, for it is in the context of the Phaedo that we can further clarify the ontological meaning of these concepts, particularly that of separation. This clarification will ultimately also enable us to understand better the philosophical meaning of the descent back down into the Cave, and more generally, the dynamic represented by the Divided Line image; that is, both images as representing an ontological necessity. It is this ontological necessity that will provide an answer to the original question of why, and from what perspective, the truly just life is happier, and more desirable, than the unjust life, even other things being unequal, as it were.^[30]

Notes

Here again, we see the danger of taking the visible-intelligible division literally. It might seem that I am closer to Ferguson, who holds that the two lower subsections of the Line are merely illustrative of the upper divisions. But this is not so, since I hold rather that the lower divisions are already images of distinctive ways of knowing, and an engagement with their appropriate objects.

Stability at this level, the acceptance of the unthinking stability of experience, is our reliance on eikasia. Literal shadows and reflections are an image or metaphor in the Line for radical dependence—and in this sense, pace Ross (68) is far from an “unimportant,” “occasional interlude.” But eikasia cannot be taken in isolation from pistis, and is a mode of perception repeated in different forms in higher divisions of the Line.

These are introduced as “common,” and used by “all modes of thought, and which everybody of necessity must learn to begin with” (322c). This indicates that we are not dealing ultimately or merely with the limited or literal sense of counting and calculation, nor with merely the sorting out of sense objects, per White, Knowledge, (94ff). Rather, the soul is already in this activity turning away from the sensible, albeit relatively unawares. The technical activity of the study per se, and as one of the five preparatory studies, makes this turning away into a conscious act, though still not as seen subsequently in the light of dialectic. But the technical or mathematical sense of calculation and counting should also be read as a metaphor or image of a broader sense of calculation—logismos as a stand-in for the capacities of the reasoning part in general, the logistikon (or more specifically, dialegesthai), implicated in the rational part’s distinguishing activity—its dunamis. And in this sense, used by “all modes of thought” more generally.

That is, the attempts at the construction of the double square and comparison of the results. These are activities of the slave-boy, however much led to them by Socrates. Further, they are activities, counting and comparing, which do not, as White suggests (Knowledge, 94, and esp. notes) “depend” on the sensible image qua sensible. See Klein, Meno, 99-107, and note 38, page 101. It is not the ambiguity of the sensible per se, whether of relation or deficiency, which calculation is called upon to resolve. Rather, the contrast is of the intuitive judgment (eikasia, or eikastic doxa) in each attempt and the contradiction that a dianoetic or dialectical skill reveals in each attempt. The necessity of the slave-boy’s responses is not brought about by the sensible image per se, but rather by what questioning brings him to look at in the images, that is to say in his thinking about the image-object. It is this to which he gives assent (both to the conclusion and to the process) in his agreeing or disagreeing with a conclusion or inference. (Cf. Klein, 104.)

Cf. Klein, Meno, 100 and note 34 on Meno 72b9-e3, on the terminology, and especially “four-sided.”

At 436e. Cf. Vlastos’ discussion in “Degrees of Reality,” in New Essays on Plato and Aristotle, ed. R. Bambrough (London: Routledge and Kegan Paul, 1965), 1-19.

Compare Gallop’s discussion of predication in Phaedo, particularly 127-29 and 192-94, with that of White Knowledge, 89ff. especially 106, note 9.

That is, in discourse as meaningful, as significant talk. Compare Ackrill “” in Vlastos, ed. Plato I, 201-08. (edit. 1978). “The dialectician makes explicit the rules in accordance with which we already talk.” But Ackrill is wrong that “mystical” Forms of the middle dialogues, “metaphysical objects of intuitive and perhaps mystical insight,” are replaced by “concepts.” The Phaedo and the Republic already both deal with the implicit presupposition of the separation of Forms as the condition for their “interweaving” (Soph.259e6) in discourse—whether the tangled web of ordinary discourse or the fine tapestry of dialectic. The Republic’s description of dialectic, let alone the demonstration of the use of analogy, exercises in separation and division within the dialogue, etc. are all hardly “mystical.”

Here the example is, of course, a literal sensible object and the perception of the senses. But I argued in the previous chapter that the senses and reflection about them, etc. must be taken as representative and illustrative, as an image of the clarification of doxa into epistēmē. This seems to be somewhat Crombie’s view as well. See An Examination of Plato’s Doctrines, vol. II. 564. “It is the senses which inhibit our knowledge of general terms. We have seen in earlier chapters that when Plato speaks of the senses in this way he does not mean literally the senses; he means rather the impression that we should get if we looked and listened uncritically” (564, cf. 86 ff.and esp. 88, 297 ff.). The production of doxai does not result from reliance on the physical senses—Crombie traces the problem to our tendency to “inductively-based impressions” from the “most evident features” rather than analysis of the object in view for what it is, and in its defining relationship to other things. And this general flaw applies to other than the physical. This is clear in his discussion of the Cave image and in reference to the discussion of imitation and sense perception in Rep. X. However, Crombie also believes Plato to have misused or over analyzed the sense of sight—see 86-89. But this is, I believe, not to appreciate how fundamental a paradigm it is, and the philosophical and/or methodological status of analogy.

10.

“At the same time” does not mean temporally at the same instant, as Vlastos seems to require for Plato to be guilty of nodding, cf. Vlastos “Degrees,” 14 ff. Vlastos recognizes that Plato means in different relations (or, in other cases, at different times) but the locution “at the same time” has as metaphor an important philosophical point; which is, that different and contradictory or opposite relations hold of the same x, and implicitly raise the question of how this is possible ontologically, and of how or whether knowledge is possible under such apparently impossible circumstances. This ambiguity of the object may force us to retreat to the separateness of the individual relations, thus a flight to the logoi. But the function of this metaphor and analysis in pointing away from the individual sensible relation is as much a pointing towards the dependency relation of instances and Forms. See Guthrie’s critique of Vlastos, in Greek Philosophy, vol. IV, 493-98. The problem as applied to the example of unit is all the clearer, since the opposite qualities of one and many qua divisibility do not even require a second object for comparison.

11.

See Gadamer, Good, 104-25 esp. regarding Philebus 11a-19b.

12.

Annas, Introduction seems to miss the irony of placating Glaucon’s need for a practical use. But rather than see the playfulness of the sop to Glaucon, and the contrasting serious analogical sense for the image of the utility to the guardian ruler, she laments Plato’s “contemplative conception of philosophy and wisdom.” Since Socrates chides Glaucon for looking in the wrong direction when it comes to astronomy it is hard to believe that the practical uses here should be at all taken seriously, at least the uses that are suggested. On the other hand, there is a real practical use if one keeps one’s eye on the reforming of the soul, and the marshalling of its forces in its interior battles. Nor is the downward path of the return to the cave without its social dimension and consequences. Annas’ criticism here is of a piece with her evident discomfort with what she considers Plato’s anti-empiricism. Per Annas, for Plato, “philosophical thinking, then, is entirely non-empirical. Its truths do not depend on experience and they are known a priori without reference to experience” (279). It is true that the thinking described for dialectic or noēsis, which “making no use of the senses” ascends to Ideas only through the use of Ideas, etc. But while truth does not depend on experience it is gotten by ascent through, and in this sense necessarily from, experience—provoked by its contradictions, in provocations that arise in the Socratic ethical elenctic as well as in the present studies in Plato’s descriptions here. And it is confirmed in application on the downward path.

13.

See Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, (New York: Dover Publications, Inc., 1992), 21-25 on practical logistic versus “theoretical logistic,” which treats of the relations themselves “namely, as homogeneous monads,” objects themselves of thinking as opposed to the numerable in the exterior world—while the numbers themselves would be the object of theoretical arithmos. For Klein’s discussion of the complex notion of “eidetic numbers” see Greek Mathematical Thought, 79-99.

14.

See also Brann, “Plato’s Theory of Ideas” in The St. John’s Review, vol. XXXII July 1980 and “Music,” 54-55, 17-25 where, following Klein, she contrasts logistikon, as a “restricted power” to the (higher) dialegesthai.

15.

Cf. Burnet, Early Greek Philosophy, 111, as cited in Shorey Republic vol. II. 167 note c. I mean the obvious division of the natural sequence of skills that every child learns, beginning with simple counting, arithmetic, geometry, etc. More specifically here, the new conventions of the educational curriculum offered by e.g., Hippias, cf. Prot.318e; also Xenophon, Symp. 462. The suggestion in the Laws (817e-818a) is that the problem is with the manner or way in which these are studied by those who take them up, the same view as is elaborated with a specific purpose in the Republic. Cf. Strauss, Laws, 105 ff. who says that it is “in deference to tradition.” The specific source for Plato’s list is often suggested as Archytas. The introduction of solid geometry may be an image of the Ideas themselves in their solidity. This would be consistent and felicitous in support of my interpretation of the separation issue.

16.

Cf. Cornford “Mathematics and Dialectic in the Republic,” (61-95 in Allen, Studies) regarding diagramma and his discussion of the use of analysis and synthesis in eliciting a demonstration of the conclusion which is “latent” in the figure and its “elements”. (69)

17.

Not, here, a contradiction between the physical figure in the sand and the eternal one, pace Grube (translation) 179, note 9.

18.

Shorey translation (Loeb Library) vol. II. 169, note f.

19.

For example of a problem and discussion, see Otto Neugebauer, The Exact Sciences in Antiquity (Providence: Brown University Press, 1957), 210.

20.

The acoustical reproduction of the ratios can be accomplished by the divisions of a single string according to the specified ratios. See Victor Zuckerkandl, Sound and Symbol. Vol. One, Music and the External World (Princeton: Princeton University Press, 1969). The extended discussion of the nature of musical tone and the dynamic of tonal “motion” is explicated in detail- see the section “Motion” 74-148, and in the second vol. Man the Musician, the section “The Musical Ear,” 83-218. For a discussion of what Plato means by the study of harmony, concord and attunement, and the relation of the mathematical to the acoustical for the Greeks, etc. see Burnyeat, “Mathematics” (2000) 47 ff. and compare Miller, 328-33 in “Beginning the ‘Longer Way’,” in The Cambridge Companion to Plato’s Republic, edited by G. R .F. Ferrari, 310-44, Cambridge: Cambridge University Press, 2007.

21.

Cf. N. R. Murphy, The Interpretation of Plato’s Republic (Oxford: The Clarendon Press, 1951) 188ff.

22.

Contrast R. Robinson’s description of dialectic with e.g., Sallis. The former, in considering various models (Dialectic, 171 ff.) for explication of the “upward path,” rejects an elenctic Phaedo model. Losing sight of the upward drive to a higher level of abstraction, resulting from looking at problems, Robinson sees instead only the elenctic examination of alternative hypotheses and/or a downward motion in the testing of hypotheses, and hence no elenctic impetus for the upward path.

23.

Compare J. C. Gosling, Plato: Philebus, translation and commentary (Oxford: The Clarendon Press, 1975), 170.

24.

Mitchell Miller, “Beginning the ‘Longer Way,’“ in The Cambridge Companion to Plato’s Republic (2007), 310-344. Compare Rowe, Philosophical Writing, 239-54, on the form of the Good, and reconciling the descriptions of the form of the Good in relation to the human good, what is “best” for the rational soul/part, with the description of a metaphysical or universal Good as a source of being, but in some way beyond it. Rowe, interpreting, as I do, texts from the Republic and Phaedo, reads the Good as “not just the principle of the system [the unity and kinship of the Ideas], but, insofar as it was ‘present’, or ‘shared in’, by every part of reality, also in a way the system itself. And it is one of the requirements of the dialectician, who will ‘see’ the good, to be ‘capable of seeing things together.’“ (253) “The form of the good will be a kind of form of the ‘best’, i.e., best for things, a common something in all cases; and in seeking to realize it, we, in common with everything else, will be seeking to partake in whatever that best might be, in whatever form it is appropriate for each kind to partake in it” (251). The link between the individual soul’s good, what is best for it, and the good in and of the cosmos, what is best for each constituent, and for the whole, is formulated by Rowe as follows: “We all desire our own end or telos (say, a kind of flourishing) , but that because of the interconnectedness of things in the universe we shall not understand what that telos is apart from a general understanding of teleology, of the ends of all kinds” (252, n.47). “[T]he human good will be an ‘application’ of a wider principle to the human sphere” (248, n.34). But then too, I would add, given the permanent limitations of being human, and hence the continued justifiable claim to Socratic ignorance, this will always remain philosophy’s ongoing project, albeit with increasing success, hence plausibility. In a sense, one might say that the eros of the true, living philosopher (not the ideal one) is mirror imaged in the erotic relationship of the forms, the form of the Good, and the cosmos.

25.

A similar point, not with respect to the soul but rather dialectic’s epistemological-ontological circle of knowledge and system, is made succinctly by Irwin: “[O]ur account of each aspect of reality would be tested by showing how it contributes to some systemic theory presenting all reality as a whole, a single coherent teleological system” (223, Plato’s Moral Theory). I am in general agreement with this description, but would see it as a project/model and have insisted on the integration of the soul’s active capacities in the account.

26.

See C. D. C. Reeve, “Blindness and Reorientation: Education and the Acquisition of Knowledge in the Republic,” in Plato’s Republic: A Critical Guide, ed. Mark L. McPherran (Cambridge: Cambridge University Press, 2010), 209-28. Reeve focuses on what the true philosophical education actually does—that is to say, the difference between the causes of blindness and the causes of real knowledge for Plato. He too stresses the need to see the journey as a process. The higher studies are thus paradigmatic and metaphorical, models of the opportunity to pose questions about the ground of their elements, the ascent. Then, in the descent, to serve as models of reason’s right use in dialectic’s explanatory ability to “remove the distorting lenses by systematically solving the puzzles and problems that cloud reason’s vision,” (222) that is, to cure it of its blindness—and to learn through practice, in particular cases, of various sorts, to take the right measure, hence “cognitive reliability.” And “talk of likenesses and originals [serves] merely a way of explaining it,” which, when taken literally and not as an image of right method, can and will “mislead,” and produce a vision of Plato as holding a “degrees of resemblance” theory that is opaque and hopelessly metaphysical. This all causes him to reject any version of the vision of the good, and the educational program that leads up to it as a sort of “vast theory that reveals or makes visible the good, while the good itself would emerge [in such a reading] as the entire structure of forms that is the theory’s ontological correlate . . . epistemological holism would result, at least where [the totality of] forms were concerned” (221). Rather, while “when the philosophers finally see the good itself, they have the infallible, unhypothetical cognitive grasp of it that is a paradigm of knowledge. But they have no knowledge of anything else until they take the road back down from it, gaining additional infallible, unhypothetical cognition in the process.” And this proper use of reason, its true work, will require as well “the reorientation of the soul’s appetitive and spirited elements” (224). For these elements, if left to their own devices, misguide the soul towards what are less real objects of either desire or disgust, hence ultimately to “misperceive the world, and to pass on perceptual [in this sense] misinformation to reason” (225). Revising one’s ontology is motivated by one’s progressive aporetic upward journey, and once having turned one’s whole soul and acquired a knowledge of the principle of order amongst what is truly ordered, reliably stable, to go to work, as it were, in making better choices—to the extent that one can.

27.

Contrast Richard Kraut, “The Defense of Justice in Plato’s Republic,” in The Cambridge Companion to Plato, ed. R. Kraut (Cambridge: Cambridge University Press, 1992), 311-37. Kraut is not out to argue the existence of the Forms, but to explore the necessary relationship between the defense of justice and the supposition of the Forms. Kraut argues that the harmony that is the justice of the soul is an imitation of the harmony, that is the order, of the Forms. But he also says that Plato “had not yet reached a firm grasp of what this harmony is in the case of Forms, and so he could not put forward a general characterization that would apply equally to the various kinds of harmony exhibited by living bodies, souls, stars, and Forms” (323). This comes as a somewhat awkward admission, if justice in the soul is to be an imitation of this order amongst the Forms, leaving aside issues surrounding the implied development thesis. See also Kraut, “Return to the Cave: Republic 519 –521, in Gail Fine (ed.) Plato 2: Ethics, Politics, Religion, and Soul (Oxford: Oxford University Press, 1999), 235-254.

28.

To understand the relation of justice and virtue in relation to this ordering is the subject of Aryeh Kosman’s “Justice and Virtue: The Republic’s Inquiry into Proper Difference” (Aryeh Kosman, “Justice and Virtue: Inquiry into Proper Difference,” in The Cambridge Companion to Plato’s Republic, ed. G. R. F. Ferrari. Cambridge: Cambridge University Press, (2007), 116-37. Kosman argues that the Republic`s concept of justice as the inner harmony of the soul and its reading of the world in its harmony should be linked to the concept of proper or appropriate difference. He begins with the very concept of a virtue itself, and justice as the virtue of a thing’s expression of its particularity in its proper function, or functioning well, hence in the case of the soul of each of its parts. Difference counts, and getting right difference, and difference right, for example of the right objects of each and the right relation to them, simply is justice. “Justice may therefore be thought to be a quite general principle of a proper agreement between being and acting, or more precisely, between dispositional and active being: between the way things are and what it is that they are busy at work being” (129). A thing’s virtue is its excellence, and that applies generally, hence to the soul, prior to any understanding of virtue in the ethical or moral sense. Justice is “one among the states or qualities of an entity that in some sense enable that entity to do well what it characteristically does” (121). That inner justice, which expresses both the harmony of the soul’s parts and their difference expressed as their respective proper/best functioning, takes priority over social or political justice in the city expresses the ontological argument of the Republic as prior to an epistemological or ethical or values interpretation. Justice as a “regulatory principle of difference” (132) underlies the proper activity of soul, soul at its ontological best functioning that I argue for as the activity or process of the interrelation of soul, world and Idea and that this activity is a two way street. The next chapter takes up this activity of the soul, and the response of being through entities.

29.

See also N. D. Smith, “Plato on Knowledge as a Power,” in Journal of the History of Philosophy, Vol. XXXVIII, Number 2, April 2000. In a personal communication (March 28, 2007) Nick Smith writes: “Please note that when Plato distinguishes different dunameis, he does so by two criteria: to what it is directed (the “objects”) and “what it accomplishes” (its “effects,” which I take to be pathemata en te psuchē)--not just by objects.

My reply, in response to his comment, and the Power article was as follows: “Yes, Socrates looks to two criteria, to what it is directed—its objects—and what it accomplishes—its effects. What I argue is that there is a reversal in the terms of the argument, and in a particularly instructive way. In brief, Plato starts by proposing that we move from different objects to different capacities, if different objects—then different capacities at 477c—but when we get to 478a the move is from “If a different capacity” then different objects—“it follows that the knowable and the opinable cannot be the same.” This switch, I argue, clearly not noticed by Glaucon, etc. means that the conclusion of Book Four is a fudge that the reader is meant to pick up, and shows the hypothetical nature of dunameis—and the role of their effects in structuring the intelligibility, or rather, readability. This move then sets us up for the continuation in the middle books to give a better ground for the conditional hypotheses—though even after the Cave and . . . Line, they remain a (dianoetic) sketch. My own suggestion is that what ”it accomplishes,” its “effects” in a sense are the objects—that is to say, the chōrismos issue is tied to the “class of realities” that are the dunameis—and how we parse these becomes not just an epistemological issue but an ontological one.” The “effects” are just what I intend by the notion of the soul’s activity in structuring its world. This is further elaborated in the chapter to follow, on the Phaedo, specifically on the notion of separation.

30.

For a more literal interpretation of just how the mathematical studies actually participate in forming the rational part of the soul in preparation for the harmonious ordering, or rule of the whole, see Myles Burnyeat, “Plato On Why Mathematics is Good for the Soul” in Mathematics and Necessity: Proceedings of the British Academy 103 ed. T. Smiley, (Oxford: Oxford University Press, 2000), 1-81. Burnyeat addresses the role of the mathematical studies in the turning of the soul, and of the consequent effect on the philosopher returned from the upward journey. Burnyeat argues that the actual “‘content of mathematics [is] a constitutive part of ethical understanding” (6). This is because “value concepts like concord, proportion, and order are also central to contemporary [Greek] mathematics,” and Plato develops this concordance. “For Plato the important task of ruling is not day-to-day decision-making, but establishing and maintaining good structures, both institutional and psychological. In both city and soul, dispositions and structures are prior to their expression in action. . . . Thus knowing what numbers are concordant, and why, has a very great deal to do with the tasks of government, because concord is an important structural value at the lower level of ethics and politics” (56). He concludes that “mathematics and dialectic are good for the soul not only because they give you understanding of objective value,” of what truly is, as it is “context invariant,” ordered within each and bound up in a “kinship” that forms a whole, “but also because in so doing they fashion justice and temperance with wisdom in your soul” (77). This direct influence of mathematics seems to me a somewhat odd conclusion. Order in the heavens, harmony and order found in the mathematical studies themselves, while they serve to turn us to the more real from the less real, and thus prepare us for study of the Ideas, does not mean that the order, harmony etc. we would find in the latter is more than analogous to the former. Secondly, the value found in the stable ordered objects of mathematics, while beautiful indeed, is not at all the same kind of value that we mean by ethical value, though both are for Plato certainly tied to “unqualified being” (19) and so on. A unified fascist state could be said to have the requisite qualities of “concord, attunement, and proportion.” What has gone wrong here? The right ordering of soul and polis is based on knowledge of the objects—hence the details of the Republic—so mathematical order and value, about which knowledge is gained, since it has rather different objects, serves only analogically. There is indeed a kinship—and the Good is responsible for both realms—but part of the turning of the whole soul, and the return of the philosopher to rule well is dependent on seeing that they are kin because they are ontological analogues—an insight that we are helped to see by the study of mathematics for the ends that Socrates sets out—but not more than that.

Education and the Mind’s Eye

The Tensions of Contradiction: The Philosopher’s Studies and the Noetic Primacy of Logismos

Turning from the Visible: The Philosophical Use of the Studies

Images of Dialectic and the Provocations of Intellect

Notes