This must be the climax; Book 7 is all about the cave allegory. In fact though, the cave allegory is just a dramatic retelling of the division between the "conditions of the soul" that we saw in Book 6. The shackled prisoners who can't turn their heads and who can therefore only see shadows on the cave wall are stuck in subsection 4: "imaging". Those who can turn around and see the statues that cast these shadows as well as the visible light of the fire are in subsection 3: "belief". Once a freed prisoner is forced out of the cave and into the bright light of day to discover real objects in the above ground world they move into subsection 2: "thought". And when they are finally able to look at the One-Good of the sun itself, they reach the highest level of philosophy, subsection 1: "understanding". The allegory just gives us a simple image for each level and makes explicit the idea that, relative to the intelligible, the visible world is underground.
The rest of the chapter unpacks various aspects of the allegory. For example, the fact that the philosopher must rise up from the darkness to the light, but then subsequently return to the cave to help the other prisoners, accounts for why he can seem so strange and useless to them.
... the eyes may be confused in two ways and from two causes, namely, when they've come from the light into the darkness and when they've come from the darkness into the light. (518a)
The round trip aspect of the allegory also explains why the philosopher king isn't much interested in ruling and does it only from a sense of duty (520e). How are you going to keep them down on the farm once they've seen Karl Hungus?
The allegory also implies that everyone has the innate capacity to see the light. They simply need to be freed from their shackles. So education isn't about giving people the capacity to see or think, nor is it about filling their heads with some specific knowledge. Instead, it's about changing our habits, about a practice that turns our whole body towards the light.
Education isn't what some people declare it to be, namely, putting knowledge into souls that lack it, like putting sight into blind eyes.
They do say that.
But our present discussion, on the other hand, shows that the power to learn is present in everyone's soul and that the instrument with which each learns is like an eye that cannot be turned around from darkness to light without turning the whole body.
...
Then education is the craft concerned with doing this very thing, this turning around, and with how the soul can most easily and effectively be made to do it. It isn't the craft of putting sight into the soul. Education takes for granted that sight is there but that it isn't turned the right way or looking where it ought to look, and it tries to redirect it appropriately. (518c)
While Plato assumes that everyone will recognize the truth when they are able to see it, revealing this turns out to be a more complex process than simply telling them about it. Learning, he seems to imply, has to actually create some sort of qualitative change in the soul. It is not just the change from the absence to the presence of knowledge.
This turns out to be an important consideration in the final phase of the philosopher's education he goes on to describe. After the compulsory study of music and poetry and physical training he laid out in Book 3, comes the voluntary study of mathematics and ultimately dialectics. These latter subjects must be studied out of a sense of curiosity and play (536e) and by means of problems (531c), and not simply because they contain some true and useful empirical facts (520b). They are meant to cultivate a taste for learning and understanding, which, we'll recall, goes beyond thought and mental images. They should, "draw one towards being" (523a).
The way Socrates describes the goal of this final phase of the philosopher's education is interesting because it explicitly contrasts the model of education as recognition with one of education as puzzlement. He claims that some experiences lead us towards understanding, whereas others do not.
I'll point out, then, if you can grasp it, that some sense perceptions don't summon the understanding to look into them, because the judgment of sense perception is itself adequate, while others encourage it in every way to look into them, because sense perception seems to produce no sound result.
You're obviously referring to things appearing in the distance and to trompe l'oeil paintings.
You're not quite getting my meaning.
Then what do you mean?
The ones that don't summon the understanding are all those that don't go off into opposite perceptions at the same time. But the ones that do go off in that way I call summoners—whenever sense perception doesn't declare one thing any more than its opposite, no matter whether the object striking the senses is near at hand or far away. (523b)
He goes on to give an example of both adequate and inadequate sense perception. Recognizing that someone is holding up three fingers falls into the first category (523d) while perceiving the bigness/smallness, thickness/thinness, or hardness/softness of the same fingers falls into the latter category. In these cases it isn't as if we're fooled into having an erroneous perception of something. Instead, when we experience a relative quality, we become puzzled as to what we really mean by hard and soft, small and large. How can something be hard (with respect to X) but simultaneously soft (with respect to Y)? We don't know what perception to have at all. When we don't simply recognize that something is what it is, we are forced to think about what it is. We saw this same structure back in Book 5. It was used as a way of justifying the idea that the Forms are the only true objects of knowledge because they are the only "things that are". When we experience a beautiful thing, we also realize that it is actually ugly relative to some even more beautiful thing (479a). That is to say that no thing is ever purely beautiful or purely ugly, but keeps flipping back and forth between these depending on how we look at it. Only the Form of Beauty would always and only be beautiful. While these impure mixtures, since they are between being and not-being, can only be objects of opinion and never knowledge, it turns out that they can be useful in spurring us towards knowledge. Basically, they are puzzles or paradoxes that set up a problem. And these problems "draw us towards being", towards the Forms, and especially towards the Good as first amongst Forms.
At first it's not at all obvious how studying mathematics is going to involve us in situations with puzzling relative qualities. But for Plato, we should puzzle over the very concept of number.
If the one is adequately seen itself by itself or is so perceived by any of the other senses, then, as we were saying in the case of fingers, it wouldn't draw the soul towards being. But if something opposite to it is always seen at the same time, so that nothing is apparently any more one than the opposite of one, then something would be needed to judge the matter. The soul would then be puzzled, would look for an answer, would stir up its understanding, and would ask what the one itself is. And so this would be among the subjects that lead the soul and turn it around towards the study of that which is. (524e)
Since we never have a direct sense experience of the concept one but only encounter it through the many ones we find in the world, it defines a paradoxical problem for us. With it and all the other numbers, we have to somehow separate out or abstract the essence of the number from the counting of particular items set in front of us. In other words, when we do mathematics we get at the number in itself, we purify it of its mixture with particular things. Learning to purify a mixture like this trains us to see the Forms or find the golden guardians mixed in with the other metals. So Socrates values math not really because of its truth, its certainty, or its rationality. Remember, mathematical forms are still images of thought, not the objects of understanding itself. Math isn't valuable because of its conclusions, but because it teaches us how to take up a problem, to purify the world into intelligible essences. So while he admits that arithmetic (525c) and geometry (527b) and astronomy (527e) may all be useful to the ruling philosopher king, the axiomatic truth and empirical success of these sciences is entirely beside the point. In fact, he chides Glaucon for defending their study on that basis (527e). An astronomy that merely correctly predicted the movements of the planets would be little more than the study of some elaborate toy mechanism designed by the gods (530a). Since it is descriptive, it wouldn't ever explain to us why the various parts move as they do. It wouldn't ascend to the level of problems (530c). It wouldn't force us to contemplate a pure and invisible ideal world that governs this one from above. Note that Plato obviously has in mind the astronomy of his day, whereas modern astronomy would come a lot closer to fitting his definition.
However, the study of all these sciences is just preparation for studying the queen of the sciences, the dialectic (534e). The reason the dialectic is so important is that it fulfills the criteria Socrates set out for true knowledge at the end of Book 6. Instead of taking its hypotheses as unqiestioned first principles or definitions, it takes them as stepping stones to reach an un-hypotetical, necessary or apodictic, first principle. The dialectic takes the Forms as hypotheses only to reach the certain conclusion that there is a natural light which illuminates them: the Good. So instead of producing the universal agreement of opinion that arithmetic or geometry inspire (we all agree that if X then Y), it reaches the self-evident certainty of knowledge.
And as for the rest, I mean geometry and the subjects that follow it, we described them as to some extent grasping what is, for we saw that, while they do dream about what is, they are unable to command a waking view of it as long as they make use of hypotheses that they leave untouched and that they cannot give any account of. What mechanism could possibly turn any agreement into knowledge when it begins with something unknown and puts together the conclusion and the steps in between from what is unknown?
None.
Therefore, dialectic is the only inquiry that travels this road, doing away with hypotheses and proceeding to the first principle itself, so as to be secure. And when the eye of the soul is really buried in a sort of barbaric bog, dialectic gently pulls it out and leads it upwards, using the crafts we described to help it and cooperate with it in turning the soul around. From force of habit, we've often called these crafts sciences or kinds of knowledge, but they need another name, clearer than opinion, darker than knowledge. We called them thought somewhere before. (533c)
It may seem odd, but Socrates doesn't actually spend a lot of time discussing how the dialectic works. We can infer from Plato's whole oeuvre that it always proceeds by question and answer, as well as by separating and unifying (as we saw in Phaedrus). Socrates explicitly makes both of these points here (435e and 537c5). To go beyond these hints though and really describe the dialectic in detail, Socrates would have to claim that he knows the Good itself, which organizes the whole dialectic. But of course, Socrates only knows that he knows nothing, so he can't claim to explain how the dialectic works -- though since he knows that he knows nothing, he can claim that it exists.
So tell us: what is the sort of power dialectic has, what forms is it divided into, and what paths does it follow? For these lead at last, it seems, towards that place which is a rest from the road, so to speak, and an end of journeying for the one who reaches it.
You won't be able to follow me any longer, Glaucon, even though there is no lack of eagerness on my part to lead you, for you would no longer be seeing an image of what we're describing, but the truth itself. At any rate, that's how it seems to me. That it is really so is not worth insisting on any further. But that there is some such thing to be seen, that is something we must insist on. (533a)
Similar to what we saw in Meno, the early Socrates' catchy claim to fame is here transformed into a whole philosophical method. His questions and answers about courage and friendship and love and justice are designed to set up problems that lead us into the confusion of contrary perceptions. From these problems we can move towards the Forms that resolve the mixture of a problem into its elements. In fact, the paradox of Socrates' motto sets up the most fundamental problem of all. How can he know and not know at the same time? This is much tougher than the paradox of the big, thick, hard ... finger (524a). But it is 'resolved' in the same way. The question itself makes us realize that there must be a difference between knowledge and ignorance, the two opposed Forms that structure knowing and not-knowing. And in the middle, we have the muddle of opinion. The Socratic method of inquiry, when it takes the paradoxes it uncovers seriously, and not as refutations in some rhetorical game (539b), exemplifies the dialectic.
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In another Deleuzian aside, the passage at 523b where Socrates distinguishes between perceptions that "summon" the understanding and those which do not is reproduced on page 138 of D&R. I actually think he uses the quote there as a sly form of humor, since the subtlety of a thought-without-image is as easy to misinterpret as Socrates' idea is for Glaucon. Deleuze goes on to make great use of the concept of problems in the rest of his book. But first he spends a few pages (up to 143) critiquing Plato's construction of that concept. Basically, while Plato clearly values problems, he ultimately expects them to be solved. In fact, the Forms are precisely the solutions to problems, the elements that allow us to resolve a mixture into its pure components. Likewise, the Good is the solution to the problem of knowledge posed by Socrates' tag line. In Plato, we resolve the problems by remembering the Forms we saw prior to our (latest) birth. Deleuze points out that this kind of remembering is precisely not 'problematic' in Plato's own sense -- it's just a form of recognizing some identity we've seen before we were born. So I guess my point here is that learning more about Plato's use of the term doesn't actually help us understand the way Deleuze employs it, unless this is by way of contrast. For Deleuze, problems are not about relative qualitative oppositions, and they are not meant to be solved but to be preserved as problems so that we are able to move completely beyond modeling thought upon the experience of recognition. It's only in this sense that he agrees with Plato -- thinking (or understanding as Plato calls it) starts only when recognition stops.
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