Theaetetus is an aporetic dialogue. There is no final answer given to the question "What is knowledge?" Although it seems that knowledge must be justified belief plus an account, none of the accounts of account pass muster. People generally assume that the answer is to be found either in (1) the notion of apprehension of the Forms as suggested in the Republic and other dialogues, or (2) the process of division in terms of species and genera described in later dialogues such as the Sophist, Philebus and Statesman, or (3) some sort of causal explanation as suggested in the Phaedo. I want to suggest that there is internal evidence within the Theaetetus for another solution.
However let's begin outside the Theaetetus with what we might think is required for a knowledge that is deep. Deep knowledge in our culture is often referred to as wisdom. We will distinguish here between having wisdom in general and having wisdom in something. We will deal with wisdom in general later. To have wisdom in something it is not sufficient just to be able to name all of the parts or even to give a definition of a thing of that sort. Deep knowledge requires being able to make predictions in that domain and being able to solve problems. Deep knowledge of car mechanics would require that we know about the parts and how they function in relation to each other. An even deeper knowledge of car mechanics would require knowledge of how cars respond to surrounding conditions, for example driver handling, road conditions and weather. Deep knowledge requires knowing something as an organic whole AND knowing how this thing functions in larger wholes to which it belongs. If we are trying to understand a car it is not sufficient just to know the definition of "car" and whether this car meets this definition. Nor is it sufficient to know this plus knowing all of the car's parts or even knowing all of the cars parts and their functional relations with each other.
Deep knowledge that is wisdom also requires contextualizing ones knowledge within the wider domain of culture and life itself. I person could have deep knowledge of torture on one level, i.e. in knowing the techniques, how to get confessions, etc., but fail to recognize the ultimate inability to synthesize this practice into the operations of a good society. So even an expert in torture would fail to have wisdom in the wider sense.
All of this I think, so far, is implicit in Plato. Another problem however is one that he raises and never adequately addresses. It is the problem of knowing particulars, for example knowing a particular person, say the person Theaetetus. (I am thinking of 208d-210b) To really know a person, let us hypothesize, is to know his or her essence. But Plato never talks about essences of particulars. This is a deep problem for Plato nonetheless. The philosopher-king, as described in the Republic, should be able to apply the knowledge he has gained from leaving the cave and perceiving the Forms to the world. But this requires being able to see, for example, the Good, in particulars.
The problem is that in order to really see the Good as it is present in the particular one must not only know the Good but also know the particular, and to know the particular is to know that which makes it unique, what distinguishes it from every other particular. The philosopher-king will not be able to do his job unless he is able to to do this. Similarly, following the Phaedrus, the rhetorician needs to be able to know his or particular audience in order to fit argument to audience. It helps to break possible audiences into types. But, as we are finding today with medicine, an individual-based approach has its own advantages. Moreover, the relationship between the Lover and the Beloved, or between the wise Teacher and his Student, is one that requires awareness of the particular. Socrates himself in the Theaetetus recognizes this as part of the role of the philosophical midwife: to deal with the particular idea that the subject gives birth to, and to match the particular student with his most appropriate teacher.
This is where aesthetics comes in, although in a way that, at first seems not to much involve the arts. (Later I will point out that it does very much involve them.) Two points need to be made. First, Plato seems on first sight to be the great anti-aesthetic philosopher. For example in the Republic he attacks lovers of sights and sounds. He consistently attacks most of the arts, for example in book X of the Republic. And yet aesthetics does play an important role in Plato, mainly by way of his erotics. In Diotima's theory of love as set forth in the Symposium Diotima requires that the philosopher begin with love of a particular body. This is followed by love a particular soul. Now it may be true that all of this is forgotten when one gets to beauty itself. But also remember that the process of moving up the ladder, as also reiterated in the Phaedrus through the allegory of the wings, requires both the Lover and the Beloved. They stimulate each other through appreciation of the beauty of the Beloved and the reflection of that in the Lover. They also stimulate each other through their shared discourses about virtue, i.e. about the excellences of things Plato's system just does not work without appreciation of beauty in particulars. Even when the Lover perceives the vast sea of beauty he perceives this, presumably, as a sea of particular things that are beautiful.
The Theaetetus starts off with questioning whether knowledge is perception. The term translated by "perception" is "aisthesis" from which we get the term aesthetics. This path to knowledge is ultimately rejected. But one has to wonder whether Plato is fully behind this rejection. Some of the arguments are just too quick. Perhaps he believes that perception does play an essential role in knowledge even though it can never be the whole story.
Moreover, when Plato describes, in the famous interlude, the philosopher as someone who, like Thales, just does not know where he is going, one thinks that this is precisely not the sort of person one would want for a philosopher-king. How could Plato be serious about the philosopher as described by Socrates in the interlude? This "philosopher" is incapable of handling himself in court or in the political realm. This would not be a person to make the pilot of the ship of state. There must be some irony here.
So, cutting to the quick, Platonic interpretation aside, here is my take on deep knowledge that can be called wisdom. Such knowledge is justified truth belief where the justification requires not only definition, classification, predictability, taking into account functional relations between parts, taking into account relations between the object known and surrounding context, but also knowledge of the essences of particulars, a kind of knowledge which is essentially aesthetic.
Earlier I suggested that the arts do not play a role in this. But in fact they do. Who are the experts in aesthetic cognition (a term first developed by the originator of aesthetics, Baumgarten)? They are the artists. Sophocles, a great artist, focuses not only on Oedipus as representing man but also as a particular person. Ironically, this works best for us in the audience because we too are particular humans as well as being human. Paying attention to the particular requires paying attention to sensual characteristics, i.e. the particular nature of Oedipus who does particular things that can be seen and recounted, for example putting out his eyes. Art engages us in the universal in the particular, and this distinguishes art from philosophy and science.
Wisdom goes beyond philosophy and science to gives us also knowledge of the universal in the particular. It has, therefore, a deeply practical or pragmatic dimension. Again, the philosopher- king will only be of use if he is able to perceive the Good as manifested in the particular thing and this requires knowing the particular thing. In short, wisdom requires synthesis of Art and Science, neither, by itself, being sufficient.
All of this is inspired by something said by Socrates as the end of the dialogue at 208d to 209d. Notice that we return here to a point at the very beginning of the dialogue which, in fact. did refer briefly to a visual art, and, in fact, refers first to music and then to visual art: "If you and I had each had a lyre, and Theodorus had told us that they were both similarly tuned, should we have taken his word for it straightaway? Or should we have tried to find out if he was speaking with any expert knowledge of music?" (144e Levett and Burnyeat). If then we are interested in faces being alike, for example Theaetetus and Socrates, we should go to someone with knowledge of drawing. And Theodorus is no artist, he is a Geometer. (145a) One would think that, on some accounts of Plato, Theodorus, being a Geometer, would be given some priority over the artist, for the Geometer has a better understanding of circles than a drawer of circles. But this is not the case. Somehow, the imitator of the bed who draws the bed, the very person who is degraded in the Republic, at last gains some respect here.
So, now at the end of the end of the dialogue we have a paradox. As Socrates puts it, "if you get hold of the difference that distinguishes a thing from everything else, then, so some people say, you will have got an account of it." Yet "so long as it is some common feature that you grasp, your account will be about all those things which have this in common." (208d) This poses, I believe, a deep problem with the theory that knowledge is true belief plus an account in terms the Forms. An account of the difference that marks out the uniqueness of an individual like Theaetetus should be something that is ineffable: it must not be in language since language always deals with the common. Or, alternatively, it can be in language, but language understood as being so flexible that the meaning of a word can be different in every case in which it is used. (I think this is the better option.)
Socrates' problem with this account of account is "how...did it come about that you were the object of my judgment and nobody else?" (209b1) Even when we mention the very visual characteristics that Theaetetus shared with Socrates, snub nose and prominent eyes, we have not distinguished him from everyone else because we have not distinguished him from Socrates. So Socrates asks Theaetetus, "Shall I even then be judging you any more than myself or anyone who is like that?" (209c1)
Then Socrates says ""It will not...be Theaetetus who is judged in my mind until this snub-nosedness of yours has left imprinted and established in me a record that is different in some way from the other snub-nosednesses I have seen...." (209c4) "and this will remind me, if I meet you tomorrow, and make me judge correctly about you," i.e. that you are indeed Theaetetus. (209c6) Isn't it interesting that this sort of talk takes us back to the discussions of the wax and aviary analogies which were brought up in relation to the issue of memory which was itself brought up in response to the defense of the perception theory of knowledge, that perception includes not only immediate perception but also remembered perceptions? The aesthetic theory of knowledge had been rejected, but now it comes back when talking about an account, and most importantly when talking about the last and therefore most viable account of account found at the end of the dialogue.
Yet Socrates draws out the argument in a different direction, saying, "then correct judgment must be concerned with differentness of what it is about," which leads to the idea that in correct judgment we already have an idea of how a thing differs from other things and so we do not need an addition of an account in terms of knowledge of differences. And if adding an account means getting to know the differences, not merely judging them correctly, then we are absurdly defining knowledge in terms of knowledge. And that is the end of the dialogue.
But what I have concluded is that there is an essential aesthetic component to knowledge, an aesthetic component which also contains an erotic component, and thus is essentially connected both to the love of beauty and to loving relations between humans. That is, I do not think Plato's whole story about knowledge can be told (a story which I find pretty close to being correct) unless the erotic and the epistemological dialogues are synthesized.