In the face of Socrates's doubts about the existence and usefulness of the "science of science" that constitutes temperance, Critias seems equally perplexed. Noting (to the reader) that Critias has a reputation to maintain, Socrates tries to keep the discussion going. He dismisses, for the moment, the question of whether such a science exists, and asks instead, if it did exist how would it lead us to distinguish what we know and what we do not know (i.e., how would it lead us to self-knowledge)?
Critias replies that, just as he who has swiftness is swift, and just as he who has knowledge knows, he who has this knowledge that knows itself will have self- knowledge. Socrates asks why having self-knowledge necessarily means that one would enable one to differentiate between what one knows and what one doesn't. Critias replies that the two things (self-knowledge and the differentiation) are the same thing. Socrates explains his objection further. First, he reminds Critias that knowing about knowledge and ignorance is not the same kind of thing as knowing about the knowledge or ignorance of something specific, like health or law. But if one comes to know about specific areas of knowledge (like health) through specific practices (like the art of medicine), how can one come to know about knowledge itself (without any specifics)?
Thus, it seems that he who has wisdom (knowledge of pure knowledge) may know that he knows, but cannot know what he knows. By the same token, the wise man (on this model) will not be able to distinguish the true physician from the false, nor will the wise man be able determine whether anyone who claims to know something really knows it. What's missing from this kind of wisdom, argues Socrates, is precisely the "subject matter" that distinguishes one science from the other. Thus, the wise man can have no knowledge of bad or good medicine unless he is a doctor himself (and so practices on the subject matter of medicine, which is health and disease).
In fact, Socrates continues, what we were searching for, in defining wisdom as a knowledge which can distinguish when it (or any other person) knows or does not know, is a kind of content-less ideal that would make everything under its control proceed perfectly, without error, from the head of the state on down. This science of science without any subject matter, this abstract knowledge that prevents all error, is simply "not to be found anywhere." Critias agrees. Socrates argues, then, that what he and Critias should really conclude about wisdom is that it is a knowledge of knowledge that is valuable because it gives extra insight into specific areas of knowledge, facilitating concrete learning and inquiry. Wisdom thus allows one to "see the science" in, for example, the science of medicine. Thus, it may be simply that Socrates and Critias have been asking too much of wisdom.
In the previous section, the dialogue reached two closely related crisis points, both stemming from the difficulty of conceiving a self as fundamentally relational. In the first case, Socrates corrects Critias's assumption that the refutation of a point doesn't make one person the winner and the other the loser; as paradoxical as it may seem, Socrates claims that his refutation of Critias's point is as much an examination of himself as it is of Critias. In the second case (at the end of the section), Socrates calls into question the whole, seemingly paradoxical notion that wisdom can be defined as that which knows both itself and its own absence (i.e., what it does not know): not only are we left unsure whether such a self-knowledge (a "science of science") can exist, but we do not even know if, given its existence, it would be of any real use.
In this section, Socrates drops the issue of whether or not wisdom as a "science of science" could exist—he'll assume it can—and proceeds to ask what goods it might create. In fact, as we shall see, this is a crisis point from which the dialogue will never recover. Not only have the participants agreed simply to skip over the vexing issue of the possibility of self-knowledge, but they have decided to address a further paradox that seems impervious to explanation: how could knowledge of knowledge possibly have any effect on anything specific, since it is defined by its transcendence of other, concrete sciences?
In short, then, the problem is that knowledge of knowledge has no "subject matter." Further, since this absence is what defines knowledge of knowledge, it is unclear how such a definition could ever be linked to subject matter without compromising itself. One intriguing way in which this abstract problem is made clearer is through Socrates's description of the ideal city-state that wisdom or temperance would supposedly rule (if it existed). Such a state would be perfect in every detail, since, with knowledge of both knowledge and ignorance at the top of the hierarchy, no one beneath it would ever act without knowing exactly what they were doing.
Socrates uses the absurd perfection of such a state to suggest that this pure ideal of reflexive knowledge is just as much of a pipe dream—the unreality of this ideal state shows how firmly entrenched the "knowledge of knowledge" definition is in unreachable idealism. Such a move is illustrative, and does not logically prove the impossibility of such knowledge. Nonetheless the point is well taken. It is curious, however, that the ideal state, governed by wisdom, should here be used as an example of impossibility, since Plato will later write the entirety of the Republic on just that idealized entity.
This consideration of idealism leads to the only semi-viable solution to the intractable separation of ideal self-knowledge from actual, practical beneficial knowledge: Socrates suggests that wisdom is really something that, even while defined by a kind of abstract knowledge about knowledge, has its effects in the facilitation of practical inquiry. This seems intuitively right, even though the exact mechanism may be a little unclear. In particular, such a model seems to match up quite well with the Socratic method itself, allowing for the pursuit of concrete knowledge to be guided by a set of meta-rules about how that pursuit should proceed (in Socrates's case, such a process is the elenchus).