Critias objects to Socrates's questioning the new definition of temperance as the "science of a man's self." None of the sciences, Critias argues, is like the others, and this includes wisdom; the products of geometry, for example, are nothing like the products of medicine or architecture. Socrates counters that each science is similar in that it has a product that is not the same thing as the science itself; in that case, what is the product of the "science" of wisdom? Critias objects again to this seeking of similarities between the sciences. Wisdom, especially, is different from the others, he says: "wisdom alone is a science of the other sciences and of itself." Socrates, he claims, is just trying to refute him, rather than truly arguing.
Socrates replies that he is indeed simply refuting Critias, because he himself has no pre-ordained conclusion that he wants to push. "How can you think that I have any other motive in refuting you," Socrates asks, "but what I should have in examining into myself?" Critias accepts this, but reaffirms his idea that wisdom is the only science that is the science of itself and of the other sciences. Socrates replies that "the science of science will also be the science of the absence of science." On this model, the temperate man will examine both what he knows and what he does not, and will do the same for others. This, in fact, is temperance, Socrates says. Critias agrees.
Next, then, says Socrates, it must be determined whether such knowledge—both of what is known and of what is not known—is possible, and, if such knowledge is possible, whether it is useful. But such knowledge, Socrates continues, seems "monstrous" when we look at any parallel example. There cannot be a kind of vision that sees all other kinds of vision as well as vision that cannot see, nor a form of desire that desires both itself and all other forms of desire. The same holds for the other senses, for love, for fear, and for opinion—all, put into the form of a science that knows itself and its ignorance and all other sciences, seem absurd. Such a science, Socrates points out, has no real "subject matter." Despite all this, however, Socrates and Critias agree to continue investigating whether such a form of knowledge might really exist.
Socrates argues that it is in the nature of this science to be a science of something, just as it is in the nature of something greater to be greater than something which is lesser. But if something were both greater than something else and greater than itself, then it becomes both itself and its own object (both greater than itself and less than itself). The same holds true if something is the double of itself (i.e., it will be both double itself and half itself). The problem here, then, is that "that which has a nature relative to itself will retain also the nature of its object." Are there any such things, whose defining relation is not to other things but rather to themselves? In short, is there anything defined by self-relation? Cases like "the power of heat to burn," or "the power of self-motion," seem like possible cases, but Socrates is not sure; the science of science then, is beyond his powers to verify. Even if it was verified, however, Socrates would not be convinced that it is the same thing as wisdom or temperance, or that it does anyone any good.
This section sees the dialogue taking a form that depends neither on a kind of trickery or flirtation (as in the first section) nor on petty academic distinctions (as in much of the second). Here, Critias emerges as a rather formidable interlocutor, and not just because of his rhetorical skill. In this, he both foreshadows and partly exceeds formidable interlocutors, like Gorgias, who command more speech that the meek, one-phrase assents that are so often the responses of Socrates's victims in the elenchus. Critias has some profound ideas, and he is not shy in actively arguing with Socrates for them. At some points, it is almost as if Plato has invested some of his own thinking in the figure of Critias (he generally either leaves his own ideas out of the dialogues or has Socrates speak them).
This active philosophizing on Critias's part builds a noticeable tension as the dialogue proceeds, a tension that explodes, at one point, into a remarkable crisis in and redefinition of the elenchus. Critias objects twice that Socrates is simply out to refute him rather than engage in any kind of productive argument. The second time this objection is made, the dialogue grinds to a sudden halt and Socrates responds with clear emotion and indignation: "And what if I am [just refuting you]?" Socrates makes the profound claim that such refutation is no more in Socrates's favor than in Critias's; true philosophical debate is not about set opinions attached to people with egos, but rather about the advance of true knowledge through the mutual deconstruction of false knowledge (knowledge that we only "fancy" we have).
This claim to purely disinterested, primarily negative argumentation, free from personal ego-investment, takes on a great deal more complexity in light of the beginning of the dialogue, which is full of deception, emotion, and deep personal investment (recall Socrates's anxiety about talking with Charmides). If Socrates's claim here about true philosophical argument seems to be questionable in light of the dialogue's opening moments, it certainly is. At the same time, the contradiction lies more with the text itself (i.e., with Plato) than with Socrates. What might be perceived as a contradiction between personal desire and disinterested argumentation is probably, for Plato, a difference with an important point: Socrates is capable of dropping base desire for pure philosophy, capable of replacing the incontrollable loss of self that happens in lust for the elevated, noble, and purposeful loss of self that should occur in philosophical debate.
For there is a loss of self in Socrates's notion of argument, a loss of self that is not just a matter of casting pride aside: Socrates claims that his refutations of Critias are just as much refutations of himself. Thematically, this aspect of the elenchus as a kind of metaphysical manipulation of the self leads quite strongly into the profound formulations about temperance as the "science of itself" that follow soon after. The central issue here is how to conceive of a science or of a form of knowledge that is defined both by its relation to itself (because it must know itself) and by its relation to its own absence (because it must know what it does not know). This "science" called wisdom must also be a science of all the other sciences, but this is a much less difficult proposition.
Our questions from the last section, about the relationship between self-knowledge and the ordered soul (temperance), have now expanded into a deep consideration of what it might mean to have self-knowledge at all. Socrates's remarkable (and difficult) contribution here is just one, particularly dense formulation of the maxim for which he is perhaps best known: not the ancient Delphic oracle's "know thyself," but rather the more disturbing maxim, "all I know is that I know nothing."