Summary: Republic (page 6)

While mathematics deduces hypotheses from other hypotheses up to a conclusion—which remains hypothetical but is mistakenly taken for an unhypothetical principle—dialectics ascends from hypothesis to hypothesis until it reaches a truly unhypothetical principle: a universal principle that presupposes no further condition ¹, one endowed with an absolute degree of certainty.

In The Republic, Plato distinguishes between ascending and descending dialectic.

While the former grasps the unhypothetical principle through the movement described above, descending dialectic consists in deducing, once this principle has been grasped, all the consequences that follow from it. It thus proceeds downward to its conclusions, relying not on any sensible data but solely on Ideas, by which it operates and to which it ultimately returns ².

What mathematical knowledge lacks, if it is to be rigorously scientific, is an independent foundation—one that is not merely posited.

Book VII

Plato continues his argument with the famous allegory of the cave.

We are like prisoners, chained inside a cave, who see only the shadows of the real world cast by the light outside onto the back wall. The philosopher is the one who breaks free from his chains, turns his head, and leaves the cave, discovering the real world. In the same way, the philosopher understands that the true world is not the sensible world of everyday life, but that this world is merely an imperfect reflection—a shadow—of the intelligible world, the realm of Ideas.

The exit from the cave thus stands as a metaphor for the ascending dialectic described above. On emerging from the cave, the philosopher sees real things (the Ideas), and beyond them, he eventually looks directly at the sun—the Idea of the Good, the unhypothetical principle.

Yet one must see it in order to conduct oneself wisely in both private and public life ³.

Plato returns to his original point. We were asking why the philosopher, above all others, is best suited to govern the city.

He alone has gained access—through ascending dialectic—to the contemplation of the Idea of the Good. Through descending dialectic, he can then determine whether one law is more conducive than another to establishing justice and the good in the city.

This knowledge is so valuable that the city should compel philosophers to govern. Some, after all, would be tempted to remain absorbed in the contemplation (theoria) of the Idea of the Good, clinging to a life of pure theory—like the freed prisoner who refuses to return to the cave.

Yet the law is not concerned with securing exceptional happiness for a single class of citizens; rather, it strives to achieve the happiness of the city as a whole. If it produces such men within the city, it is not to leave them free to turn in whatever direction they please, but to make them work together to strengthen the good of the state ⁴.

The Guardians must be taught ascending dialectic. Before that, they must be introduced to other sciences that serve only as preludes to dialectic but will train them to turn towards the intelligible world: arithmetic and geometry (for they concern knowledge of what always is and cultivate that philosophical spirit which lifts the eyes upwards, rather than wrongly lowering them towards things here below ⁵), as well as astronomy.

Dialectic is a science that Plato sums up as follows:

Trying, without the aid of any sense but by means of reason, to reach the essence of each thing and not stopping until we have grasped by intelligence alone the essence of the good ⁶.

One of Socrates' interlocutors asks him to explain the content of this science, but Socrates replies that this is impossible: You would no longer be able to follow me, for you would no longer see the image of what we are talking about, but reality itself ⁷.

Dialectic, however, is corrupted by the sophists, who reduce it to a mere game—an exercise in rhetorical combat, the art of brilliantly contradicting one's interlocutor.

¹ 511b-511e
² Ibid.
³ 517b-518c
⁴ 519c-520c
⁵ 527a-527e
⁶ 532a-533a
⁷ Ibid.