What was Socrates's belief on commitment to a life of reason in the Republic?
We can say that Socrates is totally committed to the pursuit of reason in the Republic. In fact, almost every dialogue in which Socrates speaks, he is pursuing reason.
One of the most memorable passages of the Republicis Socrates' allegory of the cave. In this allegory, people are tied up and they are facing the wall. There is a fire behind them and objects are moving. Of course, the people cannot see the objects. They only see the shadows on the wall. In this sense, their perception is not true.
Socrates uses this allegory to show that without reason, we only see the shadows. What we need to see is the real object. Socrates calls these the forms. It is only through reason or philosophy that we can actually come to understand the forms.