Recall that Peirce identified four methods for the fixation of belief: tenacity, authority, a priori, and science. Plato is the undisputed king of the a priori method, therefore it's worth looking into the epistemological assumptions that underlie Plato's ideal state. As you dive into "Book 1" of the Republic, it becomes immediately evident that Socrates is playing an active role in guiding the conversation toward a conclusion. Plato called this process of "questioning and answering" the dialectic, which is known today as the basis of all human inquiry. In the early going, the main question to be addressed is clearly identified: "What is justice?" Of course, the underlying assumption is that it is possible to "know" what justice "really is" and whether this knowledge has some kind of a foundation, and whether this knowledge is accessible to everyone. Throughout history, philosophers have defended one of two alternative "foundations" for human knowledge: the empirical foundation and the rational foundation. Empiricist philosophers anchor human knowledge in the observation of either: an exterior material world that resides "outside" of the human mind; or, an interior world of ideas that lies "inside" the human mind. Today, most philosophers and scientists are empiricists in the materialist tradition. Since Plato, rationalism has been associated with with the reality of this internal world of a priori ideas.
Plato grounded his rational foundation in a priori knowledge; that is, knowledge that exists "before human experience." In other words, Plato believed that the human mind comes front-loaded with a body of knowledge. Plato and subsequent rationalists tend to identify "real knowledge" with timeless universality. However, according to Plato, if you want to gain access to this vast body of timelessly, universal Truth you must possess both innate intelligence (nature) and many years of training and education (nurture). Plato argued that stable political regimes must have leaders that possess timelessly universal knowledge. Thus political science is mostly about identifying naturally intelligent leaders and educating them. One of the necessary conditions for entering the a priori world is the natural ability to learn mathematics and geometry. Therefore, before any student could enroll in Plato's school, "The Academy," they had to know mathematics and geometry.
What's so special about mathematics and geometry? Well, they both provide access to timelessly, universal truth; that is, 7+5=12 always was and always will be true...everywhere in the universe! Therefore, knowledge of mathematics and geometry provides students with an initial glimpse into this internal world of a priori truth. Now...back to justice! If you want to answer the question "What is justice and how do you come to know it?" There are two possibilities. You can "look out" at instances of justice in the material world, or you can "look in" at the idea of justice. Obviously, if you look at material world, you'll probably find more "injustice" than "justice," therefore that's not a very promising strategy. Therefore, Socrates chooses to explore the ideas of "justice" and "injustice." Socrates argues that the concepts of "justice" and "injustice" possess a timelessly, universal form that transcends all human languages. Unfortunately, not all of us are capable of knowing the difference between justice and injustice, therefore we must acknowledge (and trust) the authority of those naturally gifted, well-trained experts. So in an ideal political regime, authority must be based on the possession of privileged, timelessly universal knowledge that only a few can possess. Followers must be taught to submit to the authority of this naturally gifted, carefully groomed "class" of "philosopher kings."
Now in Book 1, Thrasymachus argues that "justice is in the interest of the stronger." Most of his arguments, and Glaucon's and Adeimantus' subsequent arguments are based mostly on empirical observation of how powerful humans, in fact, behave in the "material world." But does the apparent fact that the unjust tend to fare better than the unjust, necessarily imply that injustice is preferable to injustice? Interestingly, over the next nine books, Plato attempts to prove that the unjust do not, in fact, fare better than the rest of us. So...by now you're you're probably wondering, "What is justice?" Well, you'll have to read the rest of the book. As you read, pay close attention to "Gyges Ring" and the "Allegory of the Metals" (gold, silver, and bronze).