국가 9권에서 소크라테스와 글라우콘은 '참주의 삶은 철학자 왕과 비교했을 때 얼마나 불행한가?'라는 질문을 던집니다. 해당 부분을 재구성하고 어떤 점에서 틀렸는지 재미로 써봤습니다. 억까까지 합쳐서요 
Socrates and Glaucon address the question: "How much more unpleasant is the tyrant's life compared to the king's?"
Argument Reconstruction
P1. According to previous arguments, "the tyrant is at a third remove from the oligarch" in a plane figure.
P2. Similarly, "the oligarch, in turn, is at a third remove from the king" in a plane figure.
C1. Therefore, Socrates concludes that "the tyrant is removed from true pleasure by a numerical value of three times three." (From P1 & P2)
P3. Socrates makes the comparison "on the basis of squares and cubes," since he wants to compare the size of the king's pleasure versus the tyrant's pleasure.
C2. Socrates claims: "The king lives 729 times more pleasantly, while the tyrant suffers the same amount more painfully." This follows from the calculation:
(3x3)^3 = 729.
(From C1 & P3)
Refuting Socrates' Conclusion
However, C1 is incorrect, which means C2 is also false. From P1 and P2, we should instead infer that "the tyrant is removed from true pleasure by a numerical value of three plus three, minus one," in a plane figure. This is because:
- When calculating distance, we should add, not multiply.
- We must subtract one because the oligarch appears twice in the sequence, causing an overlap.
In other words, the tyrant is only five times removed from the king.
Thus, Socrates should have concluded that "the king lives 125 times more pleasantly than the tyrant," since:
5^3=125
What Does This Mean?
By refuting socrates' argument, We can know that Socrates and Glaucon are not good at calculation, despite their self-praise: "That's an extraordinary calculation of the difference between the two men —the just one and the unjust one— in terms of their pleasure and pain!"
By the way, Socrates himself states that "calculation, geometry, and all the preparatory education serve as preparation for dialectic." Moreover, he asserts that dialectic is a prerequiste for being a philosopher, as he says: "Don't you call someone a dialectician when he is able to grasp an account of being of each thing?" Since philosopher is defined as someone who can grasp an account of being of each thing, dialectic is essential to philosophy.
All of this means that socrates is not good at calculation, then he is also not good at dialectic —means that he is neither a philosopher nor a true dialectician. As a result, the truth of the arguments developed through dialectic in Republic as a whole comes into question.
