Socrates is not successful at refuting Thrasymachus’s point because he is making assumptions on what he believes is supposed to happen in a society.
He believes that any knowledgeable person in a craft will not try to surpass his fellow craftsman. He asks, “Do you think that a musician… wants to outdo another musician…?” (349e) This is weak because it contradicts Socrates’s own argument by discussing the unjust man and his actions, which implies that there is indeed a possibility of unjust acts to get ahead in society. Thrasymachus explains that when people act justly it is a disadvantage to them because the unjust are at an advantage, even though his argument is complex it is more sensible than Socrates
arguments.
Socrates also tries to argue that justice is desirable because it is a virtue of the soul, which I believe further unsuccessfully refutes Socrates argument against Thrasymachus argument. (353e) Against this interpretation, it might be said that Socrates has a valid point about the desirability of justice. It would seem that people would choose to be just, to have a better soul, for the sake of their happiness, but even if it should be this way, that does not mean that people will be just! People will be unjust if it means they can advance or get ahead in life, which is what Thrasymachus argues. This is a valid argument because it is our natural human desire to want more, whether it be power, luxuries, lust, etc. Thrasymachus is being reasonable and demonstrating a sense of reality while Socrates continues to be naive and try to paint a perfect city. He asks Thrasymachus “Will the city that becomes stronger than another achieve this power without justice, or will it need the help of injustice?” (351c) Socrates makes it seem as if there is only one right answer to this question, that supports his argument, but I differ. A city can become stronger through unjust acts to get ahead!
Socrates does not give a good argument in proving that being just is in our own interest and something we naturally want to do. His arguments are unreliable because he is arguing that justice is innate and inevitable, which is not a strong argument because there are no facts to prove it. Throughout Book I, Thrasymachus looks at justice in a broad, objective field while Socrates looks at subjectively, which leads to his weak arguments and unsuccessful rebuttals. Socrates has only advanced weak arguments instead of arguing on his own view of justice. Just maybe if he would have focused more on his own view of justice to support his points, it would have probably helped him in attempting to successfully refute Thrasymachus’s argument.