What are your impressions of Nash in the opening third of the film while he is attending graduate school at Princeton?
I think that the overall impression that the film gives of Nash is, like people of his generation and his colleagues at Princeton, that the work he does has direct and quantifiable impact on the betterment of society. The professors that teach Nash and his colleagues impart to them the importance of Mathematics in post- World War II American society:
Mathematicians won the war. Mathematicians broke the Japanese codes and built the A-bomb. Mathematicians like you.
These words cast a very large impression on Nash, who believes that his purpose is to make the next "great" discovery like those who came before him. In the pen ceremony at the start of the film, Nash looks on with envy, wondering if he, too, will ever have a chance to be honored in a similar manner. The race to establish the "next great Mathematics truth" is what animates Nash and his colleagues. The overall impression that we get of Nash is that he is driven by the authenticity of math, convinced that through its earnest pursuit, overall benefit to society and to Nash's own name can be achieved.