Why do we see Socrates in Jacques-Louis David's The Death of Socrates, and Plato in Raphael's School of Athens pointing up with their index finger? What does this gesture symbolize? My initial conjecture was that it may be a sign of achieved wisdom or an indication of authority.
Looking at the philosophy expounded by Plato, one thing comes to mind: the Ideas, set in another realm, above the material, quite abstract. Thus, it could mean that the raised index finger points to this very realm of Ideas, where the highest principles of virtue lie and to which one must aspire.
A similar explanation goes for Socrates, to which we can add that by doing so, he holds firm to his belief, even on his death bed. His posture, straight, indicates the same thing.