The two major themes in Proof by David Auburn are genius & madness and love & trust. The first, madness & genius, revolves around the comparison of Catherine and her father Robert with Hal and Claire. Through Catherine and Robert, Auburn depicts genius as being unconducive to emotional stability in a complex world built upon mundane, day-to-day tasks and understanding. Auburn suggests that the fine edge of genius is not suited to survival in a practical world in which concern for food and housing and careers must come before the luxury of extraordinary achievement built from abstractions.
Hal and Claire are representatives of intelligent people--not geniuses--who have adapted to the everyday world of accomplishment. Hal understands genius and covets it for himself, knowing that, on his own, the stroke of genius will always allude him. Claire sees the presence of genius in her father but doesn't understand its nature or even its worth. Her view of Catherine is that she is simply emotionally unstable and in need of looking after; she doesn't see genius at all.