In Act 1, Scene 1, Robert and Catherine are celebrating Catherine's birthday. It's late into the night, and Robert brings up the fact that Catherine should be celebrating with her friends. When her father asks her why she isn't out with her friends, Catherine responds, "Because in order for your friends to take you out you generally have to have friends." Not only is Catherine friendless, but she's talking to a ghost. At the end of their conversation, Robert admits, "I'm also dead." Catherine is utterly alone, her only confidant a mere figment of her imagination.
Robert's mental illness is isolating in itself, given that he cannot communicate and relate to others. Catherine tells Hal that "[I] tried to listen when he talked. Talked to people who weren't there..." The inability to relate to the real world alienates Robert and divorces him from the people around him, including his own daughter.
There is a constant question of whether Robert's old notebooks have anything useful in them. Mostly they are the writing of a madman who cannot communicate effectively and translate the ideas in his head to the outside world. Hal and Catherine often talk about whether there is anything inside the notebooks. Hal admits with disappointment in Act I, Scene III: "The point is, that book - I'm starting to think it's the only lucid one, really. And there's no math in it."
Isolation is also developed through the lack of trust among the characters. Claire doesn't believe Catherine when Catherine claims that she completed the proof in her dad's notebook. Claire says in Act II, Scene II: "Catherine, I'm sorry but I just find this very hard to believe." Hal and Claire both doubt Catherine's mathematical genius until the very end of the play, throwing her statements into question and assuming she is lying or mentally unstable.
Isolation and madness go together in Proof. Catherine is in a somewhat chosen isolation because she wanted to take care of her father. Now that he is past she remains isolated because she is fearful she is also going mad. Hal and Claire seem to be “normally” socialized and come into this world of Catherine’s from the outside. The play takes place completely at Catherine’s father’s old house and is a symbol of Catherine’s own isolation from the rest of the world. She does not leave the house, and the only connection she has is to her father’s ghost.
ROBERT: Kid, I've seen you. You sleep till noon, you eat junk, you don't work, the dishes pile up in the sink. If you go out, it's to buy magazines. You come back with a stack of magazines this high-I don't know how you read that crap. And those are the good days. Some days you don't get up, you don't get out of bed.It is only through Hal and Claire that Catherine is connected to the outside world. It is these connections that allow Catherine to not go mad from her isolation. (Act I, scene 1, p8)
We get the sense that Catherine’s father, when alive was also very isolated toward the end. Was he isolated because of his growing madness or was he mad because of his isolation? Probably a bit of both. His dementia caused him to isolate himself because he was ashamed he was no longer the brilliant mathematician of his former youth. But because he chose isolation, this may have contributed to his dementia growing worse at a quicker rate.