Gwyneth Paltrow -- in sad, lonely, fragile, forlorn mode -- plays Catherine, a bright young woman (actually, Paltrow is a little old for the part) who puts a promising career in mathematics on hold to take care of her schizophrenic father, Robert (Anthony Hopkins). She has no friends, she doesn't like her sister, she has only Dad to talk to. And she does, telling him that she's worried she might go crazy, just like he did. "You won't," he tells her in the opening scene. And there's warmth in his voice to go with his smile.
There are hints that Catherine may be going insane -- then again, maybe it's just worry about her father and the impending visit by her pushy sister (Hope Davis).
That's what this intense and talkative film keeps asking. The answer is eventually supplied, as are most of the answers to most of the questions that wend their way into the script. There are flashbacks to earlier, easier days in the father-daughter relationship, and these show the always great Hopkins in a clipped and one-dimensional performance. It's later in the film, during the heights of his illness, and the brief moments of clarity it allows him, that he lets loose, his character coming out from under a cloud and, as he puts it, catching on fire.
Paltrow's performance seems limited in that she's playing a very confused person who's stuck in a very deep rut. But after an amazing monologue at the funeral, and definitely by the end, it's safe to say that her acting here is of the powerhouse variety. The film becomes a series of studies of relationships between her and different people -- her obnoxious, polar opposite sister (who might as well be the character's mother, and a manipulative one at that), her dad (who pops in and out of sanity) and Jake Gyllenhaal's Hal (her father's doctoral student, who has his eyes on her, even though she says she's "a little out of practice").
In order for Catherine to reclaim her own life, first she must come to terms with some hidden emotions. Proof reaches its heights when part of that struggle turns into a full-fledged mystery that must be solved. The ending is optimistic, but it's certainly not of the cookie-cutter variety that Hollywood so often likes to dole out.