Gwyneth Paltrow is a serious actress. She's won respectable prizes, played Sylvia Plath, and performed on stage. And here she is again, demonstrating her seriousness, in the role she played on London's West End. Her character Catherine faces a dilemma in "Proof" that is both like and unlike Paltrow's. Catherine, like the Paltrow you might guess exists, is also serious. And she is concerned with her work and the way others perceive her. But her concern is also more fundamental, having to do with family, legacy, and self-understanding. "Proof" is hard to come by in these realms, and that's the point of this play-turned-movie. Proof is not only external and visible and documentable. Proof is also a matter of faith.
Catherine's question begins with her father, renowned and brilliant mathematician Robert (Anthony Hopkins), recently deceased and for a long time before that, crazy. That's his word for it, and he spends the first few minutes of the film appearing to Catherine and discussing definitions of insanity. "Crazy people," he argues, can't be aware that they are crazy, they can't even ask whether they are crazy, because they are so immersed in the minutiae that delimits their craziness -- their obsessions with daily details, paranoid fantasies, or the architecture of their subjective, fabricated worlds. Catherine is skeptical of this assertion: after all, her father is crazy, and yet here he is, able to ask the question of himself. Right, he agrees, without an answer. Except that, as he admits, he's also dead.
Based on David Auburn's play, John Madden's movie begins with Catherine's efforts to deal with her father's funeral and clean up his house, following her efforts to keep him together over years of madness. A genius who completed his own math-world-shattering work while in his 20s, Robert spent much of his later life writing out elaborate proofs about winter weather and his internal chill.
And while Catherine's sister Claire (Hope Davis) -- who might best be described as "chipper" -- has moved to New York City, married, and had a career, she stayed behind, within their father's sphere, tending to him, worrying about him, and keeping her own ambitions at bay. She completed some university classes, showing her own mathematical aptitude and a little lack of focus, enough to frighten her and Claire into thinking that Catherine might have inherited her father's brilliance and also his tendency to insanity.
And so "Proof" takes up questions of trust and doubt, grief and guilt, ambition and selfishness, as well as the sisters' competition and resentment. Catherine is also presently dealing with her father's ex-student, Hal (Jake Gyllenhaal), an aspiring mathematician and drummer in a local band. He comes by to sort through Robert's papers, hoping, perhaps, to find a lost instance of genius, something recorded during a rare lucid moment, a last sign that his madness was not utter and all-consuming, even as it may have seemed that way.
The film's central, concrete problem is the revelation of a proof, not quite elegant but exceptional and potentially math-world-changing (again), hidden away inside a locked drawer in Robert's home office. Flashbacks reveal that he and Catherine spent some time working on a problem together, separately, but at the same time, each writing out pages of proof, working late into nights and bent over desks and tables in deep concentration. As their handwriting is similar, it's unclear whether the newly discovered proof is Robert's or Catherine's. She claims it is hers, but neither Hal nor Claire quite believes it. And so the film presents a knot of questions, twisted up inside a knot of delicate performances and a fragmented narrative.
And yet, for all the potential nuance in this knotting, the film leaves the sisters caught up in a familiar conflict, in Claire's (villainous) controlling inclinations and Catherine's tendency to drift off or explode in hyper-articulate, theatrical frustration. Claire wants Catherine to move East with her ("It would be much easier for me," she says, not-so-sideways, "to set you up in a place in New York"), but Catherine believes she is not crazy. At least for a minute. Then she begins to disbelieve herself, having spent so much time with her dad, observing the sorts of delusions he was able to run on himself. Maybe, she worries, she is incapable of asking herself whether she's crazy, which might be all the sign she needs that she is.
"Proof" means to be serious, and it asks serious questions. It probes the very concept of sanity, it wonders about the possibilities for categorizing subjective states. It suggests that sanity is a set of norms and expectations as much as it designates mental or emotional health. It suggests the distinctions between sanity and insanity might be fine and shifting, not fixed, and certainly not orderly.
But as it asks you to believe in Catherine's claim to brilliance (visually, it tends to support her perspective, in her flashbacks as well as in her present interactions with Claire and Hal), "Proof" leaves little to your own figuring, not quite trusting you to be comfortable with irresolution or inelegance.