I confess that I loved this film. Paltrow hits the nail on the head with so many aspects of her character: the fear of being crazy, the burden of love that she undertakes when caring for her mentally ill father, the way in which her older sister treats her as fragile and subadult. Frankly, when Paltrow goes on a rant at her sister in the middle of the movie (the one in the street that starts with her pointing to Jake Gyllenhall and saying "Harold! Dobbs!"), I was cheering for her. Paltrow and the director got this spot on for me, and the whole film did an amazing job of highlighting the importance of love and trust.
The proof of the title is not only about the mathematical writings, of a genius gone mad; it is also about the fact that trust is a requirement of relationships, and that having to prove one's trustworthiness is a death blow to trust itself. Once the element of doubt is let in, the relationship becomes about feeding the doubt by disproving it; and our sense of security in the world is threatened. Especially for those who have security issues, and the mentally ill are no stranger to these, a single lie or fostering of doubt calls the entirety of the relationship into question: Doubt of others blossoms into self doubt, and soon we don't have any idea of what we know.
