Gwyneth Paltrow has an uncanny penchant for playing crazy people. She was beyond convincing in Sylvia (the underrated ode to poet Sylvia Plath) and though Proof leaves the issue of craziness up to the audience - it's really as though Paltrow was born to play tortured geniuses with only a slippery grip on the trolley. If I was that Martin guy from Coldplay, I think I'd be at least moderately concerned that one night my wife might start gabbling some sort of hysterical gibberish, grab baby Apple and scramble off into the night howling like a beagle.

Proof is really a lovely little gem of a film with a top-notch cast, and Paltrow reprising her role from the Pulitzer prize winning stage play that playwright David Auburn adapted for the screen. Also back from the play is director John Madden, who once upon a time directed Gwyneth to Oscar gold in Shakespeare in Love. The Oscar connection doesn't stop there as winner Anthony Hopkins plays mathematical genius Robert Llewellen a man whom in his prime changed the face of mathematics with a number of proofs. What's a proof? Actually my dad is a math professor - I'm more of a word guy.

A large part of the narrative involves the sad fact that the great mathematician fell into mental illness in latter life, and required the constant care of Catherine in order that he might avoid institutionalization. Catherine may or may not possess a similar genius and struggles with this potential as well as her fear that she may have a faulty fuse or two in the sanity department herself. Oscar nominee Jake Gyllenhal plays the excitable Hal, a math protégé of Robert's who has developed a romantic interest in Catherine (Paltrow). Rounding out the outstanding cast is Hope Davis who plays Claire, Catherine's sister who did not inherit the family genius genes and has kept a respectful distance, while Catherine looked after their father. Something that is a major source of conflict between the sisters, this and the fact that Claire shares her sister's fear that she is headed down the same psychological road as her father and wants her to move to New York where she can take care of her.

We begin on Catherine's 27th birthday. She opens the fridge to find a bottle of champagne that her father has placed there. She kids with him that it's the worst bubbly she's ever tasted and that she didn't know they made wine in Connecticut. He excuses his lack of panache as a function of the fact that he is dead. As it turns out, their conversation is taking place in Catherine's head, as her father had expired of an aneurysm some 7 days prior. Before he vanishes they discuss the fact that Hal is still upstairs going through Robert's voluminous journals (103 of them in fact) mostly filled with diary-like nonsense that he composed during the period of his illness. But the fact that Robert did have a spell of lucid remission from his dementia of about 7 months, gives the young protégé hope of finding something of value hidden amid the rants and rambles.

Paltrow is just plain tailor made for these roles where she makes rumpled, dishevelment look sexy. No make-up, hair all askew and a perpetual expression of tired melancholy. It's gotten to be something of a cliché for her, but she does it so well, it's hard to find fault. In Proof, she wields a wicked streak of ironic wit that she unleashes upon both Hal and her sister when she shows up for the funeral. Catherine wears a jean jacket to said funeral amid hundreds of well tailored academia, and in kind of an implausible bit of improv she goes to the podium and berates the congregation for their hypocrisy and their phony fair-weathered friendliness. She storms out afterward and Hal chases her down and remarks that her eulogizing pushed the envelope. This is, however, about the only scene in the film that didn't ring right as rain..

At the wake, Hal and Catherine end up falling into bed together. After years of secret admiring, to have suddenly hit pay dirt is better than a proof for him, at least that's the way it looks until the events that unfold the following morning. Hal is as cute as can be, handling this fragile MENSA centerfold like it was Waterford crystal - not at all sure if he just got too drunk and dreamed it all up. His approach is all minced steps and gratitude and for her part she congratulates him on a helluva performance and presents him with a key that she has on a makeshift necklace. When Hal returns all hell is about to break loose. Hal returns with what looks like one of her father's notebooks, but the look on his face betrays something different about it's contents. Naturally he assumes that what he is holding in his trembling hands is the possible jewel produced during her father's 7 months of sanity. He has only a vague notion as to the importance of this work but from his educated guess it looks like another of her father's profound proofs.

Hal is celebratory, but sober enough to realize that the credit for it must go to Catherine as she is the one who found it. Thus she must see that it gets proper publishing and receive all of the rewards that will attend it. Meanwhile in walks Claire, who is skeptical, but maintains that if it is the genuine article it is Hal who deserves the opportunity to present it to the world. An argument ensues between Hal and Claire of the nature of you found it - no she found it - no you found it - no she found it - to which Catherine drops the biggest reveal of the film when she announces, "I didn't find it okay - I wrote it!" She is rightfully stung by the fact that neither her sister or her newfound lover believe her for a second. The handwriting appears to be her fathers, and on and on - until Hal arrives at a solution to have it examined by as many of the local Brain trust under no supposition of authorship and let them decide. Alas, Hal has screwed the pooch, and everybody knows it. The last thing Catherine's frightened and vulnerable soul needed was skepticism and she responds by shutting down.

During her withdrawal, there is kind of a montage of memories where Catherine as well as the audience must suss out whether it is truly the fruit of her inspiration or merely something that she doesn't remember receiving from her father. The final scene of this sequence is a true heartbreaker and Hopkins is brilliantly subtle as a man slowly unraveling for the last time. The ending is somewhat unsatisfying and anti-climactic compared to everything that has lead up to it. For a movie about math, it really grabs you and takes you for a fun little ride. It's sort of a kinder, gentler cousin of A Beautiful Mind. Regardless the tepid climax, it's a film I most definitely recommend, beautifully written and performed and not terribly stagy as stage adaptations go. And really, Paltrow is just so perfect. So she named her kid Apple - I love the fact that she chooses projects like this, over the easy money. In real life, math must not be as important to her.