Keanu Reeves has never enjoyed a reputation as an intellectual giant but, even so, the directors Larry and Andy Wachowski exploited his pretty vacancy mercilessly in The Matrix. The moment when Reeves turned to the screen after an intensive virtual-reality training session and announced, "I know kung-fu" triggered immense hilarity in cinemas, where audiences found it hard to believe that he knew how to tie his shoelaces, let alone absorb complicated martial arts. There are moments in John Madden's Proof where you feel a similar incredulity as you are asked to believe that Gwyneth Paltrow is a prodigiously talented mathematician whose ability to whip up a quick world-beating equation is matched only by her ability to look pale and interesting as she slopes around in ugly - yet strangely alluring - cardigans and no make-up.
Naturally, charitable folk would say that this is known as acting. After all, a performer who can play Sylvia Plath without reducing her to a collection of hysterical tics is perfectly capable of impersonating an intellectual. Yet Paltrow is not the only thing that fails to ring true about this adaptation of David Auburn's Pulitzer prizewinning play. Originally staged in Manhattan in 2000, Proof transferred in 2002 to the Donmar Warehouse in London where, under Madden's direction, Paltrow took on the part of the troubled young mathematician Catherine. Unfortunately, four years later, you can still almost hear the rustle of programmes and politely stifled coughing just off-screen as the film strides stiffly from scene to stagy scene, never quite making the transition to a piece of breathing cinema.
Proof's present-day storyline is set in Chicago, and starts off with Catherine drinking alone on what is both the eve of her father's funeral and her 27th birthday. Her father, Robert (Anthony Hopkins), was a mathematician of genius who redefined his field before he was out of his mid-twenties, whereupon he began a slow descent into madness. Through flashbacks, the audience discovers how Catherine, his younger daughter, gave up her promising studies at Northwestern University to look after him, nursing him back to a point where he was able to have one last lucid year. Meanwhile, back in the present, Robert's research student Hal (Jake Gyllenhaal) is sorting though his notebooks hoping to find one last piece of mathematical brilliance that he can use for his own ends. Things come to a head on the day following Robert's funeral, when Hal and Catherine embark on a sexual relationship, Catherine's financial whiz of an elder sister Claire (the fantastically brittle Hope Davis) arrives to take charge of the family's affairs, and Catherine makes a revelation that will decide the course of her abandoned life.
Auburn is clearly not interested in making a Shine or a Beautiful Mind-style tear-jerker about tragically doomed genius. He focuses on the complex geometry of human relationships, the equations of trust, love and respect that add up to the most unfathomable bonds of sexual and familial closeness. At its best, Proof turns this emotional calculus into a kind of thriller as the audience is left wondering whose version of events actually adds up. Claire, who is almost a caricature of list-making and straight-living materialism, clearly believes that Catherine has inherited their father's madness, and slowly insinuates that her sister cannot be trusted to look after herself. Meanwhile Hal looks like a good guy (Gyllenhaal's big, "tickle-me-Elmo" eyes make him seem utterly unthreatening), but you have to wonder about a man who seduces a woman so soon after her father's funeral.
Hopkins's performance as Robert is pretty much a ruddy, just-add-bellowing portrayal of madness - the main evidence of his supposedly terrifying insanity is that he sits in the snow without a coat. Paltrow, despite her lack of credibility as a future Nobel Prizewinner, gives a more interesting performance as Catherine. The audience, like her sister, cannot be certain about her sanity, whether her intense grief, jealousy and desire to succeed could not lead her into a world of frantic delusion, and this is a bit of advanced mathematics Paltrow is equipped to pull off. Using her pallid good looks and her thin voice, which can sound as whiny and cold as a broken fridge, to conjure up Catherine's inner turmoil, she portrays a woman who has never been allowed to bloom, who is so convinced there is something wrong with her that she has somehow become wrong.
Occasionally, Proof gives a nod to the world outside Catherine and Robert's decrepit house: the terrible paranoia of academic life; the desperate, headlong hurtle to succeed before the next guy (as this film makes clear, it's very rarely the next girl). However, it still seems as dry and dusty as a row of maths textbooks. Maybe it's the stilted theatricality, the tight precision of the structure and the pained angst that characterises every relationship, but Proof seems like no more than a fraction of a cinematic experience, a string of emotional calculations that fails to provide a satisfying solution.