Proof (12A)

For an industry obsessed with numbers, Hollywood makes very few movies about mathematicians - and practically none about accountants. The last time we saw a maths whizz on screen was Russell Crowe playing John Nash in A Beautiful Mind, and it wasn't his advanced theories the film was interested in but the more dramatic possibilities of his incipient schizophrenia. Proof, adapted from the stage play by David Auburn and Rebecca Miller, also focuses upon a mathematician in mental turmoil, and proposes a similar equation of the creative and the cracked. Or, as the poet had it, "Great wits are sure to madness near allied/ And thin partitions do their bounds divide."

Just how thin that partition gets is a question of some urgency to Catherine (Gwyneth Paltrow), who has spent the last five years in Chicago looking after her father, Robert (Anthony Hopkins), a legendary maths prof whose mind is now on the teeter-totter. Though he still has periods of lucidity, he has neglected himself, and his work has gone to pot. When he first appears, all seems well, and Catherine is impressed that he can discuss his condition so calmly; crazy people don't tend to ask "Am I crazy?". But it turns out that, like Hamlet, she is conversing with her father's ghost: Robert died from an aneurysm the previous week. Catherine already seems to have inherited his genius for numbers. The question is, has he also handed on to her his madness?

To take one look at Paltrow, you'd think she's not mad so much as very, very miserable. She seems to be wilting beneath the same thundercloud of depression as her tragic suicide in Sylvia, only without the consolations of poetry to lighten the load. I timed it at approximately 45 minutes into the film before she permitted herself to smile. Perhaps Catherine is glum because she's just turned 27, a dangerous age for a mathematician: according to the stats, at 23 you're supposed to reach your intellectual peak, and by 26 you're a burn-out. This information comes courtesy of Hal (Jake Gyllenhaal), a maths geek who's been trawling through Robert's papers - 103 notebooks, to be precise - in the hope of finding some late, great work the ailing genius may have produced.

I think geeks must have really sorted themselves out since I was at college: back then they used to wear jackets knitted from hedgerows and haircuts previously fashionable around the time of Chaucer. Their defining characteristic was that they used to hang out exclusively with other geeks. They didn't have gym-pumped muscles or play drums in the college band, and they didn't look like Gyllenhaal.

And they certainly did not attract women who looked like Paltrow. "I thought you seemed... not boring," Catherine says to Hal after they spend the night together, not exactly the most ringing words of endorsement but probably as good as it gets in her melancholy world view.

Hal's determination to coax Catherine out of her shell coincides with the arrival from New York of her older sister Claire (Hope Davis), a smiley fusspot who's apparently convinced that Catherine is a chip off the old block as far as sanity goes. After the latter storms the podium at the funeral service to deliver an acid denunciation of her late pa's fairweather friends ("I'm glad he's dead," she bitterly concludes) you wonder if Claire might have a point.

The plot of the movie turns on a mathematical proof that Hal finds in Robert's desk. According to Hal, it's awesome, blindingly brilliant, a thing of wonder. According to Catherine, it's not by her father - it's by her. This, despite the handwriting that looks like her father's, and despite the fact that this etiolated young woman seems barely capable of getting out of bed, let alone wowing the maths fraternity.

As Hal and Claire try to determine the truth, the movie increasingly betrays its dramatic roots, ie people have the kind of long, impassioned arguments one only ever sees on stage. Perhaps when pressed up close to the action at the Donmar, or wherever it began life, you would be singed by Proof's rhetorical heat; removed to the colder climate of film it sounds shrill, hysterical and overworked. Director John Madden opens it out by shuffling flashbacks into the narrative, though aside from some wonky camera tilts there's not much spin on the material. While Hopkins holds back from baroque displays of derangement, one doesn't feel the tragedy of a mind in meltdown.

What the story deals in is a misplaced kind of self-righteousness. It sets up a contrast between shy, virtuous Catherine and brisk pragmatist Claire, whose chief motivation is apparently to uproot her wallflower sister and carry her back to New York. This is portrayed as the interference of a control freak, yet all Claire has done is buy Catherine some nice clothes, encourage her to eat properly and offer to mollycoddle her for a while. We should all have such considerate sisters.

Even more obvious than its favouritism, however, is the emptiness at its centre. What underlies this tale of ambition, rivalry and madness is the rarefied nature of mathematical contemplation, the voodoo of numbers and formulas. Now, nobody wants a maths lesson, but the resolute avoidance of the simplest abstract thought - one nifty metaphor might have done - leaves a giant hole. "The proof is very hip!" enthuses Hal. And the audience is very thick, it seems to follow. Four characters, plus multiple scenes of brittle emoting, minus the stage. You do the math.