Proof stars Gwyneth Paltrow, Anthony Hopkins (barely), Jake Gyllenhaal, and Hope Davis and pretty much nobody else. This is because it is a film based on a Pulitzer Prize-winning play, and plays generally have a few main characters and no supporting cast to mention.
Paltrow is decent as Catherine, the frayed daughter of Robert (Hopkins), a famous mathematician who appears mainly in flashbacks designed to demonstrate how darn tough life is when a parent you idolize loses his marbles. And lose them he does, albeit in pretty short scenes that don't really require Anthony Hopkins in the role. Catherine fears that, like his talent for math, she may have inherited her father's penchant for lunacy.
Jake Gyllenhaal plays
Hope Davis comes into play as Claire, Catherine's sister, who wants to sell the house that Catherine and Robert have lived in for years and get them both the psychiatric help that they need. She also gets drunk and recommends hair care products.
Everyone acts their little hearts out, here, but it's all for naught.
For starters, they all act like they're on stage performing a play, which leads to some annoyingly phony moments where characters will interrupt one another right where the "--" surely is at the end of the first speaker's line. Several of the scenes featuring back-and-forth dialogue come off like poorly-staged and unfunny versions of a Gilmore Girls exchange.
It doesn't help that Gyllenhaal and Paltrow have very little chemistry and their first scene together ends with such a dumb, dumb plot point that I wanted to smack myself in the head.
Also quite noticeable is the weirdly predictable storytelling structure that it inherits from the stage version. "THIS PART IS A FLASHBACK" might as well flash on the screen for all the integration that the act or so we spend in flashback has with the modern part of the story. It probably doesn't help that, despite the fact that this flashback is supposed to be several years before and at least a year or two in duration, Paltrow and Hopkins appearances change very little from the modern storyline.
There are two "revelation" scenes in the movie, conveniently located at the end of Act II and the end of Act III, respectively. The first one is mildly interesting, but the second is something you can see coming from a mile away, as the climax (if you could call it that) can only go two ways, (and if you're paying attention even a little, it can really only go one way.)
But really, the heart of the problem with Proof is that it is dull, dull, dull. I know this is probably a shock, considering it's a movie about math, but I am not just saying this to be confrontational. There are a grand total of two scenes in which anyone even raises their voice, and Paltrow playing a miserable suffering misunderstood, possibly-crazy-yet-still-sexy-in-a-sad-way math genius doesn't go nearly as far as you might think.