A superb cast provides the common denominator that allows this extraordinary story of human fallibility to play out against the backdrop of mathematical absolutes.

Gwyneth Paltrow imbues Catherine with both brilliance and a fragility of mind, as she confronts her father’s death and her own 27^th birthday. Her great fear is that she will inherit the mental illness that destroyed her genius father’s mind. Robert (the always-wonderful Anthony Hopkins) was a math professor and former head of his department at the University of Chicago. He made profound contributions to the study of mathematics before he reached the age of 27¾ as Catherine keeps reminding herself. Robert’s deterioration of mind began not long after that and continued, slowly, until his death at age 63. Though he had years when his "machinery" (as Robert referred to his brain) worked well. Even within the several months leading to his death Robert was lucid and working, and mentoring students, usually from his home.

One of those Robert helped was Hal (a buffed Jake Gyllenhaal, of "Donnie Darko" and "The Good Girl"). Now Hal asks Catherine if he can go over the more than 100 notebooks that Robert has crammed in his upstairs office. Perhaps Hal can uncover some dynamic new theory that Robert might have discovered, that will be his last legacy to leave the world. Of course, Catherine’s overbearing older sister, Claire (played annoyingly well by Hope Davis), raises the question as to whether Hal might claim the theories as his own. Catherine acknowledges that Hal could write his ticket to tenure at any university with such a find.

Catherine does trust Hal, however, with very important notes that could hold the key to unlocking a spark of genius-thinking left within the pages. She also opens herself to him with long buried passion. Catherine takes a quantum leap, even as her father did by working fiercely whenever his brain would allow.

If you doubt that watch carefully as Robert exhibits a burst of fiery passion for his work, as he sits at a table in his garden in winter and pores over his notes. How difficult it must have been for him to realize the folly of his ramblings as he insists Catherine read his last journal entries. Though they are not math, there is a poetic structure and logic in what he has written. He might also have been pushing Catherine to her potential, even as he falls deeper in the end. He loves her.

Claire has always recognized the symbiotic relationship their father shared with Catherine. They are both brilliant and both mathematicians. Hope Davis portrays Claire as a polished and capable attorney who lives in New York and has supported the Chicago household for some years, unknown to Catherine, and probably disregarded by her father. She is an anal personality, given to checklists that she ticks off - a controller but disengaged emotionally. In an effort to ensure that Catherine will return to New York with her, Claire has even sold the house back to the university, just days after Robert’s death, and without consulting her sister.

In a failed attempt at consoling Catherine, one of her father’s colleagues, commenting on Robert’s mental decline says, "After all, where’s the great man without his greatness?" Catherine is so unsure of her own talents and greatness that she’s afraid to claim what she has created, afraid to have her brilliance dimmed, as her father’s was.

The entire story illuminates the ebb and flow and even the beauty of delving deeper into the source of a fundamental truth of math, much as one realizes the logic and fire of an emotional relationship.

John Madden and Gwyneth Paltrow both won Oscars for their work together in "Shakespeare in Love," Best Director and Best Actress. Madden also directed Paltrow in the London stage production of "Proof," David Auburn’s Tony Award winning play. Auburn co wrote the screenplay with Rebecca Miller ("Ballad of Jack and Rose," "Personal Velocity").

Chicago looks great in this film…and there was no stand-in city for the exteriors. You can almost smell the musk of the books and the walls of the wonderful old Hyde Park house used in the filming, though a number of the interiors were recreated in a London studio.

"Proof" is a thought provoking film that could add up to more than one Oscar nod.