Fear
Catherine is plagued by fear. In the first scene, we learn that she is afraid that she will never accomplish anything in her career in mathematics, as she is 25 and has yet to do any significant work. She also fears that she will ultimately end up like her father, having mental health issues that debilitate her. Other characters experience a great deal of fear as well. Hal fears that he is inadequate as a mathematician and we see his need for approval from Robert in flashback scenes. Claire fears that she no longer knows how to take care of her family, nor how to be there for her sister. Robert, in his declining years, fears that his best days are behind him. Each of the characters is grappling with fear in their own ways.
Sacrifice
This is an important theme in the play as we see that Catherine sacrifices her education in order to take care of her father after his mental breakdown. She could have easily taken him to a hospital and let him live out the rest of his days there, but she didn't want to see him suffer in an institution. Her love for her father leads her to forego the opportunities in her life to ensure he has an adequate quality of life.
Claire and Catherine's conflict as sisters has a lot to do with the fact that neither of them will recognize the sacrifices that the other has made. Claire feels that Catherine is ungrateful for the financial sacrifices she has made on Claire and Robert's behalf, while Catherine feels unappreciated for the quality time she spent with her father, and the sacrifices she made to be with him in his final days.
Proof
The title of the play becomes one of its major themes. We see that the story is about having proof that Catherine wrote the mathematical proof. When Hal and Claire do not believe that it is her work, she is disappointed to find that she must prove herself. Rather than take her word for it, she needs to show some evidence that she is capable of the work.
Once Hal realizes that the work is Catherine's, he tries to apologize to her and show that he was wrong. She is miffed about the fact that he did not take her word for it when he could have, that he doubted her. The fact that she had to prove her sanity and her reliability is a huge betrayal. In the wake of this betrayal, Hal must prove to Catherine that he believes in her and that he is an ally to her.
Mental Illness
Another major theme in the play is mental illness. We quickly learn that Robert suffered from graphomania, in which he obsessively wrote things down, but lost his grasp on reality. Because of her father's illness, Catherine spends the play worried that she has inherited his illness—a fear which is only made worse by her sister's conviction that she ought to speak to a doctor.
The play looks at the ways that people perceive mental illness. Catherine saw her father's illness as a reason to take care of him and spend time with him, but Claire saw the illness as something that would be better dealt with at an institution. The play raises questions about ethical ways to handle and heal mental illness, as well as the ways that mental illness can alter the trajectory of someone's life.
Math
The play centers around a famous mathematician, his daughter, and his former student. Math is the discipline that connects them all, and which excites them the most. Catherine feels best when she is working on a math problem, and her connection to Hal, and to the memory of her father, is through her understanding of and proficiency with complex mathematical principles. The proof that Catherine writes is a huge contribution to the field, and the entire plot hinges on its discovery and planned future.
Connection
Whether Catherine suffers from the same delusions her father did is unclear, but she is undeniably depressive and antisocial. Her decision to move back in with her father and put her education on hold is a decision that cuts her off from her peers and from the rest of the world. She hardly sees anyone apart from her father, and she develops a rather thorny and unfriendly attitude. Throughout the course of the play, Catherine reckons with her own animosity towards others and tries to find some connection in a world from which she feels cut off. Hal is the main catalyst for Catherine's opening up and her willingness to find connection with others, and through their romantic and sexual bond, Catherine begins to trust others again and find some happiness.
Gender
On the porch on the night of the post-funeral party, Catherine and Hal talk about the lack of female mathematicians in history. Catherine tells Hal about Sophie Germain, a famous 19th-century mathematician who had to pretend she was a man in order to disseminate her mathematical theories. Catherine's own journey towards recognition as a brilliant mathematician mirrors Sophie Germain's, and while it is not explicitly stated, this seems to have a great deal to do with her gender. When she first tells Claire and Hal that she wrote the proof that Hal found in Robert's desk, they think she is lying or delusional. They cannot fathom that she would have written it, and insist that there must be some other explanation. It is implied that their doubt stems from their preconceptions about gender and about women in mathematics. Thus, the play stages the sexism of intellectual discourse—the fact that women must work harder than men not only to gain recognition for their ideas.
