In Proof, there is a contrast presented between the abstract and tangible aspects of life. Claire's life revolves around everything that is practical while Catherine relies more on things that are theoretical. These choices cause problems in each sister's life in that they are both disconnected from the real world and neither can relate to other people, including each other.
Catherine focuses on the theoretical, letting her life revolve around things that are not tangible, things that other people cannot necessarily see. Early on in the play, Catherine figures out that the number of days she has wasted because of her depression is a mathematically significant number. Math is a very abstract science and this scene shows how mathematically minded Catherine is. It also shows how easily she is able to think abstractly. Her father helps her mathematically manipulate this number right before he admits that he is, in fact, dead at the time of this conversation. Again, Catherine is relying on something intangible, the mental representation of Robert. While Robert was ill, Catherine stayed with him to take care of him emotionally. She did not typically wash dishes, clean the house, or pay bills, Catherine was taking care of her father's emotional state. In her mind, she was making sure he stayed well by having someone to rely on for the intangible aspects of life.
Because Catherine lives in the abstract, she is unable to relate to people who live in the real world. Because of this and other reasons, she has no friends. She tells her father, "in order for your friends to take you out you generally have to have friends." Most 25 year olds would go out with friends on their birthday; the fact that she has no friends is odd and causes the audience to worry. In the end of Act 1 Scene 1, Catherine calls the cops to keep Hal from stealing one of her father's notebooks even though, as she admits in the beginning of Scene 2, she didn't really want them to come. The cops come back the next morning and are not happy. This shows that she does not relate well with the practical world and its consequences.
Unlike her sister, Claire is overly practical, completely engrossed in material aspects of life. While Catherine is at home with their father, Robert, Claire moves to New York to continue her education, get her own place, and have a job. She pays all the bills from New York, but does not involve herself with other aspects of taking care of their father besides encouraging Catherine to put him in a full time care situation. Bills are very material and concrete, showing how much Claire relies on tangible aspects of life and how well she can deal with these things. When Claire is visiting Catherine for their father's funeral, she tries to get Catherine to try a conditioner she likes. When Catherine asks Claire for some scientific facts about the Jojoba in the conditioner, Claire replies, "it makes my hair feel, look, and smell good. That's the extent of my information...