Proof (Symbol)
The proof that Catherine writes is a multifaceted symbol. Firstly, it is a symbol of her genius: the fact that she is an especially good mathematician, as brilliant as her father. The fact that she keeps it locked away in a drawer and does not tell anyone about it is representative of the fact that she does not believe in herself, that she is depressive to the extent that she does not want to show people her gifts. Then, after she reveals herself to be its author and no one believes her, the proof represents the fact that people do not believe in her and have sexist preconceptions about mathematical genius. At the end, Hal acknowledges that the proof is her work and it is once again a symbol of her brilliance. As they look over it and seek to revise it to make it more "elegant" we see that the proof has always represented Catherine's life itself, a brilliant and clumsy mess that is wholly unique. Their revision of the proof becomes symbolic of the promise that Hal and Catherine will rebuild her life.
The Porch (Motif)
All of the action in the play takes place in the same setting: on the porch of Catherine's family home. We see it in different seasons and at different points in Catherine's journey. It is the home that Catherine was raised in, but it is also a kind of self-imposed exile, a place where she has secluded herself in order to take care of her ailing father. The house itself represents both a safe place—a place where she is loved and supported by her father—as well as a place to flail and fail. Its safety holds Catherine back in many ways. The porch, as a place in between the indoors and the outdoors, represents a threshold to the real world, and is the perfect setting for Catherine to start making the decision about whether to stay in the safety of her old life of caretaker or spread her wings and emerge into the world as her own brilliant self.
Notebook (Symbol)
Hal drops one of Robert's notebooks out of his jacket and Catherine believes he is stealing it, but he is actually taking it home to wrap it and give it to Catherine as a birthday gift. The notebook is a symbol that Hal wants to connect with Catherine, that he is a good guy who is caring and kind. For a moment it seems to symbolize the fact that Hal is greedy and just trying to something to benefit himself, but it actually represents his generosity and the fact that he wants to help Catherine.
Jojoba Shampoo (Symbol)
When Claire first arrives in Chicago, she aggressively recommends a shampoo with jojoba in it to Catherine. The shampoo represents the fact that not only does Claire have a taste for the finer things in life, being a wealthy New York yuppie, but she also likes to push her preferences on her sister. Catherine has no interest in the shampoo, insisting that hair is dead, so it would be impossible to make it healthier. The shampoo stands in for the ways that the two sisters have very different priorities. Claire loves the finer things and caring about her appearance, while Catherine has no interest, preferring the life of the mind.
Math (Symbol)
Math itself becomes a symbol for a way of approaching larger existential questions and making a difference in the world. From the start of the play, Catherine is hung up on the fact that her father made huge innovative strides in mathematics by his mid-twenties, but she has not accomplished much. Hal also wants to be a great mathematician, and he shares with Catherine an intoxication with the elegance and beauty of math. Making strides in mathematics thus becomes symbolic of a life well lived, of seeking out an elegant way of living.
