What is the inciting incident in the play Proof?

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Think of an inciting incident as a kind of jumping off point for a narrative's plot. If you can identify the one moment, early on, that all the subsequent action stems from, you've found the inciting incident.

In the case of the play Proof by David Auburn, arguments could be made for several different moments being the inciting incident. One might argue that the inciting incident is the death of Catherine's father, Robert, which occurred before the play begins and is revealed by Catherine's vision of him in act 1, scene 1. Certainly, his death shakes his daughters to their core, causes Catherine's sister Claire to come to town, exacerbates Catherine's doubts about her own mental stability, and leads to the discovery of Robert's brilliant mathematical proof, in that it allows those left behind to search through his things for signs of genius among the rambling.

There is also an argument to be made that the inciting incident is when Hal reveals to Catherine that he's been going through her late father's work, early in act 1, as this leads her to give him the key to Robert's desk, in which the titular proof is found.

Finally, as mentioned in the other answer, one could view the act of giving Hal the key as the inciting incident. Even though it comes a bit later in the play, towards the end of act 1, it is a clear turning point in the drama.

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The "inciting incident" is the moment or occurrence that starts a problem in a work of literature or play (or in a movie). Up until that point, the audience has been introduced to the backstory, and the "inciting incident" brings up the dilemma that is going to be resolved in the rest of the work.

In the play Proof, the inciting incident takes place in Act I, Scene IV when Catherine, after kissing Hal, gives him the key to her father's desk. In the desk, Hal finds a proof, and he says, "It looks like it proves a theorem...a mathematical theorem about prime numbers, something mathematicians have been trying to prove since...since there were mathematicians, basically." Hal is not only excited about the proof because it is found after the death of Robert, Catherine's father and a famous mathematician, but also because the proof was written at a time when most people assumed Robert was far too mentally ill to be productive professionally. Just as the curtain drops to mark the end of Act I, Catherine says of the proof, "I didn't find it. I wrote it." This marks the inciting incident because there is no real proof that Catherine wrote the proof, pun intended.

Act II begins with a flashback to Robert and Catherine, and then, in Act II, Scene II, the plot continues to revolve around whether Catherine wrote the proof or not. Hal does not believe that Catherine had the capacity to write the proof, and Claire, Catherine's sister, thinks the handwriting in the proof is her father's. At the end of the play, Catherine says that even though Hal believes that the proof uses several new mathematical techniques that Robert, who was suffering from mental illness, couldn't have known, "it doesn't prove anything." Hal and Catherine begin working on the proof, collaborating, which is the greatest amount of proof Hal and the rest of the world are ever going to get that Catherine is the author of the proof. This process seems to resolve the questions that were brought up in the inciting incident to some degree. 

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