Division by Zero is a 1991 short story by Ted Chiang. It's a short read that you can find here. It's about a mathematician named Renee, her husband named Carl, and a formalism that breaks math. The story begins with Renee being released from a mental hospital into the care of her husband. Carl dutifully jumps through the hoops that the hospital puts in front of him. The whole process is made slightly awkward by his admission that he also attempted suicide a long time ago. This isn't a problem. He and Renee head home and Renee begins clearing all of the papers out of her office while trying to avoid looking at them. Carl is ruminating on his feelings about the whole situation; especially his inability to empathize with his wife.
Renee was brilliant and hers was a brilliance pulled along by the perfectly inevitable certainty of mathematical logic. While various institutions vied to employ her she only ever cared for the relentless pursuit of truth. In this pursuit she finds an operation that allows her to equate any two numbers without violating any rules. She takes this to a colleague in the hope that he can find the error. He finds no error. She begins a serious effort to eliminate this vexing anomaly. Several days into the task and Carl notices his wife's agitation. He confronts Renee about her increasingly worsening mood. She snaps at him and begins writing out the explanation, filling a page with symbols that end with the statement 1=2. She's discovered the problem with the formalism. Its issue is that it's true. Arithmetic is demonstrably inconsistent.
This breaks Renee and Carl cannot relate. He is trying to relate and she is systematically explaining how he's failing. Carl is grasping at comparisons to times that science had to adopt new paradigms and Renee is trying to get across that paradigms have failed and there is no way forward. They don't bridge the gap. Time passes, mental wounds fester, and Carl tries to cheer her up. Renee isn't having it. She's too busy compulsively equating numbers to have fun. She claims that she's literally losing the ability to differentiate all quantities from one another. She bluntly dismisses his fumbling attempts to grasp the extent of her loss and they part acrimoniously. Three days later he stops her suicide attempt. It's at this point that Carl has to admit something to himself. He can no longer empathize with Renee. Their positions are a warped reflection of the attempt he made before he ever met her and yet her pain is too alien. After her stay in the psych ward, after she's stable, he's leaving. Carl isn't happy about his choice, he thought he was a better person, but the person he fell in love with has changed. In the final scene Renee apologizes for all she's put him through driving Carl to deeper shame over his intention to leave her.
Interspersed through the narrative are short mathematical explanations which cover why you can't divide by zero and how Godel's Incompleteness Theorem ended the hope that mathematics can be both complete and consistent. It's one of those topics that is weirdly controversial if you dive into the background philosophy. Are numbers an invention or a discovery? Do the laws of mathematics follow from our choice of axioms or do our choices follow from mathematics? Are questions like these even answerable? At its core math is very effective at predicting reality and we have a very hard time explaining why. We take for granted that it can't just break but we already know that if you allow division by zero equivalence goes out the window. How many more problems like that are waiting to ensnare us?
Division by Zero is a story about such a snare and the human fallout. It's also about how people can love each other only to find out that the other is someone far stranger than they can handle. Mathematical geniuses going mad is a common trope. Such stories rarely take the time to have us empathize with the character. Few stories have empathy disappearing in the moments that it's most needed. This is a story about contradictions both logical and emotional. About two suicide attempts that should have created common ground and instead widen a rift beyond repair. We tell ourselves that the world is a particular way, that we are a particular way. Division by Zero asks the awful question: what if we're wrong? How do we pick up the pieces when our assumptions shatter?
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