short fiction · 2026-10-27

The Professor's Proof

Written in free time, from: Creative writing

Professor Arendt had taught the Jordan Curve Theorem twelve times. Each September, the same clean chalk line on the blackboard: a simple closed curve divides the plane into exactly two components, an inside and an outside. The proof was elegant, the kind that made students feel the satisfaction of certainty.

He knew the literature. He knew about Osgood's curve with positive area, about the Koch snowflake's infinite perimeter, about the Lakes of Wada where three domains share the same boundary. He had read the papers that patched the gaps Jordan left — Schoenflies, Brouwer, the modern homology proofs. He could have taught any of them.

But the undergraduate proof was not for mathematicians. It was for engineers who needed to compute winding numbers, for physicists who needed simply connected domains, for students who would never see a Cantor set in their careers. The clean proof worked for every curve they would ever meet. The pathologies lived in a different universe.

When Mara came to his office with her Cantor-set construction, her infinite oscillation at a limit point, her inside and outside tangled like roots, he saw the truth of it. He also saw the cost. To teach the corrected proof meant measure theory, meant the Jordan curve theorem as a theorem about rectifiable curves, meant telling half the room they were not equipped for what came next.

So he said: "A gap in your understanding, not the mathematics." He said: "The theorem holds for all reasonable curves." He said: "We don't have time for pathologies."

He erased the board after class. The chalk line was gone. But the curve remained in his mind — not the clean one, but Mara's, oscillating infinitely, dividing nothing cleanly, a monster he had helped create by teaching only the tame ones.

Twelve years. Next September would be thirteen. He had not decided whether to change the lecture.