This unusual piece combines equal parts fashion industry and math research, with a dash of fantasy and just a pinch of homoeroticism. Grant does a favor for his old partner, Duncan, by modeling his new proof of Gödel's Theorem. As it turns out, the proof is not quite right and Grant has to fix it, impromptu, on the runway (to the wild cheers of his PhD students who have come to watch). Then, he must decide whether to collaborate on the P versus NP problem even though he feels that Duncan slighted him after their original joint paper.
To be honest, I would have been surprised to learn that there was an anthology of fantasy stories about fashion. Period. That a book so marketed would also contain a rather good work of mathematical fiction is almost beyond belief. However, Bloody Fabulous: stories of fantasy and fashion, the book which reprints many works of fantasy fashion fiction along with this previously unpublished mathematical one, proves me wrong.
For the most part, this story succeeds quite well. One has to get caught up in the feeling of it, since it doesn't really make sense, but I like the way it refers to "theorem houses" and even the way the pieces of a proof morph on the model as he walks down the runway. I was just a bit confused that it seemingly makes no distinction between new proofs of previously proved theorems on the one hand (e.g. Gödel's Theorem and Fermat's Last Theorem) and the use of proofs to discover new truths that were not known before (as a proof of P=NP would be). Obviously, in mathematics they are very different and a new proof of a known theorem would hardly generate the excitement we see on the runway in the opening scene of Incomplete Proofs. But, I really love the analogy the story makes between an unfinished garment and an incomplete proof. It finds more common ground both in description and in feeling between these two things than I would have thought possible:
(quoted from Incomplete Proofs)
Grant forced himself to match Duncan's gaze. "The first Duncan Banks collection got published in all the major journals and sold to all the major buyers. Everyone wanted to work with you. You didn't need me anymore and I might as well not have existed."
"You're never going to forgive me." Duncan seemed to deflate a little. "I'm not who I "
"Make your damn offer."
When the semester ended in a few month,s Grant would be out of a job. Besides, his legs felt like marble. Otherwise, he'd have walked out.
"I have the outline of a solution for P=NP. Flesh it out with me. Please?"
Grant sighed. Whether P=NP was one of the remaining great unsolved problems. Proving that P=NP meant biologists could quickly compute the structure of a protein rather than guessing its structure then checking for correctness. It meant computationally tractable ways to find optimal solutions to all sorts of packing and scheduling problems. No industry would be unaffected. Grant and Duncan would be heroes for the ages.
"Show me." Grant tried to sound bored.
The proof of ten as a solitary number transformed into pieces of muslin. They changed shape as they slid around Duncan's body. A lemma around Duncan's back fortified two results on his shoulders. What covered his chest seem[ed] to stay there out of sheer faith that someday something might hold it in place. He'd built it on conjectures Grant didn't recognize. After a minute, Duncan wore something that fit roughly on him, pinned together by hope and determination more than it was stitched together by mathematical theorems and logic.
...
Grant let the outline inhabit his mind. Not enough hung on Duncan to prove anything. Grant wasn't even sure what it actually proved, but the bits that were actually stitched together dazzled. The intermediate results, if verifiable, would advance mathematics nearly as much as the conclusion. What was missing defined the structure as much as what was there.
Grant's hands gripped the chair. He forced himself not to engage with proof. Math hadn't excited him much in years. But was it worth being burned by the sun and shocked by lightning again? They had never been and could never be, just about math.

According to the brief description in Bloody Fabulous, "John Chu designs microprocessors by day and writes fiction by night. His fiction has appeared in Boston Review." (Presumably, he is not the same man who directed that Justin Bieber movie, but as you can see from my several "surprises" above, I may well be wrong about that.)
