Proof by Induction

Paul Gifford is a waiting-for-tenure professor of mathematics at a university. His father, a professor-emeritus of mathematics at the same university has just passed away. This death has come at a very inconvenient time for Paul (not that there is a convenient time for a father’s death…) because the two were collaborating on solving something called, “The Perelman hypothesis”, an unspecified, fictional hypothesis of the real-world mathematician, Grigori Perelman (he who solved one of the Millennium Problems, the Poincare Conjecture, as well as the Thurston Geometrization Conjecture, and in the aftermath, turned down a Fields Medal…). The premise of the story is that for a few minutes after clinical death, a digital snapshot - the “Coda” - of the mind may be taken which can be used in a simulation to interact with a limited version of the dead person. Paul takes the help of his father’s coda to finish the proof. Along the way, the story describes his struggle with obtaining a tenure, the behavior of tenure-committee members and the research process, his father’s emotional detachment with his family, the thought that perhaps mathematical abilities may be inherited (hence, “induction” in the story title) and other such issues which could have been developed if the author had written a novel instead of a short story. As such, the story’s impact gets limited by its short length and limited elbow-room, though it does deliver a few interesting paragraphs. Convincing mathematical jargon is strewn across the story to give it a heavy mathfiction feel.

At one point, the father says something which I found a little perplexing:

“[The Jagdish-Rajput conjecture] is that hyperbolic equations correspond to node forms. They’ve tested several hundred terms using a supercomputer and they’ve all checked out.”
His father shook his head. “How’s that help us?”
“Node forms converge. Supposing we can prove their conjecture, we can use that to prove Perelman.”
“This isn’t math. This is grasping at straws. A supercomputer says it works—so what? That’s not theory. Where’s the proof?”
Paulie capped the marker, even though he suspected it could not dry out. “Don’t you see? If the correspondence holds, then—”
“Are you trying to give me a heart attack in the afterlife? Do Jagadish and Rajput have the basis for a theorem, or just a coincidence they can’t explain? Even Euler had conjectures disproven after three hundred years!”

I found the lines about “grasping at straws” and “heart attack” not very believable. In a number of real-life examples, matching of series expansion terms (say in Perturbation theory) acts as a good initial guide for an attack on a given problem. Of course, it is also true that matching a few or even many terms in an expansion does not a proof make, but then no one makes that claim anyway. The matching simply gives more confidence toward the conjecture in question. An accomplished mathematician dismissing this heuristic sounds very odd to me. In that context, I was reminded of the Langlands Program which rests on connecting seemingly unrelated propositions / conjectures as an example of using non-intuitive connections for proofs. Also related might be a recollection of the Littlewood-Skewes theorem as a well-known counter-example to the “matching a few terms should give confidence”.