There is a particular silence at the debugger when you realize the clever line of code you’ve been staring at is yours.
It does five things at once. At the time, that seemed like a virtue. Now one of those things has gone wrong, and your first task is to persuade the line to do them one at a time.
I thought of that while reading Konstantin Kakaes’s “Mathematicians race to simplify AI proofs” in Quanta. Mathematicians have been reworking an AI-generated proof that Landau–Siegel zeros do not exist. Kevin Barreto’s version came in at three pages. He says he was aiming to use more elementary, classical tools; the brevity was a by-product.
There are at least two ways to simplify a proof. You can replace a difficult technique with tools your reader already knows. That may take more ink and still make the argument easier to follow. Or you can leave out steps you trust the reader to supply. That can make a proof beautifully concise for someone who knows the terrain. For another reader, the omitted step may be the very place where the argument turns.
Three pages don’t tell us how much work Barreto removed and how much he left to the reader. To find out, we’d have to set the proofs side by side. I’d start wherever I got stuck. Can I tell which assumption I need? Can I work out why it applies here? If it fails, can I see what else falls with it?
A good proof gives you places to stop. It lets you put a finger on one step and ask why it holds without having to hold the whole argument in your head. Sometimes that takes an extra sentence. Sometimes it takes a better idea, and an entire page falls away.
An earlier entry in this Codex asked what it means to possess an answer we cannot yet explain. This one asks what we owe the next person once we can. I think we owe them enough of the reasoning to find their own mistake—or ours.
The finest shortcut leaves you a way back.
