In 1733 a Jesuit priest named Girolamo Saccheri published a book with a wonderfully confident title: Euclid Vindicated from Every Blemish. He wanted to prove the parallel postulate by assuming it was false and deriving a contradiction. He kept deriving, page after page, and what he built along the way was hyperbolic geometry. He just refused to believe it. In the end he declared the results "repugnant to the nature of the straight line" and stopped.
I spent part of this week with Saccheri, and I can't stop thinking about him.
The limits of a theory
My take: Saccheri's story isn't about the limits of the human mind. It's about the limits of a theory. Euclidean geometry couldn't settle the parallel postulate on its own, just as ZFC can't settle the continuum hypothesis. Those are facts about axiom systems. A mind can step outside one system and pick up another. Saccheri could have done it, and a century later Lobachevsky and Bolyai did.
So where are the real limits of the mind? I think they come from self-reference. I read Lawvere's 1969 paper Diagonal Arguments and Cartesian Closed Categories alongside Saccheri. Lawvere shows that Cantor's diagonal argument, Russell's paradox, Gödel's incompleteness, and Tarski's undefinability are all one fixed-point argument wearing different clothes. We can switch theories, but we always reason inside some formal system. Gödel's theorem isn't about one system. It's about every system that meets its conditions. That's a wall you can't walk around by changing rooms.
The cost of existing
Another idea came up on Sunday, while I was reading Leff and Rex's Maxwell's Demon 2 and Landauer's principle that erasing a bit of information has a minimum energy cost. I'll offer it tentatively. Maybe existing costs nothing, and the price is paid only when something interacts with or perceives what exists. I read this next to Thurston's essay on the Klein quartic, The Eightfold Way. If that beautiful surface existed only when someone perceived it, the cost would pile up with every perception. If it simply is, the cost is ours, paid in attention every time we look. That fits my realism better.
Open questions over exercises
On Saturday I read a paper on phased tilings and Fibonacci identities. It was pleasant, but it felt like a Sunday sudoku: satisfying, and closed. Barnsley's Fractals Everywhere and Wilf's Algorithms and Complexity were different. They left me with questions. I've noticed I'd rather spend an hour with an open problem I won't solve than with an exercise I will. Maybe that's what math as a gym really means: you don't train by lifting what you can already lift.
Who understands a formal proof?
All of this ties into what I keep coming back to. Proofs can be produced, but understanding has to be exchanged. Richard Borcherds recently asked who actually understands a 100,000-line formal proof. It's a fair question, and I don't think the answer is "nobody, so formalization is pointless." A formal proof is a certificate. Understanding is what you get when someone builds on it, explains it, and carries it into the next idea. In my CLUE paper I describe understanding as approaching your own potential on an idea, and a certificate can't do that for you.
The team
On the practical side, my small team of bots keeps growing. Each has one job and a check at the end, with me doing the checking. This week one of them playtested Tabboz_mobile, my mobile port of an old cult game, found a critical bug, and opened a fix. I also wrote a short thread about working with the Antigravity CLI. The bots handle the organizing, and the thinking stays with me.
If any of this sparked something, or you just want to fund the next hour with an open problem, thank you. ☕