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Transcript

Claude Proved Fermat's Last Theorem in Eleven Days — and the Mathematician Who Had Five Years and a Million Pounds to Do It Checked the Work Himself · Minds, Bodies, and Terawatts

• 0:00
  [Host] This is Minds, Bodies, and Terawatts, the show where the margin finally got wide enough. Anthropic said on Friday that a swarm of its Claude agents wrote a complete, computer-checked proof of Fermat's Last Theorem in eleven days.
• 0:13
  [Guest] And the mathematician who had a million pounds and five years to do that same job read the email at a music festival and wrote it off as a crank.
• 0:24
  [Host] Start with the theorem, because I know the name and that's about it.
• 0:29
  [Guest] Centuries ago Pierre de Fermat scribbled in a book's margin that he had a marvelous proof, and the margin was too narrow to hold it. Nobody found one until Andrew Wiles did, back when that mathematician at the festival was a graduate student.
• 0:44
  [Host] So it's been proven for decades. What did Anthropic actually add?
• 0:48
  [Guest] Not new math, because Wiles wrote for humans, who skip the obvious steps. Dozens of Claude agents, working in parallel, rewrote the whole argument in Lean, a language a computer can check line by line, and the finished file runs to thirteen million lines.
• 1:04
  [Host] A company says its machine did something nobody can read. How would anyone know it's real?
• 1:10
  [Guest] Lean either accepts the proof or it doesn't, and Anthropic put the file where anyone can run the checker. The person with the most reason to find a hole did exactly that.
• 1:22
  [Host] Who had the most to lose?
• 1:24
  [Guest] Kevin Buzzard at Imperial College London. Britain's science research council gave him a million pounds over five years to formalize this same theorem. His blog post the same day carries the title, Anthropic has beaten me to it.
• 1:37
  [Host] How did he find out?
• 1:39
  [Guest] He was at the Green Man festival in Wales with barely any phone signal when an email from a stranger appeared, subject line, end-to-end Lean formalization of Fermat's Last Theorem. He assumed it was a crank, and only learned the truth afterward, digging through the mail that had piled up.
• 1:57
  [Host] Late to the biggest news in his own field.
• 2:01
  [Guest] And not for the first time. He writes that when Wiles first announced the proof, he sat through the opening lecture, understood nothing, and skipped the rest for a holiday in Ireland with his new girlfriend. He heard the theorem was proved only when he got back. Same girlfriend at the festival this summer.
• 2:20
  [Host] And then he checked the work of the thing that beat him.
• 2:24
  [Guest] He compiled the whole code base, then ran a tool confirming the statement proved really is Fermat's, resting on nothing beyond the standard axioms of mathematics. His verdict, it checks out.
• 2:36
  [Host] Does he sound bitter about it?
• 2:38
  [Guest] Sharp, more than bitter, and he writes that the proof adds nothing. It follows the known literature faithfully, and he was already ninety-nine point nine percent sure the theorem was true.
• 2:49
  [Host] So the machine did paperwork on something everyone believed. Why is that news, then?
• 2:54
  [Guest] The speed, and he says so himself. If that much of the literature can be formalized end to end by an AI swarm in eleven days, we will start to see modern research formalized on the fly. His grant still has work, because Claude formalized the original argument from Wiles's day, while he is formalizing the modern proof and writing the version a human can actually read.
• 3:18
  [Host] Back in August we covered OpenAI's Astra cracking ten old problems with this same kind of certificate. Which is the bigger deal, new results with a certificate or an old one at this scale?
• 3:30
  [Guest] The scale is where it gets uncomfortable. Anthropic says the run burned about six billion output tokens, the word-pieces a model is billed by. Buzzard, holding a million-pound grant, wonders whether they spent more.
• 3:43
  [Host] Can anyone check that part?
• 3:45
  [Guest] No, Buzzard could only compile the file. The eleven days, the six billion tokens, how much a human steered, all of that is Anthropic's account of its own process. The proof is verifiable, and the story of how it was made is not.
• 4:00
  [Host] Does that split matter outside a math department?
• 4:03
  [Guest] It's the split every profession with a checker is about to live in. You never read your accountant's working, you checked the letters after their name, because checking the work cost more than trusting the person.
• 4:16
  [Host] And math just made checking nearly free.
• 4:19
  [Guest] For this kind of output, yes, because Lean doesn't care who the author is. So Buzzard's job this week wasn't writing the proof, it was being the one person whose word about the file the rest of us would take.
• 4:33
  [Host] What about someone without a research grant? Is this locked inside a lab?
• 4:38
  [Guest] Less than you'd think. Anthropic says a smaller famous theorem was formalized in days on three ordinary Claude Max subscriptions, a consumer plan, not a data center.
• 4:47
  [Host] So what's the human still for?
• 4:50
  [Guest] Anthropic's own read-me says the code was written to be checked, not read, and that no tool can confirm each step means what its name says. That part is left for a reader to judge.
• 5:02
  [Host] So the margin finally holds the proof. Is what's left for us just the note that explains it?
• 5:09
  [Guest] That, and a name to vouch for the parts no checker can read, so for more on verifiability, credentials, and what happens when checking gets cheap, head to unscarcity dot ai. The book is on Amazon, the audiobook is on Spotify. This has been Minds, Bodies, and Terawatts.