Posted by jlebar 14 hours ago
the project: https://imperialcollegelondon.github.io/FLT/
>>Along the way, it wrote 13 million lines of Lean and proved 29,500 intermediate theorems
Did a human check the 13 million lines of code? How does QA'ing this type of work works?
So, all you have to verify is the formalization of the theorem, and believe that the proof checker is free of bugs. You don't have to read the actual proof.
https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...
But how do you know you told it what you intended to tell it?
https://leodemoura.github.io/blog/2026-8-1-postmortem-for-ke...
https://leodemoura.github.io/blog/2026-8-24-postmortem-for-t...
...I'm not saying this FLT result is compromised. I suppose things depend on your perspective where we are on the spectrum of "finding more bugs means there are fewer left to discover" vs. "finding more bugs probably means there are still unexplored corners out there".
Note to other users: don’t downvote this kind of comment, answer it.
encode mathematical reasoning in a way that can’t be fooled.> The first coordinate of the polynomial X^2 (X^3 + X + 1 ) is equal to the prime factorization of 30 .
We defined polynomials as their coefficient functions in my algebra class, and it makes sense that you'd define a prime factorization as a function from primes to N, which naturally extends to a function N->N. So this junk theorem is part of normal math too. It just says in an obtuse way that they're both the function that's 1 at 2, 3, and 5, and 0 elsewhere.
Notably, junk theorems are true. Nobody would debate that the junk theorem is true. The main thing people would say is that junk theorems, while being true, are sensitive to precisely how you encoded mathematics, so despite being true, they are perhaps not conceptually meaningful.
As an example of a junk theorem, sasy you use the definition of the natural numbers using von neumann ordinals
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_na...
Then for any natural numbers n, m, they're implicitly sets. So n \intersect m = min(n,m). This is the wrong way to think about natural numbers. You should not use this ever in proofs. But this isn't because your proofs would be false, but instead because it is a fundamentally confusing way to think about the natural numbers. It is in this sense it is a "junk theorem".
False negative = could not find a proof of a true theorem.
False positive = erroneous proof of a theorem.
I've tried the various intros to Lean multiple times (even before Lean 4 came out) and something about the way Lean proofs are written does not align with how I think about proofs. My very brief attempts at Isabelle / RCoq feel more natural.
I think it's a pity that the future of proofs is Lean. I'd love for someone to come up with a more digestable proof language!
Theorem. $\sqrt{2}$ is irrational.
Proof.
Assume that $\sqrt{2}$ is rational. Then there are integers $a, b$ such that $a^2=2b^2$ and $(a,b)=1$. Hence $a^2$ is even. Therefore $a$ is even. So there is an integer $c$ such that $a=2c$. Then $4c^2=2b^2$ and $2c^2=b^2$. So $b$ is even. Contradiction.
Qed.
Or, say Isar in Isabelle/ZF [2].
There is an interesting discussion on MathOverflow titled "Are we stuck with Lean?" [3]. The conclusion seems to be yes, they are.
[1] https://ceur-ws.org/Vol-448/paper10.pdf
[2] https://isarmathlib.org/UniformSpace_ZF_1.html
[3] https://mathoverflow.net/questions/513742/are-we-stuck-with-...
Lean is not for humans.
Wiles’s proof will remain a mystery to me.
I'm just old enough to remember Paul Erdo"s and his notion of 'The Book', which he defined to be a book the "Supreme Fascist" (God) had which held the most elegant proofs of mathematical theorems.
https://en.wikipedia.org/wiki/Paul_Erdős#Personal_life
It would be interesting to see how Erdo"s would name such a huge proof by Claude using Lean.