For math illiterate people like me, my understanding is that FLT was already proven, but the proof was beyond complex, certainly for mere mortals like me, and now Claude has codified it, correct?
Formalizing Fermat's Last Theorem
361–370 of 526 posts
Re: Formalizing Fermat's Last Theorem
#362Re: Formalizing Fermat's Last Theorem
#363Earlier quoted context omitted.
Or we could invest in a ton of other non AI related research we're underinvesting in.
Like? I feel breakthroughs that can be found via AI might help us more in the long term where even previously non AI fields can be helped by AI. So you have specific non AI research in mind that we're underinvesting in? Because the USA is already spending crazy anyway for healthcare and I don't feel like funding is the issue but better incentives, reforms etc
Funding for basic research is being slashed by the current administration. Our society is underinvesting in basic scientific research. And, AI will not fill the gap.
Re: Formalizing Fermat's Last Theorem
#364Earlier quoted context omitted.
Like funding education. Let's build up human intelligence instead, they seem to have made great breakthroughs in every single field! The US doesn't pay too much to healthcare, they pay too much to health insurance. Too much for too little value
But US also spends too much on education as well. The issue doesn't seem to be funding but the educational reform like in mississippi, where they increased student performance without increasing their budget too much. That's why you see bad k12 educational outcomes compared to the budget spent in blue states. It's all about efficiency. Give AIa chance in few years as I feel it can make great strides.. it's hard to im…
You see funding of chatgpt as a panacea for progress.
I see funding of chatgpt as one of small part of a history where governments and industry fund basic science and moonshot programs, not to generate revenue, but to explore what is possible.
LLM funding is not aimed at improving our understanding of the world, it's aimed at making people reliant so that they may extract wealth through subscriptions for shareholders.
Americans don't get good healthcare and education because that's what they vote for, in elections and wallets. I am hopeful that that changes, but we shall see.
Re: Formalizing Fermat's Last Theorem
#365So I don't know Lean or Mathematics to any degree to really be able to say this with any level of confidence, but speaking from a pure software engineering backgrouand, how do we know that 13 MILLION lines of Lean code are bug-free? It seems to me that for a mathematical proof, bug-free would be an absolute requirement. Maybe the structure of Lean imposes that, I don't know, but that seems highly unlikely to me. That…
The structure of Lean does impose that. The code isn't being run, it's being type checked. And that's it. The overwhelming majority of Lean code is never run. It exists only to be type checked (because type checking is equivalent to verifying the proof).
You could imagine the typechecker has bugs (and indeed another comment mentions examples of bugs!). Crucially though anytime the typechecker has a bug fixed you could rerun the typechecker on the code to see if it still type checks.
This is the whole promise of formal verification. It reduces the problem of verification purely to the typechecker. If the typechecker is correct, then the proof is verified, no matter how many lines of code the proof is. As a sibling comment puts it, the chance of bugs mainly scales with the number of lines of code in the typechecker, not in the amount of lines of Lean code.
Your question is akin to asking, "yes this spellchecker ran fine on your essay, but are you sure it runs fine on War and Peace? That's 1000x more words!" To which the answer is the number of words doesn't matter if the spell checker is correct (which it might not be! And longer passages might reveal more bugs! But you can always rerun it). The main source of bugs is more lines of code in the spell checker, not in number of words in the text.
Re: Formalizing Fermat's Last Theorem
#366Earlier quoted context omitted.
Because of the ongoing training costs. They are certainly making a healthy profit margin on inference.
Never really a sound argument. It's like having new solar panels installed every week. Sure you're "profitable" on the $0.20/kWh you're selling your "free" energy at when you ignore the cost of the solar panels you're buying every week.
It is a sound argument in the context of trying to estimate what it costs them to generate this specific output. They have training cost eather way.
Re: Formalizing Fermat's Last Theorem
#367So I don't know Lean or Mathematics to any degree to really be able to say this with any level of confidence, but speaking from a pure software engineering backgrouand, how do we know that 13 MILLION lines of Lean code are bug-free? It seems to me that for a mathematical proof, bug-free would be an absolute requirement. Maybe the structure of Lean imposes that, I don't know, but that seems highly unlikely to me. That…
In lean, a theorem is specified by a type (in their highly complex "dependent type system") and proof is specified by a code that produces a term of that type. If the compiler certifies that the code indeed produces a term of that type, then the proof is correct. So, only need to trust: (1) That theorem statement is correctly encoded (FLT has a very short 1 liner description really) (2) Lean compiler is correct
> That theorem statement is correctly encoded (FLT has a very short 1 liner description really)
As someone not very familiar with Lean, does it really just depend on the entry point / theorem being correctly encoded? Can intermediate statements ever be mis encoded or misinterpreted, or is this what would count as a “bug in the Lean compiler”?
Re: Formalizing Fermat's Last Theorem
#368Looking forward to the 5 billion LoC proof of the Riemann hypothesis.
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Re: Formalizing Fermat's Last Theorem
#369Earlier quoted context omitted.
In lean, a theorem is specified by a type (in their highly complex "dependent type system") and proof is specified by a code that produces a term of that type. If the compiler certifies that the code indeed produces a term of that type, then the proof is correct. So, only need to trust: (1) That theorem statement is correctly encoded (FLT has a very short 1 liner description really) (2) Lean compiler is correct
> That theorem statement is correctly encoded (FLT has a very short 1 liner description really) As someone not very familiar with Lean, does it really just depend on the entry point / theorem being correctly encoded? Can intermediate statements ever be mis encoded or misinterpreted, or is this what would count as a “bug in the Lean compiler”?
It is the latter. If you are certain your theorem is stated correctly, and you believe that the Lean kernel against which you validate is correct, your proof is correct.
This is how the theorem for FLT looks in the particular proof we discuss here:
theorem fermat_last_theorem (n : ℕ) (hn : 3 ≤ n) (a b c : ℕ) (ha : 0 As long as this statement is correct, and the kernel is correct, the proof could be trillion lines of code, and if the kernel says it is correct, it is correct.
This proof was checked against TWO independently built kernels. So you would need TWO kernels to have the same bug to mistakenly accept an incorrect proof.
(Not impossible: such a bug indeed was recently discovered (and patched))
Re: Formalizing Fermat's Last Theorem
#370Looking forward to the 5 billion LoC proof of the Riemann hypothesis.
If AI manages to prove, or disprove, I wonder what would Clay Foundation do for the prize.
Who cares about some billionaire paying another billionaire a million dollars?
WHat matters is our understanding of maths, and whether this sort of thing makes us smarter or stupider.