Live data from Hacker News

Formalizing Fermat's Last Theorem

anthropic.com

461–470 of 527 posts

Re: Formalizing Fermat's Last Theorem

#461

Earlier quoted context omitted.

Buffer overflows are trivial to check for at runtime (~proof-checking-time) and Lean does this. Just like Java does it. I’d wager a million gazillion bucks that this is not the case.

So would you also say no chance of a stack overflow or any type of surreptitious storage overflow anywhere in the runtime do you think?

I would say it’s very unlikely to be the case here at least.

Of course some bugs in Lean may exist (I don’t have deep insight into Lean’s implementation and there have been bugs before), but I find it unlikely to be systematical or in a format that could affect the proof.

As I understand it, Lean is implemented in Lean and emits/compiles to C. In that C code, I’d be very surprised if any buffer overflows or stack overflows exist.

Such overflows are not difficult or expensive to detect, so if any were there, it should cause a crash instead of an incorrect result.

It’s not as in handwritten C where you can forget or omit a bounds check.

I’d say it is even less likely than seeing an overflow in the Core of Java cause an incorrect result (i.e. corruption instead of a crash) - because Lean uses the De Bruijn principle of reducing to a very small Core, that is easier to keep correct (others in this thread have expanded on this I better than I can I think).

Out of pure curiosity: Do you believe otherwise or have a reason to think I am mistaken?

Re: Formalizing Fermat's Last Theorem

#462

Earlier quoted context omitted.

Maybe society has the wrong values. Maybe society needs to rethink incentives. Maybe society is somewhat antisocial.

yes society has let the trillion dollar company down

You can say that American society made OpenAI and Anthropic possible. No other current society would have. Suddenly, formalisation of math is becoming cheap. That's not a problem, that's the goal, and it is here much earlier than expected. That's not antisocial. That is scientific progress.

(I swear, did not use an LLM for this)

Re: Formalizing Fermat's Last Theorem

#463

Earlier quoted context omitted.

It simply does have functions. According to ZFC, a function is a set whose members are pairs, such that no two different pairs have the same first element. I mean this quite seriously: have you considered reading any first course in set theory?

> According to ZFC, a function is a set whose members are pairs, such that no two different pairs have the same first element. Can you cite where did you get this?

As I have said a few times now, you should read any first course in set theory. I’m quoting my third-year notes from Cambridge there, but essentially every intro to set theory will say the same. (I’m sure someone will find a single counterexample that does it somehow differently.)

Re: Formalizing Fermat's Last Theorem

#464

Earlier quoted context omitted.

Eh? Any first course in set theory will present ZFC as a one-sorted theory with ten axioms (/schemas) in first order logic (inheriting an equality symbol, forall, implies etc) with one binary predicate (namely set membership), or will present a theory that is equiconsistent with a usual ZFC presentation. Honestly I’m not sure how you simultaneously claim to be a PhD in formalisation and also not be aware of the exist…

> Honestly I’m not sure how you simultaneously claim to be a PhD in formalisation and also not be aware of the existence of Isabelle/ZF, for example. I am aware, also I am not sure why you wrote all of this. Your unknown to me "first course" claims to be some authority of formalization purity?

Because you wrote:

> what are exactly rules, which could be separate topic of research, this detail is skipped

I am now confident you’re a troll, though, so I am going to bow out.

Re: Formalizing Fermat's Last Theorem

#465
post #406

Earlier quoted context omitted.

It's the same way you don't need to have GCD in stdlib to say that you can compute GCD in C++. You can make your own using parts given. You don't need to add any axioms, you just build some sets to represent numbers and make operations that act the same way as arithmetic, define some equality relations. Then you derive rules of arithmetic for your handcrafted arithmetic using ZF axioms and you're good. You get axioms…

> you just build some sets to represent numbers and make operations that act the same way as arithmetic which is already "just" some non trivial problem(there is no "operations" in set theory), and we are discussing if it is achievable.

You make relations and functions out of sets and prove theorems about them, reducing definition of things in terms of belonging to a set. This isn't particularly complicated.

Re: Formalizing Fermat's Last Theorem

#466

Earlier quoted context omitted.

> According to ZFC, a function is a set whose members are pairs, such that no two different pairs have the same first element. Can you cite where did you get this?

As I have said a few times now, you should read any first course in set theory. I’m quoting my third-year notes from Cambridge there, but essentially every intro to set theory will say the same. (I’m sure someone will find a single counterexample that does it somehow differently.)

Your third year notes from Cambridge has very low authority to me

Re: Formalizing Fermat's Last Theorem

#467

Earlier quoted context omitted.

> Honestly I’m not sure how you simultaneously claim to be a PhD in formalisation and also not be aware of the existence of Isabelle/ZF, for example. I am aware, also I am not sure why you wrote all of this. Your unknown to me "first course" claims to be some authority of formalization purity?

Because you wrote: > what are exactly rules, which could be separate topic of research, this detail is skipped I am now confident you’re a troll, though, so I am going to bow out.

I referred to specific definition in wikipedia. Your "first course notes" are irrelevant here, they can't be reviewed, they not proofread and unlikely can be considered as any reasonable quality if we are talking about real formalization of math.

Re: Formalizing Fermat's Last Theorem

#468
post #465

Earlier quoted context omitted.

> you just build some sets to represent numbers and make operations that act the same way as arithmetic which is already "just" some non trivial problem(there is no "operations" in set theory), and we are discussing if it is achievable.

You make relations and functions out of sets and prove theorems about them, reducing definition of things in terms of belonging to a set. This isn't particularly complicated.

No, once you start formalize this, it becomes complicated. There is a reason why looks like there is no "peano can be derived from zfc" theorem which would close dispute, and my opponents need to throw links on bro math from stackexchange in this discussion.

Re: Formalizing Fermat's Last Theorem

#469
I used to attend Kevin's Number Theory seminars at Imperial College many years ago and he's both a first rate mathematician and a very nice person. His blog has got me interested in maths again. Considering how pro AI he is and that he's been working on this problem for such a long time I'm a bit disappointed that Anthropic didn't involve him directly in this work. However, I think it's important to remember that without all of the work Kevin and people like him have done, the machines wouldn't be able to do this.

Re: Formalizing Fermat's Last Theorem

#470

Earlier quoted context omitted.

yes society has let the trillion dollar company down

You can say that American society made OpenAI and Anthropic possible. No other current society would have. Suddenly, formalisation of math is becoming cheap. That's not a problem, that's the goal, and it is here much earlier than expected. That's not antisocial. That is scientific progress. (I swear, did not use an LLM for this)

You're stating it's not a problem- but I'm giving you a reason why it is. This is serendipitously mirrored by a recent post from Terence Tao on Mastodon (https://mathstodon.xyz/@tao/117207856734787448)

In most cases in pure mathematics, the problems are posed not because we desperately want the solution to these problems in and of themselves, but because we have seen from past experience that human-directed efforts to solve these problems tend to spur further development of the field through the efforts to solve such problems, and then to digest any partial or complete solutions that emerge for further insights. Prematurely solving the problem by purely AI-powered methods - particularly without full transparency into the solution process - can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.

Post reply on HN