jibal

P.S. Also false -- it simply is not true that many people believed that the Jacobian conjecture was almost certainly not false--why would they? OTOH, the Collatz conjecture has been confirmed for all integers up to 2.36 *10^21, and Terence Tao has proved that it is true for "almost" all numbers: https://www.quantamagazine.org/mathematician-proves-huge-res...

Again, this is all non sequitur, because the context was a statement that most mathematicians believe the CC to be true, in which case there would be no "clean up".

It overturns the Jacobian conjecture (i.e., speculation) for dim >= 3, which we now know was an overgeneralization. Tao characterizes it as "can be viewed as an assertion that local invertibility implies global invertibility". It was already widely suspected to be false. Assuming that it was true was never warranted, so this really doesn't change anything. The significance is that an AI was able to find a relatively simple counterexample. Its "chain of thought" would be very interesting to see.

There are mathematical statements that have been shown to be undecidable (no proof for or against is possible) and such a demonstration is a huge deal.

having the wisdom

Ahem.

On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it.

It was a parallel effort ... we don't know how many people were working on it that weekend. And since the professor wanted it to be true and presumably believed that it was true, why the heck should he wait for students of unknown number and ability to find a counterexample that he didn't think existed?

https://en.wikipedia.org/wiki/George_Dantzig

During his study in 1939, Dantzig solved two unsolved problems in statistics due to a misunderstanding. Near the beginning of a class, Professor Neyman wrote two problems on the blackboard. Dantzig arrived late and assumed that they were a homework assignment. According to Dantzig, they "seemed to be a little harder than usual", but a few days later he handed in completed solutions for both problems, still believing that they were an assignment that was overdue.[4][6] Six weeks later, an excited Neyman eagerly told him that the problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[7] This story spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time, some facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the 1997 film Good Will Hunting.[6]

The Gödel sentence (which is not otherwise of any interest) is true but unprovable within that axiomatic system. The Continuum hypothesis (which is of great interest) is true or false only by stipulation.

The fact is that neither Gödel's theorems nor the Halting Problem have any actual real world consequences (outside of people talking about them). People say "oh, we can't write a verifier because of the Halting Problem", but that's simply not true since all of our programs are actually FSMs with physically limited data, and the HP is solvable for that subset of TMs. The real limitations are time and space, so this is an engineering problem -- and people who aren't suckered by Halting Problem Hysteria find engineering solutions that work on real programs, bailing if memory or time thresholds are exceeded.

You're missing the point. The counterexample to the Jacobian conjecture is valid regardless of how it was discovered ... elsewhere on this page people are even suggesting that the Anthropic mathematician may not have actually used Claude and came up with the counterexample himself.

Trusting AI math is not an issue here. It's as if you had asserted that no primes when divided by 35 produce a remainder of 6 and the model said that 41 is a counterexample and then you complained about not being able to trust AI math.

P.S. As someone else noted, your AI query was malformed ... no prime is divisible by 35, nor is any integer divisible by an integer with a remainder of 6 -- divisibility implies a remainder of 0. So perhaps the AI simply took what you wrote literally.

What you think "personally" isn't relevant. Turing proved that there are statements that are true in a consistent axiomatic system (of sufficient power for arithmetic) that cannot be proven within that axiomatic system.

That episode of TOS is a silly show and is completely irrelevant.

Why do you trust a random stranger so much?

Why do you make false claims and attack strawmen so much?

non sequitur

non sequitur

I know what the policies are and I followed them. I also read the entire discussion on the talk page before making my change ... I doubt that you did. WP:BASICMATH says that this is not OR.

If this were OR then even the "claimed" statement would have required RS. No one was willing to remove the counterexample altogether, so my edit to remove the "claimed" weasel word was perfectly valid.

See https://en.wikipedia.org/wiki/Wikipedia:No_original_research...

It's a counterexample, not a proof. A schoolchild can confirm it.

I removed the "claimed" weasel wording, and now others have followed up. The editors that don't understand math and don't realize how easily this counterexample can be confirmed have lost the debate -- not that there really ever was one.

Did you read what you responded to? The Collatz conjecture is almost certainly not false, so no "clean up" is possible.

They gave their "logic", such as it is ... and it's utterly irrational.

Note that the "they" who published the counterexample on X is some rando mathematician (Levent Alpöge) working for Anthropic, not Anthropic the organization. He posted the counterexample in a tweet -- reason enough for "not disclosing the LLM chat session". There's no reason to think that it won't provided if asked for, but it hardly seems relevant.

the fact that “will this AI cause harm” is, mathematically, an unanswerable question.

No it isn't -- this is a fundamental misunderstanding. What is unanswerable is "For all x where x is an AI, will x cause harm?" ... but there is an infinity of specific AIs that provably will or won't cause harm.

Likewise this a very common misunderstanding of the Halting Problem -- Turing proved that there is no TM that can prove whether m will halt for all possible TMs m ... but there are myriad TMs that provably do halt or provably don't halt.

Finite machines such as FSMs (finite state machines) are a subset of TMs, so yes they are actual Turing Machines. What they aren't is UTMs ... universal Turing Machines that can emulate every TM.

Turing proved that no TM can determine whether all TMs halt ... not that no TM can determine whether some specific TM halts ... the difference a common misunderstanding of the proof. Analogously, while we can't prove that no AI is or isn't harmful, we can prove that certain AIs are or aren't harmful.

Your meaning was clear ... Jtsummers bizarrely misparsed it.

Please offer proof of that obviously false claim.

It depends somewhat on the complexity of the language.

Nim uses an interpreter but Nimony, which is destined to becomes Nim 3.0, uses your approach. It will be interesting to how hassles and performance play out there and whether they keep the compile-and-run approach or go back to an interpreter.

TFA discusses this in detail. The point is memory errors in the generated code. The generated code is an output the compiler. Thus the OP counts these against the compiler--after all, that's his project, those bugs are real, and those bug reports must be dealt with. His point is that the Rust borrow checker doesn't help at all with these bugs, and that matters when weighing the pros and cons of writing the compiler in Rust vs. Zig -- the languages are a wash for that category of error.

I do think this is a bit off, because miscompilations can produce all sorts of errors, not just memory errors, so it doesn't make much sense to categorize them as memory errors -- they are logic errors in the compiler. But the basic idea holds -- the choice of implementation language doesn't matter.

Whether a text was written by a human or not is just a single bit of information. So you can't rule out its detectability a priori, since even the shortest text contains more information than that.

This is word salad, a complete non sequitur.

to write texts humans wouldn't want to write if they could help it (that's why they're getting an LLM to do it, after all)

Er, that's obviously not true.

The pompous response from zahlman is at least as bad ... accusing me of "casting aspersions" and then saying that I was "called out" by a comment that calls me a coward and tells me to "take a seat"? Can one possibly be more lacking in evenhandedness?

I forget that my HN frontend has a mute feature ... applying it to both of these.

Oh ... never mind! (In the voice of Gilda Radner.)

Gawd only knows who this guy thinks he's quoting in the nonsensical response.

So, bad arguments followed by even worse arguments.

I'll just point out that "so what" is not a counterargument.

It really is.

Speed runners or chess champions or pi digit calculators don't confuse themselves with noble researchers advancing the frontier of human knowledge.

Yow.

Boomers? I'm a boomer who retired 8 years ago. The youngest boomers hit retirement age in 3 years. It's not boomers who are worried.