3 ms·

I have no idea what you are even arguing anymore. I made the "bit flip" comment specifically to show that even with the most credible proof, there are still alw

by anyfoo 7y ago

I have no idea what you are even arguing anymore. I made the "bit flip" comment specifically to show that even with the most credible proof, there are still always at least some little (and possibly completely trivial) implicit assumptions made about the actual correctness of the proof. One of those trivial assumptions is that the authors ran their code enough (most likely just as part of just writing it and getting it ready for publishing) more than once. Or that anyone else did, as you say. And so I simply don't get why you're so hung up on Erdős's assumption of credible mathematicians making a credible proof. Are you seriously insinuating that Paul Erdős, of all people, was not aware of everything you're saying here?

MaxBarraclough 7y ago
> even with the most credible proof, there are still always at least some little (and possibly completely trivial) implicit assumptions made about the actual correctness of the proof This is true of any proof, not only computerised proofs. Sometimes maths journals publish incorrect proofs because a person made a mistake. I doubt an incorrect proof has ever been published on account of a hardware bit-flip. I'd be more worried about the kind of deterministic failure than manifests in every run, such as a bug in the CPU (the famous Intel floating-point bug, say), or in the operating system, etc. Could be mitigated by using a variety of platforms I suppose.
- pfdietz 7y ago
  And the silly hardware bit flip complaint is easily addressed: run the program N times, and reduce the chance of the error by a factor that's exponential in N. Large randomly generated primes are produced by a random algorithm like this, with a step that (when run on a composite number) shows that it is composite with probability p = a constant < 1. The chance of error there can also be made exponentially small. As you say, deterministic, or at least highly correlated, errors would dominate the chance that the proof was wrong. But that's true of human proofs too.
  - MaxBarraclough 7y ago
    > that's true of human proofs too. I'm not sure that's right. The peer review process works well (e.g. [0]) precisely because human mathematicians are likely to make (at least somewhat) different mistakes from each other. [0] https://en.wikipedia.org/w/index.php?title=Wiles%27s_proof_of_Fermat%27s_Last_Theorem&oldid=895642634#Announcement_and_final_proof_(1993%E2%80%931995) https://en.wikipedia.org/w/index.php?title=Wiles%27s_proof_o...

by anyfoo 7y ago

MaxBarraclough 7y ago

> even with the most credible proof, there are still always at least some little (and possibly completely trivial) implicit assumptions made about the actual correctness of the proof This is true of any proof, not only computerised proofs. Sometimes maths journals publish incorrect proofs because a person made a mistake. I doubt an incorrect proof has ever been published on account of a hardware bit-flip. I'd be more worried about the kind of deterministic failure than manifests in every run, such as a bug in the CPU (the famous Intel floating-point bug, say), or in the operating system, etc. Could be mitigated by using a variety of platforms I suppose.

pfdietz 7y ago
And the silly hardware bit flip complaint is easily addressed: run the program N times, and reduce the chance of the error by a factor that's exponential in N. Large randomly generated primes are produced by a random algorithm like this, with a step that (when run on a composite number) shows that it is composite with probability p = a constant < 1. The chance of error there can also be made exponentially small. As you say, deterministic, or at least highly correlated, errors would dominate the chance that the proof was wrong. But that's true of human proofs too.
- MaxBarraclough 7y ago
  > that's true of human proofs too. I'm not sure that's right. The peer review process works well (e.g. [0]) precisely because human mathematicians are likely to make (at least somewhat) different mistakes from each other. [0] https://en.wikipedia.org/w/index.php?title=Wiles%27s_proof_of_Fermat%27s_Last_Theorem&oldid=895642634#Announcement_and_final_proof_(1993%E2%80%931995) https://en.wikipedia.org/w/index.php?title=Wiles%27s_proof_o...