The question of whether there is a missing finite simple group is a precise question. But what does it mean for a natural language proof to be valid? Typically a proof contains many precise lemmas and one could ask that these statements are correct (though this leaves the question of whether they prove the theorem), but lots of math papers contain lemmas that are false as stated, but where the paper would be considered salvageable if anyone noticed.
Similarly, the Riemann hypothesis is widely believed due to a combination of partial results (a positive fraction of zeros must be on the line, almost all zeros must be near the line, the first few billion zeros are on the line, a random model of the Mobius function implies RH, etc.)
This is a very standard list of evidence, but I am skeptical that it reflects how mathematicians judge the evidence. I think that of the items you mention, the random model is by far the most important. The study of small zeros is also relevant. But I don’t think that the theorems about infinitely many zeros have much effect on the judgement.
This is a very standard list of evidence, but I am skeptical that it reflects how mathematicians judge the evidence. I think that of the items you mention, the random model is by far the most important. The study of small zeros is also relevant. But I don’t think that the theorems about infinitely many zeros have much effect on the judgement.
I agree with this, but would I cite the empirical truth of the RH for other global zeta functions, as well as the proof of the Weil conjectures, as evidence that mathematicians actually think about.
I’d be interested in corresponding a bit — shoot me an email if you’d like
I agree with this, but would I cite the empirical truth of the RH for other global zeta functions
An interesting thing about the GRH is that at oft neglected piece of evidence against it is how the Siegal zero seems to behave like a real object, i.e., having consistent properties.
An interesting thing about the GRH is that at oft neglected piece of evidence against it is how the Siegal zero seems to behave like a real object, i.e., having consistent properties.
I’m not sure what you mean here. If it didn’t have consistent properties we could show it doesn’t exist. Everything looks consistent up until the point you show it isn’t real. Do you mean that it has properties that don’t look that implausible? That seems like a different argument.
The question of whether there is a missing finite simple group is a precise question. But what does it mean for a natural language proof to be valid? Typically a proof contains many precise lemmas and one could ask that these statements are correct (though this leaves the question of whether they prove the theorem), but lots of math papers contain lemmas that are false as stated, but where the paper would be considered salvageable if anyone noticed.
This is a very standard list of evidence, but I am skeptical that it reflects how mathematicians judge the evidence. I think that of the items you mention, the random model is by far the most important. The study of small zeros is also relevant. But I don’t think that the theorems about infinitely many zeros have much effect on the judgement.
I agree with this, but would I cite the empirical truth of the RH for other global zeta functions, as well as the proof of the Weil conjectures, as evidence that mathematicians actually think about.
I’d be interested in corresponding a bit — shoot me an email if you’d like
An interesting thing about the GRH is that at oft neglected piece of evidence against it is how the Siegal zero seems to behave like a real object, i.e., having consistent properties.
I’m not sure what you mean here. If it didn’t have consistent properties we could show it doesn’t exist. Everything looks consistent up until the point you show it isn’t real. Do you mean that it has properties that don’t look that implausible? That seems like a different argument.
If we can easily prove a conjecture except for some seemingly arbitrary case, that’s evidence for the conjecture being false in that case.