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Douglas_Knight comments on The Use of Many Independent Lines of Evidence: The Basel Problem - Less Wrong
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The Euler example raises an issue: when should be more confident about some heuristically believed claim than claims proven in the mathematical literature? For example, the proof for the classification of finite simple groups consists of hundreds of distinct papers by about as many authors. How confident should one be that that proof is actually correct and doesn't contain serious holes? How confident should one be that we haven't missed any finite simple groups? I'm substantially more confident that no group has been missed (>99%?) but much less so in the validity of the proof. Is this the correct approach?
Then there are statements which simply look extremely likely. Let's take for example "White has a winning strategy in chess if black has to play down a queen". How confident should one be for this sort of statement? If someone said they had a proof that this was false, what would it take to convince one that the proof was valid? It would seem to take a lot more than most mathematical facts, but how much so, and can we articulate why?
Note incidentally that there are a variety of conjectures that are currently believed for reasons close to Euler's reasoning. For example, P = BPP is believed because we have a great deal of different statements that all imply it. 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.), but how confident should we be in such conjectures?
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.