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?