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JonahSinick comments on The Use of Many Independent Lines of Evidence: The Basel Problem - Less Wrong
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I guess we'll have to agree to disagree here :-). I find Euler's evidence for the product formula for sine to be far more convincing than what was available to Cauchy at the time.
Edit: I say more here, where I highlight how different the two situations are.
Are you sure you aren't suffering from hindsight bias?
Not 100% sure, but pretty sure. The situation isn't so much that I think that the evidence for the limit of a continuous function being continuous is weak, as much as that the evidence for the product formula for sine is very strong.
The result (and its analog) imply two formulas for pi that had been proved by other means, and predicts infinitely many previously unknown numerical identities, which can be checked to be true to many decimal places. What more could you ask for? :-)
And did Euler check them?
Polya reports on Euler performing such checks. I don't know how many he did – one would have to look at the original papers (which are in Latin), and even they probably omit some of the plausibility checks that Euler did.