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AlephNeil comments on Bayesian Epistemology vs Popper - Less Wrong Discussion
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I don't know what Popper himself would say, but one of his more insightful followers, namely Lakatos, argues for exactly that position.
I read Proofs and Refutations too many years ago to say anything precise about it. I remember finding it interesting but also frustrating. Lakatos seems determined to ignore/deny/downplay the fact of mathematical practice that we only call something a 'theorem' when we've got a proof, and we only call something a 'proof' when it's logically watertight in such a way that no 'refutations' are possible. Still, it's well-researched (in its use of a historical case-study) and he comes up with some decent ideas along the way (e.g. about "monster barring" and "proof-oriented definitions".)
Yes, Lakatos does argue for that in a certain fashion, (and I suppose it is right to bring this up since I've myself repeatedly pointed people here on LW to read Lakatos when they think that math is completely reliable.) However, Lakatos took a more nuanced position than the position that curi is apparently taking that math advances solely through this method of criticism. I also think Lakatos is wrong in so far as the examples he uses are not actually representative samples of what the vast majority of mathematics looks like. Euler's formula is an extreme example, and it is telling that when one wants to give other similar examples one often gives other topological claims from before 1900 or so.