In what other field of substantial complexity is there any expectation that errors never occur? I don't think it's defeatist to accept that there will always be problems. It seems to me in any complex field (medicine, aviation, finance, etc.) the thing that maximizes human success is to design processes so that problems are exposed, treated, contained, and prevented from re-occurring as easily as possible. Looking for something shaped like "problems will cease to exist" seems like a distraction to me.
Software is different than any other field because it's supposed to stand on pure logic.
Errors due to hardware failing because of physical properties are expected and unavoidable, but a logical statement no matter how big is supposed to be provable to be correct (except maybe weird self referencing statements, but that's another topic). I think that's what makes our expectations higher in that field as opposed to any other ones.
'Algorithms' can be proved, 'implementations' necessarily need tests.
Algorithms, are mathematically able to be proved, but any real implementation has to involve physical hardware at some level, these implementation details cannot (reasonably) be proven mathematically therefore need sufficient tests.
In the absence of a proof, tests can cover both algorithms & implementations.
If that's true to some extent, is it because Mathematics is built much more slowly, with higher standards of "inclusion" of new elements? Not the same pressure to deliver.
