Powerful algorithms that are just too hard to implement-- how to be sure they are right?
Several years ago, there has been a (rather harsh) debate on a topic similar to this. It all started when several complex proofs turned out to be incorrect, and some researchers started to raise doubts on the very nature of proofs (well, I should have said "provable cryptography," but for the sake of generality, I didn't). Both sides of the controversy accused the other to misunderstand the concepts. Here's a link for more info.

Proofs are our (mathematical) tools to prove theorems/algorithms are correct, but when they became too complex, we may slip and prove wrong things to be right. The recent 100-or-so-page proof on P≠NP is an excellent example. However, this does not rule out the very nature of proofs: Nothing's wrong with them.

One last point: I think studying the philosophy of science will give us more insight on this. (Under the given link, see the bullet "How do we know whether a mathematical proof is correct?")