TITLE: How rare is it that a theorem with published proof turns out to be wrong?
QUESTION [15 upvotes]: There is a story I read about tiling the plane with convex pentagons.
You can read about it in this article on pages 1 and 2.
Summary of the story:
A guy showed in his doctorate work all classes of tiling the plane with convex pentagons, and proved that they are indeed all possible cases.
Later, riddle was published in popular science magazine to find all these classes. One of the readers found a tiling that did not belong to any of the classes, and so the claim and the proof turned out to be wrong.

Reading this made me think about some questions.
Is it rare when a theorem was proved and the proof was published, and later it turned out that the theorem is wrong?
Can we somehow guess how many theorems out there that we think are right, but actually are wrong? I bet that if in our case, the theorem was about tiling in $R^3$ nobody would ever notice.
What can be the effect of such theorems on mathematics in general? Can it be a serious issue for mathematics even if the wrong theorems were not very important, but still, some stuff was based on them?

REPLY [6 votes]: I think Carl Mummert's answer is spot-on. One other thing worth mentioning is that more often than being wrong, or even having a recognizably flawed proof, are papers which are just really hard to understand. (This is sometimes compounded by language barriers... more than once I've had to deal with papers that were written in Russian, then either never translated or translated incoherently.) Some of these proofs are not complete, others are complete but hard to understand, some are mostly correct but have a few errors, and so on...
So people generally take a cautious approach. One good piece of advice my PhD adviser in grad school told me, never use a result whose proof you don't understand. Some people rely on proofs that they don't understand but that are accepted and many people understand, so they figure it's an acceptable risk. Others (I'm one of them) won't use any proof they don't understand. This may be more doable in some fields than others. So while there are wrong theorems out there, the effect is minimized by people recognizing this fact and proceeding with caution.