Mathematics doesn't collapse like a house of cards because the more "important" a proof is to the foundations of math, the more frequently it is tested. A counterexample to a proof is an easy way to find errors up the chain.
In the other direction is that many things that might be considered "errors" in the foundations could be looked at more like choices. Others mentioned Euclidian vs non-Euclidian geometry. There was a sort of error of the idea that Euclidian geometry was the only geometry which was fixed not by tearing it down but creating new geometries.
Mathematics doesn't collapse like a house of cards because the more "important" a proof is to the foundations of math, the more frequently it is tested. A counterexample to a proof is an easy way to find errors up the chain.
In the other direction is that many things that might be considered "errors" in the foundations could be looked at more like choices. Others mentioned Euclidian vs non-Euclidian geometry. There was a sort of error of the idea that Euclidian geometry was the only geometry which was fixed not by tearing it down but creating new geometries.