I think it depends how widely used that branch of maths becomes. In fact, I'd say that the word "branch" is a bit misleading - for many theories it's much more

I think it depends how widely used that branch of maths becomes. In fact, I'd say that the word "branch" is a bit misleading - for many theories it's much more of a "knot", with influences tying them to many many other theories across the mathematical landscape. And those theories are themselves tied to others, etcetera. It would be a very strange situation if the foundation fell apart logically without any ramifications in the rest of this "knot". A huge swathe of free-floating mathematics that's completely internally consistent but for this one error? Difficult to imagine for me in the case of cohomology from the article. I guess strictly speaking this is more of a philosophical stance - I like to believe that a lot of the current mathematics has been discovered "naturally" in some sense =)