Prompted by this bit of news, http://www.wired.co.uk/article/fmri-bug-brain-scans-results where a bug in MRI software has the potential to nullify up to 40,000 published papers. Has anything analogous happened in mathematics? Obviously not on this scale, but was that ever a bug which, once discovered, caused a large number of follow-up results to become falsified as well?
One I have heard of is the Italian school of algebraic geometry, 1885 to 1935.
I would risk to state that in mathematics such things (on large scale) do not happen:-) There are indeed some published and "accepted" false proofs, and false statements. The link given in the comment of Wojowu gives plenty of examples. However, I am sure that the list of results which depend on these incorrect results and thus are false will be very small.
The reason is that most mathematicians (at least most of those who prove significant results:-) usually take care to check what their proofs rely on. This is in the nature of mathematics: if you use a fact that you do not really understand, you do not understand what you are doing yourself.
I would state this as a bit ambitious principle: "You have to know the proofs of all results that you use". Of course this is not always so in practice, but this is an ideal, a goal. The result is that more some theorem is used, more times it is checked by various people. And the theorems which are really used much are checked so well, that one can be reasonably sure that they are correct.
I mean that none of the examples listed in the answers to Wojowu's question produced a chain of false results.
By the way, this is one reason why "computer-assisted proofs" are considered less satisfactory by many mathematicians than real "human" proofs.