"Originally the Kolyvagin-Flach method only worked under particularly constrained circumstances, but Wiles believed he had adapted and strengthened it sufficiently to work for all his needs. According to Katz this was not necessarily the case, and the effects were dramatic and devastating. The error did not necessarily mean that Wiles's work was beyond salvation, but it did mean that he would have to strengthen his proof. The absolutism of mathematics demanded that Wiles demonstrate beyond all doubt that his method worked for every element of every E-series and M-series."