[Stasis5001]

Sure, Wiles obviously understands the main thrust of his proof. But one could argue that Wiles' result depends on lots of other results, which in turn depend on other results, and so on through decades and decades of work, ultimately going back to the foundations of mathematics. Neither Wiles nor anybody else can claim to rigorously understand all of it. You can imagine this as a tree, with Wiles' work as root, and his dependencies as ancestors, and so on. An error at a lower level of the tree could, in theory, invalidate the root node.

I do agree with Buzzard that it's hard to be sure. I've definitely read papers where a critical argument isn't well written or what is written seems wrong. However, if there are low-level errors, I suspect that with some work things could be patched up.

[hyperbovine]

Right, speaking as a lapsed mathematician, I definitely see errors or gaps in published work. Wiles’ original FLT proof had one. And yes these can generally be patched up. I’m not quite as alarmed as the author is, because generally a major false result would have all sorts of alarming ripple effects and implications which would be pretty easy to spot. FLT is an extreme example where literally anyone with a calculator could in theory disprove it. The fact that no one has suggests to me that it’s likely true.

[Ceezy]

You don t need to understand everything. People spend there all lives creating new demonstrations. Each one is a new prospective on a subject. And if the first one was made of milions of nodes an other one can be made of two nodes only. Take the index theorem there are proofs that have nothing to do with each othere some are very long some aren t.

And finally from the beginning of math people misunderstand their on theorems, doesn t mean that there students won t do better.