| ▲ | kaffekaka 7 hours ago | |
In math (and other subjects too of course) many things simply cannot be understood without deep prior knowledge. I came across the Wikipedia page for the modularity theorem a while ago and it reads truly satirical to me: https://en.wikipedia.org/wiki/Modularity_theorem Not a single sentence conveys any knowledge to me. I have a theoretical physics degree so I am not afraid of math but still. | ||
| ▲ | leonidasrup 4 hours ago | parent [-] | |
It took many years for Andrew Wiles and Richard Taylor to prove the modularity theorem for semistable elliptic curves. Sometimes even Fields Medalists are not sure if a complex theorem about a complex mathematical object is 100% correct. From Peter Scholze: "— I spent much of 2019 obsessed with the proof of this theorem, almost getting crazy over it. In the end, we were able to get an argument pinned down on paper, but I think nobody else has dared to look at the details of this, and so I still have some small lingering doubts." https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-e... Vladimir Voevodsky started research program centered on formalizing Homotopy theory, because he feared possible bugs in his proofs. "This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake. And it clearly was not an accident. A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail. But this is not the only problem that allows mistakes in mathematical texts to persist. In October 1998, Carlos Simpson submitted to the arXiv preprint server a paper called “Homotopy Types of Strict 3-groupoids.” It claimed to provide an argument that implied that the main result of the “∞-groupoids” paper, which Kapranov and I had published in 1989, cannot be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the fall of 2013 (!!)." | ||