Jordana Cepelewicz in Quanta:
In 2012, the mathematician Shinichi Mochizuki claimed he had solved the abc conjecture, a major open question in number theory about the relationship between addition and multiplication. There was just one problem: His proof, which was more than 500 pages long, was completely impenetrable. It relied on a snarl of new definitions, notation, and theories that nearly all mathematicians found impossible to make sense of. Years later, when two mathematicians translated large parts of the proof into more familiar terms, they pointed to what one called a “serious, unfixable gap” in its logic — only for Mochizuki to reject their argument on the basis that they’d simply failed to understand his work.
The incident raises a fundamental question: What is a mathematical proof? We tend to think of it as a revelation of some eternal truth, but perhaps it is better understood as something of a social construct.
More here.