TITLE: Replication crisis in mathematics
QUESTION [63 upvotes]: Lately, I have been learning about the replication crisis, see How Fraud, Bias, Negligence, and Hype Undermine the Search for Truth (good YouTube video) — by Michael Shermer and Stuart Ritchie. According to Wikipedia, the replication crisis (also known as the replicability crisis or reproducibility crisis) is

an ongoing methodological crisis in which it has been found that many
scientific studies are difficult or impossible to replicate or
reproduce. The replication crisis affects the social sciences and
medicine most severely.

Has the replication crisis impacted (pure) mathematics, or is mathematics unaffected? How should results in mathematics be reproduced? How can complicated proofs be replicated, given that so few people are able to understand them to begin with?

REPLY [14 votes]: Has this crisis impacted (pure) mathematics, or do you believe that
maths is mostly immune to it?

Immune to the replication problem, yes. But not immune to the attitudes which cause scientists to do unreplicable research in the first place. Some mathematicians will announce that a particular theorem has been proven, harvest the glory based on the fact that they have proved things in the past, and then never publish their results. Rota's Conjecture is one notorious example. Now we are in a situation where (a) nobody knows whether it is true and (b) nobody has worked on it for seven years, and probably (if it turns out that no proof actually exists) will not work on it for at least another decade.

How should results in mathematics be reproduced?

In science, it would be ideal if people dedicated research time to replicating published experimental results. This doesn't happen much because there is no glory to be gained by doing it.
The analogue in mathematics would be for people to publish new proofs of existing results, or expositions of existing proofs, which is happily much more common. I don't mean copying out well-known results in new language (Tom Leinster, The bijection between projective indecomposable and simple modules), I mean expository papers like this (Cao and Zhu, A complete proof of the Poincaré and geometrization conjectures, Asian J. Math. 10 (2006) pp. 165–492).
Even more noble are the people using proof assistant software to verify existing mathematics.

How can we expect that increasingly complicated proofs are replicated
when so few people can understand them in the first place?

I think our best hope is proof assistant software. Perhaps by the end of this century, we will he living in a world where no mathematician can replicate any reasonably cutting-edge proof, yet research is still happily chugging along.