Mathematics. No problems there because the wisdom of the ancients is still true.
Actually, a surprisingly large amount isn't. For example, the entire use of infintesimals had to be rethought during the mid nineteenth century and replaced with rigorous constructions over the real numbers. It wasn't until a century later that a rigorous use of infitesimals was constructed and it looked pretty different from the version used by Newton and the people after him.
Similarly, the question of how polyhedra's Euler characteristic behaved advanced through a series of proofs followed by counterexamples to the "proofs." (Although my understanding is that it wasn't quite as extreme as what occurs in Lakatos's "Proofs and Refutations.")
Nicomachus in his treatise on perfect numbers (from around 100 CE) made a number of incorrect statements that took almost a thousand years to be shown to be wrong.
Someone deserves a large hattip for this, but I'm having trouble remembering who; my records don't seem to show any email or OB comment which told me of this 12-page essay, "Epistemic Viciousness in the Martial Arts" by Gillian Russell. Maybe Anna Salamon?
It all generalizes amazingly. To summarize some of the key observations for how epistemic viciousness arises:
One thing that I remembered being in this essay, but, on a second reading, wasn't actually there, was the degeneration of martial arts after the decline of real fights—by which I mean, fights where people were really trying to hurt each other and someone occasionally got killed.
In those days, you had some idea of who the real masters were, and which school could defeat others.
And then things got all civilized. And so things went downhill to the point that we have videos on Youtube of supposed Nth-dan black belts being pounded into the ground by someone with real fighting experience.
I had one case of this bookmarked somewhere (but now I can't find the bookmark) that was really sad; it was a master of a school who was convinced he could use ki techniques. His students would actually fall over when he used ki attacks, a strange and remarkable and frightening case of self-hypnosis or something... and the master goes up against a skeptic and of course gets pounded completely into the floor. Feel free to comment this link if you know where it is.
Truly is it said that "how to not lose" is more broadly applicable information than "how to win". Every single one of these risk factors transfers straight over to any attempt to start a "rationality dojo". I put to you the question: What can be done about it?