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TobyBartels comments on Scientific Self-Help: The State of Our Knowledge - Less Wrong
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A good place to deconstruct my own argument.
The math comparison lacks in one important piece: Math is clearly defined, and has standard textbooks. If you ask around for recommendations on how to learn math you get similar responses, and will end up learning similar things - up to a certain degree. There is only one type of math! In general people agree on what math is, and what not.
PU as well as PD is a very broad, not clearly defined subject, that contains a mash-up of many other topics. It is contradictory. Done by amateurs who generally do not care about scientific results. You get advice that goes against those transported by the mainstream (which we on LW are somewhat used to in other contexts.) But you also find the statement that the subjects of your interest will generally give you bad advice and do not even know what works for them. As will your peers, your family, potential natural friends, the media, and anyone else who you could possibly ask. That makes for a very bad heuristic in regards to its truthfulness.
And then there is the annoying property of PD advice, that it is not only difficult to actually get, but that it also hurts. Sometimes we carry gaping holes that really hurt our social life, and no one has the guts to tell us, since they are afraid of a bad reaction.
One easy to understand example is trying to tell a colleague or friend that he needs to do something about his smell.
I am not aware of a safe way to navigate this. It would be interesting to see real scientists, or science minded people undertake this exploration. But there are way to many possibilities to have it go wrong.
PU does contain basilisks. So handle with care. And do not believe any one particular source completely.
The second sentence here is true, but the first one is false. There is mainstream math, and then there are alternatives. Of course, there are insane crackpot ideas, but there are also alternative forms of mathematics that are studied by serious researchers who earn tenure for it and prove valid theorems. Buzzwords to search for include "intuitionism", "constructive mathematics", "predicatvism", "finitism", and "nonclassical mathematics" generally.
This mostly only affects things from after the 19th century, however, so nothing significant about the mathematics that most people learn in school. Even going on to more advanced material, there is a very definite mainstream to follow, so this doesn't really affect your point; this is just a hobby horse of mine.
And Though shall get a geek point for it. I was kind of waiting for someone to point this out.
And I should have mentioned "experimental mathematics", which is really different! This term can be interpreted in weak and strong ways; the former, in which experiments are a preliminary to proof, is normal, but the latter, in which massive computer-generated experimental results are accepted as a substitute for proof when proof seems unlikely, is different. The key point is that most true theorems that we can understand have no proofs that we can understand, a fact that can itself be proved (at least if if you use length of the text as a proxy for whether we can understand it).