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TobyBartels comments on Scientific Self-Help: The State of Our Knowledge - Less Wrong
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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).