The lesson is to not let rare pathological cases ruin useful generalizations (at least not outside of formal mathematics).

By the way even in formal mathematics (and maybe especially in formal mathematics), while pathological cases are interesting, nobody discards perfectly useful theories just because the theory allows pathologies. For example, nobody hesitates to use measure theory in spite the Banach-Tarski paradox; nobody hesitates to use calculus even though the Weierstrass function exists; few people hesitate in using the Peano axioms in spite of the existence of non-standard models of that arithmetic.

Nitpick: I would consider the Weierstrass function a different sort of pathology than non-standard models or Banach-Tarski - a practical pathology rather than a conceptual pathology. The Weierstrass function is just a fractal. It never smooths out no matter how much you zoom in.

Bad Concepts Repository

by moridinamael 1 min read27th Jun 2013204 comments

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We recently established a successful Useful Concepts Repository.  It got me thinking about all the useless or actively harmful concepts I had carried around for in some cases most of my life before seeing them for what they were.  Then it occurred to me that I probably still have some poisonous concepts lurking in my mind, and I thought creating this thread might be one way to discover what they are.

I'll start us off with one simple example:  The Bohr model of the atom as it is taught in school is a dangerous thing to keep in your head for too long.  I graduated from high school believing that it was basically a correct physical representation of atoms.  (And I went to a *good* high school.)  Some may say that the Bohr model serves a useful role as a lie-to-children to bridge understanding to the true physics, but if so, why do so many adults still think atoms look like concentric circular orbits of electrons around a nucleus?  

There's one hallmark of truly bad concepts: they actively work against correct induction.  Thinking in terms of the Bohr model actively prevents you from understanding molecular bonding and, really, everything about how an atom can serve as a functional piece of a real thing like a protein or a diamond.

Bad concepts don't have to be scientific.  Religion is held to be a pretty harmful concept around here.  There are certain political theories which might qualify, except I expect that one man's harmful political concept is another man's core value system, so as usual we should probably stay away from politics.  But I welcome input as fuzzy as common folk advice you receive that turned out to be really costly.