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.
I agree that the Weierstrass function is different. I felt a tinge of guilt when I included the Weierstrass function. But I included it since it’s probably the most famous pathology.
That being said, I don’t quite understand the distinction you’re making between a practical and a conceptual pathology. The distinction I would make between the Weierstrass and the other two is that the Weierstrass is something which is just counter-intuitive whereas the other two can be used as a reason to reject the entire theory. They are almost antithetical to the purpose of the theory. Is that what you were getting at?
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.
I agree that the Weierstrass function is different. I felt a tinge of guilt when I included the Weierstrass function. But I included it since it’s probably the most famous pathology.
That being said, I don’t quite understand the distinction you’re making between a practical and a conceptual pathology. The distinction I would make between the Weierstrass and the other two is that the Weierstrass is something which is just counter-intuitive whereas the other two can be used as a reason to reject the entire theory. They are almost antithetical to the purpose of the theory. Is that what you were getting at?