Mathematics students are often annoyed that they have to worry about “bizarre or unnatural” counterexamples when proving things. For instance, differentiable functions without continuous derivative are pretty weird. Particularly engineers tend to protest that these things will never occur in practice, because they don’t show up physically. But these adversarial examples show up constantly in the practice of mathematics—when I am trying to prove (or calculate) something difficult, I will try to cram the situation into a shape that fits one of the theorems in my toolbox, and if those tools don’t naturally apply I’ll construct all kinds of bizarre situations along the way while changing perspective. In other words, bizarre adversarial examples are common in intermediate calculations—that’s why you can’t just safely forget about them when proving theorems. Your logic has to be totally sound as a matter of abstraction or interface design—otherwise someone will misuse it.
While I think the reaction against pathological examples can definitely make sense, and in particular there is a bad habit of some people to overfocus on pathological examples, I do think mathematics is quite different from other fields in that you want to prove that a property holds for all objects with a certain property, or prove that there exists an object with a certain property, and in these cases you can’t ignore the pathological examples, because they can provide you with either solutions to your problem, or show why your approach can’t work.
This is why I didn’t exactly like Dalcy’s point 3 here:
Mathematics students are often annoyed that they have to worry about “bizarre or unnatural” counterexamples when proving things. For instance, differentiable functions without continuous derivative are pretty weird. Particularly engineers tend to protest that these things will never occur in practice, because they don’t show up physically. But these adversarial examples show up constantly in the practice of mathematics—when I am trying to prove (or calculate) something difficult, I will try to cram the situation into a shape that fits one of the theorems in my toolbox, and if those tools don’t naturally apply I’ll construct all kinds of bizarre situations along the way while changing perspective. In other words, bizarre adversarial examples are common in intermediate calculations—that’s why you can’t just safely forget about them when proving theorems. Your logic has to be totally sound as a matter of abstraction or interface design—otherwise someone will misuse it.
While I think the reaction against pathological examples can definitely make sense, and in particular there is a bad habit of some people to overfocus on pathological examples, I do think mathematics is quite different from other fields in that you want to prove that a property holds for all objects with a certain property, or prove that there exists an object with a certain property, and in these cases you can’t ignore the pathological examples, because they can provide you with either solutions to your problem, or show why your approach can’t work.
This is why I didn’t exactly like Dalcy’s point 3 here:
https://www.lesswrong.com/posts/GG2NFdgtxxjEssyiE/dalcy-s-shortform#qp2zv9FrkaSdnG6XQ