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:
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