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:
There is also the reverse case, where it is often common practice in math or logic to ignore bizarre and unnatural counterexamples. For example, first-order Peano arithmetic is often identified with Peano arithmetic in general, even though the first order theory allows the existence of highly “unnatural” numbers which are certainly not natural numbers, which are the subject of Peano arithmetic.
Another example is the power set axiom in set theory. It is usually assumed to imply the existence of the power set of each infinite set. But the axiom only implies that the existence of such power sets is possible, i.e. that they can exist (in some models), not that they exist full stop. In general, non-categorical theories are often tacitly assumed to talk about some intuitive standard model, even though the axioms don’t specify it.
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
There is also the reverse case, where it is often common practice in math or logic to ignore bizarre and unnatural counterexamples. For example, first-order Peano arithmetic is often identified with Peano arithmetic in general, even though the first order theory allows the existence of highly “unnatural” numbers which are certainly not natural numbers, which are the subject of Peano arithmetic.
Another example is the power set axiom in set theory. It is usually assumed to imply the existence of the power set of each infinite set. But the axiom only implies that the existence of such power sets is possible, i.e. that they can exist (in some models), not that they exist full stop. In general, non-categorical theories are often tacitly assumed to talk about some intuitive standard model, even though the axioms don’t specify it.
Eliezer talks about both cases in his Highly Advanced Epistemology 101 for Beginners sequence.