alt Hacker NewsMaybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.
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bananaflag • yesterday at 10:15 PM
You mean proof by contradiction, which is something different.
It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along!
But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that?
That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding?
Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.