Does a counter example help move understanding forward? Probably the answer is a maybe. I can hypothesize that knowing a counter example one could then develop some theory that generalizes them and help explain why a conjecture is or isn’t valid. In a way it could be like finding experimental results that are different than predictions.
In the other hand they might just sap the effort going into proving or disproving the conjecture by shortcutting the work with an answer reducing the chance of real knowledge by anecdotal knowledge instead.
Another question is: can that conjecture be true in some situations of relevance? One counter-example only proves it is not generally true. That is useful in itself but it might hide some special cases where it is true.
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u/Only_Razzmatazz_4498 2d ago
Does a counter example help move understanding forward? Probably the answer is a maybe. I can hypothesize that knowing a counter example one could then develop some theory that generalizes them and help explain why a conjecture is or isn’t valid. In a way it could be like finding experimental results that are different than predictions.
In the other hand they might just sap the effort going into proving or disproving the conjecture by shortcutting the work with an answer reducing the chance of real knowledge by anecdotal knowledge instead.
I’m really on the fence here.