[mb7733]

Maybe I was unclear. The part that I meant a constructivist would disagree was the subjective part: "This kind of proof (existence of a counterexample) provides more insight into the problem than simply finding a counterexample."

I find it hard to believe that a constructivist would agree that across the board, proof of existence of X is more interesting than a construction of X. Isn't that the point!?

[Jweb_Guru]

A direct counterexample is more "informative" in a very literal sense (its truth value doesn't collapse). But the extra proof relevant content we can use here is not that large -- all it means in this case is that we can directly compute the object and its Jacobian, a well as two points evaluating to the same result. That's nice, but it's not that interesting by itself unless I can use the exact constructed form to prove other interesting stuff (and we can! Most of the followup results that immediately followed from the disrpoof come from being able to directly transform this object into counterexamples to other conjectures; if we didn't have constructive proofs of thoe counterexamples, we wouldn't have such procedures). But being more interesting than a completely uninformative counterexample still doesn't mean it's inherently interesting or enlightening. If anything I'd indeed argue constructive arguments are generally less mysterious and magical than nonconstructive proofs -- in some sense, the constructive proof pulls back the curtain and shows you where the trick is.