As a constructivist: we don't disagree :) We just distinguish between "don't disagree" and "agree." Constructive mathematics says it's fine if you want to claim that there's not no counterexample -- you just can't use that in a situation that demands an actual counterexample (like an algorithm that produces a result). This tends to guide people towards looking for results that don't require this kind of indirection, since they apply more broadly and in more kinds of logics -- orthodox constructive results are kind of a lowest common denominator of consistent truth and remain broadly compatible with most axioms, while nonconstructive results often fail in particular models. Which seems like a pretty sane stance to me, but maybe I'm too thoroughly indoctrinated to see how unreasonable it is :P

(Note that this is about excluded middle. There ARE constructive logics with interpretations of excluded middle, e.g. some forms of classical linear logic, but they do not play as nicely with other logics. Constructivists often reject even weak forms of choice for largely the same reasons--there are some forms of choice that are constructively valid in some logics, but these results often fail to hold true in more conventional logics. And the same is true for a whole host of related notions that proof assistants like Rocq reject by default, propositional extensionality (which says that two proofs of the same proposition are equal) and function extensionality (which says functions are equal whenever their results are equal on all the arguments in their domain -- which might seem obviously acceptable until you realize that it's false in most programming languages!) being prominent but much less discussed examples. It's all about remaining broadly compatible with lots of different types of reasoning, not because people think the reasoning is invalid per se).

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mb7733 3 hours ago | parent [-]

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 43 minutes ago \| parent [-]
		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.