There are many cases where it's possible to prove that counterexamples must exist, without identifying a specific example. This kind of proof provides more insight into the problem than simply finding a counterexample.
There are many cases where it's possible to prove that counterexamples must exist, without identifying a specific example. This kind of proof provides more insight into the problem than simply finding a counterexample.
Constructivists would surely disagree!
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).
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!?
In a constructive system, it’s often possible to refute a universal proposition without exhibiting a counterexample, by proving that the proposition implies falsehood.
The constructivist will still object that you can’t, from that, conclude that “…therefore a counterexample must exist,” without actually providing a counterexample. But the general principle I was describing still applies - a proof often gives you insight that an example by itself doesn’t.
To roughly repeat myself from a sibling comment: I may have been unclear. I didn't claim that the only way to disprove something in a constructive system is to produce a counterexample.
The part that I was referring to was the last statement from the OP: that "a proof of existence of a counterexample necessarily provides more insight than a counterexample". I can't imagine a constructivist would agree with that in general.
I'm the OP. In the cases where a proposition can be refuted by a proof that doesn't involve counterexamples, by its nature that proof will tell you something about the reason that the proposition is false.
Whereas a counterexample, on its own, proves the proposition false but doesn't necessarily tell you anything else.
The real difference in the constructive case is that there are fewer classes of proposition for which a proof without witnesses is possible.
I think the constructive position is basically that people's entire issue with lack of excluded middle being absent is just that people like being able to say "P" instead of "~~P" because it sounds better, considering you can prove ~~P for all the classical propositions that use excluded middle.