[mb7733]

Are you thinking of proof by contradiction, which is rejected by constructionism?

[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?

[SpicyLemonZest]

It’s very straightforward, but it often doesn’t (and here didn’t) fully satisfy the curiosity that was embedded in the original problem. Why did the Jacobian conjecture seem to be true? Is there some underlying symmetry that’s very slightly broken? Or perhaps there’s all kinds of counterexamples, and the intuitive pattern is only real for certain kinds of functions which happen to predominate in our intuition. Then how should we adjust our intuitions to better capture the space of possible polynomial functions?

[mb7733]

Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.

[SpicyLemonZest]

I would frame it differently. The existence of compact counterexamples to a true-seeming conjecture suggests that there’s some deeper understanding waiting to be discovered. Fuzz testing for theorems, if that makes sense. I hope mathematicians in 2036 will be able to explain in detail why the Jacobian conjecture was false and identify which similar, true conjectures the community’s intuition was pointing towards.

[ChadNauseam]

We can take a simpler example. Let's say someone conjectures that all linear maps are isomorphic if they have the same domain and codomain*. A counterexample is easy to find, but true insight would be to notice that all linear maps with the same domain and codomain that are not isomorphic map some non-zero elements to zero. That is much more interesting than just finding a counterexample. Although, that isn't to say that finding a counterexample is not very interesting.

*statements only apply to maps whose domain is finite-dimensional

[nopinsight]

I suspect an AI, possibly a successor to current LLMs, will achieve that by the early 2030s. It might help illuminate many mysteries in math and beyond for us all.

[antonvs]

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.

[mb7733]

Constructivists would surely disagree!

[Jweb_Guru]

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).

[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!?

[antonvs]

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.

[mb7733]

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.

[Jweb_Guru]

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.

[murphyslaw]

Isn't the fact that you now know that the conjecture is false a huge help? At least it will help convince people to look at the conjecture more closely, no?

[inigyou]

You, right now, have the ability to spend a few weeks studying the Jacobian conjecture and its counterexample and write up a blog post about what you think is special about this counterexample.