I think parent’s point is that every false conjecture can cost a lot of time to be spent on futile affirmative proofs. So if we “clean up” a bunch of false conjectures, then more effort can be spent on interesting proofs of the others. (Probably a rather naive view of the value of conjectures but I’m just offering an alternative interpretation of the comment.)
The opposing argument there is that the hope is that solving these problems reveals other interesting maths knowledge along the way. Finding a counter example all but ensures that won't ever happen.
No, it opens up a whole new suite of questions. Now we can ask, for example: what conditions do we need for the result to hold? What dimensions does it hold in? And many more...
The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the first few billion integers or so.
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
I haven't seen any thorough analysis of how "hard" it would have been to find by brute force
Pretty hard. I asked Fable and it gave an estimate of 10^46 candidates in the counterexample's "reference class", and that's assuming you know how many distinct terms there are (as opposed to searching all polynomials of degree 7/6/4 for the three coordinates, which it estimates at 10^334).
P.S. Also false -- it simply is not true that many people believed that the Jacobian conjecture was almost certainly not false--why would they? OTOH, the Collatz conjecture has been confirmed for all integers up to 2.36 *10^21, and Terence Tao has proved that it is true for "almost" all numbers: https://www.quantamagazine.org/mathematician-proves-huge-res...
Again, this is all non sequitur, because the context was a statement that most mathematicians believe the CC to be true, in which case there would be no "clean up".
