[beering]

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.)

[nommynommynom]

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.

[c32c33429009ed6]

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...

[pegasus]

I guess the question of why counterexamples are so rare and/or hard to find is still left to be investigated.

[aureate]

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.

[orangecat]

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).

[jibal]

Did you read what you responded to? The Collatz conjecture is almost certainly not false, so no "clean up" is possible.

[aleph_minus_one]

> The Collatz conjecture is almost certainly not false, so no "clean up" is possible.

Many people believed the same about the Jacobian Conjecture.

[jibal]

non sequitur

[jibal]

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".