[tombert]

In my free time, I have taken to trying to prove the Collatz conjecture.

People much smarter and more educated than me have failed at this quest, so I will nearly certainly fail at it, but that's not really the point in my mind. Even if I'm not the one to actually prove it, I can at least try and contribute to the body of work towards proving it. Mathematics is, more than nearly anything else, the result of generations building upon previous generations work. It's never "done", always growing and refining and figuring out new things to look at.

I have a few ideas on how to prove Collatz that I have not seen done anywhere [1], and usually (at least for me) that means it's a bad idea, but it's worth a try.

One of the greatest things about humans is our willingness to have multi-generational projects. I think maybe the coolest thing humans have ever done was eliminate smallpox, and that took hundreds of years.

[1] Which I'm going to keep to myself for now because they're not very fleshed out.

[cubefox]

A related thing occurs in academia for very niche topics on which only very few people are working. Perhaps nobody for most of the time. A paper might "reply" to another paper from years or decades ago, and receive itself a reply only years later, but from a different author.

The cool thing is that you can easily become the current world leading expert on such a niche topic, because there aren't that many papers. So it's easy to know every single one of them, and the few experts are spread out in time rather than space.

It's like a web forum thread on a very obscure question, where only every few years someone contributes a new comment, likely never to be read by most of the previous authors, but read by all that come later.

[pavel_lishin]

> A related thing occurs in academia for very niche topics on which only very few people are working. Perhaps nobody for most of the time. A paper might "reply" to another paper from years or decades ago, and receive itself a reply only years later, but from a different author.

Reminds me of certain parts of "Anathem".

[Davidzheng]

I do want to say often math papers have gaps, purely explained parts and sometimes mistakes which can make it quite hard to understand a topic of literally no one else still remembers it though. However the overall advancement of math sometimes helps in this regard

[wwweston]

And it’s not only never done, it’s always on the verge of dying off. Like Bill Thurston said, mathematical understanding basically lives in communities of mathematicians, every one of them a cell in the superorganism that is the field. You’re part of the distributed filesystem providing persistence as well as the possibility of new understanding.

https://mathoverflow.net/questions/43690/whats-a-mathematici...

[7373737373]

Interesting new contender for simplest to state unsolved problem: The Antihydra

Does this program halt?

  a = 8
  b = 0
  while b != -1:
      if a % 2 == 0:
          b += 2
      else:
          b -= 1
      a += a//2

(// being integer division, equivalently a binary shift one to the right: >> 1)

https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html

https://bbchallenge.org/antihydra

[tombert]

Interesting, I hadn't heard this one.

I should see if I can model this in Isabelle or something and see what happens.

[7373737373]

for reference, the statement has been formalized in Lean in Deepmind's open problem database: https://github.com/google-deepmind/formal-conjectures/blob/e...

[gowld]

Is that also the simplest unsolved state problem?

[ygritte]

How does overflow behave?

[IshKebab]

It doesn't overflow.

[ygritte]

Too bad, that makes it harder.

[fragmede]

Fwiw, ChatGPT is able to say that it doesn't. I wonder what other classes of programs it's able to state if it halts?

[schoen]

The math community surely expects a proof of that, and ChatGPT surely doesn't (yet) have one. (Maybe some day it will, as Kevin Buzzard and others are experimenting with asking language models to produce formal proofs.)

You could get LLMs to opine on many unresolved math conjectures, but I doubt much credence should be given to their responses, when not accompanied by a proof.

[yifanl]

Tom from the pub says that it does.

[7373737373]

Most LLMs I've tried come up with invalid reasoning, many confuse empirical evidence (of simulating it for a few steps and it 'most probably not halting') with definite proof that it never does, some create invalid probabilistic mathematical arguments to the same effect

Others I've tried are caught in a loop of trying to prove the same, insufficient approach over and over again, lacking explorative and "creative" behavior

Generally it seems that LLMs lack the 'motivation' to actually try to solve unsolved problems especially if they know that they are unsolved or difficult

[IshKebab]

ChatGPT is able to say anything it wants. Surely you know this by now ...

[SirChud]

What ChatGPT says has no relevance to whether it halts.

[snarf21]

Reminds me of Stewart Brand and the Clock of the Long Now (and other longer time horizon projects they are working on).

Reminds me of a statement he made during a Tim Ferris interview that I think is quite profound for our mental health. ".... being proud is the most reliable source of happiness that I know."

[saulpw]

Proud of your work, not proud of yourself. The latter is quite a reliable source of unhappiness, I've found.

[snarf21]

In the full quote, he is talking about fitness and being able to lift things and being proud of your abilities to do so. I'm not saying it works for everyone but it is nice to have a "thing" that you can hang your hat on. The whole interview is quite interesting and worth a read.

[saulpw]

That's all fine and good. But nothing lasts. And then when you can no longer lift things, the humiliation you suffer is proportional to the pride you take in it.

Saying "it is nice to have a thing to hang your hat on" sounds like a very pleasant form of personal pride that is easy to let go of. But Hubris is a mortal sin for a valid spiritual reason, and pride is a slippery slope. The advice I gave--pride for your accomplishments, not for yourself--is similar to the advice to how to praise a child: for their efforts, not for their talents.