system
Given the following post, respond as if you're commenting on Reddit:
human
Why does the Collatz Conjecture (and problems like it) matter?

I was watching a video from one of the 3628468 youtube channels that the bearded englishman has and it mentioned this problem and I started to think, why does such a thing matter? Seems like just a funny maths thing for a child to play with but in the video the host mentioned that people have "ruined" their careers trying to solve the problem.

What would happen if the problem was proved false/true? Is this a check on if maths as a whole work or something?

If there is a simple and obvious answer, I do apologize. I'm just a lowly car mechanic and a bass player on top of that.
gpt
The problem is hard because it digs into how multiplication and addition interact with each other. We have lots of number theory that focuses on multiplication - this is where a lot of the importance of primes comes from and there are lots of methods that help us solve equations - and we have slightly less number theory that explores how numbers add together. It's not that addition doesn't happen in multiplicative number theory, or that multiplication doesn't happen in additive number theory, it's just that the kinds of questions you ask and methods you use are unique to each type.

There are a few questions that cross the boundary, however. For instance, Goldbach's Conjecture says that every even number can be written as a sum of two primes. Primes are necessarily multiplicative, but asking what numbers you get from sums is additive and so this question crosses the boundary and is therefore very hard. The ABC-Conjecture is another such question that remains unsolved (despite insistence by some otherwise), and it basically asks how information about primes persist through addition; if A+B=C, then how do the prime properties of A and B determine (or not) the prime properties of C?

Collatz Conjecture is another in this family. It mixes properties of 2,3 and addition in this dynamic kind of thing. Plus, heuristically, adding one is, in ways, the most destructive to multiplicative information, so the Collatz Conjecture has a lot of difficulties to overcome. (For instance, to get brand new primes, multiply all primes that you know together, add one, and every prime that you get from this new number is a new prime. Or, there are lots of primes like 2^(p)-1 whereas 2^(p) has the highest power an integer of that size can have.)

Solving the Collatz Conjecture will not really do anything on its own. But, because it is this kind of problem that crosses the boundaries between multiplication and addition, a solution should help us navigate that realm better. As with almost all unsolved math problems, we don't really care about the actual solution or answer, what we're really interested in are the new methods that will almost surely need to be invented to solve them. We can use those methods elsewhere and to ask new questions. And that's really what matters.