So is this (BB) then just an exercise demonstrating our inability to reasonably solve the problem?
I suppose what I’m asking is what is its significance? Or why does this matter?
As a self taught programmer, I realize I have definite gaps in the more theoretical realms of computer science.
The collatz conjecture isn't a problem that we think is unsolvable, it probably (maybe even definitely?) has a solution. It is a problem that a lot of really good mathematicians have sunk a lot of time into and made basically no progress. Collectively the mathematical community has more or less decided that we don't know how to even approach solving this kind of problem yet, and probably won't be able to solve it in the foreseeable future.
This article is saying that solving for BB(3, 3) requires solving a collatz-conjecture like problem, so it's probably also beyond our current abilities in mathematics. It's not saying it's unsolvable, just that we probably won't figure out the solution anytime soon.
As for why solving BB(3, 3) matters... it doesn't really. It's just an intellectual curiosity. We've figured out some busy beaver numbers. We've proven some upper bounds on the biggest ones we can figure out, but there's a big gap between what we've solved and what we've shown we can't solve for. Making that gap smaller is something of a game.
The Busy Beaver problem sits somewhere on the range from "intellectual curiosity" to "lens that allows us to view the edges of uncomputability". I would guess that the majority of people doing work here are hobbyists (including myself). In fact, when Tibor Rado first introduced BB, he introduced it as the "Busy Beaver Game", so it has had a playful energy since the beginning :)
