The Busy Beaver problem is one of the most theoretical, one of the most purely mathematical, of all theoretical computer science questions. It is really about what is possible in a very abstract sense.

Working on it did make people cleverer at analyzing the behavior of software, but it's not obvious that those skills or associated techniques will directly transfer to analyzing the behavior of much more complex software that does practical stuff. The programs that were being analyzed here are extraordinarily small by typical software developer standards.

To be more specific, it seems conceivable to me that some of these methods could inspire deductive techniques that can be used in some way in proof assistants or optimizing compilers, in order to help ensure correctness of more complicated programs, but again that application isn't obvious or guaranteed. The people working on this collaboration would definitely have described themselves as doing theoretical computer science rather than applied computer science.

It is purely academic... or is it?

G. H. Hardy's autobiography was titled "A Mathematician's Apology" because he, somewhat jokingly, wanted to apologize for a life of pure math... totally academic and completely useless.

Then a few years after he passed away, his math was key for The Manhattan Project to build the first successful nuclear bomb. His math lead to the device that changed the world and affected every aspect of human life for a century.

It's only "useless" until someone finds a "use".

P.S. The book documents his time with Ramanujan. If you liked the film "The Man Who Knew Infinity", you should read Apology

The research into BB numbers is purely academic and unlikely to be more than that, unless some other part of mathematics turns out to be wrong. Our current understanding is that the numbers are essentially guaranteed to be useless.

This particular proof is doubly-academic in the sense that the value was already known, this is just a way to make it easier to independently verify the result.

It's a part of a broader movement to provide machine proofs for other stuff (e.g., Fermat's last theorem), which may be beneficial in some ways (e.g., identifying issues, making parts of proofs more "reusable").

Some "natural algorithms" as they call them emerged from busy beavers. And some of them surely will be useful sooner or later, with tangible benefits such as improved databases or graphics or anything really.

Purely academic, there are no direct applications or anything unlocked by knowing the value; the benefits are all in what was learned along the way.

Purely academic, at least if you mean the result itself. People are bringing up all sorts of apologetic stuff about how it might be useful in the future, and as an ex-maths-phd I can sympathise, but to me this seems more like a large-scale bookkeeping exercise than a conceptual breakthrough. It's a very nice result though =)

100 years ago you could be asking if number theory or non-euclidean geometry has any tangible benefit of is purely academic. For the bb problem, the answer right now is no. But nobody knows what's down the road when finding useful applications in pure maths.

It's a beautiful achievement of humanity. Like going to the moon. Someone might say there are some mining or rock-related applications around going to the moon, but it's mainly not about that.

I don't like to call this "purely academic" because that sounds dusty and smells moth-chewed and utterly boring. It's purely spiritual, purely heart-warming, purely brain-flexing.

It's like asking a mountain climber whether reaching K2 has practical benefits or if it's "academic".

Researching methods to solve busy beaver problem also helps in exploring and developing methods to tame halting problem in various applications, such as static analysis.

Define tangible benefits and give example of any pure mathematics in last 200 years having tangible benefits.

One benefit is that it is related to foundations of mathematics. If you can find busy beaver number 745, you can prove or disprove Maths. If you can find busy beaver 6, you can prove few collatz like conjecture which is an open problem.

Most core research doesn't help directly, but through indirect means. e.g. This would have led to finding the areas of improvements in Lean due to its complexity. I know for fact that Lean community learns a lot from projects like Mathlib and this. And Lean is used in all sort of scenarios like making AWS better to improving safety of flights.

I am also unfamiliar, but a naive guess I can make is helping solve other math proofs, and maybe better encryption for the public. Basically everytime I see some progress in math, it usually means encryption in some way is impacted.