Comment by defrostr

Don't let specialists detract from your enjoyment of Quanta-level articles. This one is well-written, makes no egregious errors, and only omits one important fact. And that fact is ...

All this talk about exponentiation, tetration and pentation have their roots in Peano arithmetic, which for reasons of logical clarity, defines precisely one function -- the "successor function":

s(n) = n+1

Using just this function, Peano defines addition, multiplication and exponentiation with great clarity. Since Peano's time, people have been making adjustments, like including zero among the counting numbers, and by extending exponentiation into tetration, pentation and a few more exotic operations fully discussed in the linked article and elsewhere.

I personally would have liked to see a more complete exposition of the logical roots of the Busy Beaver challenge, and I think the missing parts would have made the article a better read, even for non-specialists. Maybe especially for those readers.

But Quanta articles are perfectly suited to their audience, people who may be inspired to look for more depth elsewhere.

There isn't really anything better. The Notices of the AMS has one feature article each issue, but those are sometimes too technical.

Anyway, this article seems fine to me. The "exponentiation" comment seems like a bizarre misreading. The article is just trying to explain how big BB(6) is. Before that, it explains what BB(n) is. To think it's solely about exponentiation you have to skip that entire section.

I don't think there's anything that would put some new, esoteric math concept in your mailbox every week, although there's plenty of books that cover recreational mathematics in an accessible way (Martin Gardner, Ian Stewart, etc). And for QM articles, I recommend searching the web - you can often find better explanations on some 1999-style blog somewhere.

The problem with this particular article is simple: busy beavers numbers aren't interesting because they're big. They don't break mathematics because of that; you can always say "+1" to get a larger number. There's also nothing particularly notable about Knuth's up-arrow notation, which is essentially a novelty that you're never gonna use. Instead, the numbers are interesting because they have fairly mind-blowing interactions with the theory of computability.