[xenotux]

There is a downvoted comment that reads "ah yes the totally new math of exponentiation". The snark is uncalled for, but that's actually the essence of this article: it talks about repeated exponentiation as if it were some profound mathematical discovery.

It isn't. The article neglects to explain what makes busy beaver numbers interesting in the first place. And I think it's symptomatic of Quanta Magazine articles that feature on HN several times a week. A profoundly-sounding title and pleasant writing, but not much essence beyond that.

[oersted]

Indeed, I would have at least liked to get a rough understanding of the tricks they use to classify and discard Turing machines, and how they construct these enormous runtime calculations. They are clearly computing something but they are obviously not actually running the machines normally for those numbers of steps.

[ysofunny]

but because that may actually be useful

you gotta pay for that. or dig through the academic publisher archives; which you also gotta pay for unless you believe in digital piracy and evil copyright infringement which may or may not fund terrorism

like they used to say: information wants to be expensive so pay to be free

[defrostr]

I usually get drawn into their posts also and didn’t realize they were low value; my IQ must not be high enough to differentiate. What would you recommend as alternative sources?

[lutusp]

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.

[QuesnayJr]

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.

[xenotux]

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.

[sfn42]

It's crazy to me that we're now writing articles about the fact that a large number is large.

Hey guess what, I can imagine a number even larger. It's BB(6) + 1, isn't that amazing and interesting? Wow imagine BB(6) multiplied by a googolplex, amazing. Wow, such number.

What's the point? Numbers are infinite, what else is new?

[frozenseven]

>What's the point?

What's the point of your comment? To suck the joy out of everything?

Turing machines are a fundamental object of study within the theory of computation. The complexity and wild behavior that arises from even the simplest machines is a cool discovery. BB(6) was thought to be a lot smaller, but it turns out to be really huge. The Busy Beaver game is also interesting to those who work on combinatorics and theorem provers. And of course many many people in the space of recreational math & volunteer computing love this challenge.

You don't like it? Ok, then. You don't have to.