
Who Can Name the Bigger Number? (1999) - lelf
https://www.scottaaronson.com/writings/bignumbers.html
======
dang
A bit from 2018:
[https://news.ycombinator.com/item?id=18508471](https://news.ycombinator.com/item?id=18508471)

A bit from 2016:
[https://news.ycombinator.com/item?id=11596059](https://news.ycombinator.com/item?id=11596059)

Quite a lot from 2015:
[https://news.ycombinator.com/item?id=9058986](https://news.ycombinator.com/item?id=9058986)

2013, despite subtle mistitling:
[https://news.ycombinator.com/item?id=5085463](https://news.ycombinator.com/item?id=5085463)

2010:
[https://news.ycombinator.com/item?id=2024576](https://news.ycombinator.com/item?id=2024576)

[https://news.ycombinator.com/item?id=1539538](https://news.ycombinator.com/item?id=1539538)

A tiny bit from 2009:
[https://news.ycombinator.com/item?id=951095](https://news.ycombinator.com/item?id=951095)

~~~
tomjakubowski
Not a discussion of Scott Aaronson's blog, but a relevant bit from Mr. Show,
"24 Is The Highest Number."

[https://www.youtube.com/watch?v=RkP_OGDCLY0](https://www.youtube.com/watch?v=RkP_OGDCLY0)

~~~
dang
I love Mr. Show!

------
neonate
He wrote this as a teenager. Then a new version of it in 2017:
[https://www.scottaaronson.com/blog/?p=3445](https://www.scottaaronson.com/blog/?p=3445).

"That essay might still get more views than any of the research I’ve done in
all the years since."

~~~
IHLayman
Thanks for pointing this out. He mentions in this one that he didn’t have time
to get into Graham’s number, but I still think TREE(3) is worth mentioning.

~~~
ThreeFx
Fun fact: The Kruskal Tree Theorem has been extended to undirected graphs
(they form a quasi-wellordered set wrt the graph minor relationship), giving
rise to an even faster growing sequence called the "Friedman's SSCG function".

The theorem is interesting for other reasons though: It allows us to define
graph families by forbidding certain substructures.

------
mnemonicsloth
> And in Go, which has a 19-by-19 board and over 10^150 possible positions,
> even an amateur human can still rout the world’s top-ranked computer
> programs.

Surprising, in retrospect, how fast that changed.

~~~
gqcwwjtg
It was surprising at the time, too.

~~~
im3w1l
When I heard of the breakthrough in CNN's on imagenet the first thing I
thought of was "wow I wonder if you can use them on Go". Of course I never
executed on this idea. I'm mentioning to show that it was not that non-
obvious.

------
IHLayman
Maybe I missed it, but it was interesting that tetration was mentioned without
talking about Graham’s number g_64[0] or Kruskal’s tree theorem and
TREE(3)[1], both have been quite popular huge numbers for quite a while now.

0:
[https://en.wikipedia.org/wiki/Graham's_number](https://en.wikipedia.org/wiki/Graham's_number)
1:
[https://en.wikipedia.org/wiki/Kruskal's_tree_theorem](https://en.wikipedia.org/wiki/Kruskal's_tree_theorem)

~~~
maze-le
I think the most fascinating bit about Grahams Number is, that is is used as
an upper bound in a proof about colored subgraphs in a completely connected
graph of dimension [n]. But this is only an upper bound... the actual number
of dimensions [n] that satisfies the proof is unknown, but it could be as low
as 13 for all we know...

------
MatthewWilkes
The joke at the start is fun, I'd only heard a variant in Black Country
dialect.

Aynuk: What do yow reckon the biggest number is?

Ayli: Ten thousand?

Aynuk: Worrabout ten thousand and one?

Ayli: Ar, I were close though, wor I?

------
xamuel
OP is about natural numbers, but if the challenge is generalized to computable
ordinal numbers, then it actually becomes a very interesting measurement in
the area of AGI. There's no bound on how big of a natural number any
particular AGI could name (because if an AGI can name N, then it can name N+1,
and by repeating that process, all the naturals are covered). But the set of
codes of computable ordinals is a non-computable set, so no AGI could ever
name them all, every AGI necessarily would have an upper limit on how large of
a computable ordinal it could ever name.

See my slides on the subject here:
[https://semitrivial.github.io/MeasuringIntelligence2019.pdf](https://semitrivial.github.io/MeasuringIntelligence2019.pdf)

~~~
lonelappde
How is this relevant to AGI?

~~~
xamuel
How would you measure the intelligence of an AGI? Say you've got HAL in one
corner of the room and The Terminator in the other, and you can ask them
questions and ask them to perform computations, and they'll obey you. How can
you make sense of which one is smarter?

A naive attempt would be to ask them to name the biggest (natural) number, but
that's no good: HAL says 1000, Terminator says 1001, but of course HAL also
knows 1001 is a number so Terminator only wins by dumb luck.

We can salvage the idea though by switching natural number for computable
ordinal number, and instead of having them name a single number, have them
enumerate computable ordinal numbers indefinitely. The set of codes of
computable ordinals is Turing non-computable, so neither machine will succeed
in enumerating all of them, and by comparing the size of the ordinals
enumerated by the two machines, you get an elegant, parsimonious, not-too-
contrived notion of which machine is (mathematically) smarter.

Read the slides I linked!

------
phonebucket
MIT ran this challenge as the 'Big Number Duel':
[http://web.mit.edu/arayo/www/bignums.html](http://web.mit.edu/arayo/www/bignums.html)

Worth a read just to see the winner's solution, which I thought was rather
ingenious.

------
tjchear
Here's another treatment on the topic, and in particular, Graham's number:

[https://waitbutwhy.com/2014/11/1000000-grahams-
number.html](https://waitbutwhy.com/2014/11/1000000-grahams-number.html)

------
amelius
Regarding tetration, I wonder: is it ever used in physics/engineering
contexts? If not, then why not? Is our universe too simple for more than 3
levels of arithmetic operations?

------
vinnyglennon
[https://en.m.wikipedia.org/wiki/Graham%27s_number](https://en.m.wikipedia.org/wiki/Graham%27s_number)
number may be larger?

~~~
klank
From the link you provided:

> As there is a recursive formula to define it, it is much smaller than
> typical busy beaver numbers.

While I can't argue the specific details, I'm not surprised. BB numbers are
not computable. Grahams number is.

------
symplee
Further ideas/concepts:

[https://en.wikipedia.org/wiki/Large_numbers](https://en.wikipedia.org/wiki/Large_numbers)

------
eindiran
Rather than the BB sub n definition proposed in the article, I propose a new
notation BB^n( x ), which is to the Busy Beaver sequence what Knuth up-arrow
notation is to the hyperoperation sequence. So you don't need to rely on the
magic of a "super duper machine" to define the number, just recursive use of
the Busy Beaver function. It's kind of tenuously on the boundary of the rules,
since it uses a made-up notation.

------
ccvannorman
There is a community of googologists who study, record and define the largest
numbers ever described. If you want to bend your mind a bit around gargantuan
numbers I suggest taking a casual walk down their list.

[https://googology.wikia.org/wiki/List_of_googolisms/Uncomput...](https://googology.wikia.org/wiki/List_of_googolisms/Uncomputable_numbers)

------
fastaguy88
Knuth wrote an interesting article on the topic, in 1976, because computers
had become so powerful.

[https://science.sciencemag.org/content/sci/194/4271/1235.ful...](https://science.sciencemag.org/content/sci/194/4271/1235.full.pdf)

------
jes5199
he mentions population growth falling due to birth control as an unlikely
prediction, but it seems to be where we’re actually headed! There’s a book
with a lot of research on this called “ Empty Planet: The Shock of Global
Population Decline” - they say we’ll hit peak human population in 2050 +/\-
10years

~~~
EliRivers
Read that last month; actually very readable, I found, and the way the book
tackles different parts of the world and suggest them to be essentially be on
different parts of the same glidepath is not unconvincing.

------
cryptica
This game is more about business/negotiation than math.

I'd only play this game if I get to reveal my number first and the opponent
agrees to read aloud both numbers in decimal form one digit at a time. Then
I'd write down Graham's number and leave the room.

------
pachico
Sorry, but this reminds me of Mr. Show's sketch "the highest number
"[https://m.youtube.com/watch?v=RkP_OGDCLY0](https://m.youtube.com/watch?v=RkP_OGDCLY0)

------
mhh__
That's numberwang?

------
shouyatf
In fact, BB(k) is roughly equivalent to asking "what's the biggest non-
infinity namable number in k characters".

~~~
baddox
That doesn’t sound right, because “BB(k) plus 1” is larger than BB(k) and has
fewer than k characters for sufficiently large k.

~~~
dandanua
What you say is a Berry paradox. To be consistent we fix a programming
language before any such definitions. So you have to count all characters in
the BB(k) subprogram. Hence the total program for “BB(k) plus 1” definitely
will have more than k characters.

~~~
baddox
I don't think _my_ example (for example, "BB(11111) + 1") demonstrates the
Berry paradox. The original comment "the biggest non-infinity namable number
in k characters" is the Berry paradox, which is what I was pointing out.

------
techmaster7b
All of this work on numbers and yet the incorrect use of the word decimated.

------
logicallee
here's what I came up with in a few seconds:

let g(x) mean "x to the googolplex power"; my number is g(g(...g(googolplex)
with a googolplex number of g's.

~~~
gerdesj
h(x): g(x) ^ g(x)

etc

