
Who Can Name the Biggest Number? - pizza
http://www.scottaaronson.com/writings/bignumbers.html?repost3yearslater
======
aroman
Great read, and some really clever explanations of Turing's work.

While this wasn't mentioned in the post, I thought this seemed relevant:
<http://en.wikipedia.org/wiki/Graham%27s_number>

This bit always blew my mind:

"Graham's number is unimaginably larger than other well-known large numbers
such as a googol, googolplex, and even larger than Skewes' number and Moser's
number. Indeed, the observable universe is far too small to contain an
ordinary digital representation of Graham's number, assuming that each digit
occupies at least one Planck volume. Even power towers of the form are useless
for this purpose, although it can be easily described by recursive formulas
using Knuth's up-arrow notation or the equivalent, as was done by Graham. The
last ten digits of Graham's number are ...2464195387."

I'm pretty sure that's the biggest number. Or maybe Graham's Number + 1 ;)

~~~
jerf
Graham's number is typically cited as the largest number that has ever been
seriously used in a mathematical proof, as the Wikipedia article says, but
Busy Beaver passes it very quickly.

One of the nasty little characteristics of BB that Aaronson describes, but can
stand to be reiterated since I've noticed many people don't seem to quite
follow it, is that any mathematical algorithm you can encode into a Turing
Machine in some number of rules X, is therefore at most the top-end bound of
BB(X), and more likely (which in this case is serving as
mathematician's-understatement-speak for "certainly"), not even close. And for
all that the TM "programming language" isn't that efficient, certainly almost
anything you can describe in a reasonable English paragraph is, say, well less
than 100 states in a TM. For all of Graham's Number's stonking size, in
algorithmic terms, it still isn't that complicated.

Generally speaking, human attempts to create Big Numbers by fancy recursively-
applied algorithms are still quite simple to put into a TM. This is why it's
an interesting point that we don't even understand what exactly a BB(5)
solution is doing; a mere five states and the human mind is already lost,
lost, lost. I think of Busy Beaver as showing in its own way just how
thoroughly we have _not_ mastered math, and never will master more than a tiny
little island that happens to be relatively easy to follow, because BB shows
us that we don't have to swim out into the great sea of "all math" very far at
all for there to be monsters.

~~~
btilly
Whether or not BB(n) is even defined is an interesting philosophical question.

If n is a modestly large number, it can be proven that in no consistent axiom
system below a certain size can any explicit number written out in base 10
EVER be proven to be an upper bound for BB(n). If we make n something like 100
million, that encompasses all possible axiom systems that are human
comprehensible.

A "finite" number with no provable upper limit - how much sense does it make
to say that is well-defined? It is enough to make you take up constructivism!
(Or quit math for something more sensible. Like politics.)

------
lisper
Here's another mind-blowing result: there are about 10^80 subatomic particles
in the known universe. If every one of these particles were a computer with a
clock cycle of the Planck time (10^-43 seconds) and was running continuously
for the life of the universe (10^17 seconds) the total number of computable
states would be only about 10^140, which is the number of states that can be
represented in about 20 bytes of memory. The total number of states that will
actually be computed by physical computers will, of course, be much, much
smaller than this. So make every cycle count :-)

~~~
Someone
2 log(10^140) / 8 ~= 58.134, so it's more like 58 bytes. Irrelevant for the
argument? Definitely, but I had to double-check the moment I thought '140
decimal digits in about 160 bits? That cannot be right.'

Also, the number of states that the universe has time for to compute may be a
lot smaller than the number of possible states. For example, if each particle,
at each step, made a two-way choice, that single particle could run through
that 10^140 potential states in no time flat. Because of that, you cannot say
"64 bytes ought to be enough for anybody"

~~~
kzrdude
by "2 log" you mean log_2

~~~
Someone
Apologies. I thought that would be evident from context, but I should have
explained it. Also, it would have been better if I had left of the space.

To make matters even worse, I just realize that the custom to write the
logarithm in base 2 of x as ²log x (superscript digit two, 'log', 'x') may not
be used world-wide and may even be specific to the Netherlands
([http://nl.wikipedia.org/wiki/Logaritme#Natuurlijke_of_neperi...](http://nl.wikipedia.org/wiki/Logaritme#Natuurlijke_of_neperiaanse_logaritme))
That may have made this even more incomprehensible.

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ColinWright
It's intriguing following the submission history of this item, and similarly
interesting comparing the comments here with the comments three years ago. I
idly wonder how much they reflect the change in the HN readership in that
time.

We all know about sentiment analysis, I wonder if a similar sort of analysis
could be done between this thread and one or both of the others. Find some
sort of change, then look for similar changes in other items, discussed twice
with a long gap.

<http://news.ycombinator.com/item?id=47408> 1971 days ago

<http://news.ycombinator.com/item?id=492615> 1426 days ago

<http://news.ycombinator.com/item?id=951095> 1157 days ago

<http://news.ycombinator.com/item?id=1539538> 912 days ago, 68 comments

<http://news.ycombinator.com/item?id=2024576> 761 days ago, 31 comments

<http://news.ycombinator.com/item?id=2531994> 620 days ago

<http://news.ycombinator.com/item?id=3262788> 425 days ago

<http://news.ycombinator.com/item?id=3615532> 334 days ago

------
jliut
Relevant: <http://djm.cc/bignum-results.txt>

This is a precise version of the challenge, to write a large number in 512
bytes of C code.

~~~
kzrdude
more interesting than it sounds. The challenge uses a hypothetical version of
C with arbitrary big memory and integer limits.

------
Xcelerate
I actually got fussed at for my answer to a variant of this question this past
semester. I have a professor who talked about when he was in college and his
instructor had the challenge: what is the biggest number you can make with
three nines?

His answer was 9^(9^9). Of course, I thought this question warranted further
discussion and I mentioned that Knuth's up arrow notation allows an even
bigger number and demonstrated it on the board. The professor seemed annoyed
and I asked him about it later; I believe he mistook my interest in
contributing to the discussion as trying to one-up him (which was NOT my
intention at all).

I've been much quieter in that class since :(

~~~
jrockway
9! (9! up arrows) 9!

Of course, you can also have 9!! and 9!!! and 9!!!! or even 9! factorial signs
after the 9, etc.

The original problem statement of "write this in 15 seconds using standard
notation" is less easy to game. But I think the right answer is to just have 9
(up arrow with 9 under it) 9 and then make the 9s bigger using factorials
until your 15 seconds run out.

How pointless.

~~~
splat
Keep in mind, though, that you would want to write it (9!)! or ((9!)!)!. 9!!
is actually the double factorial, which is defined as 9!! = 1 * 3 * 5 * 7 * 9.
So 9!! < 9!.

~~~
jrockway
I've never heard of that, but sure, do (9!)!. Since the problem as stated by
the OP (not the article itself) specifies that only 9s are limited, it doesn't
make a difference.

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pazimzadeh
I love that this essay was so easy to follow in the beginning, before
ascending farther that I could have imagined.

I assume that there are so few comments because everyone is still reading!

------
DanielRibeiro
Nice read. I was epxecting it to mention Knuth's arrow notation though[1]

[1] <http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation>

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InclinedPlane
Define: the oh-holy-crap number, d, as follows:

a = the largest number for which there is a concise definition in the English
language version of wikipedia

b = the 2nd largest number defined on wikipedia

c = the 3rd largest number defined on wikipedia

Then, let d = c -> b -> a (using Conway chained arrow notation)

~~~
yuliyp
This works until you put the definition of d into the English language version
of Wikipedia

------
diziet
I always had a fascination with large numbers and expressing them in
mathematical notation. It's fun to use the next "step" to outdo someone you
talk to, ie tetration for exponentials, knuth up arrows, ackermann function,
etc.

I'd gone even as far as put on our startup's team page an offer to give a
discount to anyone who can give interesting answers to a similar question as
the one proposed in the article!

------
jlgreco
Oh wow. This is the essay that solidified my decision to go into/stay into
computer science. I had forgotten about it, great to read it again.

------
designatedInit
I really enjoyed reading this. It covers so many different topics and is
relatively simple to understand.

------
aurelianito
As usual, xkcd has something related to the topic: <http://xkcd.com/1162/>

~~~
dazmax
Or the third panel of <http://xkcd.com/207/>

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threepipeproblm
Who can name the least integer not nameable in fewer than nineteen syllables?

~~~
SanderMak
Did you also read Rucker's 'Infinity and the Mind'? Contains a fascinating
discussion of this problem.

~~~
threepipeproblm
I just remembered a philosophy prof. mentioning it. The phrase "the least
integer not nameable in fewer than nineteen syllables" is a direct quote from
Berry, I think.

------
SeanDav
I always thought that a googolplex was a googol to the power of a googol.

