Comments on: Graham's Number
http://www.metafilter.com/121668/Grahams-Number/
Comments on MetaFilter post Graham's NumberFri, 09 Nov 2012 03:58:38 -0800Fri, 09 Nov 2012 03:58:38 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Graham's Number
http://www.metafilter.com/121668/Grahams-Number
"If you actually tried to picture <a href="http://myscienceacademy.org/2012/09/30/grahams-number-numberphile/">Graham's number</a> in your head, then your head would collapse into a black hole." SLYT discussion of a number beyond "stupidly, stupidly big." <a href="http://www.metafilter.com/89927/Very-large-numbers">Previously</a>.post:www.metafilter.com,2012:site.121668Fri, 09 Nov 2012 03:02:54 -0800zannigraham'snumbergrahamsnumberbignumberblackholenumberphileBy: memebake
http://www.metafilter.com/121668/Grahams-Number#4674963
Graham's number + 1comment:www.metafilter.com,2012:site.121668-4674963Fri, 09 Nov 2012 03:58:38 -0800memebakeBy: edd
http://www.metafilter.com/121668/Grahams-Number#4674972
I'd think that's an understatement. You're talking black holes the size of a stupidly big number of universes, I'd have thought.comment:www.metafilter.com,2012:site.121668-4674972Fri, 09 Nov 2012 04:08:35 -0800eddBy: sammyo
http://www.metafilter.com/121668/Grahams-Number#4674977
I had the opportunity once to hear Ron Graham give a talk about stuff computers could never ever be big enough to handle. Great fun, I was clearly the dumbest guy in the room but it was cool to listen to the audience very quietly go "ooh, ahhh.... huh" over and over.comment:www.metafilter.com,2012:site.121668-4674977Fri, 09 Nov 2012 04:11:14 -0800sammyoBy: The Card Cheat
http://www.metafilter.com/121668/Grahams-Number#4674979
If I'd known "= stupid big" was a valid math answer I would have used it on tests.comment:www.metafilter.com,2012:site.121668-4674979Fri, 09 Nov 2012 04:14:49 -0800The Card CheatBy: InsertNiftyNameHere
http://www.metafilter.com/121668/Grahams-Number#4674988
OK, not to rain on these guys' parade, but I despise when maths folks replace a perfectly good notation (for us common folk) aka the exponent thing and then just plop an up arrow in it because the up arrow is clearly superior, but they never tell us plebs why they need to do that! Also, the fact that they needed parentheses to specify the order of operations tell me that your up arrow is lacking in information.
Also, screw Graham's number, I much prefer the eminently more useful Graham Cracker. That's just me, however.comment:www.metafilter.com,2012:site.121668-4674988Fri, 09 Nov 2012 04:32:07 -0800InsertNiftyNameHereBy: samworm
http://www.metafilter.com/121668/Grahams-Number#4674991
Several laugh out loud moments in this video.comment:www.metafilter.com,2012:site.121668-4674991Fri, 09 Nov 2012 04:33:45 -0800samwormBy: Eideteker
http://www.metafilter.com/121668/Grahams-Number#4674993
<em>"Graham's number + 1"</em>
Try a kiloGraham.comment:www.metafilter.com,2012:site.121668-4674993Fri, 09 Nov 2012 04:42:36 -0800EidetekerBy: salmacis
http://www.metafilter.com/121668/Grahams-Number#4674995
The point about replacing the exponent with the arrow is that the arrow notation can be expanded in a way that exponents cannot easily.
3^3
3^^3 = 3^(3^3)
3^^^3 = 3^^(3^^3) = 3^(3^(3^(3^(3^3))))
And you're already going beyond what you can easily write with exponents.comment:www.metafilter.com,2012:site.121668-4674995Fri, 09 Nov 2012 04:45:32 -0800salmacisBy: DU
http://www.metafilter.com/121668/Grahams-Number#4674998
<i>just plop an up arrow in it because the up arrow is clearly superior, but they never tell us plebs why they need to do that!</i>
<a href="http://en.wikipedia.org/wiki/Up_arrow_notation">Up Arrow Notation</a> (aka "Knuth Up Arrow Notation") is to exponentiation as exponentiation is to multiplication and as multiplication is to addition.
2 + 2 + 2 +...10 times => 2 x 10
2 x 2 x 2 x...10 times => 2<sup>10</sup>
2<sup>2<sup>2...10 times</sup></sup> => 2↑10
(what a terrible up arrow--what's the real unicode up arrow notation arrow?)comment:www.metafilter.com,2012:site.121668-4674998Fri, 09 Nov 2012 04:49:23 -0800DUBy: tommasz
http://www.metafilter.com/121668/Grahams-Number#4675009
I'm afraid to even think about thinking about it.comment:www.metafilter.com,2012:site.121668-4675009Fri, 09 Nov 2012 05:03:16 -0800tommaszBy: Reasonably Everything Happens
http://www.metafilter.com/121668/Grahams-Number#4675011
I tried to picture it.
I know what you're going to say..."watch out for the black holes!"
Well, I'm just a rebel like that. A big math number imagining rebel.comment:www.metafilter.com,2012:site.121668-4675011Fri, 09 Nov 2012 05:03:40 -0800Reasonably Everything HappensBy: edd
http://www.metafilter.com/121668/Grahams-Number#4675016
<i>Up Arrow Notation (aka "Knuth Up Arrow Notation") is to exponentiation as exponentiation is to multiplication and as multiplication is to addition. </i>
Not quite. The continuation of that chain from addition through multiplication to exponentiation is tetration. The UAN is more general, and starts earlier in the sequence than you suggest. The single arrow is exponentiation, the double arrow is tetration, and the triple arrow continues from there and so on.comment:www.metafilter.com,2012:site.121668-4675016Fri, 09 Nov 2012 05:09:09 -0800eddBy: Faint of Butt
http://www.metafilter.com/121668/Grahams-Number#4675022
If you possess a stupidly big quantity of cheese, you should serve it with Graham crackers.comment:www.metafilter.com,2012:site.121668-4675022Fri, 09 Nov 2012 05:15:02 -0800Faint of ButtBy: King Bee
http://www.metafilter.com/121668/Grahams-Number#4675030
<a href="http://www.metafilter.com/121668/Grahams-Number#4674988">></a> <i>I despise when maths folks replace a perfectly good notation (for us common folk) aka the exponent thing and then just plop an up arrow in it because the up arrow is clearly superior, but they never tell us plebs why they need to do that!</i>
Isn't it almost immediately obvious? I mean, sure, nobody is going to <i>really</i> use 3↑3 when they can just write 3<sup>3</sup>. But 3↑↑↑3? That's a tower of 3s which is 3<sup>3<sup>3</sup></sup> high. You need 7.6+ trillion symbols for that, whereas up arrow notation gets it done in five.comment:www.metafilter.com,2012:site.121668-4675030Fri, 09 Nov 2012 05:28:22 -0800King BeeBy: GenjiandProust
http://www.metafilter.com/121668/Grahams-Number#4675132
<em>Also, screw Graham's number, I much prefer the eminently more useful Graham Cracker.</em>
Graham's number is useful! It keeps you regular <em>and</em> deters masturbation, much like the cracker, but it has a more amusing notation system.comment:www.metafilter.com,2012:site.121668-4675132Fri, 09 Nov 2012 06:55:54 -0800GenjiandProustBy: erniepan
http://www.metafilter.com/121668/Grahams-Number#4675187
<a href="http://en.wikipedia.org/wiki/Kruskal's_tree_theorem#Friedman.27s_finite_form">TREE</a>(Graham)comment:www.metafilter.com,2012:site.121668-4675187Fri, 09 Nov 2012 07:21:26 -0800erniepanBy: Egg Shen
http://www.metafilter.com/121668/Grahams-Number#4675189
3^^^3 is an exponent tower of 3s that is 7,625,597,484,987 levels high. If it took you a second to calculate each new level of the stack, it would take more than 240,000 years to finish. Call that number X. This is already face-meltingly big.
3^^^4 is an exponent tower of 3s that is X levels high. 3^^^5 is an exponent tower of 3s that is 3^^^4 levels high. When you get to 3^^^X, that is 3^^^^3. Even God couldn't visualize this number and it took only 4 arrows to get there.
But 3^^^^3 is only G1. G2 is 3^^...^^3 with G1 arrows. G3 is 3^^...^^ with G2 arrows. <em>G64</em> is Graham's number.
And as a final kick in the teeth, that number is (an outer limit for) the <em>dimensions of a hypercube</em> whose vertices caught your interest.
Mathematicians frighten me.comment:www.metafilter.com,2012:site.121668-4675189Fri, 09 Nov 2012 07:22:03 -0800Egg ShenBy: FirstMateKate
http://www.metafilter.com/121668/Grahams-Number#4675241
Question they never answer in the video :
Graham's number is supposed to be the largest number that can be used to satisfy this problem, yes? Why? I don't fully understand what the number is supposed to be doing [and venturing to wikipedia just makes me feel dumber].comment:www.metafilter.com,2012:site.121668-4675241Fri, 09 Nov 2012 07:45:25 -0800FirstMateKateBy: symbioid
http://www.metafilter.com/121668/Grahams-Number#4675274
Based upon a probably badly informed reading - they're dealing with graph theory of some sort. How connections are formed. I'm not sure exactly how this relates to hypercubes, exactly, but I'd imagine it's something to do with the vertices of a hypercube. On preview, as Egg Shen said what they're trying to do is find a way to determine the number of dimensions of a hypercube given the number of vertices. At least, that's how I'm reading Egg...
What I'm curious is why 3, exactly? What happens if you said n=x, let n b the number of arrows. so 2^^2, 3^^^3, 4^^^^4, 5^^^^^5, etc... what's the pattern there.
That said, I love the black hole thing because the concept of entropy and blackholes and the holographic principle is fucking fascinating and it's crazy to see a number that exceeds the entropy of the brain.
They say it's impossible to know what it is. How long would it take to calculate it? I imagine greater than the age of the universe, but still...comment:www.metafilter.com,2012:site.121668-4675274Fri, 09 Nov 2012 08:00:37 -0800symbioidBy: charlie don't surf
http://www.metafilter.com/121668/Grahams-Number#4675317
<em>Mathematicians frighten me.</em>
Mathematicians giving patronizing explanations of abstract concepts to non-mathematicians frightens me. There is no accepted mathematical notation of "stupidly big."comment:www.metafilter.com,2012:site.121668-4675317Fri, 09 Nov 2012 08:19:10 -0800charlie don't surfBy: symbioid
http://www.metafilter.com/121668/Grahams-Number#4675321
"Space," it says, "is big. Really big. You just won't believe how vastly hugely mindboggingly big it is. I mean you may think it's a long way down the road to the chemist, but that's just peanuts to space. Listen ..."comment:www.metafilter.com,2012:site.121668-4675321Fri, 09 Nov 2012 08:20:26 -0800symbioidBy: Space Kitty
http://www.metafilter.com/121668/Grahams-Number#4675346
I learned from QI that 7 is the final digit in Graham's number, but I'm entirely too innumerate to know if they were joking.comment:www.metafilter.com,2012:site.121668-4675346Fri, 09 Nov 2012 08:34:33 -0800Space KittyBy: ostranenie
http://www.metafilter.com/121668/Grahams-Number#4675371
^_^comment:www.metafilter.com,2012:site.121668-4675371Fri, 09 Nov 2012 08:41:43 -0800ostranenieBy: phl
http://www.metafilter.com/121668/Grahams-Number#4675543
<em> I'm entirely too innumerate to know if they were joking.</em>
That apparently <a href="http://en.wikipedia.org/wiki/Graham%27s_number#Rightmost_decimal_digits">is actually a thing</a>, Space Kitty.comment:www.metafilter.com,2012:site.121668-4675543Fri, 09 Nov 2012 09:44:41 -0800phlBy: albrecht
http://www.metafilter.com/121668/Grahams-Number#4675575
<em>I learned from QI that 7 is the final digit in Graham's number, but I'm entirely too innumerate to know if they were joking.</em>
Not only that, but if you turn it upside down, the last seven digits spell "BOOBIES". Try it!comment:www.metafilter.com,2012:site.121668-4675575Fri, 09 Nov 2012 09:53:46 -0800albrechtBy: RobotVoodooPower
http://www.metafilter.com/121668/Grahams-Number#4675675
<a href="http://en.wikipedia.org/wiki/User:Sligocki/Beating_Graham's_number">Busy Beaver vs. Graham's Number</a>
I don't know enough to tell if this is a valid proof, but assuming the Busy Beaver is an uncomputable Honey Badger then I think it has a fair chance.comment:www.metafilter.com,2012:site.121668-4675675Fri, 09 Nov 2012 10:33:25 -0800RobotVoodooPowerBy: straight
http://www.metafilter.com/121668/Grahams-Number#4675733
There's no 'L' in 'BOOBIES,' albrecht.comment:www.metafilter.com,2012:site.121668-4675733Fri, 09 Nov 2012 10:49:01 -0800straightBy: evilangela
http://www.metafilter.com/121668/Grahams-Number#4675739
One of the things I like about Graham's Number is that it shows you just how far you are from being able to comprehend "infinity".
Unlike something like a Busy Beaver number, Graham's Number is very straightfoward in how it's calculated. You can comprehend the process. Yes, it doesn't take long to realize the that the numbers are ungodly big. But you can get a feeling for the magnitude of the number at first - and realizing that there aren't enough atoms in the universe to even represent the number of DIGITS in the number informs you of that scale.
Just that little bit of comprehension you get from trying to do this gives you the feel of a number that is likely far, far bigger than you ever thought of when you hear "infinity". And then you realize just how little you can understand that concept.comment:www.metafilter.com,2012:site.121668-4675739Fri, 09 Nov 2012 10:50:48 -0800evilangelaBy: Halloween Jack
http://www.metafilter.com/121668/Grahams-Number#4675975
I sort of lost track of the explanation around the time they started talking about committees, due to the progressively-irritating cutting between the two of them. However, I remain confident in my comprehension of <a href="http://www.youtube.com/watch?v=LWN-glGMDkc&feature=fvwrel">one being the loneliest number that you'll ever do.</a>comment:www.metafilter.com,2012:site.121668-4675975Fri, 09 Nov 2012 12:00:53 -0800Halloween JackBy: albrecht
http://www.metafilter.com/121668/Grahams-Number#4676070
<em>There's no 'L' in 'BOOBIES,' albrecht.</em>
<small>Yeah, I meant the last seven digits after you flipped it over... </small>
Let me try to make amends for my lulzy comment by saying something that I think is seriously interesting about Graham's Number, which is that I think it shows the difference in the ways that we think we "understand" numbers. That is, we have different regimes for conceptualizing numbers based on their sizes: small numbers (on the order of 10) we can think of in terms of picturing groups of objects; slightly larger numbers (maybe 100-1000) we might think of in terms of their arithmetic properties, whether they're primes or powers or whatever; larger numbers still (the thousands up into the trillions and quadrillions) we might have descriptive names for; and at some point we are only accustomed to really thinking about numbers in their decimal expansions. What Graham's Number shows is that even that fails us at some point when we venture into the realms of numbers with more <em>digits</em> than we can even comprehend, which is rightly to be expected when we start compounding exponential operators together (the number of digits being basically the log of a number base 10). You don't have to go out that far to get numbers with more digits than you can imagine.
So the question is: what does it really mean to know a number that big? The video makes the point that the entropy of Graham's Number is greater than the entropy of a black hole, but that's a really flawed way of thinking about entropy, treating the number as a random sequence of digits (or, more accurately, bits). What's probably more appropriate is to consider its <a href="http://en.wikipedia.org/wiki/Kolmogorov_complexity">Kolmogorov Complexity</a>, which is something like the length of the shortest computer program that could generate the number. We know that such an algorithm exists that's fairly short, because we can describe the number in terms of constructing it, defining the up-arrow operators and iterating some number of times, etc. So in a sense, we know the number would fit in our heads because it's already there. What's really amazing about it is that it comes about organically as a solution to a fairly simple problem.comment:www.metafilter.com,2012:site.121668-4676070Fri, 09 Nov 2012 12:30:57 -0800albrechtBy: fantabulous timewaster
http://www.metafilter.com/121668/Grahams-Number#4676370
<blockquote>"If you actually tried to picture Graham's number in your head, then your head would collapse into a black hole."</blockquote>
Well, I tried, but I couldn't picture the number, and I haven't collapsed into a black hole (at least as far as I can tell). Does that mean that I actually failed to try?
I think that being able to increase the mass-energy density of my brain by thinking very hard would be a great superpower.comment:www.metafilter.com,2012:site.121668-4676370Fri, 09 Nov 2012 14:50:35 -0800fantabulous timewasterBy: LogicalDash
http://www.metafilter.com/121668/Grahams-Number#4676997
Infinity is defined to be greater than all finite numbers. Brains, like computers, can only perform arithmetic on finite numbers, and arithmetic here includes what you do to get an estimation of the number of digits for instance. Right out the gate, infinity is defined in such a way that trying to get an idea of how big it is would be totally pointless.
So I don't quite get the cult about it. Is it just a bunch of numerologists saying "my number is bigger than yours"?comment:www.metafilter.com,2012:site.121668-4676997Fri, 09 Nov 2012 23:59:10 -0800LogicalDashBy: klausman
http://www.metafilter.com/121668/Grahams-Number#4676999
I'll always remember when Ron Graham came to lecture at my university when I was in grad school. We had, of course, read about his number and the whole up-arrow notation. But when he came to spend some "informal time" with faculty and grad students, it erupted into an hour-or-so long juggling fest (he a former president of the <a href="http://en.wikipedia.org/wiki/International_Jugglers%27_Association">International Jugglers' Association</a>). He taught us about <a href="http://en.wikipedia.org/wiki/Siteswap">site swapping</a>, introduced us to <a href="http://www.amazon.com/exec/obidos/ASIN/0387955135/metafilter-20/ref=nosim/">Polster's book</a>, and taught my friend Ander to juggle. That was a really great afternoon, and not what I expected.
As for Graham's number, I agree that it's stupidly big. I was trying to get a feel for how big it is, and I discovered the <a href="http://en.wikipedia.org/wiki/Ackermann_function">Ackermann function</a> which is related in that it creates very large numbers by recursion. It's actually a cool exercise (for folks new to recursive functions) to try to follow that definition and see how the function grows. If you get stuck, Wolfram|Alpha will evaluate the function by typing Ackermann(a, b) where a and b are according to the definition in the article. It's a short process, though, as Ackermann(3, 2) is 29, but Ackermann(4, 2) has over 19,000 digits.comment:www.metafilter.com,2012:site.121668-4676999Sat, 10 Nov 2012 00:02:40 -0800klausmanBy: King Bee
http://www.metafilter.com/121668/Grahams-Number#4677181
<i>Infinity is defined to be greater than all finite numbers. Brains, like computers, can only perform arithmetic on finite numbers, and arithmetic here includes what you do to get an estimation of the number of digits for instance.</i>
OK...
But <a href="http://en.wikipedia.org/wiki/Cardinal_number">how many</a> integers are there? Infinity? How many even integers are there? Half that, or the same amount? How many real numbers are there between 0 and 1? More or fewer than the number of integers, or is it all the same? I think you're playing a little free and loose with the word "infinity."comment:www.metafilter.com,2012:site.121668-4677181Sat, 10 Nov 2012 07:20:10 -0800King BeeBy: LogicalDash
http://www.metafilter.com/121668/Grahams-Number#4678549
Yes, I'm aware of the different cardinalities of infinity. I even think it's cool to know that you can't construct a well-ordering of the real numbers. But it seems like the expected response on mentioning aleph-one is "WHOA, bigger than infinity??" That, I don't get.comment:www.metafilter.com,2012:site.121668-4678549Sun, 11 Nov 2012 12:47:21 -0800LogicalDash