Large Numbers and Humanity
This eight-year-old article from Scott Aaronson is thoughtful on many levels. Aaronson starts off innocently enough, with a "game" in which contestants must name the largest number they can in 15 seconds.
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
This exercise leads to a journey through human thought with unfortunate ramifications for our future.
One answer is to write down as many 9s as you can on the notecard. Of course, this answer would be shattered by anyone who wrote down a bunch of 9s raised to the power of a bunch more 9s. This is, in turn, blasted by anyone who writes 9^9^9^9^9^9, and so on. In fact, each of these improvements so far defeats the previous, that you could give your competitor some extra time, and still win. Instead of 15 seconds, you could give your opponent as long as the universe has so far existed, and you'd be plenty safe. Even 9^9^9 is vastly more than the number of particles in the known universe.
Aaronson goes on to show us that even these exponentials are child's play compared to strange entities such as the Ackermann series and, still crazier, something called "Busy Beaver" numbers that have to do the longest timea theoretical computer could work on a problem without working infinitely long.
All of this really gives perspective that the winner of this contest is the one who has the highest order paradigm. After you read about Busy Beaver numbers, you almost feel sorry for the poor saps who are still doing 9^9^9.
Aaronson also looks back through history, showing how far we've come. The bible said that pi = 3, and made numerous references to the number of stars and number of grains of sand on a beach being "infinite." That kind of thinking is what you'd expect from someone who just wrote three or four 9s on the notecard, lol. Archimedes came along and said the number of grains of sand must certainly be less than 10^63, therefore it is finite. Shocking stuff for his time. Incidentally, the church threatened death to Pythagoras of Samos for saying that the square root of 2 is an irrational number (its decimal form has infinitely many, non-repeating digits) and Galileo died for his scientific beliefs. Only in pockets of the world where reason had a chance to flourish did humans compute more and more digits of pi, and come up with higher-order paradigms of large numbers, and so on.
Why are people so afraid of large numbers? Why should anyone care in the first place?
Physics professor Albert Bartlett said, "The greatest shortcoming of the human race is our inability to understand the exponential function."
He's talking about the idea that if the human population continues to double every 40 years as it has so far, that the mass of all humans in the year 3750 would equal the mass of the Earth. Clearly this will be stopped short by famine, disease, nuclear war, or something. But the notion of "sustainable growth" of only a few percentage points per year is the crazy-talk of someone who doesn't realize the math. Radiation levels, population levels, CO2 omissions and many other things operate at large scales and are very real problems. They can't be addressed unless they can be understood in the first place, and that means it's important for the average person to have some grasp of exponential numbers.
Aaronsons's article also has a very fascinating section about WHY people can't grasp large numbers. Apparently, when you are asked to just guess "which of these two answers is probably closer" in a simple math problem, this uses a different part of the brain than when you have to compute something exactly. The estimation part of the brain is great for telling if there are about 3 or about 6 enemy tribespeople attacking you. But it's useless at the scale of exponential numbers. Humans use the language center of the brain to exactly compute these types of numbers, because no one has the intuition to deal with them.
I'll close with this quote from Aaronson:
"If people fear big numbers, is it any wonder that they fear science as well and turn for solace to the comforting smallness of mysticism?"
--Sirlin
March 28th, 2007 at 1:44 pm
lol, is dis sum thinly veiled attack on religion?
March 28th, 2007 at 2:28 pm
That wasn’t my main point. Is your comment a thinly veiled support of supernatural beliefs, rather than reason?
http://richarddawkins.net/foundation
–Sirlin
March 28th, 2007 at 2:39 pm
This is akin to the mental exercises I drop on my friends and family from time to time.
Our primitive brains are too narrow to accept and even imagine or comprehend something as infinitely vast as the universe or very large numbers (which are used to quantify the universe). Most people haven’t even the faintest grasp of a light-year or what it really signifies.
Imagine if you will walking from your computer to your front door. No problem right? You can even mentally zoom out and get a pretty detailed overview of the entire path. You can remember nearly every major object and direction change you will encounter.
Now, go bigger, imagine a quick jog down to the nearest store. Undoubtedly you can still imagine the path in it’s entirety, you may even be sharp enough to recall many of the houses along the way, but undoubtedly, some details will be missed in your mental picture.
Now, a short drive to the next town over. Suddenly, imagining the entire route becomes a step-by step process, you must make associative connections. Remembering all objects and direction changes on the route becomes very difficult.
Now imagine going to see relatives three states over. Try to pinpoint as much detail about the entire trip as you can. More than likely, only a few landmarks and memorable expanses will immediately occur to you. The distance covered is simply too large to imagine without the help of a map.
Getting back to space, the light year, and mind bogglingly large numbers, so large as to be completely ignored by the logic centers in your brain, the speed of light is approximately 186,282 miles per second. Before I continue any further, I just want you to try to contemplate that. Just for reference, the circumference of the Earth is about 24,900 miles around. In other words, if you could travel at the speed of light, you could travel around the globe almost seven-and-a-half times in one second!
Hopefully, by this time you have gotten a grasp of the relative speed of light and how signifigant it is. Now onto a light year.
A light year is the distance light can travel in one year. Remember that light can encircle the globe seven times in one second. In numerical terms of distance, a light year is approximately 5,865,696,000,000 miles. Almost 6 trillion miles. That is a single light year. As you can clearly see, even the shortest distances in space are near impossible to comprehend. Numbers in space spiral quickly out of reason.
Scientific and mathematical numbers are equally adept at quickly growing far too large for the human brain to fully understand, the aforementioned grains of sand or exponential numbers are superb examples. On a fundamental level in fact, human minds are somewhat hard pressed to accurately deal with numbers higher than 100. Think about it, try to visualize one apple, now visualize five apples, each one separate and in it’s entirety. Now ten apples, can you still see ten separate apples in your mind’s eye? you might think of them in groups of five, now go for twenty-five apples all at once, all together. How many apples can you envision before they become a jumbled mess or you have to break them down into successive groups? Without breaking it down into feet, can you estimate how long 100 inches is?
The interesting distinction you make in your article Sirlin, is the one of hard calculation versus estimation. To calculate a number is relatively easy, provided you know the mathematical rules. Estimation however, comprehending that value and being able to quantify or compare it requires a degree of creative thinking that is very close to the degree of concentration it requires to truly understand Quantam Physics and wormholes in a technical sense.
March 28th, 2007 at 2:48 pm
There are a few statements in your post that are inaccurate or misleading, so I’ll quickly discuss them.
1) Pythagoras of Samos lived in the 6th Century B.C. I’m not sure which church you’re saying threatened him with death, given that there weren’t any around then. Perhaps you mean to refer to the legend where Pythagoras (or at least the Pythagoreans) killed or exiled Hippasus, one of his own followers, for discovering irrational numbers.
2) It’s not true to say that “Galileo died for his scientific beliefs.” He was indeed under house arrest at the time of his death, but he died of natural causes.
3) The statement that the Bible “said that pi = 3″ is somewhat misleading. Nowhere in the Bible is a claim like “the ratio of the circumference of a circle to its diameter is precisely three to one,” so it’s not as if it makes any specific argument that pi = 3. This attribution comes as a result of looking at the passage I Kings 7:23 (also II Chronicles 4:2) which mentions a round vessel which has a 10-cubit diameter and a 30-cubit circumference. First, consider that the numbers given might be approximations, especially given the fact that they both have just one significant digit, so it is inappropriate to assert that the passage is giving a standard for pi approximation. Secondly, there is also the possibility that the 10 cubits refers to the outer diameter and the the 30 cubits refers to the inner circumference, which would give the vessel a thickness of about 4 inches, which is reasonable given its total size (about 15 feet across). If this is the case, then the approximation of pi that one might then take from the passage is much more accurate.
4) The Bible doesn’t say that the number of grains of sand is infinite or that it is not possible to name a number with a greater magnitude, but rather that it is sufficiently large as not to be physically countable. This is a much different claim, I think, and a quite reasonable one at that (if anyone doubts this, try counting the number of grains of sand in just one sandbox, then think about this claim again). This doesn’t seem to be a case of “ignorance and irrationality lead[ing] to fatalism concerning big numbers,” to use a phrase of Aaronson’s.
Also, it’s worth noting that members of mystic groups often have extensive mathematical training and make great mathematical discoveries. For example, the Pythagoreans had some rather advanced mathematical concepts (such as the idea of infinity) and highly valued the role of proofs in furthering their mathematical knowledge. Perhaps it might not be such a good idea after all to try and force a link between mysticism and the “fear [of] big numbers.”
March 28th, 2007 at 3:12 pm
I just had to say that the above poster is really researched in this topic. Very well-put-together argument. That is all…
March 28th, 2007 at 3:20 pm
H.D.:
1) The persecution of Pythagoras was an anecdote from one of my high school math teachers. Perhaps it’s incorrect.
2) Ok, good catch and quite a typo on my part. I meant *Socrates* was sentenced to drink poison hemlock because he “failed to recognize the gods recognized by the state.” As you rightly point out, Galileo, one of the greatest scientists of all time, was merely put under house arrest for the last 8 years of his life because he said the Earth goes around the Sun.
3) and 4) Well, I could just as well argue that the bible meant what it said on both of those cases, but there’s not much point in dwelling on what a book that old meant to say. We have a much better view of the world now, in any case.
Ess2s2: Your light-years are bending my mind. I heard my father used to walk 5,865,696,000,000 miles to school, in the snow.
–Sirlin
March 28th, 2007 at 5:11 pm
I don’t think the article got to as big of numbers as it could have. Actually, I’m entirely sure that they stopped short of the ideal. First of all, I concluded quickly that exponentials are inferior to factorals as 9!! is a whole lot bigger than 9^9 so at the very least we can go with that. You can do two factorals in the number of characters required to do one exponentiation after all, if we discount using superscripts for exponentiation as that is not really feasable after about two iterations(9!! is still bigger than (9^9^9 which is pretty much the last iteration of exponentiation that you could perhaps do with superscripts). The thing is, assuming you defined the Busy Beaver sequence:
BB(1111) would be a much worse thing to write than BB(9!!!) since 9 factoral factoral factoral is insanely bigger than 1111. If you’re worried that 9 takes too long to write, you could just do 7!!! as 7 takes no longer to write than 1, and if it let you put in an extra factorial, it would be worth it. For that matter, there’s the question of just how many characters you can fit inside the parenthesis after factoring in all the rest of the time it define the Busy Beaver sequence. Clearly the first question is a question of how many characters at minimum it takes to define the Busy Beaver characters and one of how many characters you can write per second(higher order Busy Beaver sequences don’t have enough external definition to be able to define in 15 seconds of writing, assuming you are not allowed external preparation such as publishing a paper).
BB(BB(9!!!)) if you have exactly enough time to write 12 characters after defining the BB sequence would surely surpass BB(9!!!!!!!). But how about BB(BB(9!!))! ? Just with twelve characters and the Busy Beaver sequence already defined, what is the biggest possibility? If you can write more characters, what is the theoretical ideal balance between BB(), the four characters needed to define a new imbedding of the BB sequence and !, the factorals you would tack on the 9 that is always going to be in the center of the parenthesis, and for that matter where in the maze of parenthesis is the ideal place to execute the factorals? The Busy Beaver numbers are too big to actually calculate so how would you declare a winner in the event of two different attempts to win through these methods? Of course, a smart person would define the BB sequence in such a way so that he simply must write B instead of BB so with 12 characters he would write B(B(9!!!))!! if he needed an easy way to trump the sucker who wrote BB(BB(9!!)). That begs the question of how you were defining Busy Beaver sequences as quickly as possible of course and of what the minimum number of characters needed to define the BB sequence is.
Of course, that also brings up the question of language. A clever person who was multilingual could write two Chinese symbols for the words “Busy Beaver” in lieu of the 10 English characters needed. Even if the Chinese symbols take twice as long to write as an English character, that still gives him 6 more characters for use in the actual sequence which would make winning easy. There’s also the question of which types of symbols are the fastest to write and of how legibile the handwriting needs to be(do partially completed symbols which are still readable count?). Ultimately we run into a point of subjectiveness. We’re requiring our “judge” mathematician to be able to read at least one foreign language, and perhaps we have to ask if he’s able to interpret a mix of multiple languages as we use different languages to express different parts of the Busy Beaver definition as quickly as possible. Of course, we are making every single characters as quickly as possible which means that if we can not make a part of the mark and it would remain legible that we would do so. Therefore, the theoretical ideal is one of making your writing as illegible as possible without ever crossing the magically undefined line into “unreadable”. Manual dexterity is ultimately a big question too as someone who can inherently write a small amount faster might be able to get one more character in, and simply tacking on another factoral to the end of the same number your opponent wrote is a surefire win. I think this demonstrates that this contest falls apart at higher levels, and a different ruleset is ultimately needed in order for the contest to avoid becomming degenerate. Excluding the need to define Busy Beaver sequences and instead giving them standard notation which every contestant must use for them to be recognized and then establishing character limits instead of time limits would seem to fix these problems.
I don’t understand the math well enough to be able to figure out the theoretical ideal rules by which someone should play(I am just barely grasping what a Busy Beaver sequence is really), but it would be interesting to see what the absolute max on this is.
March 28th, 2007 at 8:42 pm
Because of the conditions given: “Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.” it’s unclear that, for example, BB(11111) would qualify.
However there’s another approach to this problem, which is to use an unconventional ordering:
I’ll define x{y to be true if y is equal to 0, false if x is equal to zero, and otherwise, true if x
March 28th, 2007 at 11:16 pm
Don’t forget changing the base. 100 in base 9!!! is pretty big.
March 29th, 2007 at 2:35 am
No offense to you Sirlin, I love your articles, but hopefully most people who are interested in this topic will go ahead and read the original article.
March 29th, 2007 at 3:14 am
Ess2s2: Indeed, one light year is a whole bunch of miles. So many miles in fact it becomes impossible to comprehend the number. Which is why we convert all those miles to one (1) light year. And it suddenly becomes a lot more comprehensible.
Same reason we convert 100 000 millimeters to 100 meters.
The problem here is not a problem of numbers but rather one of reference. We have nothing around us to match the distance of a light year to. We do not travel light years away to go to work every day. That’s what makes it incomprehensible.
March 29th, 2007 at 3:31 am
Exponential numbers do not grow as fast as factorials. I’d be writing (9)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.
March 29th, 2007 at 4:57 am
Vontre, no offense to you either, but the entire point of me posting a link to the original article is so people will read it. Not sure how I would have been offended if people did read it.
Rufus: You had me, then you lost me. I’m guessing your post was accidentally cut short.
–Sirlin
March 29th, 2007 at 6:39 am
“Although computers will probably never solve NP-complete problems efficiently, there’s more hope for another grail of computer science: replicating human intelligence”
From the link.
“Probably”? LOL. That’s the whole point of NP complete problems. That statement is true by definition.
March 29th, 2007 at 6:56 am
>I don’t understand the math well enough to be able to figure out the theoretical ideal
>rules by which someone should play(I am just barely grasping what a Busy Beaver
>sequence is really), but it would be interesting to see what the absolute max on this is.
The ideal game would be something like this: Write a computer program in n bits or less to write the largest finite number possible. So you might have something like this, which prints 999
for (1..3){
print 9
}
but this next program would be illegal, because it never finishes (it just prints 11111… forever)
while(1){
print 1
}
So that’s the ideal game. And guess what? The busy beaver numbers are the answer to this very question! So as far as I can tell, the largest number you can pretty much write is the busy beaver number itself.
So if this game ever came up in real life, I would disallow the use of something like BB(10 billion). Why? Because BB isn’t computable! You can use Ackermann(10 billion) all you want — that’s computable. But unless you can give me a systematic way to calculate BB(10 billion), that’s not a legitimate entry. And if you can, you’ve proved most unsolved problems in mathematics :)
Julien
March 29th, 2007 at 9:49 am
David, you have to stop using “lol” as punctuation. Seriously, sort it out.
Otherwise, good article. I’m drunk so sorry for the crassness of this post.
March 29th, 2007 at 9:55 am
Responding to some earlier post, the difference between BB(100) and BB(10^10^10) is not too significant, because they both are so very much smaller than the analogous numbers for the imaginary stronger computers described.
I’d actually played the game before seeing the article, with a longer time limit; it is fairly amusing, although I suspect that referencing the literature might make it a little less so. The techniques for generating big numbers end up being fairly similar to the techniques used to generate comparatively large ordinals in most flavors of set theory, and I have always found this interesting to no end. If you are using a reasonably powerful axiom system (ie ZFC) you don’t have to get busy beaver numbers involved if they feel too undefined, although you obviously have to use numbers that no algorithm on a computer in the modern sense could ever write.
>> “Probably”? LOL. That’s the whole point of NP complete problems. That statement is >> true by definition.
I am not sure what you mean by the “whole point,” but “probably” seems like the appropriate word. The definition certainly does not entail intractibility, unless you understand the term “computer” in some sense other than that of a machine which computes.
March 29th, 2007 at 2:35 pm
NP complete problems cannot be solved in polynomial time. So if your idea of solving them efficiently is in solving them in polynomial time you are out of luck. I suppose it depends on what you consider efficient.
March 29th, 2007 at 3:56 pm
This eight-year-old article from Scott Aaronson is thoughtful on many levels. Aaronson starts off innocently enough, with a “game” in which contestants must name the largest number they can in 15 seconds.The above article is very interesting,if you want any further information click on the link:TonyRobbins/Motivation
March 29th, 2007 at 9:50 pm
James M:
It hasn’t been proven to be true that NP-complete problems (or other NP problems, for that matter) can’t be solved in polynomial time, and I think that this is the point that Paul was making. The P vs. NP problem is one of the Clay Mathematics Institute Millennium Prize Problems, and if you want more information about it, check out this Wikipedia article:
http://en.wikipedia.org/wiki/Complexity_classes_P_and_NP
March 30th, 2007 at 5:44 am
Yes that is certainly true. I suppose the “probably” is warranted.
March 31st, 2007 at 5:20 am
David, there is a “playing to win” alert at the Desktop Tower Defense forums.
http://handdrawngames.sencerd.co.uk/index.php?topic=199.0.
April 1st, 2007 at 2:33 am
1/0 = gg
April 1st, 2007 at 11:38 am
Lasker, this is totally off-topic, but that Desktop Tower Defense game is great.
April 2nd, 2007 at 11:00 am
……
>.>
April 2nd, 2007 at 11:07 am
Uhh…stick to games.
April 3rd, 2007 at 1:07 am
Perhaps you don’t understand the point of a blog?
April 3rd, 2007 at 11:55 am
Sirloin; Great article, I like your wit. I do believe in the ’supernatural’, but well within reason. Your remarks that ‘could’ be taken the wrong way are well placed for added humor, and thus applaudable. Hearing about the game makes me want to take a crack at it.
Anyway, nice article. Just wanted to pay kudos, and let it be known that the lighthearted tone of the religious references can be seen.
Cheers.
Buton.
April 4th, 2007 at 12:08 am
I wasted my entire night after getting off work yesterday playing Desktop Tower Defense.
April 4th, 2007 at 8:22 am
Why not just write “The number that is 1 greater than the number on my opponent’s card”? :P
April 4th, 2007 at 8:29 am
Wouldn’t 9^9 not even be a number? It EQUALS a number. But that’d be saying that 9×9x9×9x9×9x9×9x9 is a number too.
April 4th, 2007 at 10:28 pm
RPKay: “Why not just write ‘The number that is 1 greater than the number on my opponent’s card’?”
That’s covered by the rather precise definition of the rules of the game: “… by consulting only your card …”
April 8th, 2007 at 7:02 am
If you can bring up the BB sequence, Ackerman sequence, or whatnot, can I just invent a function called f of x, where x is an index card, and f is the largest number theoretically representable by that index card?
April 8th, 2007 at 11:07 am
“If you can bring up the BB sequence, Ackerman sequence, or whatnot, can I just invent a function called f of x, where x is an index card, and f is the largest number theoretically representable by that index card?”
No - that’s not well-defined because I could write down ‘The largest finite number that can be described on an index card like this one plus one.’ on a reasonably sized index card. This (clearly) leads to a completeness theorem-like problem.
April 12th, 2007 at 2:48 pm
Uhh…stick to games. wow gold
April 13th, 2007 at 3:07 pm
DO NOT USE MY COMMENTS TO PEDDLE YOUR CRAP.
April 13th, 2007 at 11:43 pm
“Aaronson also looks back through history, showing how far we’ve come. The bible said that pi = 3, and made numerous references to the number of stars and number of grains of sand on a beach being “infinite.” That kind of thinking is what you’d expect from someone who just wrote three or four 9s on the notecard, lol. Archimedes came along and said the number of grains of sand must certainly be less than 10^63, therefore it is finite. Shocking stuff for his time. Incidentally, the church threatened death to Pythagoras of Samos for saying that the square root of 2 is an irrational number (its decimal form has infinitely many, non-repeating digits) and Galileo died for his scientific beliefs. Only in pockets of the world where reason had a chance to flourish did humans compute more and more digits of pi, and come up with higher-order paradigms of large numbers, and so on.”
Um, wow Sirlin…I thought you were smarter than this.
April 23rd, 2007 at 8:24 am
Give Sirlin a break. If you guys are so eager to defend the Bible, go get busy at evilbible.com or whatever… Sticking to math here seems to be more on topic and, IMO, more interesting
April 24th, 2007 at 3:31 pm
Very interesting. Is it true about the Pi=3 thing? Did they really do that, or is it just a joke, cause I find it hard to believe. Is it truly sad that bad people in power have forced the hiding of truths just to “win” and stay at the top of the ladder. Don’t you agree Sirlin? If only people followed both the path of reason and the path of love(caring about each other as human beings) perhaps the world would be in a better state. It saddens me that the same place where “reason had a chance to flourish” at the time of the dark age, is now a place of constant bloodshed. Always remember that cold logic is not enough, never suppress your concern for the well being of your fellow man and woman.
Off topic:
Btw, as I have studied the numerous science as you have (math, physics, chemistry, biology and computer science) I have am convinced that the world was created not randomly, but rather by a very thought out design. I could write about this here, but it’s rather off topic. If you want more info you may look into:
“GOD, the evidence, the reconciliation of faith and reason in a postsecular world” by Patrick Glynn
“Symbiotic universe” by George Greenstein
“A brief history of time” by Stephen Hawking
April 26th, 2007 at 10:29 pm
As I recall, God commands someone to create a circle of radius 10 and circumference 30. That isn’t the same as saying pi is 3.0, but it is fairy suspect.
July 31st, 2007 at 12:53 pm
As I recall, the solution is that the pot wasn’t necessarily a perfect circle; it’s hard to make a perfect circle out of clay using only your bare hands.
January 12th, 2008 at 10:46 pm
This whole entry lacks something !
and that is: a call to Ackerman’s function using Graham’s number as both arguments !!!
4 gr3t justice !
ps: if you don’t know what I’m talking about, looking around the internet may lead you to the source of this quote (which is awesome ^^)