Infinity, as you may remember, is a mathematician’s fancy way of saying “I don’t know.” Unfortunately, it’s so fancy a word that some folks start to think it means something more.
Take for instance the “Infinite Monkey Theorem.” There are several variants to this, but at its simplest it states that a monkey at a typewriter given enough time will produce any sequence of characters, for instance the works of Shakespeare. While I have always considered this a rather brutal attack on ol’ Willie, the meaning is clear: a “random” stream of characters, “infinitely” long will contain within it every finite sequence.
It turns out that we have at our disposal a pretty good “infinitely” long “random” stream. The digits of Pi are, as far as we can tell, uniformly distributed, and if you buy the myth of physical infinity, then you have all the digits you could ever want.
So this means that encoded within the digits of Pi are the complete works of William Shakespeare. Now, the astute reader may think to herself (since all astute readers are female) “But the digits of Pi are numbers and Shakespeare used a very old, but still relatively similar to our own, alphabet.” Very astute, dear reader, read on.
Think back to middle school when secrets were fun and passing notes was fun and both together were exciting. You may have been tempted, in an effort to keep your teacher from easily reading your secret notes to the rest of the class, to replace letters with numbers, for instance ‘A’ would be written as ‘1’ and ‘Z’ would be ‘26’. This is, under several different meanings, a “code” and a message written with such a mapping would be “encoded.” So it stands to reason that you could, with a little effort and a little time (compared of course to physical infinity) encode the entire works of Shakespeare.
This encoded sequence of numbers, according to the Monkey Theorists, exists within the digits of Pi. And not just that encoded version of Shakespeare, but any encoded version of Shakespeare. Pick any number for any letter, or pick any number for any sequence of letters, or pick any sequence of numbers for any sequence of letters, or any other encoding you can come up with, and that encoded version of Shakespeare is captured in the digits of Pi.
But it’s not just Shakespeare’s works. The Bible, Quran, War and Peace, the U.S. Constitution, the entire Library of Congress, every tweet, status update, and comment on every forum, wall, and site of the interwebs is, in every possible encoding and in every possible order, completely present uncut and uncensored in the digits of Pi.
If you bothered to encode the entire history of the universe in any of an infinite number of possible encodings, sure enough, you’d find it in all its glory nestled safe in the decimal expansion of Pi.
I used to do an experiment with my students when I was a professor. Let’s say I have a fair coin. I’m going to flip it and if it comes up heads you give me a dollar, tails I give you one. The first flip is heads and I’m up $1. The next two flips are heads. You can stop any time, and I already have $3. Let’s investigate what you might be thinking.
One possible thought might be “3 heads in a row? Tails have to be coming up soon!” This is a common belief referred to as “The Gambler’s Fallacy.” It states that in face of independent trials, there is the belief that a recently infrequent outcome is more likely, in a sense to “balance” the outcomes. So after 3 heads in a row, you might be feeling that “tails is due.” You’d be wrong.
You might also think that a fair coin should come up tails half the time, so given that this is a fair coin, you still expect a 50% chance of getting a dollar from me on the next flip, and as that $3 isn’t all that bad of a loss, you stay in the game.
The next 2 flips come up heads. Then the next 5 do as well. I’ve made $10 in 10 flips. If you feel even more strongly that tails is due, you’re really really wrong. If you feel like you still have a 50% chance of getting a dollar the next flip, you’re wrong too.
I told you it was a fair coin. I know it’s not a fair coin. I know there’s no such thing as a fair coin. I know that a fair coin is nothing more than a thought experiment at best. Moreover, since I’m the one who proposed we gamble, I probably had every reason to expect 10 heads in 10 flips. I’m almost certainly playing with an intentionally unfair coin. But if I tell you it’s a fair coin and get you to believe it’s a fair coin, I can take your money all day long.
There’s no such thing as infinity. It’s a fancy way of saying “I don’t know.” In theory, the whole of human history is encoded within the infinite digits of Pi. In practice, I can get you to give me $10 if you believe a monkey can write Shakespeare.
The original thought experiment concerning a monkey with a typewriter was an unfair game constructed by an evolutionist. According to him a monkey, one in particular named William Shakespeare, randomly (since everything is random) wrote the complete works of Shakespeare in exactly the order they were written. So the only real problem with believing the Infinite Monkey Theorem is that you have to account for the typewriter.
Coincidentally, the problem isn’t any better with infinitely many monkeys. However, if you have one monkey for every real number, which technically isn’t known to be possible even in a world with actual infinities, then you’d be able to find any encoding of anything ever. Well, sort of. It turns out some researchers in England tried getting monkeys to type on typewriters. What they got was a bunch of broken typewriters covered in monkey poo and a very long sequence of S’s.
By the way, the guardian did no fact checking on that article. No mathematician in their right mind would assure anyone the Infinite Monkey Theorem is true. It’s simply not well defined enough. No mathematician would confuse a monkey on a typewriter with a random source. No mathematician would confuse anything real with a random source. No mathematician would ever confuse infinity with “enough time.”
As another side note, I’ve never seen a convincing argument for the belief that a random sequence contains within it every finite sequence. I’m personally of the opinion that a hypothetical consequence with non-zero probability doesn’t imply anything more than it might happen.
Posted with : Bare with Me