Infinitely many monkeys?

[borisp]

Warning: what follows may have implications with regard to how man evolved from monkeys...creationists are particularly invited to comment...

A common phrase goes something like "infinitely many monkeys, typing away at infinitely many typewriters, must eventually produce the works of Shakespeare." Now of course this has no rigorous meaning; rather it is intended to express the notion that "some successes are purely by accident," or perhaps "put enough heads together, and something interesting is bound to happen."

But let's take it rigorously, in that we assume we have the set of all monkeys M, that it is infinite, and that they all do nothing but type on their typewriters. Further assume that for any infinite subset S of M, there exists a monkey m in S such that m types exactly the works of Shakespeare.

Is the set G (of "genius" monkeys) that finish the works of Shakespeare finite or infinite? Suppose that it were finite. Then the complement of this set must be an infinite set of monkeys, none of whom have amounted to anything as playwrights. Contradiction. So G is infinite.

Is the complement of G, call it D (consisting only of "dumb" monkeys), finite or infinite? Suppose that it were infinite...oops, again a contradiction. So D is finite.

So M consists of an infinitude of genius monkeys, and finitely many dumb monkeys. Seems reasonable to me, in that it agrees with experimental data thus far.

Or, you could go with the "quantum computing" philosophy: the probability of hitting the exact works of Shakepeare is on the order of (1/26)^(a lot), so it would take at least every atom in the known universe, making a bajillion calculations per second, from the dawn of time, to achieve even a tiny likelihood that this would happen.

In any event, I like to tell this story when people utter this phrase, because "infinite" has a precise mathematical meaning, and its misuse can lead to nonsense. The same is true of the notion of "probability," which only applies to a model within a model, and not a model within reality. I think these distinctions are relevant to many discussions on this forum, and I wish more philosophy folk were aware of this.

Anyway, I apologize for all the typing. I'm starting to get in to this whole "internet forum" thing. This probably signifies the beginning of the end of its popularity.

(PS...As far as I know, I made this crap up, but maybe its one of those ghost memories that you don't remember is a memory when you remember it. I'm sure one of those Barnes & Noble's books on the "mysteries of infinity" has something analogous, although I can't remember reading any of them, so please don't accuse me of (intentional) plagiarism [img]/images/graemlins/smile.gif[/img])

[btmagnetw]

no i remember reading the same thing. whatever it is i read went on to actually perform the experiment and it ended up being like 26 pages of SSSSSSSSSSSDBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAANFIJESSSSSSSSSSSSSSSSSSSSSSSSSSSS or something of the like.

edit: here it is
http://www.vivaria.net/experiments/n...n/NOTES_EN.pdf

you also probably read that wiki article as you both happened to mention the number of atoms in the universe
[ QUOTE ]
For comparison purposes, there are only about 1079 atoms in the observable universe and only 4.3 x 1017 seconds have elapsed since the Big Bang. Even if the universe were filled with monkeys typing for all time, their total probability to produce a single instance of Hamlet would still be less than one chance in 10183800. As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…", and the statement that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers." This is from their textbook on thermodynamics, the field whose statistical foundations motivated the first known expositions of typing monkeys.[3]

[/ QUOTE ]
http://en.wikipedia.org/wiki/Typewriting_monkeys

and one final edit to say: "it was the best of times, it was the BLURST of times?!??"

[borisp]

lol, now I feel like an idiot, that is eerie. Although I did actually first write this down a few years ago (the first time I heard the quantum computing bit, summer of 2002), I honestly never have seen it. It was really weird reading that Wiki page. I guess I'll take this down as it is obv played out [img]/images/graemlins/blush.gif[/img]

Edit: New topic: how would one train a monkey to edit an original post?

[soon2bepro]

[ QUOTE ]
Warning: what follows may have implications with regard to how man evolved from monkeys

[/ QUOTE ]

Man did NOT evolve from monkeys.

We share a common ancestor.

Monkeys and Humans are roughly similarly complex living beings. The same is true for most of the lifeforms on earth, especially animals.

[borisp]

[ QUOTE ]
[ QUOTE ]
Warning: what follows may have implications with regard to how man evolved from monkeys

[/ QUOTE ]

Man did NOT evolve from monkeys.

We share a common ancestor.

Monkeys and Humans are roughly similarly complex living beings. The same is true for most of the lifeforms on earth, especially animals.

[/ QUOTE ] *golf clap*

[Piers]

The problem is unstable in that within a few billion years some of the monkeys will evolve into a species intelligent enough to understand what’s going on. At which point one of the monkeys will decide that the only way out of their slavery is to print the ******* works of Shakespeare and proceed to do just that.

So while it is possible that a monkey will type the works of Shakespeare at random, its much more likely (<-extreme understatement) that one of the monkey will be intelligent enough to know wants going on and do it deliberately. Was that what you were saying?

[m_the0ry]

Because the status of the monkey as either 'dumb' or 'smart' is a continuous random variable, it is also uncountable. Therefore the cardinality of both sets (dumb monkeys and smart monkeys) dictates that they both will be infinite if the number of monkeys is infinite.

[Borodog]

[ QUOTE ]
In truth, the Library includes all verbal structures, all variations permitted by the twenty-five orthographical symbols, but not a single example of absolute nonsense. It is useless to observe that the best volume of the many hexagons under my administration is entitled The Combed Thunderclap and another The Plaster Cramp and another Axaxaxas mlö. These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library. I cannot combine some characters
<font color="white"> . </font>
dhcmrlchtdj
<font color="white"> . </font>
which the divine Library has not foreseen and which in one of its secret tongues do not contain a terrible meaning. No one can articulate a syllable which is not filled with tenderness and fear, which is not, in one of these languages, the powerful name of a god. To speak is to fall into tautology. This wordy and useless epistle already exists in one of the thirty volumes of the five shelves of one of the innumerable hexagons -- and its refutation as well. (An n number of possible languages use the same vocabulary; in some of them, the symbol library allows the correct definition a ubiquitous and lasting system of hexagonal galleries, but library is bread or pyramid or anything else, and these seven words which define it have another value. You who read me, are You sure of understanding my language?)

[/ QUOTE ]

[vhawk01]

[ QUOTE ]
The problem is unstable in that within a few billion years some of the monkeys will evolve into a species intelligent enough to understand what’s going on. At which point one of the monkeys will decide that the only way out of their slavery is to print the ******* works of Shakespeare and proceed to do just that.

So while it is possible that a monkey will type the works of Shakespeare at random, its much more likely (&lt;-extreme understatement) that one of the monkey will be intelligent enough to know wants going on and do it deliberately. Was that what you were saying?

[/ QUOTE ]

No they won't, they will all die from lack of food and water. And then an empty room sits for eternity, until the typewriters rust away and the room is reclaimed by nature.

Oh wait, thats not part of the hypothetical?

[almostbusto]

people are getting too wrapped up in monkeys are neglecting the issue.

i think this is a starting point:

an uncountable number of trials with a nonzero probability of a success results in an uncountable number of successes and an uncountable number of failures.

I am pretty certain that statement is true. they don't really talk about an infinite numbers of trials in probability class so I guess i am not 100%, instead they focus on useful things like the law large numbers. which actually might do more for this post. http://en.wikipedia.org/wiki/Law_of_large_numbers