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 /images/graemlins/smile.gif)

[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/notes/publication/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 /images/graemlins/blush.gif

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

[Piers]

[ 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.

[/ QUOTE ]

Yes your right, that’s much more likely.

[borisp]

Just to be clear, I understand that this fairy tale has no bearing on reality. That is my point; for statements such as those that are often tossed around by armchair philosophers to be meaningful in any rigorous sense, one must usually clarify their content beyond "sound bite" status.

And "infinite" is not the same as "uncountable."

[vhawk01]

[ QUOTE ]
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

[/ QUOTE ]

I don't think any of us are getting too wrapped up in monkeys. Such a thing might not even be possible.

[almostbusto]

[ QUOTE ]

And "infinite" is not the same as "uncountable."

[/ QUOTE ]
i know, i used uncountable for a reason.

[Piers]

Its possible we agree. I think hypotheticals that are designed to make a serious point about the real world should be held to a higher standard than ones that are not.

[ QUOTE ]
Once upon a time there was a town with just husbands and wives. But, there was something magical about it. It just so happened that when a wife cheated on her husband, he would grow an ugly pair of horns out of his h

[/ QUOTE ]

For a scenario designed to put a story around a mathematical problem anything is fine.

[ QUOTE ]
infinitely many monkeys, typing away at infinitely many typewriters, must eventually produce the works of Shakespeare

[/ QUOTE ]

Here the author is using a hypothetical scenario to make a serious point about the universe; maybe a step in an argument concerning the chance of unlikely events occurring.

Nickpicks like the difficulty in feeding an infinite number of monkeys (so what is infinity here?), or controlling their breeding so they continue being monkeys (how do things drift over time) and such like are fair game. Its quite possible some such points are pertinent to the point being attempted. Certainly the author should expect to have to handle such attacks, and maybe gain understanding as a result.

[MelchyBeau]

do you mean countably infinite, i.e. There is a one to one correspondance with the natural numbers. when I hear uncountably infinite the best example I can think of is the real numbers.

[borisp]

Piers, yes I think we do agree. (Although I see the term probability applied in too loose a context more often than with infinity.)

MelchyBeau, the infinity in the original post is intended solely to mean "not finite." I think Piers is just pointing out that uncountable versus countable is a relevant distinction.

The original post is essentially:

Suppose there is an infinite set X. Defining a property P that satisfies "If a subset E of X is infinite, then there exists e in E such that P(e) is true" is logically equivalent to selecting finitely many elements from X for which P is false.

In reality, there are finitely many monkeys. So hypothesizing an infinitude of monkeys would certainly result in some strange monkeys.

[LA_Price]

A better understanding of the idea of infinity is described in the works of Benoit Mandelbrot. For instance the question How Long is the coast of England? (http://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Se lf-Similarity_and_Fractional_Dimension)

The problem with the monkey example is our own language description, concept of time, idea of "independence", etc.

A better way to phrase the statement would be if a monkey could type an infinite number of random letters in a second, they would eventually end with the works of shakespeare.

[Sephus]

[ QUOTE ]
if a monkey could type an infinite number of random letters in a second, they would eventually

[/ QUOTE ]

it would take less than a second.

[surftheiop]

If the monkeys type infinetly fast doesnt that mean its imposible to produce only one copy of shakespeares work? The istant it started typing wouldnt infinite copies be produced?

[LA_Price]

Less than a second is included under the word eventually. Also infinetely many copies would be produced, but that wouldn't really matter as whether you get one or an infinitetly many is the same under the above statement.

These are really small points to argue. I think Mandelbrot's idea of fractality is much more important as it pertains to the idea of infinity.

[almostbusto]

a better way to frame the problem in my opinion:

i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits. Therefore, at some point in the binary expansion of Pi there exists a string of ones and zeros equivalent to a Microsoft word file that contains the written works of Shakespeare. There is also a jpeg image of your high school yearbook photo too. which i think is much more mind blowing.

[bunny]

[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[Sephus]

[ QUOTE ]
Less than a second is included under the word eventually.

[/ QUOTE ]

obviously i had no issue with the literal meaning of the sentence.

[almostbusto]

[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html

doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.

[Piers]

[ QUOTE ]
do you mean countably infinite, i.e. There is a one to one correspondance with the natural numbers. when I hear uncountably infinite the best example I can think of is the real numbers.


[/ QUOTE ]

I think it would be just as tricky for me to feed aleph zero monkeys as aleph one, irrespective of your opinion of the continuum hypothesis. So I don’t think it matters, although admittedly a countable number monkeys might appear superficially more acceptable.

[bunny]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html

doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.

[/ QUOTE ]
It seems to me that he is basing his claim on the fact that the expansion of pi is infinite, non-repeating with all digits occuring with equal frequency. He seems to make an intuitive leap that this means all finite sequences will eventually occur which is just plain wrong (eq 0.123456789112233445566778899111... has the above properties but never contains the finite sequence 28).

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

a quick googling gave me this:
http://sprott.physics.wisc.edu/pickover/pimatrix.html

doesn't look like the best source. however, it does point at out that such numbers are called transcendental numbers. so even if you don't believe pi is transcendental, then just imagine some other transcendental number.

[/ QUOTE ]
It seems to me that he is basing his claim on the fact that the expansion of pi is infinite, non-repeating with all digits occuring with equal frequency. He seems to make an intuitive leap that this means all finite sequences will eventually occur which is just plain wrong (eq 0.123456789112233445566778899111... has the above properties but never contains the finite sequence 28).

[/ QUOTE ]

If every "next digit" of pi was truly random then I think a probability argument could be made that any Fixed Sequence of length N has a 1/10^N chance of coming up next. Thus, since you have infinitely many 1/10^N chances of it coming up next it almost surely comes up over and over again.

It would be like flipping a coin infinitely many times. If you did, you would almost surely see 1 googolplex of heads flipped in a row somewhere in the infinite sequence of flips. Not only that, but you would see it infinitely many times, with probability 1.

However, I don't think it's clear that every next digit of pi behaves as if it is completely random. The conjecture might still be provable but I don't think it's obvious.

PairTheBoard

[ADDboy]

[ QUOTE ]
[ QUOTE ]
i believe it has been proven that not only is Pi irrational, but that the decimal expansion of Pi contains all possible finites sequences of digits.

[/ QUOTE ]
Anyone able to provide a citation for this? It seems wrong to me.

[/ QUOTE ]

I'm pretty sure that this is open for pi, but it is known to be true for almost all numbers in the sense that if you pick a number uniformly from (0,1) then it has this property.

[bunny]

Thanks. It's kinda ironic that I have no problem believing this is almost always true for numbers selected at random as you say but doubt it's true for pi in particular. /images/graemlins/smile.gif