Infinite Monkey Theory..

johnnyskeleton

1. What we call the complete works of Shakespeare is a collection of the most commonly accepted versions of his plays. He wrote several different versions of his most famous plays, many of the editions were copied by people watching the play, and often times his plays would have evolved by as performance and audience reaction dictated. So there is no fixed combination of words which make up the works of shakespeare, it is the sentiment behind it. For example, some people say "too too sullied flesh" [Hamlet, Act 1 Scene 2] others says "too too solid flesh", and it is not enough for an editorial note to tell you which is correct, but it is down to human preference to decide which is correct. Maybe Shakespeare is a bad example.

2. There is also an element of paradox. If you had an infinite amount of monkeys typing for an infinite amount of time, could they not also produce a perfectly articulated antithesis of this argument which proves that this theory is not possible? If infinity is used to show that everything is possible, then proof of its own impossibility must also follow.

3. In any case what is the point of this theory?

Crucifix

johnnyskeleton wrote:

1. What we call the complete works of Shakespeare is a collection of the most commonly accepted versions of his plays. He wrote several different versions of his most famous plays, many of the editions were copied by people watching the play, and often times his plays would have evolved by as performance and audience reaction dictated. So there is no fixed combination of words which make up the works of shakespeare, it is the sentiment behind it. For example, some people say "too too sullied flesh" [Hamlet, Act 1 Scene 2] others says "too too solid flesh", and it is not enough for an editorial note to tell you which is correct, but it is down to human preference to decide which is correct. Maybe Shakespeare is a bad example.

Well they'd produce every possible version, so everybody's happy.

2. There is also an element of paradox. If you had an infinite amount of monkeys typing for an infinite amount of time, could they not also produce a perfectly articulated antithesis of this argument which proves that this theory is not possible? If infinity is used to show that everything is possible, then proof of its own impossibility must also follow.

Infinity isn't to show that everything is possible, but rather that everything that can happen will happen. If such a antithesis exists, well then I suppose they would, but I do not think such a thing does.

3. In any case what is the point of this theory?

I guess it's really just to get people thinking about the nature of infinity.

johnnyskeleton

Crucifix wrote:

Infinity isn't to show that everything is possible, but rather that everything that can happen will happen. If such a antithesis exists, well then I suppose they would, but I do not think such a thing does.

My understanding (and I accept that there may be different definitions of infinity depending on whether it is philosophy, maths etc) is that infinity is everything, and it is not just confined to everything that can happen. This is largely because we are (as you point out) not able to know everything that is possible. For example, in maths, I would suggest that infinity includes, for example, the square root of -1.

Now, as far as we are concerned it is impossible to have the square root of -1. We cannot define a number which, if multiplied by itself will give -1, and indeed because maths is basically an abstract human creation, we can fairly say that it can't happen.

However, it exists in potentia (because everything conceivable exists in potentia) , and if infinity did not contain potential number x, then it must be finite. If infinity was in any way limited, even limited to what can happen, it is finite (in that it includes what can happen, but excludes what cannot). Even if there is an unlimited quantity of things that can happen, the fact of there being something outside that quantity means that it is not infinite (because it is confined to a limited, but undefined, quantity).

The trouble with infinity is that its the very devil to define.

I have another question for you…If you had infinite monkeys and infinite time, could they write down the infinity of numbers? Infinity / infinity = 1, so it is mathematically possible, but what if one of those monkeys writes down infinity + 1?

Stompbox

But all the infinite monkey's will have a lifespan of how may years? They'll only have about 30 years max to complete it so it ain't possible unless you have super mega immortal monkeys!....I don't think evolution has progressed enough yet.

johnnyskeleton

Sweet wrote:

But all the infinite monkey's will have a lifespan of how may years? They'll only have about 30 years max to complete it so it ain't possible unless you have super mega immortal monkeys!....I don't think evolution has progressed enough yet.

Good point. Does one individual monkey have to write the complete works, or can one monkey write Romeo & Juliet, another King Lear, two for Hamlet (it's just that good), etc?

Son Goku

As the number of trails tends to infinity, the likelyhood tends to 1.

For instance if you flip a coin, it's 1/2 chance it won't be heads. Flip it twice and there is only a 1/4 chance it won't be heads. Flip it thrice, there's only a 1/8 chance it won't be heads, e.t.c.

The chance of it not occuring tends to zero, as the number of trails tends to infinity. Same for Shakespere's plays, it doesn't matter if there are different versions of Shakespere's plays, it doesn't matter if the monkeys are French/Co-Habiting/Pacifist, it's just a statement about how probability works.

Now, as far as we are concerned it is impossible to have the square root of -1. We cannot define a number which, if multiplied by itself will give -1

Yes we can. It's called i. i^2 = -1.

Crucifix

johnnyskeleton wrote:

My understanding (and I accept that there may be different definitions of infinity depending on whether it is philosophy, maths etc) is that infinity is everything, and it is not just confined to everything that can happen. This is largely because we are (as you point out) not able to know everything that is possible. For example, in maths, I would suggest that infinity includes, for example, the square root of -1.

Now, as far as we are concerned it is impossible to have the square root of -1. We cannot define a number which, if multiplied by itself will give -1, and indeed because maths is basically an abstract human creation, we can fairly say that it can't happen.

However, it exists in potentia (because everything conceivable exists in potentia) , and if infinity did not contain potential number x, then it must be finite. If infinity was in any way limited, even limited to what can happen, it is finite (in that it includes what can happen, but excludes what cannot). Even if there is an unlimited quantity of things that can happen, the fact of there being something outside that quantity means that it is not infinite (because it is confined to a limited, but undefined, quantity).

The trouble with infinity is that its the very devil to define.

I have another question for you…If you had infinite monkeys and infinite time, could they write down the infinity of numbers? Infinity / infinity = 1, so it is mathematically possible, but what if one of those monkeys writes down infinity + 1?

You're looking at infinity as a set, but it is a number. The number infinity doesn't contain i. It is a value, albeit a hypothetical one.

bonkey

johnnyskeleton wrote:

My understanding (and I accept that there may be different definitions of infinity depending on whether it is philosophy, maths etc) is that infinity is everything,

No, infinity is not "everything".

The set of real numbers between 0 and 1 is infinite in size. It does not contain the number 2.

Now, as far as we are concerned it is impossible to have the square root of -1.

It is no more impossible to have the square root of -1 then it is to have -1 in the first place. You can't find a real-world quantity which is sqrt(-1), sure, but then you also can't find a real-world quantity which is -1.

You can have conceptual quantities of -1, but conceptual is little different from theoretical. i exists (although we could argue whether or not thats the right term) within certain mathematical frameworks.

Of course, its possible that infinities only exist within certain mathematical frameworks as well....and do not exist in reality. I could be wrong, but to my understanding, an infinity has never been shown to exist.

(Aside - Son Goku could show me up on this one, but my understanding of singularities is that they do not technically lead to infinities.)

We cannot define a number which, if multiplied by itself will give -1,

Sure we can. its i, its well-defined, and its properties are well understood. It is no more mysterious than -1 in the first place.

If infinity was in any way limited, even limited to what can happen, it is finite

Not true.

There are an infinite number of "whole" numbers (i.e. numbers not containing a fraction. This infinity does not contain the number 3/2. However, 3/2 is a number.

If we say "we limit our set to only those numbers which do not contain fractions", then our set is still infinite in size but does not contain everything.

The trouble with infinity is that its the very devil to define.

I agree. People have gone mad trying to come to terms with infinity.

I have another question for you…If you had infinite monkeys and infinite time, could they write down the infinity of numbers?

That depends on what order of infinities we're talking about.

It is mathematically possible to have two infinities where one can show that infinity1 > infinity2 !!!

Son Goku

bonkey wrote:

(Aside - Son Goku could show me up on this one, but my understanding of singularities is that they do not technically lead to infinities.)

Singularities in physical theories usually mean a quantity is no longer well-defined rather being infinite, they mark the point were a new theory is needed, so you are correct.
Also infinity isn't an element of the Real or Complex numbers and since most physical theories use the Reals or the Complexes, they don't have access to it.

(If you add infinity to the reals (i.e. give them an extra element called infinity) you get a system called the hyperreals.
If you add infinity to the complex numbers you turn them into a sphere, but that's a side topic.)

bonkey wrote:

There are an infinite number of "whole" numbers (i.e. numbers not containing a fraction. This infinity does not contain the number 3/2. However, 3/2 is a number.

If we say "we limit our set to only those numbers which do not contain fractions", then our set is still infinite in size but does not contain everything.

bonkey wrote:

It is mathematically possible to have two infinities where one can show that infinity1 > infinity2 !!!

Believe it or not the infinite set of whole numbers is the same size as the infinite set of fractions, but both sets are smaller than the set of the Reals.

johnnyskeleton

wikipedia wrote:

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a larger set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity) and called the imaginary unit, which is defined such that i2 = −1. Using this notation, we can think of i as the square root of −1, but notice that we also have (−i)2 = i2 = −1 and so (−i) is also a square root of −1.

i is imaginary; it exists as I have said, in potentia. We start with the premise that all numbers have a square root. Because of this we deduce that there must be a square root of -1. Without a better name for this imaginary number, lets call it i.

Now, if our infinite monkeys can produce any combination of letters which will make, among other things, every version of the complete works of Shakespeare (although it will have none of the soul) then ipso facto they must also be able to create the antithesis to this theory. We start with the premise that all combinations of letters must be written. Because of this we deduce that there must be a combination of letters which will be the antithesis. Without a better name for this imaginary antithesis, lets call it porcupine's theory of non-esistencenessness.

We don't have to know what it is, we have reasoned its existence from our premise.

bonkey wrote:

The set of real numbers between 0 and 1 is infinite in size. It does not contain the number 2.

I don’t think this is correct. The set of real numbers between 0 and 1 is infinitesimal. That is to say there are infinite divisions (or infinite variations of the answer to what real number is between 0 and 1?). But it is finite in the sense that it can be confined to numbers between 0 and 1. There may be infinite divisions, but it is not unlimited.

There is a substantial difference between what we cannot practically define because it does not end, and what theoretically cannot be defined because there is nothing outside of it which we can use as a basis of comparison to delimit it. The universe is theoretically infinite because we do not believe there is a start nor a stop. However, even though there are so many things in the milky way galaxy that we cannot define them (and they are infinitesimal in that if we keep breaking it down into smaller parts we will never find an end [except in the sense that our physical reality can be broken down into particles, and sub atomic particles do not really exist for us]) it does not mean that the milky way is infinite.

If we have so many monkeys that it is impossible count them and we know of no other monkeys, that is infinite monkeys. However, if we have so many monkeys that it is impossible to count them but we know that Mojo the helper monkey is not included, then we have a finite, although unascertainable amount of monkeys. If we had infinite monkeys, we could ask the zoo keeper “is monkey x here” and no matter what x is, the zoo keeper would answer yes, he is. By the very fact that he says Mojo is not here, the quantity of monkeys is no longer infinite.

crucifix wrote:

You're looking at infinity as a set, but it is a number. The number infinity doesn't contain i. It is a value, albeit a hypothetical one.

I don’t think I am. I look at infinity as a concept. If I did look at infinity as a set, it would be fair to say that set x contains infinite variables. What I am saying is that if it does not contain variable q, then it is not infinite. It can be defined as not containing q.

bonkey wrote:

It is no more impossible to have the square root of -1 then it is to have -1 in the first place. You can't find a real-world quantity which is sqrt(-1), sure, but then you also can't find a real-world quantity which is -1.

I’ll tell that to my bank manager the next time I see him.

Perhaps I'm taking too literal an approach to the concept of infinity, but if we are talking about infinite monkeys reproducing Shakespeare, then we are talking about the kind of infinity that includes everything that could conceivably be written, irrespective of how mind bogglingly improbable it may be.

GrumPy

basquille wrote:

Yep..

"It was the best of times, it was the BLURST of times!!.. YOU stupid monkey!"

lol yeah, I musta seen that episode an infinite amount of imes....

Crucifix

johnnyskeleton wrote:

Now, if our infinite monkeys can produce any combination of letters which will make, among other things, every version of the complete works of Shakespeare (although it will have none of the soul) then ipso facto they must also be able to create the antithesis to this theory. We start with the premise that all combinations of letters must be written. Because of this we deduce that there must be a combination of letters which will be the antithesis. Without a better name for this imaginary antithesis, lets call it porcupine's theory of non-esistencenessness.

But only if it's possible to disprove. Imagine another thought experiment, say a man bouncing a ball, for an infinite amount of time, within our own scientific system; gravity, restitution etc, except the man is forever young and the ball doesn't damage or wear. The ball will never turn into an elephant, even over an infinite amount of time, because that's not possible within the constraints of our system.
So if it is impossible to disprove the monkey thing, which it should be since you started off by assuming it's true, then a real antithesis will never be produced.

I don’t think this is correct. The set of real numbers between 0 and 1 is infinitesimal. That is to say there are infinite divisions (or infinite variations of the answer to what real number is between 0 and 1?). But it is finite in the sense that it can be confined to numbers between 0 and 1. There may be infinite divisions, but it is not unlimited.

The difference between the numbers may be infinitesimal, but the amount of values is infinite. It is unlimited, in the sense that there is no limit to the amount of real numbers between 0 and 1. You can always produce a new one, no matter how many have gone before.

johnnyskeleton

crucifix wrote:

Imagine another thought experiment, say a man bouncing a ball, for an infinite amount of time, within our own scientific system; gravity, restitution etc, except the man is forever young and the ball doesn't damage or wear. The ball will never turn into an elephant, even over an infinite amount of time, because that's not possible within the constraints of our system.

Also in this example the ball will never be the ball used by Michael Jordan in a stunning display of basketball. So if the infinite monkeys are confined to only typing letters rather than producing something, even if they do type the series of letters which exactly correspond to the complete works of shakespeare, they would not have created the complete works of shakespeare in substance, only in form. Is it not then possible that they would produce a series of letters which would form the antithesis to this thesis? Neither the thesis nor the antithesis would be true or false, they would just be mere possibilities.

I'm also reminded of Douglas Adam's computation of the population of the universe. The universe is infinite, but not all planets are inhabited (therefore there are a finite number of inhabited planets). Infinity divided by a finite number is so near to 0 as makes no difference, therefore anybody who thinks he is alive is delusional.

bonkey

johnnyskeleton wrote:

I don’t think this is correct. The set of real numbers between 0 and 1 is infinitesimal.

Infinitesimal means "infinitely small". The set of real numbers between 0 and 1 is most certainly not infinitely small.

That is to say there are infinite divisions (or infinite variations of the answer to what real number is between 0 and 1?).

Infinite variations, eh?

So by your logic, this means that everything is a variation of the answer to that question. Assuming that by "answer" you mean "correct and valid answer", your logic now suggests that 2 is a real number between 0 and 1! If not, then you have used infinite to refer to a set which does not contain everything...exactly the point I was trying to make.

But it is finite in the sense that it can be confined to numbers between 0 and 1.

That does not make it finite. That makes it bounded.

There may be infinite divisions, but it is not unlimited.

I never said it was unlimited. I said the set was infinite in size but it does not contain everything. It is, to be fully accurate, infinite and bounded.

The set is infinite in size, but it does not contain every number.

I’ll tell that to my bank manager the next time I see him.

Do that. If you tell him that you couldn't possibly have a -100 Euro credit balance, your manager should tell you that you have a +100 Euro dedit balance and ask you if that makes you any happier.

Accounting balances are technically positive numbers, Debit or Credit.

The convention of displaying a negative Debit is just that - a convention. The only way you can transform it into real money is by determining who owes whom a positive amount of cash, and then obtaining that positive amount of cash and transferring ownership.

Your bank cannot give you -100 Euro to close your account. Rather, you must give your bank 100 Euro. The negative amount is, in this sense, non-real. It exists only as a concept and must be mapped to a positive amount in the real world in order to be worked with.

Perhaps I'm taking too literal an approach to the concept of infinity, but if we are talking about infinite monkeys reproducing Shakespeare, then we are talking about the kind of infinity that includes everything that could conceivably be written, irrespective of how mind bogglingly improbable it may be.

The "theory" only works at a certain level. It is an attempt to give a laymans equivalent to the mathematical notion that any non-zero probability must occur given an infinite sample-space.

(Note, if something can be excluded from the sample space, such as the number 2 from the list of reals between 0 and 1, then the probability of finding the number 2 in that sample-space is 0 and so no violation occurs).

When you start analyzing it to a mathematical level, you discover that there are caveats which must be born into account (things which will result in 0-probabilities) so we can't say for certain that the monkeys would really produce the works of Shakespeare)

An interesting one which has wrecked my head (and which maybe our resident Quantum Guru can resolve for me) is this...

If the universe in its current state is open-ended expansionary, then it will be infinite in duration. Any given quanta can spontaneously jump from one point to another. The distance and direction jumped is random, but the further the less likely. Thus, there is a vanishingly-small-but-non-zero probability that all quanta at some point will spontaneously jump, resulting in a universe which is in a state of lower entropy....and also be a state the universe has been previously in. There is a non-zero probability of everything jumping back to "Big Bang" conditions, just as there is a non-zero probability of everything jumping back to exactly the state it was in when I started typing this mail.

Teh implications of this would be that history would not be required to be what we observe it to have been....and that the same situations could (and thus perhaps would) play out an infinite number of times!!!

johnnyskeleton

Words are tricky creatures, but is there not a difference between mathematically infinite and the logical concept infinity?

The former is where the possibilites are endless, the latter is a collection of things which we cannot define because everything we could possibly think of is included.

Even if I were to look at it as mathematical infinte, then the theory can be stated as:

(any given combination of letters) is an element of ∞*(random series of letters).

The substance or meaning of (any given combination of letters) is irrelevant to the mathematical formula. The infinite monkeys theory is somewhat different in that:

(great literature) is an element of ∞*(sentient but illiterate creatures who have the ability and the volition to type letters on the typewriters)∞*(time)

although I think the second ∞ is just for show.

If the theory is purely mathematical, then the antithesis is included in (any given combination of letters) although it is devoid of meaning or substance.

However, there are problems (albeit very literal ones) with the second formula too in that we are assuing that monkeys have the ability and volition to type.

So what I'm suggesting is that either the theory is not true or it is true and the monkeys could produce any possible combination of letters.

bonkey wrote:

Your bank cannot give you -100 Euro to close your account. Rather, you must give your bank 100 Euro. The negative amount is, in this sense, non-real. It exists only as a concept and must be mapped to a positive amount in the real world in order to be worked with.

A positive number (i.e. 100 Euro) also exist only as a concept and must be mapped to a positive amount in the real world in order to be worked with. Or, all numbers are abstract concepts which we use to count up parts of the real world. Just because I know how drunk I'll be after 10 beers, doesn't make it any more real than me knowing how drunk I'll be -100,000 brain cells later. The only difference is that we use positive numbers more often.

Capt'n Midnight

Would the monkeys have any feedback ?

Even as spell checker would speed up the process enormously. Large biological molecules like RNA and Proteins that have catalytic activity self-fold to some extent while being assembled, which guides what happens next.

As for infinity , I like this. You have a hotel with an infinite number of rooms, all full, so what do you do when more people arrive ?
http://scidiv.bcc.ctc.edu/Math/InfiniteHotel.html

The obserable uinverse is finite, but it may extend further, but we don't know yet because of the speed of light.

back to the monkeys, an infinte number of them means infinnte genetic variation so you will have plenty of winged monkeys and some more inteligent ones, and you will also have ones that have supernatural powers. super nutral in the sense of being better than natural. since brain waves are electrical impulses if they had antennas in their heads they could read minds and elicite the shakespere that way.


the big question is would they do it properly ?
"You don't know shakespere until youve experienced it in the original klingon."