I think we are all aware that monkeys don't live forever, strawberryfyre, but thanks for the reminder. One monkey or a hundred or a million makes no difference. More monkeys just means parallel processing. Stuff will be typed a bit faster, that's all. The same "final" document will still be produced, in the long (infinite) run.
What original great works of art wouldn´t the monkeys randomly write while pursuing the perfect copy of a book you can find anywhere.... Please Register or Log in to view the hidden image!
The Infinity Monkey Theory So you want the mathematical perspective on the "monkeys typing" scenario? Keep in mind that this is going to be an entirely theoretical answer. As you can imagine, there are some serious practical problems with having an actual infinite number of monkeys typing on an infinite number of typewriters (e.g. where would you put them? what would you feed them?), but since we're mathematicians we can gleefully ignore such considerations. The cheap and easy answer to your question is, "yeah, they'll crank out Shakespeare's works... eventually." This is assuming they really are typing at random. The monkeys with typewriters I have personally observed (mostly of the "young human/little sister" variety) tend to bang on the same keys repeatedly, so it's hard to imagine them actually turning out Shakespeare. But again, this is math so we will ignore the real world. As large as Shakespeare's collected works are, they are still finite. If you type at random, eventually some six-jillion-letter combination you type will end up being the collected works of Shakespeare. An easier way to think about this is picking lottery numbers. Imagine you are filthy rich and decide to buy a bunch of lottery tickets in an effort to win Powerball. Since you are filthy rich, you can afford to buy six jillion lottery tickets with every possible combination of numbers that could come up, and thus you would be guaranteed to win the lottery. It's the same concept with monkeys typing. The grittiest detail in this problem is that the answer is only yes if we are talking about an infinite number of trials; that is, having an infinite number of monkeys or letting one monkey pound away for an infinite amount of time. If we are restricted to a finite number of monkeys and a finite amount of time, then the answer is no. It is entirely possible that in a finite amount of time a finite number of monkeys may type out nothing but pages upon pages of meaningless drivel. It's also possible (although unlikely) that one monkey may get it right the first time. A good way to think of this is to imagine rolling a six-sided die numerous times and waiting for a six to come up. It may come up on the first roll. It's possible that you could keep rolling and rolling millions of times without a six coming up, although you would expect it to come up within six rolls, since there is a 1/6 chance of a 6 turning up on each roll. Let's do an actual example. Since the collected works of Shakespeare are a pretty lofty goal, let's just see about how long we would expect it to take for a monkey to crank out one of Shakespeare's sonnets, for example the following: Look in thy glass and tell the face thou viewest -48 Now is the time that face should form another -45 Whose fresh repair if now thou not renewest -43 Thou dost beguile the world unbless some mother -47 For where is she so fair whose uneard womb -42 Disdains the tillage of thy husbandry -37 Or who is he so fond will be the tomb -37 Of his self love to stop posterity -34 Thou art thy mothers glass and she in thee -42 Calls back the lovely April of her prime -40 So thou through windows of thine age shall see -46 Despite of wrinkles this thy golden time -40 But if thou live rememberd not to be -36 Die single and thine image dies with thee -41 In the above sonnet I removed all punctuation, just leaving the letters and spacing--we can't expect too much; they're only monkeys, right? If my letter count is correct, this leaves 572 letters and spaces. To further simplify, we won't worry about carriage returns, capital letters, or any other such stuff. Anyhow, say we give a monkey a special typewriter that has 27 keys (26 keys for the letters of the alphabet along with a space bar). We let the monkey type 572 characters at a time, pull the sheet out, and see if it's the sonnet. If not, we keep going. We'll do some calculations on the fly here to see how long this process will take. Got a calculator handy? First of all let's find out how many 572-letter possibilities there are for the monkey to type. We have 572 characters, and 27 choices for each character, so there will be 27^572 possibilities (that's 27 times itself 572 times). Punching this into my calculator... er... okay, on second thought better use a computer....I get the following number of possibilities: 5496333784561099393693048531368044344887926194198532520694117049056247 2568424395482058851927075593679213263223991649095444601504350463483987 5025610104140864608504908534119526789608399222986117684072414622768253 6214908304427395812519474546086831288010236639735783766919573127540345 2575089566044810413932116060031762894505524988451285440971813773606694 0163946473467668970711919689863460271936750837609798272198814318196353 5086770723528603185438692855503864007605689811533968043988986405766599 4634626982653271152473969190655534329764726804924235126863461599117918 7453007805890829071114522894672065623217961791812204851353664903930975 3565419938168852881272755213408072890621434530416560019423439471934830 8488558728285338553045399661579902802268940348808763480359167736446637 8909091744053824079947245708112252748079248200721 It's a big number, about 5*10^818. Let's say our monkey can type about 120 characters per minute. Then the monkey will be cranking out one of these about every five minutes, 12 every hour, 288 per day, and 105120 of them per year. Divide that big number by 105120 and you get that it would take that monkey about 5*10^813 years to type out that sonnet. Now say we get 10^813 (that's ten followed by 813 zeros) monkeys working on the job. With that many monkeys working 24 hours a day, typing at random, one of them is likely to crank out the sonnet we are looking for within five years. If the monkeys are particularly unlucky, you may have to let them run an infinite amount of time before they crank out the desired sonnet, but chances are with this many monkeys on the job you will get results in five years. To make a long story short, if you have only a finite number of outcomes and you take an infinite number of trials, you will end up getting the outcome you are looking for. Well, forget about making a long story short, I'll give you one more mind-blowing example. A typical digitized picture on your computer screen is 640 pixels long by 480 pixels wide, for a total of 307200 pixels. Using only 256 different colors, you can get decent resolution. Now if you take 256^307200 (256 times itself 307200 times) you get... well, a pretty big number, but a finite number nonetheless. That's the number of different images you can have of that particular size. Any picture you would scan into a computer at that size and resolution will necessarily be one of those images. Therefore, contained in those images are the images of the faces of every human being who ever lived along with the images of the faces of every person yet to be born. Deep stuff, eh? I'll leave you with that thought.
Doesn't work. The size of the universe does not increase the chances. For your trick to work, you should be asking for an infinite number of planetary systems, or an infinite number of Earths. You have several hundred monkeys typing on typewriters, but they are in an infinitely large room... see where I'm going? Regarding the original post: Just as it is difficult to imagine how small the probability of hammering out Shakespeare's complete works, so is it impossible for us to grasp the enormity of the infinite. The latter overpowers the former every time.
Not that impressive, really. Sure... the number is able to be abstractly represented using power notation, but most readers might not realize how ridiculously large that number is. By comparison, there are about 10^80 particles in the known universe. The number you came up with isn't amazing for how small and encompassing it is (to hold all possible pictures at that resolution and color depth), rather it is startling for how gargantuan it is. I would be more impressed to know, abstractly, that there are a finite number of possible images at any resolution... seeing how big that number is should give more doubt to the monkey/shakespeare skeptics (of which I am not one).
Even if there are an infinite number of earths, it doesn't mean that any two of them would have to be the same, or that any conceivable scenario must exist on one of them. You can find infinitely many numbers between 2 and 3 without having any of them be the same number, and without having any of them be greater than 4.
Calculators True a finite and abstract number would be more interesting but to find that, well you have a one in 27^105210 chance of doing that alone. But more about the topic. It took me days to figure out that long number and I would like to know if there ia a real calculator that can do it in mere seconds like simpler equations. I can't find one and my thesis is about infinity and it's finite proportions. LARGE PROCESSING CALCULATORS ARE NEEDED!!!!!!!Please Register or Log in to view the hidden image!
A monkey is a sentient being - there is no guarantee that it won't just repeat what it has previously typed or simply avoid typing certain letters at all. You could only guarantee random output with a fairly powerful computer which, given an infinite amount of time, could create every way in which the alphabet and punctuation marks can be arranged. At some point it would churn out the sentences I have just finished typing.
Yes, and I think that there might be numbers with fractions that go over not all numbers from 1 to 9 also there might be numbers where there always is a random pattern, since of the way the sequence is repeated.
This isn't true. The only way you can have infinitely many numbers between 2 and 3 is if you keep changing the level of precision of the group of numbers. That is, you can only do this by adding decimal points as you go along. Any level of precision you choose for your set, you will find a finite number in that set. By defining your set as something dynamic (i.e. the rules keep changing), you are cheating. This myth is still taught in college classrooms, and it is the maddening source of the Zeno paradox, which shouldn't be a paradox at all.
If you choose specific numbers to start and stop at YES you could argue there is finite space between but the reality is there is infinity between in fractions. Because the number is irrational doesn't mean it does not exist.....You can continuously fracture an object infinitely and that is an example of infinity. As far as CHEATING by continuously fracturing a measurement infinitely smaller and smaller I don’t agree as that is no more cheating than picking and choosing when the measurement of an object starts and ends with set increments of numbers. Fracturing an object using SET increments is a practical way of measuring an object for a purpose which is why its taught in school and always used in all measurements. I’m not necessarily arguing anything other than Infinity is real it does exist and it exists everywhere and the implications of that are extremely interesting. As far as measuring an object for a practical purpose you HAVE to use set increment measurement otherwise you would never get anything done. Outside of practical purpose driven measurement you can fracture/divide any object infinitely.
No you can not. In order to do so, your unit of measurement would have to be zero, which is impossible. Again, you are falling for the never-ending process of choosing finer and finer units of measurement and pretending that this process will ever reach zero. It will not, therefore you always have a finite number of divisions in any segment. The never-ending nature of irrational numbers does not dent the argument. You can not change the "last" number of each irrational number to make it a distinct number, as there is no "last" number. Let's take the segment of the number line between zero and one. Divide it in half, and you have a smaller number of .5 and 2 segments. Divide in half again and you have .25 and 4 segments. You can do this forever and the first number will never reach zero and the second number will never "reach" infinity. This, despite the fact that the first number will forever grow in decimal precision and the second number will never stop doubling.
If you could cut an object of any size in half over and over again.....how and when do you propose that it would ever reach ZERO? when you cant see it anymore? When your instruments are no longer capable of cutting such a small object in half? When the object reches its fundamental atomical paticles that are considered the "smallest" possible units? (That we know of) Its a very perplexing concept to understand but I think the human desire/need to MEASURE everything is what is making infinity so incredibly hard to understand. When you realise because of infinity, there is no difference between the earth in the universe and a grain of sand in the desert, it makes measuring everything rather.....pointless unless of course your trying to build a condo. Then measurment becomes the "practical" thing to do. Im obviously not taking the scientific approach to this understanding as it would take far too long. My approach is philosophical. hense....the forum location.
I like this thought experiment. You are right that they will never be zero or infinity. Funny thing is, the number of divisions you perform on this segment can be infinity!!!
Nope. You can do it FOREVER. Which is an eternity. But you will never reach zero or infinity, which is what you would have to do to prove your point. As I said a few posts back, it is the eternal nature of the process of creating ever finer standard units of measurement which create the illusion that there are an infinite number of divisions in a line segment. As soon as you pick your degree of precision, you both define your unit of measurement and tell me how many of them are in the segment. If you keep sliding your hand to the right, down a growing line of decimal places, and looking up at me with a clever grin, you are fooling yourself, but not me. This resolves Xeno's paradox and also this simpler but more confounding one: I just took my finger and ran it along a ruler, passing through an infinite number of points. The answer is that points do not exist. They are the unit size=zero, which will never be reached. A figment of our imagination with theoretical uses only. I didn't run through an infinite number of anything.
Logic flaw. Aren't you assuming every possible "random" sequence of text, to be equally probable? Doesn't that like, almost never happen in nature? And No. Neither a monkey nor a typewriter would last that long.
I'll go even further to say that every possible finite sequence of letters will have been typed out an infinite amount of times.
