
#REDIRECT Infinite monkey theorem
#REDIRECT Talk:Infinite monkey theorem
[[Image:monkey-typing.jpg|thumb|According to Kolmogorov's zero-one law, given enough time, a chimpanzee like this one typing at random will eventually type out a copy of one of William Shakespeare's plays.]] The infinite monkey theorem#Footnotes says that almost surely (i.e. with probability equal to 1) a monkey hitting keys at random on a typewriter keyboard will eventually type every book in France's Bibliothèque nationale de France (National library). In the restatement of the theorem most popular among English speakers, the monkeys eventually type out the collected works of William Shakespeare; others replace the National Library with the British Museum or the Library of Congress. The name is a popular misnomer for an idea from Félix Édouard Justin Émile Borel's book on probability, published in 1909, which introduced the concept of "dactylographic#Footnotes monkeys". A popular statement of the theorem is that an infinity number of monkeys typing for an infinite amount of time will produce a given text. To insist on both infinities, however, is excessive. A single immortal monkey who executes infinitely many keystrokes will eventually type out any given text, and an infinite number of monkeys will begin producing ''all'' possible texts immediately, with no wait. ==Proof sketch== The infinite monkey theorem is relatively straightforward to prove. If two events are statistically independent, meaning neither affects the outcome of the other, then the probability of both happening is equivalent to the product of the probabilities of each one happening on its own. For example, if the chance of rain in Sydney on a particular day is 0.3 and the chance of an earthquake in San Francisco on that day is 0.8, the chance of both happening on that same day is 0.3 × 0.8 = 0.24. Now, suppose the typewriter has 50 keys, and the monkey is trying to type the word "banana". Typing at random, the chance that the first letter typed is ''b'' is 1/50, as is the chance that the second letter typed is ''a'', and so on. These events are independent, so the chance of the first six letters matching "banana" is 1/506. For the same reason, the chance that the next 6 letters match "banana" is also 1/506, and so on. Now, the chance of ''not'' typing "banana" in each block of 6 letters is 1 − 1/506. Because each block is typed independently, the chance, ''X'', of not typing "banana" in any of the first ''n'' blocks of 6 letters is ''X'' = (1 − 1/506)''n''. As ''n'' grows, ''X'' gets smaller. For an ''n'' of a million, ''X'' is 99.99%, but for an ''n'' of 10 billion it is 53% and for an ''n'' of 100 billion it is 0.17%. As ''n'' approaches infinity, the probability ''X'' limit (mathematics) zero; that is, by making ''n'' large enough, ''X'' can be made as small as one likes. If we were to count occurrences of "banana" that crossed blocks, ''X'' would approach zero even more quickly. The same argument applies if the monkey were typing any other string of characters of any length. The same argument shows why infinitely many monkeys produce a text as quickly as it would be produced by a perfectly accurate human typist copying it from the original. In this case ''X'' = (1 − 1/506)''n'' where ''X'' represents the probability that none of the first ''n'' monkeys types "banana" correctly on their first try. When we consider 100 billion monkeys, the probability falls to 0.17%, and as the number of monkeys, ''n'' increases to infinity the value of ''X'' (the probability of all the monkeys failing to reproduce the given text) decreases to zero. This is equivalent to stating that the probability that one or more of an infinite number of monkeys will produce a given text on the first try is 100%, or that it is certain they will do so. The only difficulty remaining is in locating a successful monkey. The theorem exemplifies a proposition in the probability theory called Andrey Nikolaevich Kolmogorov's Kolmogorov's zero-one law, which was published in 1933, 24 years after Borel's book cited above. ==Probabilities== Ignoring punctuation, spacing, and capitalization, and assuming a uniform distribution of letters, a monkey has one chance in 26 of correctly typing the first letter of ''Hamlet.'' It has one chance in 676 (26 times 26) of typing the first two letters. Because the probability shrinks exponential growthly, at 20 letters it already has only one chance in 2620 = 19,928,148,895,209,409,152,340,197,376, roughly equivalent to the probability of buying 4 lottery tickets consecutively and winning the jackpot each time. In the case of the entire text of ''Hamlet'', the probabilities are so vanishingly small they can barely be conceived in human terms. The text of Hamlet, even stripped of all punctuation, contains well over 130,000 letters. The mere fact that there is a chance, however unlikely, is the key to the "infinite monkey theorem", because Kolmogorov's zero-one law says that such an infinite series of independent events must have a probability of zero or one. Since we have shown above that the chance is not zero, it must be one. To consider that an event this unlikely is guaranteed to occur given infinite time can give a sense of the vastness of infinity. Gian-Carlo Rota wrote in a textbook on probability (unfinished when he died): :"If the monkey could type one keystroke every nanosecond, the expected waiting time until the monkey types out ''Hamlet'' is so long that the estimated age of the universe is insignificant by comparison ... this is not a practical method for writing plays. (We cannot resist the temptation to quote from Alfred North Whitehead, 'I will not go to infinity'.)" In ''The Nature of the Physical World: The Gifford Lectures'' (Macmillan, New York, 1929, page 72) the physicist Arthur Eddington wrote: :"If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel." In physics, then, the force of the "monkeys argument" lies not in the probability that the monkeys will "eventually" produce something intelligible, but in the practical reality that they will not. Any physical process that is even less likely than such monkeys' success is effectively impossible; this is the statistical basis of the second law of thermodynamics. ==Myth about origins== It is often reported, though highly improbable, that Borel's use of monkeys and typewriters in his theorem was inspired by an argument used by Thomas Henry Huxley on June 30, 1860. Huxley was involved in a debate with the Anglican Bishop of Oxford, Samuel Wilberforce, held at a meeting of the British Association for the Advancement of Science at Oxford, of which Wilberforce was a vice-president, and was sparked by the publication of Charles Darwin’s ''Origin of Species'' seven months earlier, in November 1859. No transcript of the debate exists, but neither contemporary accounts of it nor Huxley's later recollections include any reference to the infinite monkey theorem. The association of the debate with the infinite monkey theorem is probably an urban myth triggered by the fact that the debate certainly did include some byplay about apes: the bishop asked whether Huxley was descended from an ape on his grandmother's or his grandfather's side, and Huxley responded that he would rather be descended from an ape than from someone who argued like the bishop. It is most unlikely that Huxley would have referred to a typewriter. Although patents for machines resembling modern typewriters were granted as early as 1714, commercial production of typewriters did not begin until 1870, and a skilled debater like Huxley would hardly have let his point depend on a device whose existence would have been unknown to most of his audience. ==Infinite monkey experiments== [http://user.tninet.se/~ecf599g/aardasnails/java/Monkey/webpages/index.html#results "The Monkey Shakespeare Simulator"] website, launched on July 1, 2003, contains a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. As of January 3 2005, matches as long as 24 consecutive letters, four words have been recorded ("RUMOUR. Open your ears; 9r"5j5&?OWTY Z0d "B-nEoF.vjSqj[..." from ''Henry VI, part 2''). Due to processing power limitations, the program uses a probabilistic model (by using a random number generator) instead of actually generating random text and comparing it to Shakespeare. When the simulator "detects a match" (that is, the RNG generates a certain value or a value within a certain range), the simulator simulates the match by generating matched text. In 2003, scientists at Paignton Zoo and the University of Plymouth, in Devon in England reported that they had left a computer keyboard in the enclosure of six Macaques for a month; not only did the monkeys produce nothing but [http://www.vivaria.net/experiments/notes/publication/NOTES_EN.pdf five pages] (Portable Document Format) consisting largely of the letter S, they started by attacking the keyboard with a stone, and continued by urinating and defecating on it. == Literature and popular culture == Jonathan Swift's ''Gulliver's Travels'' (1782) anticipates the central idea of the theorem, depicting a professor of the Grand Academy of Lagado who attempts to create a complete list of all knowledge of science by having his students constantly create random strings of letters by turning cranks on a mechanism (Part three, Chapter five). In "Inflexible Logic" by Russell Maloney, a short story that appeared in ''The New Yorker'' in 1940, the protagonist felt that his wealth put him under an obligation to support the sciences, and so he tested that theory. (He had heard the British Museum version of the story.) His monkeys immediately set to work typing classics of fiction and nonfiction. The rich man was amused to see unexpurgated versions of Samuel Pepys' diaries, of which he owned only a copy of a Thomas Bowdler edition. A similar theme was struck in the story "The Library of Babel" by Jorge Luis Borges, which contains a potentially unlimited number of volumes filled with random strings of characters. The narrator notes that every great work of literature is contained in the library; but these are outnumbered by the flawed works (which are vastly outnumbered by the gibberish). Popular culture references to this theorem include ''The Simpsons'' (in one episode, Montgomery Burns has his own room with 1000 dactylographic monkeys, one of which is chastised for mistyping a word in the opening sentence of ''A Tale of Two Cities "It was the best of times, it was the blurst of times."''), ''Family Guy'' (a group of monkeys is shown collaborating on a line from Shakespeare's ''Romeo and Juliet'' in a cut scene) and ''The Hitchhiker's Guide to the Galaxy'' (Ford Prefect (HHG) and Arthur Dent, under the influence of the Infinite Improbability Drive, are ambushed by an infinite number of monkeys who want their opinion on the monkeys' script for ''Hamlet''). In the comic strip ''Dilbert'', Dogbert tells Dilbert that his report would take "three monkeys, ten minutes". The theorem is also the basis of a one-act theater by David Ives called "Words, Words, Words", which appears in his collection ''All in the Timing''. In the one-act, three monkeys named John Milton, Jonathan Swift, and Franz Kafka have been confined to a cage by a scientist until they can write ''Hamlet.'' There is a humorous short story by R.A. Lafferty entitled "Been a Long, Long Time", in which an angel is punished by having to proofread all the output text until some future time (after trillions of Universes have been created and died) when the monkeys produce a perfect copy of Shakespeare's works. In Tom Stoppard's play Rosencrantz & Guildenstern are Dead, one character says, "If a million monkeys..." and then cannot continue, as the characters are actually "within" ''Hamlet'', one possible topic of this rule. He then finishes the sentence on a different topic. In 2000, the IETF Internet standards committee's April 1st RFC proposed an "Infinite Monkey Protocol Suite (IMPS)", a method of directing a farm of infinitely many monkeys over the Internet. [http://www.wilwheaton.net/ WWDN], the blog of author and actor Wil Wheaton, uses the slogan, "50,000 monkeys at 50,000 typewriters can't be wrong." His witticism won him a Bloggies in 2002 for the category "Best Tagline of a Weblog." A rather jocular quote by Robert Wilensky on the theorem is, ''"We've all heard that a million monkeys banging on a million typewriters will eventually reproduce the entire works of Shakespeare. Now, thanks to the Internet, we know this is not true."'' Comedian Bob Newhart has a stand-up comedy routine in which a lab technician monitoring an "infinite number of monkeys" experiment discovered that one of the monkeys has typed "To be, or not to be; that is the gezortenblatt." ==References== * ''[http://news.bbc.co.uk/1/3013959.stm No words to describe monkeys' play]'' (9 May 2003) BBC News * ''[http://www.cbsnews.com/stories/2003/05/12/national/main553500.shtml Monkey Theory Proven Wrong]'' (9 May 2003) CBS News *RFC 2795 — The Infinite Monkey Protocol Suite (IMPS) ==External links== *[http://user.tninet.se/~ecf599g/aardasnails/java/Monkey/webpages/index.html Monkey Shakespeare Simulator] *[http://www.vivaria.net/experiments/notes/publication/ Real Life Parody of the Keyboard/Monkey Concept] *[http://www.wired.com/news/culture/0,1284,58790,00.html Monkeys Don't Write Shakespeare] ==Footnotes== #Footnote_1 To some lay persons, ''"infinite monkeys"'' and ''"infinitely many monkeys"'' may seem synonymous; to mathematicians, the former is incorrect because ''each individual monkey is finite''. #Footnote_2 The word ''dactylographic'' appears in the English translation of Borel's book, and seems to be an Anglicization of a French word for typewriter, but in English, ''dactylography'' has come to mean the study of fingerprints. Probability theory Thought experiments Theorems
==the answer to life, the universe, and everything==
I've had another shot at the paragraph in the article: better now? --User:Paul A 12:50, 3 Apr 2004 (UTC) I think that's great! The only phrase that bothers me is : how long it takes the virtual monkeys to produce a complete Shakespearean play I may be indulging my taste for excessive precision by mentioning it, but i find it a little confusing in that it fails to distinguish between the collective nature of the project (in the sense that ''any'' monkey can complete a play) and the individual nature of the completion of any single play by ''one'' monkey. I especially fear my excessiveness in this case, in that i have no alternate wording to suggest for making the distinction clearly (tho i'll sleep on it).
One ''question'' does occur to me, tho, and its answer might help: isn't one user of the program letting their machine simulate one monkey, and if so, might that focus offer a less tricky wording? --User:JerzyUser talk:Jerzy :Each user of the program is letting their machine simulate the ''whole'' room-full-of-monkeys; one user simulates the room for a while, then the next user simulates it for a while, and so on. (In practice, because it's a random process that doesn't depend on past events, the set-up allows different users to simulate conceptually-subsequent slices of the room's history simultaneously. Ignore that sentence if it's giving you trouble.)
Conceptually, each monkey's pages of typing are added to a single communal pile that is then submitted a page at a time to be checked for matches. (Incidentally, the matching rules require not only that the output match the beginning of a play, but that the play begin at the top of a new page. If a monkey starts typing out a play halfway down the page, it doesn't count.) Consequently, the goal is for the entire play to appear out of the combined output, not out of the output of any one monkey.--User:Paul A 08:13, 5 Apr 2004 (UTC) Wow, way f'g cool. The elaborate design described suggests more effort than i thought on the part of the origninators, is moderating my disdain for the project. Tnx for Paul's dogged effort of research and description! --User:JerzyUser talk:Jerzy 14:39, 2004 Apr 5 (UTC) ::I believe that this simulator is just a random number generator, with probabilities updated in real-time based on variables like number of monkeys, monkey years, etc. In other words, the simulator does not simulate typewriters, pages, typed texts, monkeys, and bananas. How it detects a match is like this: when the RNG generates a certain number or numbers in a certain range (the range is calculated based on the calculated probabilities), the simulator then (said that it) "detects a match" and simulates the match (generates random matched text). I also believe that the random keystroke generator is just an accessory unrelated to the simulator and has nothing to do to the RNG (it just adds level of realism). User:202.65.112.42 04:10, 31 Oct 2004 (UTC) == What about older precursors, like Galileo or the hermeticists? == The emphasis of this article on modern thinkers is pretty odd, considering that this is infact a recurring conundrum throughout human history. Not just Galileo, but the umpteen hundred beautiful names of god in Islamic tradition. Oh, I just remembered, maybe something on Arthur C. Clarke too... --Cimon ==Huxley/Wilberforce debate== I have removed the following passage about the 1860 debate between T. H. Huxley and the Bishop of Oxford (Samuel Wilberforce), because it is so far as I can see unsustainable: :Samuel Wilberforce began the debate and, after making several scientific points regarding the plausibility of Charles Darwin work, concluded with William Paley’s argument that a watch implies the existence of a watchmaker, and similarly design in nature implies the existence of a Designer. :Thomas Henry Huxley then arose and put forward his now well-known argument that six eternal monkeys or apes typing on six eternal typewriters with unlimited amounts of paper and ink could, given enough time, produce a Psalms , a Shakespearean sonnet, or even a whole book, purely by chance that is, by random striking of the keys. :In the course of his presentation Thomas Henry Huxley pretended to find the 23rd Psalms among the reams of written gibberish produced by his six imaginary apes at their typewriters. He went on to make his point that, in the same way, molecular movement, given enough time and matter, could produce Samuel Wilberforce himself, purely by chance and without the work of any Designer or Creator god . No transcript of the Huxley/Wilberforce debate exists, but typewriters were not in commercial production at the time, and though prototypes did exist and might have been known to members of the British Association, it is unlikely that Huxley would have relied on that. No mention of the infinite monkey theorem can be found in Huxley's own accounts of the debate (for excerpts, see[http://aleph0.clarku.edu/huxley/guide7.html]), and the account given here differs markedly from contemporary descriptions. In my view the association of the infinite monkey theorem with the Oxford debate is an urban myth born of the fact that there really was some by-play about apes - but this was the famous exchange in which Wilberforce asked Huxley which side of his family the ape ancestry was on, and Huxley replied that he'd sooner be descended from a creature of mean intelligence than one of high intelligence who misused it in the way the Bishop had. User:Seglea 09:38, 30 Aug 2004 (UTC) == Gian-Carlo Rota == "Gian-Carlo Rota wrote in a textbook on probability (unfinished when he died):" but later completed by a cadre of determined simians... ==Explanation of my last edit== A few months back I created the section titled "pedantic usage note" a few months ago after someone objected to the inclusion of that usage note in an earlier part of the article, and seemed to think it was unduly pedantic. I think it's useful, especially in view of some of the terminological confusion I've seen elsewhere on Wikipedia since then that could have been avoided if those who were confused had seen this fact pointed out. But now someone has said that if the article admits to being pedantic, then there's probably more pedantry "hidden [sic!]" elsewhere in it, and it should therefore not be a featured article! I think that criticism has no merit. I did not remove any pedantry, but I did remove that which may confuse the overly literal-minded: the word ''pedantic''. Here's how that section now appears: ::Usage note ::To some lay persons, "infinite monkeys" and "infinitely many monkeys" may be synonymous; to the mathematician, the former is incorrect because ''each'' monkey ''individually'' is finite. User:Michael Hardy 21:22, 21 Sep 2004 (UTC) == Arthur Eddington == What is the context for the quote by Arthur Eddington? In particular, what sort of vessel was he talking about, and what manner of extreme probabilities was he illustrating by invoking this concept? --[[User:Eequor|User:Eequor[ υωρ]]] 04:43, 23 Sep 2004 (UTC) ==Orthography -- explanation of my recent edit== The so-called theorem is not a misnomer; rather, the ''name'' give to the theorem is a misnomer. That name is "the infinite monkey theorem", including the definite article. If the first sentence were about the theorem itself rather than about the phrase, then of course I would not highlight the definite article along with the rest of the phrase, since it's not part of the title phrase. When writing about a phrase rather than using the phrase to write about what it refers to, one italicizes it (see Wikipedia:Manual of Style). Usually it's better to right about the thing that the term refers to rather than about the term, e.g. "A dog is an animal that barks" rather than "''Dog'' refers to an animal that barks." When there are divergent meanings or when for some other reason it is better to write about the term rather than about the thing, one italicizes the term. And notice that of course I included the indefinite article when writing about dogs but I did ''not'' include it when writing about the ''word'' "dog"; it would be silly to say that a dog is a word referring to an animal that barks. A dog is an animal, not a word. User:Michael Hardy 19:56, 29 Sep 2004 (UTC) I just changed it from :The "infinite monkey theorem" is a misnomer.... to :"The infinite monkey theorem" is a misnomer.... Here's why. A while back I read an assertion that said :Blogs are short for web logs. That is absurd. It could have said :"Blogs" is short for "web logs". and then it would have made sense: it is the ''word'' that ''is'' (singular!) short for "web logs". Loosely, it could have said :Blogs is short for web logs. and I wouldn't have thougth the person who wrote it didn't understand what he was writing. But to write :Blogs are short for web logs. looks like failure to distinguish between writing about the word and writing about the things themselves. Now what if he had written :"The blogs" is short for "the web logs". That would have been correct. Contrast this with :The "blogs" is short for the "web logs. That makes no sense. What is the definite article doing there if it's not part of the expression being asserted to be short for "the web logs"? For the same reason, if this article had said :"Infinite monkey theorem" is a misnomer.... it would make sense, and if says :"The infinite monkey theorem" is a misnomer.... it would make sense. But if the word "the" is not part of the expression being asserted to be a misnomer, then what is it doing there? User:Michael Hardy 01:37, 31 Oct 2004 (UTC) == Infinite monkeys == Although this is not how it was originally stated, is it not the case that if an infinite number of monkeys begin typing random characters from a finite alphabet, one of them (indeed, infinitely many of them) will immediately produce any given text? User:Dcoetzee 03:16, 31 Oct 2004 (UTC) Huh, you're right. It is guaranteed that a ''finite'' number of monkeys will produce the text given an ''infinite'' amount of time. But given an ''infinite'' amount of monkeys, then they will produce the text immediately. Might want to mention that in the article somewhere. User:Fishal 04:19, 31 Oct 2004 (UTC) No it's not right. Assuming that the monkeys each type at one character per second, it will take them twenty seconds to produce all the twenty character texts and two hundred seconds to produce all the two hundred character texts. The minimum time taken for a particular text to be typed is directly proportional to the length of the text. Thus the works of Shakespeare will take much longer to appear than the works of Dashiell Hammett. I have amended the article accordingly. -- User:Derek Ross | User talk:Derek Ross 18:58, 2004 Oct 31 (UTC) :This is exactly what I meant. It's just that it's hard to say it precisely without being verbose. User:Dcoetzee 19:51, 31 Oct 2004 (UTC) ==Dactylography== Just to shed some more light on the term "dactylography". The origin of the word is neither French or English, it is, of course, Greek. dactylo (pronounced thaktilo) means finger graphy (pronnounced grafi) means writing. daktilografo is a verb in Greek meaning typing (writing with fingers). Regards, Konstantinos == Lottery odds == ''19,928,148,895,209,409,152,340,197,376, roughly equivalent to the probability of buying 4 lottery tickets consecutively and winning the jackpot each time.'' This feels at first like a good real-life illustration of something very improbable, but it for me it always invites the question: "which lottery"? The UK national lottery, for instance, has odds of about 14 million to one against winning, and the term "jackpot" is a bit confusing, too, because the prize is often shared among several winners. I can't think of a better real-world example for now, though: randomly choosing one of the world's six billion people a certain number of times in a row, perhaps? I don't have the maths to know how many times... User:Mswake 14:43, 31 Oct 2004 (UTC) Does it really matter if you have a perfect, concrete example or not? The words and their connotations carry the point across well enough, in my opinion. However, since you want suggestions, maybe being struck by lightning X times in a row, or dropping a bag full of coins and having them all land on their sides (vertically, I mean)? Anyway. (Sorry I don't have a username, I'll get one soon enough...) User:65.94.228.36 21:10, 25 Mar 2005 (UTC) ==About the Monkey== :"A single monkey who executes infinitely many keystrokes will eventually type out any given text, and an infinite number of monkeys will immediately produce all possible texts simultaneously." Um... I would have assumed that the point of using infinitely many monkeys over a single monkey typeing for an infinite amount of time is that, eventually, the finite monkey will die. User:FuncUser_talk:Func 15:05, 31 Oct 2004 (UTC) ::It comes down to the Zero-One law again. There is a positive, though trivially small, chance that a particular monkey sitting down at the typewriter types everything he is supposed to type perfectly. If that monkey is immortal, has a typewriter that won't break down, an infinite amount of paper and ink, and doesn't have to worry about the Big Crunch or the Last Judgement, he has infinitely many independent tries at getting it "right". The Zero-One law then kicks in -- the probability that he will ever get it right, given this infinite number of tries, has to be zero or one. Since the probability isn't zero -- indeed the chance wasn't zero if he only got one chance -- the probability must be one. ::If infinitely many monkeys sit down at their infinitely many typewriters, each has an independent (miniscule) chance of getting his "assignment" right on the first try. Since that chance is non-zero, and there are infinitely many independant attempts (each of the infinitely many monkey trying once) to do it, the chance that some one of the infinitely many monkeys gets it right on the first try must be one. Worse, as long as there are not infinitely many texts, we can put every other monkey doing Shakespeare, every other monkey of the ones left doing Faulkner, every other monkey not yet assigned doing Hemingway, every other monkey not yet assigned doing the Bible, and so on for as many works as you want to produce, and be guaranteed that in one "try", one monkey in each group will have produced his assigned text. ::Diagram to explain that last bit Monkeys 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ... Assigned to Shakespeare x x x x x x x x x x x x x ... Assigned to Faulkner x x x x x x x ... Assigned to Hemingway x x x ... Asigned to the Bible x x ... Still unassigned o o ... ::Pretty mind-boggling, huh? The joys of infinity! (The quoted text above is probably confusing to the reader, though. If my explanation worked for you, I could have a go at fixing it -- or if it really worked, you could fix it yourself.) User:Mpolo 15:50, Oct 31, 2004 (UTC) == Monkey research == Iremembered a researh of which six monkeys and six typewriters are locked up in a room. The result is monkeys banging on the typewriters and one mokey typeing the same letter over and over. ("bbbbbbbbbbbbbbbbbbbbbbbbbbbb") unno whether i should put it. User:SYSS Mouse 16:41, 31 Oct 2004 (UTC) :It's already there... in "Attempts at simulation". According to the article, it was an ''s''. User:Mpolo 16:54, Oct 31, 2004 (UTC) == Zero-One law == The Zero-One law is really irrelevant here. Kolmogorov's zero-one law is a deep result having to do with (Kolmogorov's own) formulation of probabilities on sequences. Here, however, we have a trivial calculation that can be done with high school mathematics, and was known (in this way or the other) to Pascal. User:Gadykozma 17:13, 31 Oct 2004 (UTC) :I wouldn't go so far as to say the zero-one law is irrelevent; this is certainly a demonstration of it. I've added a simpler proof from basics, though, that should hopefully be very easy to understand. User:Dcoetzee 18:24, 31 Oct 2004 (UTC) ==Proof section== It should be noted that the Proof section actually calculates the probability that the monkey will type the word ''banana'' with its first letter appearing at a position divisible by six within the first 6n characters rather than just the probability that the monkey will type the word ''banana'' within the first 6n characters. While this does not affect the proof aspect of the calculation, it does make an unlikely event seem even more unlikely than it really is. -- User:Derek Ross | User talk:Derek Ross 20:05, 2004 Oct 31 (UTC) ==Wait a moment== If an _infinite_ number of monkeys were typing on typewriters at least one would have to produce the entirety of Hamlet immediately (or at least with a delay of however long it takes this monkey to type it out) according to the logic of the zero-one law. -- Anonymous Reader :That's what the article says, isn't it? User:Michael Hardy 21:42, 1 Nov 2004 (UTC) :So how long would it take you to find the monkey in question? Suppose you start to read the first few letters, discount that monkey, move to the next one. Will it take a limitless amount of time, or less, on average? User:N12345n 15:00, 2004 Nov 2 (UTC) ::It will, of course, take a finite amount of time. Assuming monkey ''n'' is the first one who types it successfully, you'll only have to examine slightly more than ''n'' letters to find it on average, since most of the monkeys will screw up on the first letter. This ''n'' would probably be very large though. User:Dcoetzee 23:08, 2 Nov 2004 (UTC) :Interesting point. Assuming the article's 50 key typewriter, 2% of the monkeys will get the first letter correct, 2% of those will get the second letter correct, 2% of those will get third letter correct, etc. So you will have to examine somewhere around (n + n / 49) characters. User:Derek Ross | User talk:Derek Ross 04:13, 2004 Nov 3 (UTC) == Not a proof == That isnt a proof. An informal explanation, but not a proof. :It's intended to be palatable to people with little formal background. I changed the name to ''Proof sketch'' to avoid suggesting it's a complete, general proof. User:Dcoetzee 23:12, 2 Nov 2004 (UTC) == Best paragraph in Wikipedia == :In 2003, scientists at Paignton Zoo and the University of Plymouth, in Devon in England reported that they had left a computer keyboard in the enclosure of six Sulawesi Crested Macaques for a month; not only did the monkeys produce nothing but five pages consisting largely of the letter S, they started by attacking the keyboard with a stone, and continued by urinating and defecating on it. Bravo. User:Tempshill 22:51, 2 Nov 2004 (UTC) == Anal-retentive 'pedia == ''NB: the animal pictured here is a chimpanzee, which is an ape, not a monkey.'' Wow. Yes, it is an encyclopedia, but come on... :Yeah, I mean, really. It's better to just leave that out so people can point at us and go "how stupid ''are'' those editors, and how can they call ''that'' an encyclopedia", right? :You should be ''glad'' Wikipedia does its utmost to be accurate. :-) User:JRM 19:26, 2005 Mar 20 (UTC) ::I don't know, considering that whether the creature doing the typing is a monkey or not makes absolutely no difference to the theorem, it might be going a little overboard. It might be better to include it much more briefly and subtlely. I'm revising to that effect. User:Dcoetzee 20:03, 20 Mar 2005 (UTC) :::Now that ''is'' subtle. We won't let anyone make a monkey out of Wikipedia. User:JRM 20:24, 2005 Mar 20 (UTC) == Infinity Generation == For a long time, I've thought about ''Infinity Generation'' - that is, the generation of every possible permutation in a given man-made system. Since such systems are by their nature, finite, they cannot have infinite permutations. However, such systems can appear to humans to be infinite, since humans are unable to perceive or comprehend infinity directly. The level to which any system could appear to humans as infinite would be known as it's ''granular limit'' - a term which I coined after thinking about how when we look at a newspaper photo from far enough away, it appears solid, but when we look close enough we see the actual dots (or granules) that form the image. The ''granular limit'' would be the limit at which we could be close enough to see and comprehend the image as one entity, but not actually perceive its granules. Coming back to the point in discussion. I first conceptualized ''Infinity Generation'' when thinking about simple images on computers. Take for example a standard Digital Camera image of 640x480 pixels (total = 307,200) pixels. A low-colour format might make use of the GIF standard of 256 colours. To calculate all possible permutations of that Digital Camera image, one takes the number of pixels and raises it to the possible values that each pixel might have (ie: 307,200 ^ 256 = 6.0232391645008958015139896503459e+1404). With today's computing power, we could create a programme which would extrapolate all of those possible permutations of that image within a reasonable timeframe. Since an image of 640x480x256 would have an acceptable granular limit, it could easily be recognisable as any kind of object, person, whatever that exists (or doesn't exist) in our modern world. After generating every image permutation, we would have effectively generated Infinity - we would have within our images, every object, etc. that has, will, does, can't or never will exist. Taking this one step further, we could apply the Infinity Generation to texts, such as Shakespeare. We would configure the computer programme to a suitable length (ie: The length of a Shakespearean Play), and after a while, we would have every possible permutation of that length of text. Not only would we have the Shakespearean Play, we would have every text that fitted into the configured space. Text would be a better place to start than images, since images are made up of millions (or billions) of colours, and text has only the usual 26 characters (52 if counting uppercase), plus the usual punctuation, white space, etc. If left to run long enough, the Generators would come up with every thing that was, will or is written. Furthermore, we could expand our Infinity Generation to Films, Music and more. At the present time, such ''Infinity Generators'' would be notoriously hard to implement without some kind of AI, since we would be like Leonardo chipping away the stone, to reveal the statue beneath - we would have to separate the 99.999999999999999 (or whatever) % of chaff, to reveal the wheat - the comprehensible stuff. What we would do with all this stuff, even I don't know. But look at it this way - amongst all the chaff, the wheat of unimaginable power would lie - a formula for nuclear fusion, a cure for diseases. Plus all the new music, films, texts etc. Truly, from Infinity comes Creativity.... -- [http://www.mattwarne.com Matt 'devolution' Warne] == Chaos Monkeys == This argument assumes that the key-mashings of the monkeys is completely random from one letter to the next. It's possible that if this experiment were tried then perhaps there would emerge discernible monkeymashing patterns that might actually preclude the creation of long, coherent, flawless texts beyond a certain threshold. Simple limitlessness is not enough; in this case you also need random variation. Maybe it would turn out that, out of every X keys hit, a monkey will always wind up hitting two keys that are immediately next to each other after a certain level of fatigue sets in. Or maybe the monkey eventually develops a preference for a certain set of keys and ignores the rest. It's possible that an infinite number of monkeys hitting a keyboard is less like pi and more like the repeating decimal .142857... It's at least possible that there are some regular patterns that would emerge that would ruin this randomness-dependent theorem. If you can get an infinite set of random key generators, the theorem is sound. But I'm not convinced that monkeys are perfect random letter generators. It may monkeybe that there is a slight flaw in this monkeyplan. User:Mr. Billion 16:26, 12 May 2005 (UTC)
See other meanings of words starting from letter:
Words begining with Infinite_monkey_theorem:
Infinite_Monkey_Theorem
Infinite_Monkey_Theorem
Infinite_monkey_theorem
Infinite_monkey_theorem
Infinite Monkey Theorem
#REDIRECT Infinite monkey theorem
Infinite Monkey Theorem
#REDIRECT Talk:Infinite monkey theorem
Infinite monkey theorem
[[Image:monkey-typing.jpg|thumb|According to Kolmogorov's zero-one law, given enough time, a chimpanzee like this one typing at random will eventually type out a copy of one of William Shakespeare's plays.]] The infinite monkey theorem#Footnotes says that almost surely (i.e. with probability equal to 1) a monkey hitting keys at random on a typewriter keyboard will eventually type every book in France's Bibliothèque nationale de France (National library). In the restatement of the theorem most popular among English speakers, the monkeys eventually type out the collected works of William Shakespeare; others replace the National Library with the British Museum or the Library of Congress. The name is a popular misnomer for an idea from Félix Édouard Justin Émile Borel's book on probability, published in 1909, which introduced the concept of "dactylographic#Footnotes monkeys". A popular statement of the theorem is that an infinity number of monkeys typing for an infinite amount of time will produce a given text. To insist on both infinities, however, is excessive. A single immortal monkey who executes infinitely many keystrokes will eventually type out any given text, and an infinite number of monkeys will begin producing ''all'' possible texts immediately, with no wait. ==Proof sketch== The infinite monkey theorem is relatively straightforward to prove. If two events are statistically independent, meaning neither affects the outcome of the other, then the probability of both happening is equivalent to the product of the probabilities of each one happening on its own. For example, if the chance of rain in Sydney on a particular day is 0.3 and the chance of an earthquake in San Francisco on that day is 0.8, the chance of both happening on that same day is 0.3 × 0.8 = 0.24. Now, suppose the typewriter has 50 keys, and the monkey is trying to type the word "banana". Typing at random, the chance that the first letter typed is ''b'' is 1/50, as is the chance that the second letter typed is ''a'', and so on. These events are independent, so the chance of the first six letters matching "banana" is 1/506. For the same reason, the chance that the next 6 letters match "banana" is also 1/506, and so on. Now, the chance of ''not'' typing "banana" in each block of 6 letters is 1 − 1/506. Because each block is typed independently, the chance, ''X'', of not typing "banana" in any of the first ''n'' blocks of 6 letters is ''X'' = (1 − 1/506)''n''. As ''n'' grows, ''X'' gets smaller. For an ''n'' of a million, ''X'' is 99.99%, but for an ''n'' of 10 billion it is 53% and for an ''n'' of 100 billion it is 0.17%. As ''n'' approaches infinity, the probability ''X'' limit (mathematics) zero; that is, by making ''n'' large enough, ''X'' can be made as small as one likes. If we were to count occurrences of "banana" that crossed blocks, ''X'' would approach zero even more quickly. The same argument applies if the monkey were typing any other string of characters of any length. The same argument shows why infinitely many monkeys produce a text as quickly as it would be produced by a perfectly accurate human typist copying it from the original. In this case ''X'' = (1 − 1/506)''n'' where ''X'' represents the probability that none of the first ''n'' monkeys types "banana" correctly on their first try. When we consider 100 billion monkeys, the probability falls to 0.17%, and as the number of monkeys, ''n'' increases to infinity the value of ''X'' (the probability of all the monkeys failing to reproduce the given text) decreases to zero. This is equivalent to stating that the probability that one or more of an infinite number of monkeys will produce a given text on the first try is 100%, or that it is certain they will do so. The only difficulty remaining is in locating a successful monkey. The theorem exemplifies a proposition in the probability theory called Andrey Nikolaevich Kolmogorov's Kolmogorov's zero-one law, which was published in 1933, 24 years after Borel's book cited above. ==Probabilities== Ignoring punctuation, spacing, and capitalization, and assuming a uniform distribution of letters, a monkey has one chance in 26 of correctly typing the first letter of ''Hamlet.'' It has one chance in 676 (26 times 26) of typing the first two letters. Because the probability shrinks exponential growthly, at 20 letters it already has only one chance in 2620 = 19,928,148,895,209,409,152,340,197,376, roughly equivalent to the probability of buying 4 lottery tickets consecutively and winning the jackpot each time. In the case of the entire text of ''Hamlet'', the probabilities are so vanishingly small they can barely be conceived in human terms. The text of Hamlet, even stripped of all punctuation, contains well over 130,000 letters. The mere fact that there is a chance, however unlikely, is the key to the "infinite monkey theorem", because Kolmogorov's zero-one law says that such an infinite series of independent events must have a probability of zero or one. Since we have shown above that the chance is not zero, it must be one. To consider that an event this unlikely is guaranteed to occur given infinite time can give a sense of the vastness of infinity. Gian-Carlo Rota wrote in a textbook on probability (unfinished when he died): :"If the monkey could type one keystroke every nanosecond, the expected waiting time until the monkey types out ''Hamlet'' is so long that the estimated age of the universe is insignificant by comparison ... this is not a practical method for writing plays. (We cannot resist the temptation to quote from Alfred North Whitehead, 'I will not go to infinity'.)" In ''The Nature of the Physical World: The Gifford Lectures'' (Macmillan, New York, 1929, page 72) the physicist Arthur Eddington wrote: :"If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel." In physics, then, the force of the "monkeys argument" lies not in the probability that the monkeys will "eventually" produce something intelligible, but in the practical reality that they will not. Any physical process that is even less likely than such monkeys' success is effectively impossible; this is the statistical basis of the second law of thermodynamics. ==Myth about origins== It is often reported, though highly improbable, that Borel's use of monkeys and typewriters in his theorem was inspired by an argument used by Thomas Henry Huxley on June 30, 1860. Huxley was involved in a debate with the Anglican Bishop of Oxford, Samuel Wilberforce, held at a meeting of the British Association for the Advancement of Science at Oxford, of which Wilberforce was a vice-president, and was sparked by the publication of Charles Darwin’s ''Origin of Species'' seven months earlier, in November 1859. No transcript of the debate exists, but neither contemporary accounts of it nor Huxley's later recollections include any reference to the infinite monkey theorem. The association of the debate with the infinite monkey theorem is probably an urban myth triggered by the fact that the debate certainly did include some byplay about apes: the bishop asked whether Huxley was descended from an ape on his grandmother's or his grandfather's side, and Huxley responded that he would rather be descended from an ape than from someone who argued like the bishop. It is most unlikely that Huxley would have referred to a typewriter. Although patents for machines resembling modern typewriters were granted as early as 1714, commercial production of typewriters did not begin until 1870, and a skilled debater like Huxley would hardly have let his point depend on a device whose existence would have been unknown to most of his audience. ==Infinite monkey experiments== [http://user.tninet.se/~ecf599g/aardasnails/java/Monkey/webpages/index.html#results "The Monkey Shakespeare Simulator"] website, launched on July 1, 2003, contains a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. As of January 3 2005, matches as long as 24 consecutive letters, four words have been recorded ("RUMOUR. Open your ears; 9r"5j5&?OWTY Z0d "B-nEoF.vjSqj[..." from ''Henry VI, part 2''). Due to processing power limitations, the program uses a probabilistic model (by using a random number generator) instead of actually generating random text and comparing it to Shakespeare. When the simulator "detects a match" (that is, the RNG generates a certain value or a value within a certain range), the simulator simulates the match by generating matched text. In 2003, scientists at Paignton Zoo and the University of Plymouth, in Devon in England reported that they had left a computer keyboard in the enclosure of six Macaques for a month; not only did the monkeys produce nothing but [http://www.vivaria.net/experiments/notes/publication/NOTES_EN.pdf five pages] (Portable Document Format) consisting largely of the letter S, they started by attacking the keyboard with a stone, and continued by urinating and defecating on it. == Literature and popular culture == Jonathan Swift's ''Gulliver's Travels'' (1782) anticipates the central idea of the theorem, depicting a professor of the Grand Academy of Lagado who attempts to create a complete list of all knowledge of science by having his students constantly create random strings of letters by turning cranks on a mechanism (Part three, Chapter five). In "Inflexible Logic" by Russell Maloney, a short story that appeared in ''The New Yorker'' in 1940, the protagonist felt that his wealth put him under an obligation to support the sciences, and so he tested that theory. (He had heard the British Museum version of the story.) His monkeys immediately set to work typing classics of fiction and nonfiction. The rich man was amused to see unexpurgated versions of Samuel Pepys' diaries, of which he owned only a copy of a Thomas Bowdler edition. A similar theme was struck in the story "The Library of Babel" by Jorge Luis Borges, which contains a potentially unlimited number of volumes filled with random strings of characters. The narrator notes that every great work of literature is contained in the library; but these are outnumbered by the flawed works (which are vastly outnumbered by the gibberish). Popular culture references to this theorem include ''The Simpsons'' (in one episode, Montgomery Burns has his own room with 1000 dactylographic monkeys, one of which is chastised for mistyping a word in the opening sentence of ''A Tale of Two Cities "It was the best of times, it was the blurst of times."''), ''Family Guy'' (a group of monkeys is shown collaborating on a line from Shakespeare's ''Romeo and Juliet'' in a cut scene) and ''The Hitchhiker's Guide to the Galaxy'' (Ford Prefect (HHG) and Arthur Dent, under the influence of the Infinite Improbability Drive, are ambushed by an infinite number of monkeys who want their opinion on the monkeys' script for ''Hamlet''). In the comic strip ''Dilbert'', Dogbert tells Dilbert that his report would take "three monkeys, ten minutes". The theorem is also the basis of a one-act theater by David Ives called "Words, Words, Words", which appears in his collection ''All in the Timing''. In the one-act, three monkeys named John Milton, Jonathan Swift, and Franz Kafka have been confined to a cage by a scientist until they can write ''Hamlet.'' There is a humorous short story by R.A. Lafferty entitled "Been a Long, Long Time", in which an angel is punished by having to proofread all the output text until some future time (after trillions of Universes have been created and died) when the monkeys produce a perfect copy of Shakespeare's works. In Tom Stoppard's play Rosencrantz & Guildenstern are Dead, one character says, "If a million monkeys..." and then cannot continue, as the characters are actually "within" ''Hamlet'', one possible topic of this rule. He then finishes the sentence on a different topic. In 2000, the IETF Internet standards committee's April 1st RFC proposed an "Infinite Monkey Protocol Suite (IMPS)", a method of directing a farm of infinitely many monkeys over the Internet. [http://www.wilwheaton.net/ WWDN], the blog of author and actor Wil Wheaton, uses the slogan, "50,000 monkeys at 50,000 typewriters can't be wrong." His witticism won him a Bloggies in 2002 for the category "Best Tagline of a Weblog." A rather jocular quote by Robert Wilensky on the theorem is, ''"We've all heard that a million monkeys banging on a million typewriters will eventually reproduce the entire works of Shakespeare. Now, thanks to the Internet, we know this is not true."'' Comedian Bob Newhart has a stand-up comedy routine in which a lab technician monitoring an "infinite number of monkeys" experiment discovered that one of the monkeys has typed "To be, or not to be; that is the gezortenblatt." ==References== * ''[http://news.bbc.co.uk/1/3013959.stm No words to describe monkeys' play]'' (9 May 2003) BBC News * ''[http://www.cbsnews.com/stories/2003/05/12/national/main553500.shtml Monkey Theory Proven Wrong]'' (9 May 2003) CBS News *RFC 2795 — The Infinite Monkey Protocol Suite (IMPS) ==External links== *[http://user.tninet.se/~ecf599g/aardasnails/java/Monkey/webpages/index.html Monkey Shakespeare Simulator] *[http://www.vivaria.net/experiments/notes/publication/ Real Life Parody of the Keyboard/Monkey Concept] *[http://www.wired.com/news/culture/0,1284,58790,00.html Monkeys Don't Write Shakespeare] ==Footnotes== #Footnote_1 To some lay persons, ''"infinite monkeys"'' and ''"infinitely many monkeys"'' may seem synonymous; to mathematicians, the former is incorrect because ''each individual monkey is finite''. #Footnote_2 The word ''dactylographic'' appears in the English translation of Borel's book, and seems to be an Anglicization of a French word for typewriter, but in English, ''dactylography'' has come to mean the study of fingerprints. Probability theory Thought experiments Theorems
Infinite monkey theorem
==the answer to life, the universe, and everything==
"attempting to create a complete list of all knowledge of science by having his students constantly create random strings of letters by turning cranks on a mechanism."
why don't we try this on wikipedia? also, has anyone ever combined random text generation (though perhaps using a dictionary) with genetic algorithms? something where each revision is human-reviewed, given marks, and then grammar, syntax and phrases from high scoring drafts are kept for future revisions, and where material from unpopular drafts is digarded?
:Yes. See [Darwinian Poetry http://www.codeasart.com/poetry/darwin.html]. User:Dcoetzee 18:21, 31 Oct 2004 (UTC)
==Various aspects==
If it does indeed take an infinite number of monkeys an infinite amount of time to hammer out Hamlet, then wouldn't it be provable that a ''finite'' number of monkeys ''couldn't'' do it -- even with an infinite amount of time? No. An infinity of infinities is still just the same old infinity. Just one infinity is enough. So one monkey and eternity, or an infinite number of monkeys and not very long at all. However, expecting an infinite number of monkeys to type on a single typewriter may not work. The same theorem of probability says any finite number of monkeys would do the job in some finite amount of time. If you specify a ''particular'' finite number of years, the probability that the job would be done before then is less than 1. It approaches 1 as time approaches infinity. The source seems to be Borel's book on probability, published in about 1910. There's also a short story based on this idea that appeared in the ''New Yorker'' c. 1940.
Borel's book also has an account of the Borel-Cantelli theorems. The title of this article may be misread as referring to "infinite monkeys," which is inaccurate. It's not the monkeys that are infinite; each monkey is finite. Rather, it is the ''number'' of monkeys that it asserted to be infinite. (Unnecessarily so, if one allows an infinite amount of time.) One should speak of infinitely many monkeys, not of "infinite monkeys."
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The title of this article as it now stands Borel's dactylographic monkey theorem does not express the concept as it it commonly known. Indeed typing the above expression into google returns zero results. If the origin of this concept resides with Borel then this should be stated within the body of the article, and the title of the article should be where people would expect to find it as per Wikipedia:Naming conventions (common names). User:Mintguy
This article is nice! I was just about to link ''dactylographic'' in the article to ''dactylography'', as uncommon words should be linked for explanation, but decided to look it up first: ''dactylography: n. Chiefly US the scientific study of fingerprints for purposes of identification.''
Why do the monkeys have to have anything to do with fingerprint identification? And if this is true, maybe a short explanation of this could be put into the article. Thanks, User:Snoyes 02:42 Mar 7, 2003 (UTC)
:It seems to be a false friend - the French term "dactylographier" means "to type." - User:Montrealais
Is the theorem really a case of Kolmogorov's 0-1 law? The law tells us that the probability infinitely many copies are typed is either 0 or 1, but doesn't tell us which. Conversely, we know the probability at least one copy is typed is nonzero, but Kolmogorov's law no longer applies since the probability something eventually occurs isn't in the tail sigma-algebra. As I learned it, this theorem was a consequence of the second Borel-Cantelli lemma. user:Kevinatilusa
Re: "grotesquely incorrect". It's just grammar, chill out. Clearly, I did not mean an infinitely large keystroke. User:Daniel Quinlan 00:00, Dec 11, 2003 (UTC)
== Counting to Infinity Before Speaking Harshly ==
I reverted an edit that changed a 'graph to
: There need not be infinity many monkeys; a single monkey who executes infinitely many keystrokes suffices. This is inherent in the concept of infinity.
by adding the link and the last sentence. First i must comment that there is a laudable insight in the sentence. Second, i must add that it doesn't belong in the article.
While i'm confident that i know what i'm talking about here, i see some obligation to bear in mind that i am proposing to correct others who have shown similar confidence, and i don't know the content of the Kolmogorov theorem in question, except by what i would called "inferred reputation". It is for that reason that i request someone who has studied it in a formal setting to check my understanding that:
: K. addresses limits as a variable approaches infinity rather than transfinite cardinals (or or transfinite ordinals).
I intend in any case to lay out the fact that there is no "concept of infinity" for that equivalence to be inherant in, but three such concepts:
# A dualist philosophical concept that has only a poetic relationship to mathematics, but is nevertheless durable and popular.
# A concept of Isaac Newton and Gottfried Leibniz, fundamental to calculus, which deals with the in-fin-ite simply in terms of the absence of end, and uses "infinity" simply as a shorthand for conditions that specify the non-existence of specific limits, and the existence and numerical value of specific limits, without positing any infitite number.
# A concept of Georg Cantor, of transfinite number, fundamental to discussion of differences in the natures of various infinite collections of abstractions, but almost independent of limit-theory.
I am convinced that Infinite monkey theorem has suffered seriously by failure to recognize these distinctions, and propose to thoroughly rework the muddled language that has called forth the idea that the seductive concept of transfinite arithmetic helps to understand this subject. But not in the next 12 hours. --User:JerzyUser talk:Jerzy 09:43, 2004 Mar 2 (UTC)
:Jerzy is right. There are multiple concepts of "infinity" (his list is less than complete, but never mind that) and Kolmogorov was writing about a limit as a finite-value variable approaches infinity, as is done in calculus. Transfinite arithmetic should not be brought in to this article. User:Michael Hardy 20:37, 2 Mar 2004 (UTC)
:... but on the other hand, I don't see that the article has suffered from exclusion of these distinctions, except when that comment about what is "inherent in the concept of infinity" was added. User:Michael Hardy 20:48, 2 Mar 2004 (UTC)
== 'Net Shakespeare Simulator ==
I am not an enthusiast of this activity; in fact, i think those who install it should dedicate their processor cycles to something more socially valuable, like downloading pornography. But we should document it in a way that does more justice to the limitations that it imposes upon itself.
The article invites the inference that the "records" of a dozen or so characters reflect the longest two or three words that both WS and the monkey-engine have put together. My strong impression (from an admittedly brief visit to the site) is that in fact they represent the longest matches between the engine and any string that ''begins'' a WS work (or is it a WS play? Do the sonnets count? (Does his "second-best bed" will count?!)) I'm not going to be the one to get the details right, or state them fluently, but i think the present description is inadequate. --User:JerzyUser talk:Jerzy 01:28, 2004 Mar 31 (UTC)
:It's just the plays; a list of exactly which plays heads the FAQ on the site. And you're right about it just being beginnings.I've had another shot at the paragraph in the article: better now? --User:Paul A 12:50, 3 Apr 2004 (UTC) I think that's great! The only phrase that bothers me is : how long it takes the virtual monkeys to produce a complete Shakespearean play I may be indulging my taste for excessive precision by mentioning it, but i find it a little confusing in that it fails to distinguish between the collective nature of the project (in the sense that ''any'' monkey can complete a play) and the individual nature of the completion of any single play by ''one'' monkey. I especially fear my excessiveness in this case, in that i have no alternate wording to suggest for making the distinction clearly (tho i'll sleep on it).
One ''question'' does occur to me, tho, and its answer might help: isn't one user of the program letting their machine simulate one monkey, and if so, might that focus offer a less tricky wording? --User:JerzyUser talk:Jerzy :Each user of the program is letting their machine simulate the ''whole'' room-full-of-monkeys; one user simulates the room for a while, then the next user simulates it for a while, and so on. (In practice, because it's a random process that doesn't depend on past events, the set-up allows different users to simulate conceptually-subsequent slices of the room's history simultaneously. Ignore that sentence if it's giving you trouble.)
Conceptually, each monkey's pages of typing are added to a single communal pile that is then submitted a page at a time to be checked for matches. (Incidentally, the matching rules require not only that the output match the beginning of a play, but that the play begin at the top of a new page. If a monkey starts typing out a play halfway down the page, it doesn't count.) Consequently, the goal is for the entire play to appear out of the combined output, not out of the output of any one monkey.--User:Paul A 08:13, 5 Apr 2004 (UTC) Wow, way f'g cool. The elaborate design described suggests more effort than i thought on the part of the origninators, is moderating my disdain for the project. Tnx for Paul's dogged effort of research and description! --User:JerzyUser talk:Jerzy 14:39, 2004 Apr 5 (UTC) ::I believe that this simulator is just a random number generator, with probabilities updated in real-time based on variables like number of monkeys, monkey years, etc. In other words, the simulator does not simulate typewriters, pages, typed texts, monkeys, and bananas. How it detects a match is like this: when the RNG generates a certain number or numbers in a certain range (the range is calculated based on the calculated probabilities), the simulator then (said that it) "detects a match" and simulates the match (generates random matched text). I also believe that the random keystroke generator is just an accessory unrelated to the simulator and has nothing to do to the RNG (it just adds level of realism). User:202.65.112.42 04:10, 31 Oct 2004 (UTC) == What about older precursors, like Galileo or the hermeticists? == The emphasis of this article on modern thinkers is pretty odd, considering that this is infact a recurring conundrum throughout human history. Not just Galileo, but the umpteen hundred beautiful names of god in Islamic tradition. Oh, I just remembered, maybe something on Arthur C. Clarke too... --Cimon ==Huxley/Wilberforce debate== I have removed the following passage about the 1860 debate between T. H. Huxley and the Bishop of Oxford (Samuel Wilberforce), because it is so far as I can see unsustainable: :Samuel Wilberforce began the debate and, after making several scientific points regarding the plausibility of Charles Darwin work, concluded with William Paley’s argument that a watch implies the existence of a watchmaker, and similarly design in nature implies the existence of a Designer. :Thomas Henry Huxley then arose and put forward his now well-known argument that six eternal monkeys or apes typing on six eternal typewriters with unlimited amounts of paper and ink could, given enough time, produce a Psalms , a Shakespearean sonnet, or even a whole book, purely by chance that is, by random striking of the keys. :In the course of his presentation Thomas Henry Huxley pretended to find the 23rd Psalms among the reams of written gibberish produced by his six imaginary apes at their typewriters. He went on to make his point that, in the same way, molecular movement, given enough time and matter, could produce Samuel Wilberforce himself, purely by chance and without the work of any Designer or Creator god . No transcript of the Huxley/Wilberforce debate exists, but typewriters were not in commercial production at the time, and though prototypes did exist and might have been known to members of the British Association, it is unlikely that Huxley would have relied on that. No mention of the infinite monkey theorem can be found in Huxley's own accounts of the debate (for excerpts, see[http://aleph0.clarku.edu/huxley/guide7.html]), and the account given here differs markedly from contemporary descriptions. In my view the association of the infinite monkey theorem with the Oxford debate is an urban myth born of the fact that there really was some by-play about apes - but this was the famous exchange in which Wilberforce asked Huxley which side of his family the ape ancestry was on, and Huxley replied that he'd sooner be descended from a creature of mean intelligence than one of high intelligence who misused it in the way the Bishop had. User:Seglea 09:38, 30 Aug 2004 (UTC) == Gian-Carlo Rota == "Gian-Carlo Rota wrote in a textbook on probability (unfinished when he died):" but later completed by a cadre of determined simians... ==Explanation of my last edit== A few months back I created the section titled "pedantic usage note" a few months ago after someone objected to the inclusion of that usage note in an earlier part of the article, and seemed to think it was unduly pedantic. I think it's useful, especially in view of some of the terminological confusion I've seen elsewhere on Wikipedia since then that could have been avoided if those who were confused had seen this fact pointed out. But now someone has said that if the article admits to being pedantic, then there's probably more pedantry "hidden [sic!]" elsewhere in it, and it should therefore not be a featured article! I think that criticism has no merit. I did not remove any pedantry, but I did remove that which may confuse the overly literal-minded: the word ''pedantic''. Here's how that section now appears: ::Usage note ::To some lay persons, "infinite monkeys" and "infinitely many monkeys" may be synonymous; to the mathematician, the former is incorrect because ''each'' monkey ''individually'' is finite. User:Michael Hardy 21:22, 21 Sep 2004 (UTC) == Arthur Eddington == What is the context for the quote by Arthur Eddington? In particular, what sort of vessel was he talking about, and what manner of extreme probabilities was he illustrating by invoking this concept? --[[User:Eequor|User:Eequor[ υωρ]]] 04:43, 23 Sep 2004 (UTC) ==Orthography -- explanation of my recent edit== The so-called theorem is not a misnomer; rather, the ''name'' give to the theorem is a misnomer. That name is "the infinite monkey theorem", including the definite article. If the first sentence were about the theorem itself rather than about the phrase, then of course I would not highlight the definite article along with the rest of the phrase, since it's not part of the title phrase. When writing about a phrase rather than using the phrase to write about what it refers to, one italicizes it (see Wikipedia:Manual of Style). Usually it's better to right about the thing that the term refers to rather than about the term, e.g. "A dog is an animal that barks" rather than "''Dog'' refers to an animal that barks." When there are divergent meanings or when for some other reason it is better to write about the term rather than about the thing, one italicizes the term. And notice that of course I included the indefinite article when writing about dogs but I did ''not'' include it when writing about the ''word'' "dog"; it would be silly to say that a dog is a word referring to an animal that barks. A dog is an animal, not a word. User:Michael Hardy 19:56, 29 Sep 2004 (UTC) I just changed it from :The "infinite monkey theorem" is a misnomer.... to :"The infinite monkey theorem" is a misnomer.... Here's why. A while back I read an assertion that said :Blogs are short for web logs. That is absurd. It could have said :"Blogs" is short for "web logs". and then it would have made sense: it is the ''word'' that ''is'' (singular!) short for "web logs". Loosely, it could have said :Blogs is short for web logs. and I wouldn't have thougth the person who wrote it didn't understand what he was writing. But to write :Blogs are short for web logs. looks like failure to distinguish between writing about the word and writing about the things themselves. Now what if he had written :"The blogs" is short for "the web logs". That would have been correct. Contrast this with :The "blogs" is short for the "web logs. That makes no sense. What is the definite article doing there if it's not part of the expression being asserted to be short for "the web logs"? For the same reason, if this article had said :"Infinite monkey theorem" is a misnomer.... it would make sense, and if says :"The infinite monkey theorem" is a misnomer.... it would make sense. But if the word "the" is not part of the expression being asserted to be a misnomer, then what is it doing there? User:Michael Hardy 01:37, 31 Oct 2004 (UTC) == Infinite monkeys == Although this is not how it was originally stated, is it not the case that if an infinite number of monkeys begin typing random characters from a finite alphabet, one of them (indeed, infinitely many of them) will immediately produce any given text? User:Dcoetzee 03:16, 31 Oct 2004 (UTC) Huh, you're right. It is guaranteed that a ''finite'' number of monkeys will produce the text given an ''infinite'' amount of time. But given an ''infinite'' amount of monkeys, then they will produce the text immediately. Might want to mention that in the article somewhere. User:Fishal 04:19, 31 Oct 2004 (UTC) No it's not right. Assuming that the monkeys each type at one character per second, it will take them twenty seconds to produce all the twenty character texts and two hundred seconds to produce all the two hundred character texts. The minimum time taken for a particular text to be typed is directly proportional to the length of the text. Thus the works of Shakespeare will take much longer to appear than the works of Dashiell Hammett. I have amended the article accordingly. -- User:Derek Ross | User talk:Derek Ross 18:58, 2004 Oct 31 (UTC) :This is exactly what I meant. It's just that it's hard to say it precisely without being verbose. User:Dcoetzee 19:51, 31 Oct 2004 (UTC) ==Dactylography== Just to shed some more light on the term "dactylography". The origin of the word is neither French or English, it is, of course, Greek. dactylo (pronounced thaktilo) means finger graphy (pronnounced grafi) means writing. daktilografo is a verb in Greek meaning typing (writing with fingers). Regards, Konstantinos == Lottery odds == ''19,928,148,895,209,409,152,340,197,376, roughly equivalent to the probability of buying 4 lottery tickets consecutively and winning the jackpot each time.'' This feels at first like a good real-life illustration of something very improbable, but it for me it always invites the question: "which lottery"? The UK national lottery, for instance, has odds of about 14 million to one against winning, and the term "jackpot" is a bit confusing, too, because the prize is often shared among several winners. I can't think of a better real-world example for now, though: randomly choosing one of the world's six billion people a certain number of times in a row, perhaps? I don't have the maths to know how many times... User:Mswake 14:43, 31 Oct 2004 (UTC) Does it really matter if you have a perfect, concrete example or not? The words and their connotations carry the point across well enough, in my opinion. However, since you want suggestions, maybe being struck by lightning X times in a row, or dropping a bag full of coins and having them all land on their sides (vertically, I mean)? Anyway. (Sorry I don't have a username, I'll get one soon enough...) User:65.94.228.36 21:10, 25 Mar 2005 (UTC) ==About the Monkey== :"A single monkey who executes infinitely many keystrokes will eventually type out any given text, and an infinite number of monkeys will immediately produce all possible texts simultaneously." Um... I would have assumed that the point of using infinitely many monkeys over a single monkey typeing for an infinite amount of time is that, eventually, the finite monkey will die. User:FuncUser_talk:Func 15:05, 31 Oct 2004 (UTC) ::It comes down to the Zero-One law again. There is a positive, though trivially small, chance that a particular monkey sitting down at the typewriter types everything he is supposed to type perfectly. If that monkey is immortal, has a typewriter that won't break down, an infinite amount of paper and ink, and doesn't have to worry about the Big Crunch or the Last Judgement, he has infinitely many independent tries at getting it "right". The Zero-One law then kicks in -- the probability that he will ever get it right, given this infinite number of tries, has to be zero or one. Since the probability isn't zero -- indeed the chance wasn't zero if he only got one chance -- the probability must be one. ::If infinitely many monkeys sit down at their infinitely many typewriters, each has an independent (miniscule) chance of getting his "assignment" right on the first try. Since that chance is non-zero, and there are infinitely many independant attempts (each of the infinitely many monkey trying once) to do it, the chance that some one of the infinitely many monkeys gets it right on the first try must be one. Worse, as long as there are not infinitely many texts, we can put every other monkey doing Shakespeare, every other monkey of the ones left doing Faulkner, every other monkey not yet assigned doing Hemingway, every other monkey not yet assigned doing the Bible, and so on for as many works as you want to produce, and be guaranteed that in one "try", one monkey in each group will have produced his assigned text. ::Diagram to explain that last bit Monkeys 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ... Assigned to Shakespeare x x x x x x x x x x x x x ... Assigned to Faulkner x x x x x x x ... Assigned to Hemingway x x x ... Asigned to the Bible x x ... Still unassigned o o ... ::Pretty mind-boggling, huh? The joys of infinity! (The quoted text above is probably confusing to the reader, though. If my explanation worked for you, I could have a go at fixing it -- or if it really worked, you could fix it yourself.) User:Mpolo 15:50, Oct 31, 2004 (UTC) == Monkey research == Iremembered a researh of which six monkeys and six typewriters are locked up in a room. The result is monkeys banging on the typewriters and one mokey typeing the same letter over and over. ("bbbbbbbbbbbbbbbbbbbbbbbbbbbb") unno whether i should put it. User:SYSS Mouse 16:41, 31 Oct 2004 (UTC) :It's already there... in "Attempts at simulation". According to the article, it was an ''s''. User:Mpolo 16:54, Oct 31, 2004 (UTC) == Zero-One law == The Zero-One law is really irrelevant here. Kolmogorov's zero-one law is a deep result having to do with (Kolmogorov's own) formulation of probabilities on sequences. Here, however, we have a trivial calculation that can be done with high school mathematics, and was known (in this way or the other) to Pascal. User:Gadykozma 17:13, 31 Oct 2004 (UTC) :I wouldn't go so far as to say the zero-one law is irrelevent; this is certainly a demonstration of it. I've added a simpler proof from basics, though, that should hopefully be very easy to understand. User:Dcoetzee 18:24, 31 Oct 2004 (UTC) ==Proof section== It should be noted that the Proof section actually calculates the probability that the monkey will type the word ''banana'' with its first letter appearing at a position divisible by six within the first 6n characters rather than just the probability that the monkey will type the word ''banana'' within the first 6n characters. While this does not affect the proof aspect of the calculation, it does make an unlikely event seem even more unlikely than it really is. -- User:Derek Ross | User talk:Derek Ross 20:05, 2004 Oct 31 (UTC) ==Wait a moment== If an _infinite_ number of monkeys were typing on typewriters at least one would have to produce the entirety of Hamlet immediately (or at least with a delay of however long it takes this monkey to type it out) according to the logic of the zero-one law. -- Anonymous Reader :That's what the article says, isn't it? User:Michael Hardy 21:42, 1 Nov 2004 (UTC) :So how long would it take you to find the monkey in question? Suppose you start to read the first few letters, discount that monkey, move to the next one. Will it take a limitless amount of time, or less, on average? User:N12345n 15:00, 2004 Nov 2 (UTC) ::It will, of course, take a finite amount of time. Assuming monkey ''n'' is the first one who types it successfully, you'll only have to examine slightly more than ''n'' letters to find it on average, since most of the monkeys will screw up on the first letter. This ''n'' would probably be very large though. User:Dcoetzee 23:08, 2 Nov 2004 (UTC) :Interesting point. Assuming the article's 50 key typewriter, 2% of the monkeys will get the first letter correct, 2% of those will get the second letter correct, 2% of those will get third letter correct, etc. So you will have to examine somewhere around (n + n / 49) characters. User:Derek Ross | User talk:Derek Ross 04:13, 2004 Nov 3 (UTC) == Not a proof == That isnt a proof. An informal explanation, but not a proof. :It's intended to be palatable to people with little formal background. I changed the name to ''Proof sketch'' to avoid suggesting it's a complete, general proof. User:Dcoetzee 23:12, 2 Nov 2004 (UTC) == Best paragraph in Wikipedia == :In 2003, scientists at Paignton Zoo and the University of Plymouth, in Devon in England reported that they had left a computer keyboard in the enclosure of six Sulawesi Crested Macaques for a month; not only did the monkeys produce nothing but five pages consisting largely of the letter S, they started by attacking the keyboard with a stone, and continued by urinating and defecating on it. Bravo. User:Tempshill 22:51, 2 Nov 2004 (UTC) == Anal-retentive 'pedia == ''NB: the animal pictured here is a chimpanzee, which is an ape, not a monkey.'' Wow. Yes, it is an encyclopedia, but come on... :Yeah, I mean, really. It's better to just leave that out so people can point at us and go "how stupid ''are'' those editors, and how can they call ''that'' an encyclopedia", right? :You should be ''glad'' Wikipedia does its utmost to be accurate. :-) User:JRM 19:26, 2005 Mar 20 (UTC) ::I don't know, considering that whether the creature doing the typing is a monkey or not makes absolutely no difference to the theorem, it might be going a little overboard. It might be better to include it much more briefly and subtlely. I'm revising to that effect. User:Dcoetzee 20:03, 20 Mar 2005 (UTC) :::Now that ''is'' subtle. We won't let anyone make a monkey out of Wikipedia. User:JRM 20:24, 2005 Mar 20 (UTC) == Infinity Generation == For a long time, I've thought about ''Infinity Generation'' - that is, the generation of every possible permutation in a given man-made system. Since such systems are by their nature, finite, they cannot have infinite permutations. However, such systems can appear to humans to be infinite, since humans are unable to perceive or comprehend infinity directly. The level to which any system could appear to humans as infinite would be known as it's ''granular limit'' - a term which I coined after thinking about how when we look at a newspaper photo from far enough away, it appears solid, but when we look close enough we see the actual dots (or granules) that form the image. The ''granular limit'' would be the limit at which we could be close enough to see and comprehend the image as one entity, but not actually perceive its granules. Coming back to the point in discussion. I first conceptualized ''Infinity Generation'' when thinking about simple images on computers. Take for example a standard Digital Camera image of 640x480 pixels (total = 307,200) pixels. A low-colour format might make use of the GIF standard of 256 colours. To calculate all possible permutations of that Digital Camera image, one takes the number of pixels and raises it to the possible values that each pixel might have (ie: 307,200 ^ 256 = 6.0232391645008958015139896503459e+1404). With today's computing power, we could create a programme which would extrapolate all of those possible permutations of that image within a reasonable timeframe. Since an image of 640x480x256 would have an acceptable granular limit, it could easily be recognisable as any kind of object, person, whatever that exists (or doesn't exist) in our modern world. After generating every image permutation, we would have effectively generated Infinity - we would have within our images, every object, etc. that has, will, does, can't or never will exist. Taking this one step further, we could apply the Infinity Generation to texts, such as Shakespeare. We would configure the computer programme to a suitable length (ie: The length of a Shakespearean Play), and after a while, we would have every possible permutation of that length of text. Not only would we have the Shakespearean Play, we would have every text that fitted into the configured space. Text would be a better place to start than images, since images are made up of millions (or billions) of colours, and text has only the usual 26 characters (52 if counting uppercase), plus the usual punctuation, white space, etc. If left to run long enough, the Generators would come up with every thing that was, will or is written. Furthermore, we could expand our Infinity Generation to Films, Music and more. At the present time, such ''Infinity Generators'' would be notoriously hard to implement without some kind of AI, since we would be like Leonardo chipping away the stone, to reveal the statue beneath - we would have to separate the 99.999999999999999 (or whatever) % of chaff, to reveal the wheat - the comprehensible stuff. What we would do with all this stuff, even I don't know. But look at it this way - amongst all the chaff, the wheat of unimaginable power would lie - a formula for nuclear fusion, a cure for diseases. Plus all the new music, films, texts etc. Truly, from Infinity comes Creativity.... -- [http://www.mattwarne.com Matt 'devolution' Warne] == Chaos Monkeys == This argument assumes that the key-mashings of the monkeys is completely random from one letter to the next. It's possible that if this experiment were tried then perhaps there would emerge discernible monkeymashing patterns that might actually preclude the creation of long, coherent, flawless texts beyond a certain threshold. Simple limitlessness is not enough; in this case you also need random variation. Maybe it would turn out that, out of every X keys hit, a monkey will always wind up hitting two keys that are immediately next to each other after a certain level of fatigue sets in. Or maybe the monkey eventually develops a preference for a certain set of keys and ignores the rest. It's possible that an infinite number of monkeys hitting a keyboard is less like pi and more like the repeating decimal .142857... It's at least possible that there are some regular patterns that would emerge that would ruin this randomness-dependent theorem. If you can get an infinite set of random key generators, the theorem is sound. But I'm not convinced that monkeys are perfect random letter generators. It may monkeybe that there is a slight flaw in this monkeyplan. User:Mr. Billion 16:26, 12 May 2005 (UTC)
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