Infinite Monkey Theorem

Random typing of infinite duration on a typewriter will almost certainly create all of Shakespeare's texts or any national library.

The infinite monkey theorem ( Engl. Infinite , infinite ', monkey , monkey' and theorem , theorem '), also German theorem of endless tip monkeys , states that a monkey infinitely around typing long randomly on a typewriter, almost certainly at some point will write all the books in the Bibliothèque nationale de France , the national library of France . In English-speaking countries it is said that this is how William Shakespeare's works will emerge at some point .

One of several variants of the theorem assumes an infinite number of monkeys typing on typewriters at the same time and claims that at least one of them will type in the works mentioned above directly and without error.

The formulation of the theorem is intended to amaze and therefore uses visual language. The theorem is of scientific origin and has a mathematically sound background. In the form of an example, it illustrates a statement made by probability theory , the Borel and Cantelli lemma . The thought experiment resulting from the theorem can be useful for the idea of ​​infinity and the classification of probabilities and is also used for these purposes.

The motifs "an infinite number of monkeys on typewriters", "a monkey who is forever typing on a typewriter" and "the coincidental creation of meaningful texts" were well received in literature and popular culture and were widely used.

Mathematical treatment

The theorem can be clearly demonstrated with the help of simple means of probability calculation. A simplified representation follows:

A typewriter has 50 keys. It is now assumed that a monkey typing on the keyboard at random presses each of the 50 keys with the same probability, i.e. that no key is systematically preferred or neglected. The probability of tapping a certain key is then 1/50. In addition, the keystrokes are independent of each other. This means (somewhat simplified) that the probability of pressing a key on the second hit is again 1/50, regardless of which key was previously pressed. For independent events, the probability that both will occur at the same time is equal to the product of the probabilities for the occurrence of the individual events.

Event A now consists in the fact that a monkey types in the word "hamlet" six times at random.

The probability that the first letter typed is an “h” is 1/50; likewise the probability that the next letter is an "a". Thus the probability for the letter sequence “ha” is equal to 1/50 · 1/50 (= 1/2500). The probability of the event A, with the first six inputs to get the letters "hamlet", that is the complementary event ( counter event ) consists in the fact that in a series of six letters not the word "hamlet" is written. It has a chance of . ${\displaystyle p_{1}}$ ${\displaystyle p_{1}=1/50^{6}.}$ ${\displaystyle {\bar {A}}}$ ${\displaystyle {\bar {p}}_{1}=1-p_{1}=1-1/50^{6}}$

Therefore, the event that in two attempts with six letters each the word “hamlet” is not written in either of the two has a probability of

${\displaystyle {\bar {p}}_{2}={\bar {p}}_{1}\cdot {\bar {p}}_{1}=\left(1-{\frac {1}{50^{6}}}\right)^{2}}$

and the event that the word "hamlet" is not written in any of the n attempts with six letters each has the probability

${\displaystyle {\bar {p}}_{n}=\left(1-{\frac {1}{50^{6}}}\right)^{n}.}$

The sequence of these probabilities tends towards 0. With increasing n, the sequence of probabilities of typing “hamlet” at least once in n attempts with six letters each tends towards 1. This in turn means that a monkey with an increasing number of attempts with against 1 aspiring probability will type the word "hamlet". ${\displaystyle ({\bar {p}}_{n})}$ ${\displaystyle (p_{n})=(1-{\bar {p}}_{n})}$

With these formulas one can also make statements about the required number of experiments with which the monkey will be successful with a certain probability. To z. For example, to have typed "hamlet" at least once with a probability of at least 10 percent, n must be determined from the inequality . So it must apply ${\displaystyle p_{n}\geq 0{,}10}$

${\displaystyle 1-\left(1-{\frac {1}{50^{6}}}\right)^{n}\geq 0{,}10}$

or changed

${\displaystyle \left(1-{\frac {1}{50^{6}}}\right)^{n}\leq 1-0{,}10.}$

By taking the logarithm (this is allowed because all occurring values ​​are positive) the condition is obtained

${\displaystyle n\cdot \ln \left(1-{\frac {1}{50^{6}}}\right)\leq \ln 0{,}90}$

and after division by ln (1−1 / 50 ⁶ ) (the inequality sign is reversed because this expression is negative).

${\displaystyle n\geq {\frac {\ln 0{,}90}{\ln(1-{\frac {1}{50^{6}}})}}.}$

It follows from this that around 1.6 billion attempts are required. And if the desired result is to occur with a probability of at least 90 percent, at least must ${\displaystyle n\geq 1.646.257.350}$

${\displaystyle n\geq {\frac {\ln 0{,}10}{\ln(1-{\frac {1}{50^{6}}})}}}$

d. H. Experiments are carried out. ${\displaystyle n\geq 35.977.876.618}$

Furthermore, you can now determine how many attacks must be made in order to achieve the desired result with a certain probability. It is assumed that after the first six keystrokes, the second attempt begins with the second letter, so that only one more keystroke has to be made. Then the monkey only needs seven keystrokes for two attempts, eight keystrokes for three attempts and generally n + 5 keystrokes for n attempts. So in order to be successful with a probability of at least 90 percent, he must make at least 35,977,876,618 + 5 attacks. At a speed of one stroke per second, that would be about 1,140 years, then there was a 90 percent probability that he would have typed the word "hamlet" at least once.

The above considerations can easily be generalized and the following formula is obtained:

The following applies to the minimum number n of attempts to obtain a letter group (in a certain order) of length m on a typewriter with t keys with a probability of at least q percent

${\displaystyle n\geq {\frac {\ln(1-{\frac {q}{100}})}{\ln(1-{\frac {1}{t^{m}}})}}.}$

The line of thought given above can also be transferred to the variant of the question why at least one of the infinite number of monkeys will certainly type in the text correctly right away. For the sake of simplicity, the text is again 6 letters long. In this case, the probability for the event is that none of the first n monkeys types the word "hamlet" correctly on the first attempt. This probability also tends towards zero, so that the probability that at least one of the monkeys types in the desired text the first time (i.e. the counter-event) tends towards one. ${\displaystyle {\bar {p}}_{n}}$

For completeness it should be mentioned: The selection of the characters of the desired sequence (here “hamlet”) is insignificant for this situation. The length of the character string (here six) is also irrelevant: If the character string is longer, the probability of the event (A) is lower and the approximation is therefore slower, but the approximation described still occurs. Finally, the thought experiment uses a symbolic monkey as a stylistic device to carry out the chance experiment , which represents the random character. It also uses the great symbolic length of the works mentioned in order to underline the amazing effect on the viewer; as already described, the length of the desired character string is not important for the statistical approximation of the secure event.

Notes on the thought experiment

The demonstration of the theorem given here for the purpose of illustration makes use of simplifications that are useful in thought experiments, but not necessarily necessary from a mathematical point of view.

In addition to the independence of the individual keystrokes, an equal distribution of the frequencies of the characters in the letter sequence was assumed. This condition simplifies the symbolic calculation and understanding, but is not a necessary requirement. The necessary prerequisite is that the probability of each letter appearing with each attack is greater than zero.

The magnitude of some probabilities

If you ignore capital letters, umlauts, punctuation marks and spaces for the sake of simplicity and assume that the letters follow a discrete uniform distribution (i.e. the same probability for each letter), then there is a probability of 1/26 for a single monkey in a single attempt, that he typed the first letter of the drama Hamlet correctly. The probability of typing the first two letters correctly in a single attempt is:

${\displaystyle {\frac {1}{26}}\cdot {\frac {1}{26}}={\frac {1}{676}}}$,

The probability for the observed event decreases exponentially , with 20 letters it is only:

${\displaystyle {\frac {1}{26^{20}}}}$ = ${\displaystyle {\frac {1}{19\,928\,148\,895\,209\,409\,152\,340\,197\,376}}}$

This roughly corresponds to the probability of winning the jackpot with six correct numbers every time with four lottery tickets with four draws in a row. In the case of the entire Hamlet text, the probability is so small that it can hardly be grasped in human terms. If the entire punctuation is neglected, Hamlet's text contains more than 130,000 letters - the probability in the idealized case would be:

${\displaystyle {\frac {1}{{26}^{130\,000}}}}$which corresponds approximately . ${\displaystyle {\frac {1}{3{,}4\cdot {10}^{183\,946}}}}$

Even if the entire visible universe were filled with monkeys the size of atoms, and they would tap to the end of the universe , the probability of ^getting Hamlet would be many orders of magnitude less than 10 ^−138,800 . As Charles Kittel and Herbert Kroemer note, “[…] the probability for Hamlet in every conceivable case is zero”, and the statement that the monkeys will eventually achieve their goal “gives a wrong impression about very, very large numbers ". They do this in their book on thermodynamics , the statistical basis of which led to the first mention of the typing monkey.

A formal proof

The fact that there is a certain - albeit very small - positive probability for the random writing of all of Shakespeare's works is the key to the proof of the Infinite Monkey theorem: Already from the zero-one law of Kolmogorow and Borel it follows that that the limit superior of an infinite series of independent events must have a probability of either one or zero. Translated, this means: either an infinite number of these events will occur almost certainly (i.e. with probability one) or almost never (corresponding to probability zero).

Although the Infinite Monkey Theorem has no formal character, a formal statement can be derived for strings in general:

• The probability that any finite character string will appear at least once in a random string of infinite length is 1. And not only that: it will almost certainly occur infinitely often. One monkey would be enough to write all of Shakespeare's works an infinite number of times in an infinitely long time.

This statement follows relatively easily from the Borel-Cantelli lemma . If the random character sequence of infinite length is arbitrarily divided into blocks of the length of the character sequence of finite length under consideration, the occurrence of each individual event from the sequence of (random, independent) events has the same positive probability. The sum over the infinitely many constant summands is infinite. ${\displaystyle (A_{n})_{n\in N}}$ ${\displaystyle P(A_{n})}$

The Borel-Cantelli lemma then says: If the sum of the probabilities is infinite and the events are independent, then the probability of the limes superior is equal to 1. ${\displaystyle A_{n}}$ ${\displaystyle A_{n}}$ ${\displaystyle A_{n}}$

Expressed formally: ${\displaystyle \sum _{n=1}^{\infty }P(A_{n})=\infty \Rightarrow P(\limsup A_{n})=1}$

The thought that when viewing infinite periods of time such an unlikely event will certainly occur, is used here to illustrate infinity .

Origin and reception in literature

The Argentine writer Jorge Luis Borges traces the origin of the thought experiment in his text "The Complete Library" (Spanish title La biblioteca total ) back to antiquity and describes the following process: In his work, Aristotle used metaphysics in the representation of the views of Leukippus , who (with his pupil Democritus ) is considered to be the founder of atomism , wrote that atoms are equal to one another and that only their arrangement can form different objects. He compared that to the way tragedy and comedy are made up of the same "atoms", the characters. Three centuries later, in his work De natura deorum ("Of the essence of the gods") , Cicero made a mocking reference to the atomistic worldview:

“Anyone who thinks this is possible will also have to believe that if innumerable letters made of gold, each representing a letter from the twenty-one of the alphabet, were thrown on the floor together, they could form the Annals of Ennius in a legible form. I doubt the possibility that chance can create a single legible verse. "

- Cicero : De natura deorum II, 37 (§ 93)

Borges follows the development of this argument through Blaise Pascal and Jonathan Swift into his day and notes that the statement has changed: In 1939, according to him, the saying was: “Half a dozen typewriters would, in some infinity, all books of the British Museum. ”Borges corrects this by adding that an immortal monkey would be enough.

Some examples are given later in Borges' text to make the contents of the Total Library imaginable: It would contain everything (“Everything would be in its blind volumes”), for example the detailed history of the future "), His own dreams and half-dreams towards the morning of August 14, 1934 (" dreams and half-dreams at dawn on August 14, 1934 "), the proof of Fermat's last sentence (" proof of Pierre Fermat's theorem "), etc. He then wrote that next to every single fact there were millions of lines full of nonsense ("but for every sensible line or accurate fact there would be millions of meaningless cacophonies, verbal farragoes, and babblings."). Borges concludes from this that all generations of mankind would perish before the shelves of the total library [...] ever rewarded them with a bearable side (“but all the generations of mankind could pass before the dizzying shelves - shelves that obliterate the day and on which chaos lies - ever reward them with a tolerable page. ")

In the short story with the Spanish title La biblioteca de Babel ( The Library of Babel ) Borges pursues the theme of the infinite library and again uses the literary and scientifically relevant topics of infinity, reality and causality .

In some places there are references to the English biologist Thomas Henry Huxley (1825–1895). Seven months after Darwin's The Origin of Species was published in November 1859, at the meeting of the British Association for the Advancement of Science in Oxford on June 30, 1860 , Huxley had a famous dispute with Samuel Wilberforce , the Anglican Bishop of Oxford and vice-president of that scholarly organization. In this dispute, Huxley is said to have made the following statement:

“Six eternal apes, randomly striking the keys of six eternal typewriters with unlimited amounts of paper and ink would be able to produce Shakespearean sonnets, complete books, and the 23rd Psalm. In the same way, molecular movement, given enough time and matter, could produce Bishop Wilberforce himself, purely by chance and without the work of any designer or creator. "

“Six eternal monkeys, randomly hitting the keys of six eternal typewriters with unlimited amounts of paper and ink, would be capable of producing Shakespeare's sonnets, complete books, and the 23rd Psalm. In the same way, given sufficient time and matter, molecular motion could produce Bishop Wilberforce itself, by pure chance and without the work of any designer or creator. "

It is debatable whether Huxley actually said this. Some authors assume that the saying Huxley was only awarded later, partly because the aforementioned typewriter ( typewriter ) was only used later and could therefore not be used by Huxley for a striking comparison:

"The story [...] is doubtless fictitious since the Huxley-Wilberforce debate of 1860 antedated the emergence of the typewriter."

"The story [...] is undoubtedly fictional because the Huxley-Wilberforce debate of 1860 preceded the appearance of the typewriter."

- Nicholas Rescher : Studies in Cognitive Finitude ; Transaction Pub (2006)

The modern picture of the theorem of the infinite number of monkeys can be found in the article Mécanique Statistique et Irréversibilité by Émile Borel from 1913. His monkeys represent as a living image the production of a large, random sequence of characters for the representation of statistics.

The physicist Arthur Eddington wrote the following sentence, which makes it clear that there are allusions to the thought experiment in many areas of science:

“If I unintentionally slide my fingers over the keys of a typewriter, it can happen that a legible sentence appears in the resulting tome. If an army of monkeys jingles on their typewriters, they can write any of the books in the British Museum. The likelihood that they will do this is significantly higher than the likelihood that all the molecules in a container will collect in half. "

- Arthur Eddington : The Nature of the Physical World: The Gifford Lectures ; Macmillan, New York, 1928, p. 72 (freely quoted from translation from English)

The last sentence is an allusion to the second law of thermodynamics : The above-mentioned collection of all molecules in a container is possible according to the rules of probability (mathematics), but according to the second law of thermodynamics (physics) in a closed system, like a container, not (apart from microscopic systems).

References to the theorem

References to areas of science, limitation of the statement

Math and infinity

The key to understanding the conclusions is the (somewhat difficult to understand) concept of infinity in mathematics.

In visual terms, a monkey can almost certainly type any text that has ever been written or will ever be written in the future, if only he is given an infinite amount of time; mathematics (Kolmogorow and Borel-Cantelli) allows this figurative conclusion .

At first glance, this symbolism gives the possibility that the monkey will write down every existing or ever known knowledge in the world. But the meaningful texts that arise by chance are created together with a disproportionately higher (infinite) number of non-meaningful texts. The monkeys would write down a viewed text together with an infinite number of versions, each with all conceivable orthographical or content-related errors - so it is not possible to distinguish the meaningful from the irrelevant versions without the correct version already being available.

A reference between the symbolism and the concept of entropy in information theory can be seen here, where mathematical means are used to delimit the information content of a message as opposed to random strings of characters.

The limitation of the symbolism of the theorem can be superficially compared with the statement of the second law of thermodynamics in physics, which (simplified) makes the following statement: "The entropy (clear disorder ) of a closed system increases continuously or at best remains the same, because only the maintenance of a certain state of order requires externally supplied energy. The restoration of an (often "ordered") initial state of lower entropy requires the use of energy or information (see Maxwell's demon ). "

Probability and evolution

Authors who are close to the idea of ​​“ intelligent design ” often argue with the theorem that the probability of life occurring by chance is extremely low. For example, Ken Wilber calls it the "silly random mutations" and deduces from this that it "cannot be chance that drives the world". Deepak Chopra writes: “The idea that creation works without consciousness is like the insane idea of ​​a room full of monkeys that hit keys at random and at some point - after millions of years - perhaps created a work that corresponds to Shakespeare's . ”The objection to this is that evolution is mainly determined by non-random selection .

Experiments on the theorem

The evolutionary biologist Richard Dawkins refers in his book The Blind Watchmaker to the idea of ​​the typing monkey, demonstrating how the interplay between mutation and natural selection achieves its effectiveness and can be distinguished from pure chance, represented by typing monkeys. His goal is to find the difference in effectiveness between “cumulative selection”, in which successful mutation steps are retained and the starting point for further mutation selection steps, and “single step selection”, in which all intermediate steps are discarded and in every step started all over again, to make it clear. Dawkins describes a computer program that produces the Hamlet line “METHINKS IT IS LIKE A WEASEL” to show the extent to which the cumulative selection differs from a hypothetical typewriter monkey (equated with single-step selection). To do this, a random text is first generated. This text is compared with the Hamlet text, whereby only those letters that already match the Hamlet text are transferred to the next step. The other letters are created again at random, the newly created text is again compared with the Hamlet line, and so on. This continues until the text matches the Hamlet text. This algorithm with cumulative selection proves to be much more efficient, that is, it requires much fewer steps than would be the case with “single-step selection”. Dawkins himself points out in his book that this thought experiment is only intended to demonstrate a partial aspect of evolution, the effectiveness of cumulative selection, and not biological evolution itself, as this is not directed towards a specially prescribed goal.

In 2003 scientists and students at Paignton Zoo and the University of Plymouth in Devon, England reported that they had placed a computer keyboard in a cage with six macaques for a month : the monkeys had done nothing useful: only five pages, with the texts mainly of the letter S passed. The monkeys had also hit the keyboard with a stone and deflated over the keyboard. The “experiment” had no scientific character and was conceived as a performance (artistic presentation).

All experiments on the theorem have in common that they work with empirical individual experiments, i.e. samples . It is not possible to draw a valid conclusion regarding an infinite population from single experiments of limited duration or number of monkeys, i.e. finitely many samples ; an equally infinite sample would have to be used as a basis. Therefore, when considering the Infinite Monkey Theorem, it must be borne in mind that empirical proof is impossible.

References to the theorem from art and everyday culture

Apart from the texts on the topic already listed in the section on the origin of the theorem and historical outline in literature , there were numerous allusions and artistic incorporations of the motifs around the theorem in literature, television and computer culture:

Literature and other texts

Jonathan Swift lets Gulliver encounter the writing monkeys in the land of Lilliput.

In a play by the British playwright Tom Stoppard called "Rosencrantz & Guildenstern are Dead" , which tells the story of Hamlet from a different perspective, one character says, "If a million monkeys ..." and then cannot go on - possibly because it is itself part of the Shakespearean universe and explained its own fictionality by pronouncing the theorem. The sentence ends on a different topic. This scene does not exist in the film of the same name. We are only talking about six monkeys that are thrown into the air and are equally likely to land on their butts or heads.

In Nevil Shute's dystopian novel "On the Beach" ( Chapter 5) , some survivors of a nuclear war in Australia received largely incomprehensible radio signals from near Seattle, USA. In about 106 hours of broadcasting time, only two understandable plaintext words were received, which an admiral commented with the words: “I don't think the words can be significant. It's probably a fortuitous transmission. After all, if an infinite number of monkeys start playing with an infinite number of typewriters, one of them will write a play of Shakespeare "(" I don't think the words could have any meaning. This is probably a random transmission. If one an infinite number of monkeys begins to play on an infinite number of typewriters, one of them will eventually write a play by Shakespeare. ”). In fact, research shows that there was no human intent behind the radio signal transmissions.

In the book " The Hitchhiker's Guide to the Galaxy " by the English writer Douglas Adams , the two main characters Arthur Dent and Ford Prefect are attacked by an infinite horde of monkeys who want to discuss a Hamlet script with them with an improbability factor of 1 to ${\displaystyle 2^{267709}}$ their improbability .

In Matt Ruff's book “ Fool on the hill ” , Mr. Sunshine has a room full of monkeys who sit at typewriters and produce stories.

In Michael Ende's “ Neverending Story ” , the people of a city who cannot find their way home from “Fantastica” have to create random letter combinations as a kind of occupational therapy, as the city guide - a monkey - explains; the point is that this is how all stories arise in infinite time. Ende expressly points out that the Neverending Story will also be among them.

In a short story by the science fiction and fantasy writer RA Lafferty called "Been a Long, Long Time", an angel is punished with the fact that he has to read the entire text production of monkeys on typewriters until one day in a distant time the monkeys have a perfect one Make a copy of Shakespeare's works.

In a Dilbert comic, Dilbert Dogbert shows a poem of his own. Dogbert says that he once heard that a thousand monkeys with an infinite amount of time could write all of Shakespeare's works. Dilbert, confused, asks what about his poem. Dogbert adds: "Three monkeys, ten minutes."

watch TV

In the episode “Last Exit to Springfield” (German title: Princess von Zahnstein) of the animated series The Simpsons (Season 4, Episode 17) Mister Burns has a piece written on typewriters in a huge room full of monkeys. Burns pulls a monkey's sheet of paper out of the typewriter and reads aloud: “No, oh well, not no. This is the plague here. "

In the episode “Battle of the Sexists” of the series Die wilden 70er (Season 1, Episode 4), Eric Forman calls out the following to his girlfriend Donna Pinciotti after she has scored a basket in basketball: “Pinciotti actually scores! Hell freezes over! A monkey types Hamlet! ” (Pinciotti throws a basket! Hell freezes! A monkey writes Hamlet!)

In “The King is Dead”, the seventh episode of the second season of the American cartoon series Family Guy , Peter Griffin responds condescendingly to Lois Griffin's understanding of art by referring to the Infinite Monkey Theorem: " Art-Schmart. Put enough monkeys in a room with a typewriter they'll produce Shakespeare. "(No way art. Put enough monkeys in a room with a typewriter and they'll write Shakespeare.)

The American comedy show The Colbert Report had a section about how many monkeys you would need for different works of art. According to Colbert , it would take a million monkeys writing to infinity to create the works of Shakespeare, ten thousand alcohol-drinking monkeys who write ten thousand years to create Hemingway's works, and ten monkeys who type three days to create the works of Dan Browns .

Computer culture

In 2000, the IETF Internet Standards Committee issued an April Fool's joke RFC on the topic of Infinite Monkey Protocol Suite (IMPS) : a collection of fictitious protocols and methods in technical language that are used to monitor and coordinate an infinite number of Monkeys on typewriters are supposed to help. The RFC is written in an entertaining manner and describes the logistics around the monkeys and their "production" on the typewriters in a way that is typical for RFCs.

The standard formatting of the C programming language in the GNU Emacs editor is often described as "worse than random" with the following words: "An infinite number of monkeys typing into GNU emacs would never make a good program."

In the event of an internal server error of type 500 , the Internet platform YouTube sends a message to the client computer with the following content: “500 Internal Server Error Sorry, something went wrong. A team of highly trained monkeys has been dispatched to deal with this situation. If you see them, show them this information: “ followed by a text block of approx. 3000 characters from a Base64- coded string.

literature

• Elmo, Gum, Heather, Holly, Mistletoe, Rowan (Makaken Monkeys from Paignton Zoo): Notes Towards the Complete Works of Shakespeare . Ed .: Geoff Cox. Kahve-Society, London 2002, ISBN 978-0-9541181-2-9 ( viaria.net ( Memento from January 20, 2013 in the Internet Archive ) - Text produced by monkeys from Paignton Zoo using a computer May - June 2002, English) . .
• Herrmann, Hans-Christian von: Literature and Entropy , Berlin 2014, ISBN 978-3-428-14012-1 .

Web links

• The media art project of the JACS group watch the monkey typing “Faust” and “Hamlet”
• Alfred Schreiber Crazy Libraries . Die Zeit online, June 30, 2006.
• Monkey Theory Proven Wrong . CBS News, May 9, 2003
• Monkeys Don't Write Shakespeare . Wired magazine (English)
• Notes Towards the Complete Works of Shakespeare . ( Memento from January 20, 2013 in the Internet Archive )
• Parable of the Monkeys "Aka The Topos of the Monkeys and the Typewriters" Collection of text references to the theorem (English)

References and comments

1. ↑ The version on Gutenberg.org comprises 132 680 letters and 199 749 characters in total
2. ^ Charles Kittel , Herbert Kroemer : Thermal Physics (2nd ed.) . WH Freeman Company, 1980, ISBN 0-7167-1088-9 , p. 53.
3. ↑ freely translated from Cicero : De natura deorum II, 37 (§ 93) in the Latin original: “[93] Hic ego non mirer esse quemquam, qui sibi persuadeat corpora quaedam solida atque individua vi et gravitate ferri mundumque effici ornatissimum et pulcherrimum ex eorum corporum concursione fortuita? Hoc qui existimat fieri potuisse, non intellego, cur non idem putet, si innumerabiles unius et viginti formae litterarum vel aureae vel qualeslibet aliquo coiciantur, posse ex is in terram excussis annales Enni, ut deinceps legi possint, effici; quod nescio an ne in uno quidem versu possit tantum valere fortuna. ” thelatinlibrary.com
4. ^ Jorge Luis Borges : La biblioteca total ("The Total Library") Trans. By Eliot Weinberger. In: Sur , No. 59, August 1939. In: Selected Non-Fictions . Penguin, 1999, ISBN 0-670-84947-2 .
5. ^ Jorge Luis Borges: La Biblioteca de Babel . (1941). In: Ficciones . Alianza, Madrid 1971. English translation: The Library of Babel . In: Borges: Labyrinths . Penguin, Harmondsworth 1970
6. ↑ Course Algorithmic Art & AI ( Memento of the original from March 5, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. of the Institute of Artificial Art Amsterdam , quote: "Bibliographic research by René Glas, Pepijn van der Meer and Chuntug Taguba" 
7. ^ Arthur Stanley Eddington: The Nature of the Physical World: The Gifford Lectures. Macmillan, New York 1928
8. ^ Nicholas Rescher: Studies in Cognitive Finitude. Transaction Pub, 2006, ISBN 3-938793-00-7 .
9. ^ Émile Borel: Mécanique Statistique et Irréversibilité. In Journal Phys. 5e série, No. 3 , 1913, pp. 189-196.
10. ↑ Ken Wilber: A Brief History of the Cosmos. Fischer Taschenbuch, Frankfurt 1977, ISBN 3-596-13397-1 , p. 48.
11. ↑ Deepak Chopra, Leonard Mlodinow: Creation or Chance ?: How Spirituality and Physics Explain the World - A Debate. Arkana, Munich 2012, ISBN 978-3-442-34106-1 , p. 60.
12. ↑ Richard Dawkins: The Blind Watchmaker: A New Plea for Darwinism . dtv, Munich 1986, ISBN 3-423-11261-1 , p. 70 ff .
13. ↑ "Methinks it is like a weasel" , Hamlet , Act 3, Scene 2 (In the translation of Christoph Martin Wieland is the seventh scene in Act 3, from the weasel is a blackbird here.)
14. ↑ No words to describe monkeys' play . BBC , May 9, 2003
15. ↑ Nevil Shute, "On The Beach", Random House, 2010, 320 pages, p. 75, https://books.google.de/books?id=Tn6qMFqRVtwC&pg=PT75&lpg=PT75
16. ^ Dilbert writes a poem and presents it to Dogbert:

    DOGBERT: I once read that given infinite time, a thousand monkeys with typewriters would eventually write the complete works of Shakespeare.
    DILBERT: But what about my poem?
    DOGBERT: Three monkeys, ten minutes.

17. ^ Family Guy - Monkeys Writing Shakespeare
18. ↑ RFC 2795  - The Infinite Monkey Protocol Suite (IMPS)

This version was added to the list of articles worth reading on September 30, 2006 .