Infinite monkey theorem
"The infinite monkey theorem" says that, with probability equal to 1, a monkey seated in front of a typewriter keyboard and hitting keys at random will eventually type every book in France's Bibliothèque nationale de France (National Library). It is a popular misnomer for an idea from Émile Borel's book on probability, published in 1909. The book introduced the concept of "dactylographic1 monkeys" . Borel exemplified a proposition in the theory of probability called Kolmogorov's zero-one law by . Strictly speaking, what Borel was illustrating was only a special case of Kolmogorov's zero-one law, the more general statement of which had not yet been given (Kolmogorov's famous monograph on probability theory was not published until 1933).
A popular statement of the theorem is that an infinite 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 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.
Subsequent restatements by other people have replaced the National Library not only with the British Museum but also with the Library of Congress; a popular retelling says that the monkeys would eventually type out the collected works of William Shakespeare.
The literary notion may have its origin in Jonathan Swift's Gulliver's Travels (1782), part three, chapter five, in which a professor of the Grand Academy of Lagado is 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.
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 grows exponentially smaller, 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 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 A.N. 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 other words, in physics 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, in effect, that he would rather be descended from an ape than 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.
Literature and popular culture
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 bowdlerized edition.
A similar theme was struck in the story "The Library of Babel" by Jorge Luis Borges, which contains billions of volumes filled with random strings of characters.
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 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). The theorem is also the basis of a one-act play by David Ives called "Words, Words, Words", which appears in his collection All in the Timing. In the one-act, three monkeys named Milton, Swift, and 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 2000, a paper was submitted to the IETF Internet standards committee as an April Fool's joke proposing an "Infinite Monkey Protocol Suite (IMPS)", a method of directing a farm of infinitely many monkeys over the Internet.
Attempts at simulation
"The Monkey Shakespeare Simulator" web site, 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 25 September 2004, matches as long as 21 consecutive letters, four words have been recorded ("Love's Labour's Lost" "KING. Let fame, that wtIA ...") (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 certain value or value within 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 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.
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."
Usage note
To some lay persons, "infinite monkeys" and "infinitely many monkeys" may be synonymous; to mathematicians, the former is incorrect because each monkey individually is finite.
References
- No words to describe monkeys' play (9 May 2003) BBC News
- Monkey Theory Proven Wrong (9 May 2003) CBS News
- RFC 2795 — The Infinite Monkey Protocol Suite (IMPS)
External links
- Monkey Shakespeare Simulator
- Real Life Parody of the Keyboard/Monkey Concept
- Monkeys Don't Write Shakespeare
Footnote
1 The word dactylographic appears in the English translation of Borel's book, and seems to be an Anglicization of a French word for typewriting, but in English, dactylography means the study of fingerprints.