Talk:Infinite monkey theorem

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Current status: Former featured article

Real Monkeys[edit]

The section Real Monkeys is one of the funniest things I ever had the joy to read. Thank you very much. (talk) 13:42, 18 November 2016 (UTC)

Much as I appreciate Shakespeare, sometimes I think the "performance art" of the real monkeys has more relevance to our contemporary world.--Jack Upland (talk) 09:17, 21 November 2016 (UTC)

I agree wholeheartedly; this entire article made me laugh, but this section was the best IntegralPython (talk) 19:20, 7 December 2018 (UTC)

Date Correction Needed?[edit]

Is the date for the real monkeys incorrect? Wikipedia says the experiment occurred in 2003, but the copyright date in the linked PDF says 2002. Where did the 2003 date come from? (talk) 04:45, 30 August 2020 (UTC)

Swift and Pascal bits are missing[edit]

The history section mentions Swift and Pascal in passing, but unlike the Aristotle and Cicero bits, the Swift and Pascal references are never explained or quoted. I found the Swift bit in the popular culture article (which I am working on at the moment), but don't know where to find the Pascal bit. The Swift bit should be added:

  • 1782 - 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): although his intention was more likely to parody Ramon Llull.

And someone should find and add the Pascal bit. Thanks. Carcharoth 14:27, 12 August 2007 (UTC)

Hi. I have found a book by Swift published 1774 (there are many editions - I found this particular edition on JISC Historic Books). The works of Jonathn. Swift, D.D.: D.S.P.D. with notes historical and critical. pg 176-182 (By J. Hawkesworth, L.L.D. and others. Printed for J. Williams, Dublin Library : Bodleian Library (Oxford) - Accessed via In this is the Part III Chapter 5. The machine created by the professor in the academy has words, pronouns, punctuation and all the parts of speech written on individual tablets. The tablets are all put into the machine and the handle is turned. The students of the professor then read off the sentence that is formed and it is written down. "..whereby by his contrivance the most ignorant person at a reasonable charge and with a little bodily labour might write books in philosophy, poetry, politics, law, mathematics, and theology, without the least assistance from genius or study.".
There is a note to this chapter in this edition by Lord Charles Boyle, Earl of Orrery:
"The project for a more easy and expeditious method of writing a treatise in any science, by a wooden engine, is entertainingly satirical; and is aimed at those authors who, instead of receiving materials from their own thoughts and observations, collect from dictionaries and common-place books, an irregular variety, without order, use or design: "ut nec pes nec caput uni, reddatur formae". Orrery." (The latin is a reference to Horace "where the feet and the head have no relation to the other parts").
The addition of the quote from Orrery in this edition of Swift's Works is interesting. It puts Swift and Bentley in the same group. Bentley's Confutation of Atheism (1692) was part of the Boyle Lectures. (See my section below History - references preceding Borel). And according to the page, Gullivers Travels, the book was read by almost everyone when it was published in 1726. Thus almost everyone would have this idea of a machine pumping out sentences from random assortments of words and letters.
In the same edition is Johnathan Swift's essay, A Tritical Essay upon the Faculties of the Mind, about which Swift's footnote reads, "in a farcical, satiric light, designed purely to expose the folly and temerity of those brainless, illiterate scriblers (sic), who are eternally plaguing their contemporaries with a parcel of wild, incoherent, nonsensical trash. Swift." It is in this essay that Swift talks about the jumbling up of letters - but in a similar vein to Bentley without the monkey:
"And if this be so, how can the Epicurean's opinion be true, that the universe was formed by a fortuitous concourse of atoms; which I will no more believe, than that the accidental jumbling of letters of the alphabet could fall by chance into a most ingenious and learned treatise of philosophy. Risum teneatis amici? Horace (latin: "Could you refrain from laughing, my friend?" from the same passage of Horace as Orrery quotes.)
In fact another book by William Wotton Reflections upon ancient and Modern Learning, Wotton complains about what he considers to be Swift's satirical attack on himself and Richard Bentley in Swift's Tale of the Tub. Zorgster (talk) 16:11, 23 January 2012 (UTC)

Later history section?[edit]

Is there a possibility for a rigorously sourced selection of examples from the 'in popular culture' article being integrated to this article under the title 'Later history' or 'Recent history'? At the moment, I'm rigorously sourcing the examples and re-ordering them in date order. I'm also turning up papers that mention not the mathematics or the early history and development of the idea, but rather of the current history and usage of the idea in literature and elsewhere. In other words, rather than being a "here are some examples", it would become "here is a history of the later use of the idea". Carcharoth 14:34, 12 August 2007 (UTC)

New paper on this topic - Monkeying Around with Text[edit]

Please see Monkeying Around with Text, Terry Butler, University of Alberta, Computing in the Humanities Working Papers, January 2007. I've used this as a reference over at Infinite monkey theorem in popular culture, and I think it will be useful here as well. Carcharoth 18:26, 12 August 2007 (UTC)

Expanded summary for popular culture section[edit]

I've now added a summary for the popular culture section, based on my rewrite of the daughter article Infinite monkey theorem in popular culture. See my edit here. Carcharoth 12:11, 16 August 2007 (UTC)

Stephen Ballentyne[edit]

Editors of this page may wish to be wary of including material by the philosopher of mathematics Stephen Ballentyne until there is evidence that such an individual exits and has been published. --Mark H Wilkinson (t, c) 19:12, 16 September 2007 (UTC)

Yes, even though Ballentyne seems to exist, publishing is key. The edits to this article are confused, and if they are based on a published source, I would be very interested in learning which publisher endorsed it and what exactly it said. Melchoir 04:41, 17 September 2007 (UTC)
Apparently this person is now teaching Religious Studies at Uppingham School.[1] Unless he publishes in a reputable journal, and his point of view is discussed by multiple independent published sources, his opinions are just as non-notable and unencyclopedic as those expressed at the local pub.  --Lambiam 07:04, 17 September 2007 (UTC)
Or that could be another individual entirely. The Ballentyne edits have been introduced by two new user accounts, apparently set up for the sole purpose of pushing this issue ([2], [3]).
At least we're getting a better class of vandal. --Mark H Wilkinson (t, c) 09:03, 17 September 2007 (UTC)
It's a shame that the Internet appears to be the sole source of verification for published source materials and that no evidence of Ballentyne's published material exists (as yet) on the Internet. My fellow students and I will strive to correct this omission by writing to the editors of the journals and books he has contributed to. It is sad that material about monkeys urinating on typewriters and repetitive computerised random number experiments are present in this article, instead of the dynamic range of mathematical philosophies that exist on the subject. --Merisalis 04:33, 18 September 2007 (UTC)
Funny thing about the internet is that the work of noted academics tends to turn up in Google searches. For example: Prof JF Toland is exceedingly easy to find. --Mark H Wilkinson (t, c) 10:41, 18 September 2007 (UTC)


Excuse me, what is the relevance of this to the article?

"Various Christian apologists on the one hand, and Richard Dawkins on the other, have argued about the appropriateness of the monkeys as a metaphor for evolution." —Preceding unsigned comment added by (talk) 14:23, 18 September 2007 (UTC)

Evolution is probably the most common and most important context for the infinite monkey theorem in modern popular culture, and there is a section of the article describing how. The Wikipedia:Lead section provides a one-sentence summary of that section.
I've reverted a bunch of recent edits to the lead that had little basis and removed information. I've also reverted the "However" paragraph from the "Real monkeys" section; the section already states that monkeys are not random number generators, and we don't need arguments that experiments are "unnecessary". Melchoir 01:00, 3 November 2007 (UTC)

I don't get it... :P[edit]

Infinity is an unending period of time. Why does the term "almost surely" apply? Shouldnt it just be "certainly"? After all, there are no limits. -- Anonymous DissidentTalk 09:42, 21 December 2007 (UTC)

Imagine that a fair coin is tossed infinitely often. Denoting the two sides as 0 and 1, this gives an infinite sequence of bits, something like
(space limitations do not allow to show the full sequence). What is the chance of getting exactly this sequence? In fact, all possible sequences are just as likely, such as the sequence
or, for that matter, the sequence
So if it is certain that the last sequence will not occur, it is just as certain that any other sequence, such as the two above it, will not occur, including the one that actually is the result of tossing the coin infinitely often. Under the normal meaning of the word certain, that is a contradiction.  --Lambiam 18:56, 21 December 2007 (UTC)

I had the exact identical question back in the section entitled (appropriately enough) "Question". Refer to Dcoetzee's answer there. —Preceding unsigned comment added by (talk) 17:35, 29 March 2008 (UTC)

This looks like a case of approaches infinity vs. infinity. Similarly p(x)->1 not p(x) identical to 1. (talk) 12:30, 19 June 2008 (UTC)

It is definitely not surely. If a monkey presses a button at random, that means it is possible that it can repeatedly press the same button each time. The probability of this continually happening is very small, but it can happen. For this reason it is impossible to 'guarantee' that the works of Shakespeare would ever be produced. It only becomes more likely as more time is given. —Preceding unsigned comment added by (talk) 14:31, 27 June 2008 (UTC)

"Huxley" &c.[edit]

Does it not seem a little silly to put a picture of Huxley up when the article does not actually owe much to the man? He simply was attributed wrongly, and had a nice little quip about the bishop. The only other illustration contained in this article illustrates the article beautifully. Huxley, however, has little to do with this. I'm just being picky, but it threw me for a second. It seems like unnecessary emphasis on an only mildly meaningful character (in the context of this article). (talk) 19:35, 2 February 2010 (UTC)

The "Evolution" section currently begins:

In his 1931 book The Mysterious Universe, Eddington's rival James Jeans attributed the monkey parable to a "Huxley", presumably meaning Thomas Henry Huxley. This attribution is incorrect.

The footnote for this bold statement includes the following citation: Padmanabhan, Thanu (2005). "The dark side of astronomy". Nature. 435: 20–21. doi:10.1038/435020a.. However, I cannot figure out how that article relates to the Infinite Monkeys, let alone to whether or not "Huxley" invented that formulation. Am I missing something, or has there been an error?


Borges follows the history of this argument through Blaise Pascal and Jonathan Swift, then observes that in his own time, the vocabulary had changed. By 1939, the idiom was "that a half-dozen monkeys provided with typewriters would, in a few eternities, produce all the books in the British Museum."

Interestingly, when Borges says this, he too attributes it to a "Huxley":

Siglo y medio más tarde, tres hombres justifican a Demócrito y refutan a Cicerón. En tan desaforado espacio de tiempo, el vocabulario y las metáforas de la polémica son distintos. Huxley (que es uno de esos hombres) no dice que los "caracteres de oro" acabarán por componer un verso latino, si los arrojan un número suficiente de veces; dice que media docena de monos, provistos de máquinas de escribir, producirán en unas cuantas eternidades todos los libros que contiene el British Museum.

I would be interested in knowing the history of this attribution. Is there a kernel of truth in it? I guess I'll have to check Respectfully quoted: a dictionary of quotations (the other authority cited in the aforementioned footnote) when I get home, but in the meantime can anyone elucidate this? It seems like we should have some firm sources if we're going to categorically state that the "attribution is incorrect," right? --Iustinus (talk) 23:44, 28 December 2007 (UTC)

Should have specified: since Borges wrote in 39, he's presumably repeating the claim from Jeans. But it's still very interesting that hementions it. --Iustinus (talk) 00:50, 29 December 2007 (UTC)
I think the attribution to Huxley is correct. To Julian Huxley, not Thomas Huxley. Six monkeys, infinite years... (one source: Universal Book of Mathematics). Eddington, Jeans, Huxley were all contemporaries (see God and the Universe) Zorgster (talk) 20:05, 16 January 2012 (UTC)
I have found an article from Irishman's Diary (The Irish Times, May 18, 1939. Accessed via ProQuest Historical Newspapers: The Irish Times (1859-2007)). There is a short piece titled "Monkeys by the Million", the author reports listening to a radio program in which the host mentioned a friend from Aberdeen told him "if you had a sufficiently large number of monkeys thumping the keys of an equally large number of typewriters for an indefinite period of time, you eventually would produce all Shakespeare's plays.". A Scottish friend of the author disputes this, "..because if you covered the earth with typewriters and monkeys, with a patch three-foot square in the middle for the Editor of the Irish Times to sit on - if it would hold him - and to take observations; and if you set these said monkeys knocking the keys of each typewriter about a line a minute, say 40 characters, the chances that at end of the year one of them had produced 'To be or not to be, that is the question' is about one in a million million million million million million million". This therefore differs from the comment above that "by 1939, the idiom was" - it is highly likely that by 1939 the idiom had taken a multitude of forms. Zorgster (talk) 17:28, 21 January 2012 (UTC)
From The Manchester Guardian Sept 12th 1928, pg 5, The British Association:
The article discusses the talk given by Professor Frederick George Donnan at the Annual Meeting. He is discussing a discovery made by physiologist A V. Hill regarding cell respiration and the constant need for oxygen. It is not clear whether the Guardian reporter is making this comment, or whether Prof. Donnan said it:
"... but according to the statistical theory of probability if we waited long enough anything that was possible, no matter how improbable, would happen. If six monkeys were set before six typewriters it would be a long time before they produced by mere chance all the written books in the British Museum, but it would not be an infinitely long time." (Guardian, Sep 12 1928, pg 5) Zorgster (talk)
From The Guardian - Feb 5th 1927 - A Definition of Extreme Improbability:
This article discusses a talk on Feb 4th 1927 by Arthur Eddington at the 2nd Gifford Lectures titled The Nature of the Physical World (of which the book is referenced in the main article), in which he discusses entropy in the universe. Eddington's metaphor for entropy is of air spreading out in a box and the article adds (or reports) that the chance of all the air ending up in one half is very low... "The chances against this are greater than that an army of monkeys drumming on an array of typewriters should by accident compose all the books in the British Museum.". This attribution is earlier than that cited in the main article, but I cannot confirm it is the words of the reporter or Eddington. Zorgster (talk) 19:11, 21 January 2012 (UTC)


It must be made very clear here what we take on faith in our definition of "infinite". Even if "infinity" as a logical abstraction is comfortable and acceptable in the exact branches of mathematics, bringing it into statistics raises innumerable difficulties, not all of which are mathematical. --VKokielov (talk) 05:05, 4 February 2008 (UTC)

One of the worst Wikipedia articles ?[edit]

And it also has an error - in the Direct Proof section, please note the while the individual keystrokes are independent (by assumption), the blocks of 6 letters are NOT, since they overlap. and if they not overlap, there is the possibility of |1QpBAN| |ANAhlp|, which is not accounted for. Either way, the proof is incorrect.

zermalo (talk) 20:02, 17 March 2008 (UTC)

Your claim is not constructive. Please point out which particular sentence you believe is incorrect by directly quoting it here. Melchoir (talk) 01:32, 18 March 2008 (UTC)
I think this is covered pretty well by footnote 1. Does it need to be in the body of the article? Algebraist 14:29, 5 May 2008 (UTC)

Whether or not the text is organized into blocks is irrelevant to the proof. Organizing the text into blocks of, say, 6 implies that we will be making 1/6th as many samples which in turn implies that the time needed will be 6 times longer. The samples are still independant. Let us examine the alternative. No blocks: There are two variations to this problem. Case 1: Will "poem" be replicated. In this case, overlapping samples can be considered independant. 'Nuff said. Case 2: searching for the first instance of "poem". This case is harder, (as samples demonstrate dependance), but we can still fall back on the blocks (or even case 1). (talk) 12:48, 19 June 2008 (UTC)

The proof is correct, because if the probability of BANANA occurring on a block boundary is 1, then any more likely event has the same probability. There's no need for a precise analysis here. Dcoetzee 01:45, 28 June 2008 (UTC)
Oh, dear! I hadn't noticed the claim. It is wrong, but the errors also really do not matter. See the "Definitely wrong but morally right" section infra. JoergenB (talk) 20:20, 7 October 2008 (UTC)

The Brainiac Experiment[edit]

When I came to this page I immediately thought of an experiment conducted on Brainiac testing this idea. They sat several monkeys and several "Brainiacs" down at computers, and the closest they could get to Shakespeare's works was when one monkey typed "alas" in the whole six hours. I would put this on the page, but I don't think I'd do a very good job of it and I'm relatively new to Wikipedia, so please could someone else who knows more about what they're doing add this? Thanks:) Lowri (talk) 17:27, 22 July 2008 (UTC)

Do you have a reliable source for this story? Surely anyone of the "Brainiacs" could have typed To be or not to be; in fact almost any English-speaking "Joe Shmoe" would know that much Shakespeare. What have the computers to do with it?  --Lambiam 00:04, 27 July 2008 (UTC)
Is the story itself not reliable enough? Well, it doesn't matter anyway as their experiment really didn't prove anything, they probably just did it for the "lulz". (talk) 21:24, 31 July 2008 (UTC)

How tiny is very tiny?[edit]

In a recent edit, the sentence

"The probability of a monkey typing a given string of text as long as, say, Hamlet, is so tiny that, were the experiment conducted, the chance of it actually occurring during a span of time of the order of the age of the universe is minuscule but not zero."

was changed to:

"The probability of a monkey typing a given string of text as long as, say, Hamlet, is very tiny, but not zero."

The edit summary stated:

"This is redundant and wrong ".

I don't see what is wrong with this, assuming the age of the universe is not significantly more than 10100000 years and that the monkey does not type significantly faster than 10100000 characters per second. Please enlighten me.  --Lambiam 23:41, 12 August 2008 (UTC)

You might want to knock a zero off the second exponent (I'm not sure of the average word length of Hamlet, so I'm being conservative), but otherwise You're right. Algebraist 23:52, 12 August 2008 (UTC)
I see the full edit summary was This is redundant and wrong, especially interesting to be wrong only 2 lines after explaining the perils of reasoning in this exact way...:), referring to 'the perils of reasoning about infinity by imagining a vast but finite number, and vice versa'. Perhaps Diza was imagining an infinite age of the universe? Algebraist 23:55, 12 August 2008 (UTC)

Yes, the shortened version, while true, is a bit of an empty statement - it's a bit obvious. But the original is also vague - we've no idea what "the chance of" is actually referring to, or even what "the experiment" is. The theorem is only meaningful in an idealised, thought-experiment sense. I suspect any re-write eliminating this vagueness would render it too wordy for an introductory paragraph. I can't imagine calculations about the universe or the laws of physics having any place here - there's plenty of scope for them in the main body. I'd say the sentence is both redundant and vague, and the section works better without it.Bob D (talk) 00:14, 13 August 2008 (UTC)

The antecedent of "it" in the original is obviously "a monkey typing a given string of text as long as, say, Hamlet". In the preceding sentences of the lede it has just been explained that the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum, which in an infinite amount of time will almost surely produce a given text, such as the complete works of William Shakespeare. At least to me, it appears so obvious that the experiment is to let an abstract device produce a random sequence of letters for an indefinite amount of time that I don't think this needs to be explicated.  --Lambiam 13:05, 17 August 2008 (UTC)
Since 'it' has no reference to a time scale or rate, so there is no meaning to the 'chance of it occurring' within a given time.Bob D (talk) 06:28, 18 August 2008 (UTC)
The "chance of it occurring" in question is, of course, a non-constant function of the typing rate. This does not make it meaningless, nor does it prevent us from observing that it is miniscule in any experiment. Melchoir (talk) 07:01, 18 August 2008 (UTC)

130.000 (or actually more) ?[edit]

Take Hamlet from and pipe it through

perl -pe 's/\[.*?\]//; s/[\s\,\.\-\;]//g'|wc

and you'll see it's just a little less than 130.000 characters.

Not that it matters much, but still :) Rkarlsba (talk) 03:55, 3 September 2008 (UTC)

Definitely wrong but morally right[edit]

The section Direct proof contains a fallacious statement (whence of course its proof also is not quite correct). At the same time, the error is of no importance for the thesis of the article; instead of the proposed exactly exponential expression, you get an approximately exponential expression, not with the same but with a rather similar base, and thus the conclusion is not at all involved. Thus, the argument ought to be rephrased. I'm afraid there is no doubt about the results in themselves; they are the most simple cases in studies of growths of dimension sequences related to finitely presented algebras, which were studied in the '70's e.g. by Victor Ufnarovski, Warren Dicks, and myself. The simple, purely combinatoric situations, independently later have been rediscovered by combinatorians (coming to the same conclusion, of course); and I suspect that the simple "one forbidden word" enumeration problem also has been covered by independent discoveries several other times, both before and after the first publication of it that I know about (by Govorov in 1972).

Here is the result, and an outline of the proof. A much more general result, covering any finite number of "forbiodden words", was proved and puublished by V. E. Govorov in the Mat. Zametki 12 (1972), pp. 197-204. I concentrate on "BANANA" and 50 letters, however.

Assume given an alphabet of 50 letters, and among these the letters A, B, and N. For any ("European") natural number n, let an be the number of strings of length n in the 50 letters, that do not contain BANANA as a subword; let us call "BANANA" forbidden, and the strings without an occurrence thereof allowed. Then, for , clearly . For , fulfils the recursion formula


The reason is simply this: We may prepend any letter to any one of the allowed strings of length , and in that manner we get strings. Obviously, these strings include all the allowed strings of length n. However, some of them are forbidden; namely those beginning with "BANANA". For any one of these forbidden strings, the last letters will form an allowed string, say S (since otherwise already the string ANANAS would be forbidden, before prepending the initial "B"). Thus, out of the candidates for allowed strings, exactly fail.

Now, the most neat way to sum up these recursive properties are by the generating formal power series (as it ought to be called), alias the generating function (as it usually is called). In fact, it is not hard to see that


by means of the usual methods taught in an elementary coursis including combinatoric enumeration by means of formal power series. However, our purpose here is slightly different; we want an estimate of the probability for finding or not finding "BANANA" as a subword of an arbitrary length n string. Assuming that the letters are chosen i.i.d. and with equal probabilities, all the strings are equally probable; whence the probability of not having "BANANA" as a subword is exactly . Now, as also taught in these elementary enumeratoric combinatorics courses, a rational expression for a formal power series, such as the one supra, may be converted to a polynomial expression for the coefficients, in terms of the roots of denominator polynomial of the rational expression; or, more precisely, of the associated "auxiliary equation"


A sixth degree equation is a bit hard to solve by elementary means (as Abel proved); but it is easy to see that this one does not have double roots (by taking the g.c.d of it and its derivative). This yields, that there are constants , such that for any we have


where the are the roots of the auxiliary polynomial. The absolutely largest of these roots, say , is a positive real number; and thus already for moderately large n the quotient will be rather close to . However, as you may see by simple means, the auxiliary equation does not have any rational root, whence a fortiori


Now, I know that arguments involving probabilities often invoke hot sentiments; and I also know that nobody likes being told that someone else knows much more about these subjects. I'm really not trying to bully anyone; but I did write part of my ph.d. thesis about these things, and have published a few articles about this later, whence I would be lying if I told you that this just is a guess. I've really tried to explain why you get the results, perhaps too lengthily; I'm absolutely prepared to discuss them further in detail, here or on my user talk page; but, as I said, the differences are actually not that important. The base of the exponential expression is a number somewhat smaller than 1, that's all that matters here, actually. I'd like to rewrite the text slightly, weakening the claims a bit (and not including the Govorov et al. precise results with proofs, if you don't mind), but providing a correct and sufficient estimate. However, I won't try this, if there are too many "unresolved issues" about the mathematics left. JoergenB (talk) 21:50, 7 October 2008 (UTC)

What's wrong with the direct proof as it stands? Algebraist 11:54, 8 October 2008 (UTC)
The entire point of the simple proof is just to demonstrate a simple case of the theorem and why it's true in that case. It doesn't need to provide an accurate estimate of the probability, only to bound it from below, and then show that that bound tends to 1. A more accurate estimate may be relevant, but for this initial example it's overkill and defeats the point of having a simple example. Dcoetzee 21:45, 8 October 2008 (UTC)
I believe JoergenB meant we should use the phrase "Sketch of proof" if we don't meant to have the exact details to be correct. K61824 (talk) 05:10, 10 May 2009 (UTC)

Link 24: the monkey Shakespeare simulator[edit]

This site bombarded me with Java errors and made Firefox crash. Is this a common issue with the site? Should the link be removed if the site is unusable because of this? - (talk) 22:21, 25 October 2008 (UTC)

might be just you. worked fine on ieFirl21 (talk) 15:13, 3 September 2009 (UTC)

Kolmogorov Complexity and monkeys typing on a computer[edit]

One of the most interesting sidenotes to the infinite monkey theorem is the fact that the monkeys would be significantly faster if they used a computer instead of a typewriter. I first heard about this in Cover and Thomas, "Elements of Information Theory".

Given that the Information entropy of the English language is about 1.5 bits per character, Shakespeare's works could probably be compressed by a factor of at least three. Thus, the probability that the monkeys come up with a compressed version of Hamlet (or a computer program which prints Hamlet) is much higher than the probability that they produce the full text.

I just find that interesting... what do others think about it? If there are enough people that motivate me (write to j dot b dot w at gmx dot ch) I'll create a Wikipedia account and write it down properly. —Preceding unsigned comment added by (talk) 11:00, 19 December 2008 (UTC)

Yes, that's interesting - I've added your paragraph (with minor changes) to the Probabilities section as a footnote to the text.Bobathon (talk) 13:48, 11 January 2009 (UTC)
It may be appropriate to create a section for this somewhere within the article - it has a significant bearing on all calculations relevant to this subject (though clearly not on the final "finite but non-zero" conclusion)Bobathon (talk) 13:54, 11 January 2009 (UTC)


There are some broken equations in the Solution section. Could somebody more knowledgeable than I fix that? Thanks! –Juliancolton Tropical Cyclone 02:53, 25 February 2009 (UTC)

Weasel misconception[edit]

I have corrected a misconception in the article which stated that "Dawkins has his Weasel program produce the Hamlet phrase METHINKS IT IS LIKE A WEASEL by typing random phrases but constantly freezing those parts of the output which already match the goal". The bold text is incorrect; in the context of this article, it's not a big deal, but it is part of the toolkit used by those who oppose evolution, and is plainly wrong.

The current Weasel program article does not clearly address the issue, but it is well covered in the discussion. The correct letters appear to lock because the program chooses the best match from mutated progeny (so mutations which make a good letter bad will usually not be the next parent). However, a video shows that the program does not lock correct letters, and the words used by Dawkins in The Blind Watchmaker make it obvious that his simulations apply random mutations to all locations. Johnuniq (talk) 08:55, 2 May 2009 (UTC)

How was the specific probability calculated?[edit]

Just wondering, how was the number 3.4 × 10183,946 obtained? This needs to be explained somewhere, or else the number should be removed for being unverifiable original research. ··gracefool 18:54, 8 May 2009 (UTC)

It's just 26^130000. Algebraist 20:00, 8 May 2009 (UTC)
How do you work that out? ··gracefool 10:38, 9 May 2009 (UTC)
It was clear from the immediately preceding text that 26^130000 was the number that should have appeared there, so I just checked (using Google calculator) that it was. Algebraist 18:37, 9 May 2009 (UTC)
To calculate n = 26^130000 we take log of both sides: log(n) = 1300000*log(26) = 183946.5352
Therefore n = 10^0.5352 * 10^183946 = 3.429 * 10^183946 Johnuniq (talk) 00:29, 10 May 2009 (UTC)

Thanks. So including punctuation the figure is about 10^360783 (26 letters x2 for capitalisation, + 12 for punctuation characters = 64, log(64)*199749 characters). That makes a big difference to the number! ··gracefool 01:18, 10 May 2009 (UTC)

I've added this stuff to the article. ··gracefool 05:36, 25 June 2009 (UTC)

I once slapped the keyboard at random and got "iloveyouall". Professor M. Fiendish, Esq. 04:51, 13 September 2009 (UTC)

"Almost surely"[edit]

The phrase "almost surely" has ABSOLUTELY NOTHING to do with the monkey metaphor! It's a precise mathematical term with a precise meaning that has NOTHING to do with metaphors. Please fix the article so that it makes sense! —Preceding unsigned comment added by (talk) 18:04, 15 August 2009 (UTC)

Broken external link[edit]

The first external link (to the Baltimore Examiner) is broken, and just redirects to the homepage of the Washington Examiner. (talk) 09:52, 9 November 2009 (UTC)

You meant the first link in the External Links, correct? I couldn't find it either. Looks like it got lost when baltimoreexaminer merged into washingtonexaminer. Lots of googling no go. Worse news, see here:*/ Looks like baltimoreexaminer never let IA archive their stuff, so it'll be real hard to find again. :-( —Aladdin Sane (talk) 10:43, 9 November 2009 (UTC)

Probability section fails[edit]

That whole section is wp: or, not to mention wrong. It assumes that the monkey is only striking keys that produce letters the monkey could strike any of the function keys or numbers etc. It cites no sources except the bottom where it takes a quote. The main trouble is that this section falsely presents a low probabilty by restricting the origional terms of the theory to the life of the known universe rather then infinity which is how it is supposed to be. Any probability repeated with a time frame of infinity will be come 1, the life of the universe isnt even a warm up phase compared to infinity. This section needs to be removed entirely it is not helpful Smitty1337 (talk) 23:04, 21 April 2010 (UTC)

I agree that OR is an issue but I support the section (WP:IAR or whatever) since it is verifiable and useful to readers. The restriction to 26 letters is just a commonly-made simplification – if the monkey had more than 26 choices the probability would be lower than the effectively zero value shown in the article. No one is denying that in some metaphorical sense randomly striking keys would eventually produce a sentence, but it is valuable to learn that in practical terms the outcome is impossible. The concept has sometimes been used to assert that freak things will eventually occur by blind chance (i.e. physical things in a practical universe). Whereas that is true for many very rare phenomena, it is not true in this universe for an event such as randomly typing a particular book. Also, note that there is a very reliable reference that supports the conclusion of the section. Johnuniq (talk) 02:02, 22 April 2010 (UTC)
this section shows the odds at 3.4 × 10183,946 . if we assume 1 letter per second and 130,000 total seconds thats 2166.66 hours or 90 days roughly 1 quarter of a year, per full book attempted (and this is generous because it doesn't assume failure on letter 3 throws the book out and starts over right there). if the 3.4 × 10183,946 attempts required to get 1 right is all that's statistically probable, then the monkey should write one copy every 76.5 x 10183,946 years which divided by infinity will happen infinite times so in 76.5 x 101839460 years we'd have the 10th copy (collectors edition i presume). I'm being absurd to get my point across, this theorem is meant to prove a point, not be taken literally, the point is that anything that has a probability that is not zero no matter how excessively large the denominator is on that fraction, the concept of infinity makes that number minisucle to the point of if it "can" happen it will happen repeatedly in 345235324532452345234523453245234523452345234532 years (not even .00001% of infinity of course). this is all OR of course which is why i'd never push to say such on the article, but neither should some silly probability section give a false notion of unlikelyhood when infinity makes the chance 100% (eventually) Smitty1337 (talk) 10:33, 22 April 2010 (UTC)
From the reliable source: As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…". The numbers used for input to the calculation are shown in the "Direct proof" section. Don't you find it interesting that the chance of randomly typing Hamlet is essentially zero, even if you have as many monkeys as there are particles in the universe, and each types 1,000 keystrokes per second for 100 times the life of the universe? Johnuniq (talk) 11:20, 22 April 2010 (UTC)
The trouble with making that statement is that it is without context. The probability may be operationally zero, as the sourcer says, and that is infact true, but the problem is that any probability given infinite repetition will become 1, and if not stopped upon successful completiion, then technically the book will be made and remade infinite times. the statement is true but by itself its misleading because its just the probability without moving to the next logical step of infinite repeatition, and everything above it is wp: or Smitty1337 (talk) 00:43, 23 April 2010 (UTC)

From a Darwinian approach to the question "How long would monkeys type the works of Shakespeare"?, the solution is the period intelligent man has evolved from primates. A monkey , trapped on earth a million years ago with nothing but to mutate itself into an intelligent being, discover words, concoct writing, invent the typewriter and finally produce among its descendants a literary genius would take a few million years. This period has been historically tested and is much much less than the predicted statistical outcome with the assumption that the monkey will genetically remain stagnant without hope for increased brain capacity but just random act in his eternal life. —Preceding unsigned comment added by (talk) 20:16, 1 September 2010 (UTC)

You have missed the point entirely, as the article says right in the lead, its a metaphorical moneky, as in one monkey, not a series of generations, there is no evolution involved in this article and its only meant to imply a concept of probability and infinity. Smitty1337 (talk) 18:29, 2 September 2010 (UTC)
Sorry to engage in forum talk, but while of course you are correct, I think probably knows that, and the point they made is interesting because it highlights how human commonsense can fail when extrapolated too far. Our brains provide a model of the world whereby we are happy to talk about an ideal monkey typing for millions of years (which is fine), but in the time frames under discussion lots of things will change in ways that we struggle to appreciate (the Atlantic ocean is expected to disappear in 200 million years or so, due to plate tectonics; can't find it on Johnuniq (talk) 02:35, 3 September 2010 (UTC)

None of this is right at all. A correct neuron configuration is required to type out a William's Shakesphere Play. Given infinite time, the neuron configuration of the brain will never reach precisely the right locations and synapses as that so the monkey will be typing out an entirely play by chance because it is deterministically a zero probability. In order for an action to occur, a neuron must be fired, and certain neurons specifically. —Preceding unsigned comment added by (talk) 13:33, 20 November 2010 (UTC)


"Popular interest in the typing monkeys is sustained by numerous appearances in literature, television, radio, music, and the Internet. In 2003, an experiment was performed with six Celebes Crested Macaques. Their literary contribution was five pages consisting largely of the letter 'S'."

  1. This wrongly suggests that it was a scientific experiment, rather than a student art project.
  2. I don't see that this is significant enough to mention in the lead. Feezo (Talk) 03:05, 8 February 2011 (UTC)
    I agree that the sentence should be removed from the lead. I suppose the mention in the "Real monkeys" section is warranted, although it adds little of value to the article other than to show that the topic is of general interest. Johnuniq (talk) 05:46, 8 February 2011 (UTC)

Weird sentence in the lede[edit]

"The theorem illustrates the perils of reasoning about infinity by imagining a vast but finite number, and vice versa. " - what's "perilous" about it? What, I'm gonna get shot if I think about infinity by imaging a vast but finite number? What the hell is this sentence even supposed to mean? Nonsense.Volunteer Marek (talk) 21:24, 11 April 2011 (UTC)

And what for monkey's sake does the "vice versa" refer to? Reasoning about "finity" by imagining a minuscule but infinite number? Seriously.Volunteer Marek (talk) 21:26, 11 April 2011 (UTC)

almost surely[edit]

"almost surely" implies the change the money wont type the play is considerable, but the chance is actually infinitesimal or zero. (talk) 22:35, 14 April 2011 (UTC)

It's a technical term; it's not up to us to change it. It's explained in the text, which is really all we can do. --Trovatore (talk) 22:45, 14 April 2011 (UTC)
"Almost surely" is the wrong term. It is not a technical term at all, but colloquial English. The concept is that an infinite amount of time, or monkeys, is available to write the complete works of Shakespeare. Given the unlimited extent of infinity the works of Shakespeare WILL be written. Not "almost surely". (talk) 04:51, 20 September 2014 (UTC)
No, "almost surely" has a rather strict mathematical definition. In my calculus-oriented mind, it means that the probability of not typing a given work has limit 0 as the amount of typing n tends to infinity (i.e. for any possible positive probability there always exists an n where the probability of not typing the work is less than that given positive probability); the linked article has a better definition. But there exists no n for which the probability of not typing a given work is actually 0. To assume otherwise implies the gambler's fallacy (if the typing is truly at random).--Jasper Deng (talk) 04:58, 20 September 2014 (UTC)

typing for infinity[edit]

If the apes could type for infinity, and there was a small amount of probability that they could type the works of some author, is it also possible there is a small chance they write something that hasn't been written yet (assuming nobody saw the writings and somehow copied them)? Could the typing somehow predict a future? If they could, they could also predict another piece of writing that will never happen, or never has happened, creating something new, couldn't they? I may be asking something that has been gone over already, but I don't think so. — Preceding unsigned comment added by (talk) 03:47, 3 July 2011 (UTC)

anything about the article ? Arjuncodename024 05:21, 3 July 2011 (UTC)
For the record, there is no "predictive" value to monkeys typing for billions of years, because any process which found some coherent text in the randomness (say, a collection of poetry about probability) would in effect be "writing" that text by searching for it. Overall, readable random texts are way way way less frequent than unreadable ones, unless you have an evolution-like process that keeps the readable stuff and ditches the unreadable, in which case it's a somewhat different puzzle.
However, if we're talking about an infinite length of time, then yes, even one monkey will indeed produce every possible finite sequence of text with probability 1, including readable books that no human had ever written — an infinite number of such books, in fact. There's nothing special about Shakespeare here, it's just used to illustrate this surprising idea. ± Lenoxus (" *** ") 00:32, 13 October 2011 (UTC)
Nothing surprising about it when you consider how long infinite time is. ··gracefool 03:22, 3 November 2011 (UTC)
Lenoxus has it right, but in other words: You can't predict the future with randomness, because to recognize it as something apart from randomness, you have to already know the information in question. If there was no Shakespeare, you couldn't recognize Hamlet as being something special, any more than any of the other millions of readable books likely to be typed before that. If there were a specific prediction, eg. "The World Trade Center will be destroyed on September 11, 2011", how would you tell its accuracy over a million other predictions of the same thing happening on a different date? ··gracefool 03:22, 3 November 2011 (UTC)

The picture in the article is wrong also[edit]

That's a chimpanzee, not a monkey. And that's a camera, not a typewriter (I think). — Preceding unsigned comment added by (talk) 05:45, 10 September 2011 (UTC)

It's a very old typewriter: zoom in on it to see the detail (the base, the keys, the paper holding structure at the top). As for the ape historically 'monkey' was used for both apes and monkeys. Certainly when this phrase arose it would be common usage. Even now it is still used informally like that.--JohnBlackburnewordsdeeds 12:15, 26 September 2011 (UTC)
Typical. When the monkeys produce a work of literature, it's credited to an ape!--Jack Upland (talk) 09:19, 21 November 2016 (UTC)

Title of first (non-lead) section[edit]

Why is the first section under the TOC headed Solution? Solution of what? Problems have solutions, but the theorem is not stated as a problem. I've looked through the history and it appears to have been that way for quite a while so I don't want to change it rashly, but surely we can do better than that. Suggestions? Maybe the inelegance of this word indicates a more structural difficulty with the article, and suggests moving the proof sketch further down? --Trovatore (talk) 20:13, 12 October 2011 (UTC)

Yes, it is odd, although I don't think moving it would be particularly helpful. Perhaps "Analysis"? Also, in "proof of this theorem", the "this" should be replaced with "the infinite monkey". Johnuniq (talk) 00:31, 13 October 2011 (UTC)

Similar concepts[edit]

Why aren't there links to similar concepts? Someone could include a link to Bogosort or something, which I believe is a great example. (talk) 09:09, 15 November 2011 (UTC)

Good question! I'll add some later, if I can remember to. In the meantime, feel free to be bold. Evanh2008, Super Genius Who am I? You can talk to me... 09:43, 15 November 2011 (UTC)

Why the link at the top about "not to be confused with..."?[edit]

Why exactly is there a link at the top of the article staying this should not be confused with the hundreth monkey effect? They are totally unrelated things and in my opinion, are not easily confused. If this statement stays, then I think we should also add the following statement: "Not to be confused 12 Monkeys." I mean, there is a number at the front of the statement and it says monkeys... a reader might be confused. Krohn211 (talk) 19:09, 1 December 2011 (UTC)

I agree: if this title were ambiguous it should link to a disambiguation page but it clearly isn't. Further I can't see anyone confusing this with the hundredth monkey effect which seems not to be widely known: from the article it's a discredited crank theory, not part of mainstream science. I've moved it to 'See also' but I have no objection to it being removed altogether.--JohnBlackburnewordsdeeds 19:35, 1 December 2011 (UTC)
I am not a heavy wikipedia user so I don't know the goal of the "See Also" section. From my understanding, this section is meant for related topics and I don't believe this is a related topic. If it were up to me, I'd remove all references to it from this article. Looking at the history I can't tell who put this there in the first place. Since I didn't put it there, I don't want to be the one to remove it. But if I had a vote, I'd say remove it altogether. Krohn211 (talk) 03:53, 2 December 2011 (UTC)

The above issue was fixed in this edit by JohnBlackburne. Johnuniq (talk) 06:45, 2 December 2011 (UTC)

Unnecessary paragraph[edit]

"Primate behaviorists Cheney and Seyfarth remark that real monkeys would indeed have to rely on chance to have any hope of producing Romeo and Juliet. Monkeys lack a theory of mind and are unable to differentiate between their own and others' knowledge, emotions, and beliefs. Even if a monkey could learn to write a play and describe the characters' behavior, it could not reveal the characters' minds and so build an ironic tragedy"

Summary: monkeys are too dumb to write Shakespeare.

This is taking stating the obvious to a new level; it reads like a tabloid article or something from a waiting room magazine. Perhaps this somehow meets a guideline I don't know about, but surely a fact that no reader is ever likely to not already be aware of expressed in 73 words with "sciency" language and name-dropping of behaviourists does not belong on Wikipedia. Kombucha (talk) 00:05, 23 December 2011 (UTC)

We should lose the name-drop at the least. Kombucha (talk) 00:11, 23 December 2011 (UTC)
Feel free to remove the whole "Real monkeys" section because it is just unrelated commentary (i.e. it is nothing to do with the actual "theorem", and is essentially nonsense, no doubt expressed in impressive language in the original). Johnuniq (talk) 01:49, 23 December 2011 (UTC)

History - references preceding Borel[edit]

I am fairly new to editing Wikipedia articles, not new to Wikipedia. I have been researching this topic to find an earlier reference to "the likeliness of monkeys writing great works". I found a reference by Richard Bentley in a sermon (originally 1692/3) regarding the probability that a monkey scribbling away could ever write Hobbes' Leviathan as a comparison to the probability of creation. I dispute that Borel is the first person to use the idea of monkeys typing works by chance in the context of probability. (As an aside, in the French, I have also found 'singe' with the meaning of 'Mime Artist' or 'people who mimic' ("les singes de Balsac (sic)" appeared in a dictionary to denote the plethora of authors all trying to mimic the style of Balzac.)) Bentley's prose is in the context of probability (albeit in terms of creation and not mathematics). True there were no typewriters in 1692, but this merely means that Borel had re-framed the concept of 'monkey scribbles' to 'les singes a frapper' on a typewriter (if it had not already been re-framed previously to him). Also in the re-framing one monkey becomes one million monkeys.

My edit was made in haste... and was removed as 'misplaced and original research'... I would like to discuss this. (Source: Richard Bentley's The folly and unreasonableness of atheism) Zorgster (talk) 04:41, 15 January 2012 (UTC)

It was this edit that was reverted. I haven't wanted to take the time to investigate this issue (I saw your edit and decided to leave it). However, my guess is that the editor who reverted your edit thinks that we would need a secondary source that makes an association between the information you found and the topic of this article. If writing an article at some other place, it would appear under the writer's name and any views in the article would clearly be the views of the author. However, there is no author here, and all statements need to be verifiably related to the topic, rather than likely associations that an editor has located. I have not formed any firm views on the issue, and mention this for background. Johnuniq (talk) 06:06, 15 January 2012 (UTC)
Hi.. thank you for taking the time to explain that. I see what you are saying. I could only speculate myself. ...that the sermon, which was part of the Boyle Lectures and so re-published in 1737 (A Defence of Revealed and Natural Religion), in 1809 (Eight Sermons, Oxford), in 1838 (The Works of Richard Bentley, Vol 3), was possibly used in arguments against atheism - or to strengthen religion. It would have been well circulated amongst scientist and clergy alike. Bentley's 'Confutation of Atheism' also discusses the ideas of Isaac Newton. I would suggest that any scientist, including Huxley, would be familiar with this text. And the argument about monkeys scribbling Hobbes would have been read by many an academic. Huxley may have used it in his crossings with Owen and Wilberforce (in Oxford, 1860) - in debates of Darwinism and religion. The paragraphs around Bentley's mention of the monkey in the 1838 edition, talk a great deal about probability and chance... and as is said in the philosophy of history... we tend, when reading historical prose, to frame the usage of words in the past using our understanding of the present... The human body was seen then (1692) as the creation of the divine... A body was often compared to a book... and the comparison here is that you can deny the 'hand of god' in the creation of man, as much as you could conceive of a monkey ever scribbling Leviathan out of pure chance (Kristine Haugen's Richard Bentley: Poetry and Enlightenment). Again, one can only speculate... and one needs to spend time looking into it... I've used my quota of spare time, too :-) Zorgster (talk) 06:31, 16 January 2012 (UTC)

Possible FAR[edit]

Referencing on this article is still sub-par and has been tagged as such for close to 5 months (Criteria 1c); the prose in some parts, such as "in popular culture", is rough (Criteria 1a). This should be fixed, if possible. — Crisco 1492 (talk) 23:58, 31 May 2012 (UTC)

numbers of bibles etc[edit]

I'm sorry if someone has already mentioned this but has anyone considered how many Bibles that contain one or more errors would be produced in order to produce one without errors. One way of looking at this would be if the monkeys started at the same time and typed at the same rate how many monkeys would we need before all the material in the universe had been turned into faulty bibles. And many correct letters would there be in the one copy that was correct.

Secondly I thought one of the Bernoulli's said that where a probability was exceedingly low, as in this case, one could completely ignore it because the probability of almost anything else, for example, the existence of a god with a sense of humour, explaining the result would be astronomically higher.

john f (talk) 11:48, 2 May 2013 (UTC)

The Simpsons' Did it![edit]

The Simpsons TV show parodied this idea in some episode, with what I believe to be one of the cleverest lines in the show. Mr. Burns goes into a special room of his which houses a number of monkeys dutifully (and fearfully I believe, thanks to Mr. Burns' reputation) typing away at their assigned computers. Mr. Burns walks over to see the result of a random monkey's work and reads out "It was the best of times, it was the BLURST OF TIMES!", and smacks the monkey or something. 'Worst' was spelled incorrectly, and the line is not Shakespeare's, but still pretty good for monkeys. This could be added under a 'cultural references' section, since it's so damn funny. Also, a picture of Mr. Burns reading the transcript should be included, for completeness. Jake Papp (talk) 14:14, 7 August 2013 (UTC)

That is already on Infinite monkey theorem in popular culture; it does not need to also be in the main article. meshach (talk) 16:41, 7 August 2013 (UTC)

Why is there a whole article when there is already a section for popular culture on the original article's page?


The beginning paragraphs (before contents) need more citations to back up the claims they make. In fact, the whole article could use more citations backing up their claims, especially the Direct Proof section. Bibliophile scribe (talk) 08:23, 3 December 2013 (UTC)

Twitch Plays Pokemon[edit]

More than 50 thousands of players are playing the same videogame. Can this be a practical approach to the theorem? — Preceding unsigned comment added by (talkcontribs) 19:00, February 19, 2014 (UTC)

No is is just a coincidence meshach (talk) 06:21, 20 February 2014 (UTC)

Proof sources? Original research?[edit]

Are there sources for the mathematical proofs, and other information, in the § Solution section? Because it doesn’t really seem to cite any. Someone not learned in probability theory would have to take Wikipedia’s word for it on basically that whole section, which seems to go against the whole idea of WP. —Frungi (talk) 07:23, 13 March 2014 (UTC)

On the other hand, the mathematics presented is pretty straightforward. The combined probability of n independent events is the product of their individual probabilities, just as it is stated in the section, and the conclusions are just the result of simple algebra. Personally, I would prefer an example using only 27 possible keys (26 letters plus a space), but it does not change the idea behind the math shown. We have to assume some level of reader competence, and the section does have links to more detailed math articles for readers who want more information about probability. — Loadmaster (talk) 23:14, 13 March 2014 (UTC)

Relevance of picture?[edit]

I do not think the top page image should be there. I think that the infinite monkey theorem is an extremely important scientific principle and that a picture of this novelty drags the article down. Discuss. Mackatackastewart (talk) 12:17, 23 March 2014 (UTC)

The picture is fine. It is entirely in keeping with an article titled "infinite monkey theorem". Johnuniq (talk) 00:43, 24 March 2014 (UTC)
It should at least be of a monkey, not a chimp.-- (talk) 11:32, 30 August 2018 (UTC)

Wilberforce debate summary bias[edit]

Is the conclusion to the wilberforce debate bias? It cites an uninternettable paper by Nicholas Rescher. Obviously the person is worth citing but I don't know enough about them to say whether they may have a slant worth mentioning. It seems to question the historicity. Other mentions are here Thomas_Henry_Huxley#Debate_with_Wilberforce and the main article is 1860_Oxford_evolution_debate — Preceding unsigned comment added by (talk) 00:27, 27 March 2014 (UTC)

Wrong definition.[edit]

It *should* read more like this:-

The infinite monkey theorem states that the complete works of William Shakespeare can be produced by getting an infinite number of monkeys to type randomly on a typewriter keyboard.

It's an "infinite monkey" theorem, not a single monkey for an infinite amount of time. — Preceding unsigned comment added by (talk) 15:21, 19 June 2014 (UTC)

See the third paragraph:
Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters.
In any case, whether it is one monkey or infinite monkeys, if they are typing for an infinite amount of time, the effect is the same. Although of course if you have infinite monkeys you can do without infinite time, you only need an amount of time equal to the shortest possible time it could take a monkey to type the works of Shakespeare. ··gracefool 10:43, 22 June 2014 (UTC)
With an infinite number of monkeys, all it takes is a single random keystroke from all of them at once to produce every written work, including the entire Shakespeare corpus. This is simply because a single random keystroke from an infinite number of monkeys instantly produces an infinitely long set of random characters. Somewhere within the set of all infinite typed characters is a subset for any given work (as well as any given unpublished work, or any given sequence of random gibberish). This is true even if you order the monkeys in a sequence, i.e., assign each monkey a specific place (number or index) within the sequence of all monkeys; somewhere among that sequence is a subsequence of characters exactly matching any chosen text (assuming completely random typing). In fact, any given (finite) subsequence will occur an infinite number of places within the complete sequence. — Loadmaster (talk) 22:57, 21 September 2014 (UTC)

Off-topic peeve alert: Please don't say "infinite monkeys" when what you mean is infinitely many monkeys. Infinite monkeys would be more than one monkey, each of which is infinite. What it means for a monkey to be infinite, I'm not sure, but that's another discussion. --Trovatore (talk) 23:39, 21 September 2014 (UTC)


How about a simple general formula calculating the odds that at least one of m monkeys (effectively random-character generators) on typewriters with k keys each (the size of the alphabet) will eventually, after typing i keystrokes at random, turn out producing a string of the length n with a probability of p? That would make it easier for the lay reader to follow the examples. --Florian Blaschke (talk) 16:33, 14 December 2014 (UTC)

Recent edits[edit]

Some recent edits by myself and others were block-reverted on the the grounds that they hadn't been discussed[4]. I think my changes had resulted in an overall improvement, but I'm not greatly attached to the revisions I made to the lead. I have restored the edit I made to the reference to Eddington in the Statistical Mechanics section as I have no doubt that this clarifies his point. Any suggestions or comments? DaveApter (talk) 12:45, 17 April 2015 (UTC)

I'm OK with the clarification of Eddington's meaning. I'm not OK with characterizing the entire subject of the article as a "misunderstanding". The theorem as stated, and with the elaborations on "monkey" and so on in the following paragraph, really is true; it's not a misunderstanding. Physical realizability is not really the point; we're talking about Platonistic mathematical truth, not physics. --Trovatore (talk) 18:24, 17 April 2015 (UTC)


The more I look at this article, the more I find it totally confused. Is there actually a reliable source that refers to the assertion as a theorem, rather than for example a postulate or an illustration? The reference to infinity is poorly framed and sloppily presented. In so far as mathematics treats the notion of infinity, there are various orders of infinity, and the enumeration of the set of natural numbers is the lowest order. It does not follow that an infinite sequence of symbols contains every conceivable sub-sequence. It could not contain the complete sequence of the decimal representation of pi for example, or even the representation of the square root of 2.

The mathematically rigorous way to express the so-called "proof" would be to state that the limiting value of the probability of any given sequence appearing approaches unity as the sample size increases without limit. This does not imply that the probability ever reaches unity. Even if it did, it is a misconception to treat 'Probability of 1' as being equivalent to 'certain to occur'.

Finally since there are not even an infinite number of atoms in the universe, much less an infinite number of monkeys, the premise is counterfactual. In formal logic, a false premise entails any conclusion, true or false. Therefore this "theorem" is trivially true only in the sense that "If 1 = 0 then I am the Pope" is a true (but vacuous) proposition. DaveApter (talk) 12:23, 14 May 2015 (UTC)

  1. Hmm. On the naming issue you may have a point — I don't really know whether "infinite monkey theorem" can be said to be a standard name for the result. ("Postulate" certainly does not seem to be an improvement, though.)
  2. No one claims that an infinite sequence necessarily contains every possible subsequence. If there is such a claim in the article, please point out where it occurs, so that it can be corrected.
  3. The distinction between "probability 1" and "certainty" is explicitly mentioned in the article.
  4. A mathematically rigorous presentation does not in fact necessarily need to mention limits. A probability measure can be defined for the entire space of possible outcomes, each considered as a completed infinite totality. However, limits may be a more accessible way of describing the result for most readers, and I don't exclude that it might be an improvement to mention it.
  5. The situation is obviously intended as a counterfactual, and from the point of view of conceptual analysis, that does not in fact make the result vacuous. For a false assertion p, the claim "if p then q" is automatically true, but the counterfactual "if p were true then q would be true" is not (though the interpretation of the latter is obviously not truth-functional; it's something more complicated).
  6. However, in point of fact, no one knows whether the universe contains a finite or infinite number of atoms, or even monkeys. See shape of the universe#Infinite or finite. You can find lots of claims in print about the "number of atoms in the universe", but almost always, these are (rather sloppily) using the term "universe" to mean observable universe. This point is somewhat of a digression and probably not all that relevant to the article, so we shouldn't belabor it. --Trovatore (talk) 14:54, 14 May 2015 (UTC)

Regarding point 2: surely this is the heart of the matter - if the sequence representing the works of Shakespeare does not of necessity occur in the infinitely long sequence produced by the monkeys, the surely the 'theorem' is false? More to the point, what do you see as being the insight which is being given by the assertion of this theorem? DaveApter (talk) 15:05, 14 May 2015 (UTC)

It does not necessarily occur. But it does occur almost surely. As to what insight I see, that's a bit off-topic, because the article is not about my insights, but it may be of interest that, given an infinite (or even merely unlimited) amount of time or number of trials, extremely unlikely occurrences become almost guaranteed to happen. --Trovatore (talk) 15:20, 14 May 2015 (UTC)

Apologies for not having expressed myself clearly; I wasn't asking about your personal insights, I was querying what point is being illustrated by this so-called theorem. It is not controversial that the probability of any very unlikely event can be increased to any given value by a sufficient number of repetitions of the trial. However, it appears to me merely to promote misunderstanding to refer to events so unlikely that they could not be elevated to high levels of probability even in timescales orders of magnitude greater than the age of the universe. DaveApter (talk) 15:50, 14 May 2015 (UTC)

Well, I think the whole point is that your sentence starting "[i]t is not controversial" really does apply even to events that are just that unlikely. That's not a "misunderstanding". That's just true. --Trovatore (talk) 16:53, 14 May 2015 (UTC)

It might be more illuminating to illustrate the probability of some specific character sequence (eg "Tomorow and tomorrow, and tomorrow, creeps in this petty pace from day to day.") being generated by a random-character generator operating at say one character per microsecond for the timespan of this universe. This would still not equal unity, so speaking of infinite time spans (or infinite numbers of generators) is merely speculation about counterfactual hypotheticals. DaveApter (talk) 10:36, 15 May 2015 (UTC)

No speculation involved. It's a theorem. You can prove it. It's true. --Trovatore (talk) 14:51, 15 May 2015 (UTC)

The "proof" given is entirely invalid since it relies on the assertion that one divided by infinity equals zero, a proposition that no mathematician would incorporate into any rigorous proof. The entire article contains a great deal of original research, editorialising, undisciplined speculation and muddled thinking. Is there even any reliable source that even describes this statement as a "theorem"? The overall thrust is misleading, as it seems to be implying that it is possible for random processes to produce a structured output, whereas the context in which this image is used - for example by Eddington - is precisely to illustrate the absurdity of such a suggestion. DaveApter (talk) 09:49, 29 June 2015 (UTC)

Uff. No, it does not "rely on the assertion that one divided by infinity equals zero". The notion of what constitutes a zero probability is all perfectly standard and rigorous. It's part of measure theory, and you need to learn something about it.
As to whether this result in particular is called a "theorem" in reliable sources, that may actually be a valid criticism. But that goes more to the article name than to the content. I don't know what is the best name for the article; it's a genuine issue.
As for your last sentence, the point is that it is possible for a random process to produce a structured output, and that is in fact true. --Trovatore (talk) 19:26, 29 June 2015 (UTC)
I do not agree with Trovatore that the "infinite monkey theorem" really is a specific theorem; I think it allows a range of different interpretations that produce different mathematical statements. I don't think all who cite this "theorem" have the same exact interpretation in mind, but I do believe that, when calling it a theorem:
  • they do have assumptions in mind on how the "monkeys" behave that cause the resulting statement to be a theorem (even if those assumptions are not the same for everybody), and
  • those assumptions are not a description of how they would expect real monkeys to behave.
A reasonable further specification of "typing randomly" would be to assume memorylessness: that is, to assume that the choice of the next key and the time until that key is hit are both completely independent of what has happened until then. This gives you a precise mathematical interpretation of the problem; under that interpretation, the problem is much simplified and it is easy to prove that for every string its probability of being produced converges to 1.
Another reasonable interpretation is to assume that the monkeys type at a fixed rate, and (as before) that each key has the same probability of being chosen each time.
You may argue that these interpretations are too strict, that we cannot assume that the monkeys hit each key with the same frequency or that their typing speed remains constant. But these assumptions can be relaxed quite a bit before the theorem no longer holds; and if you relax them beyond that, you need a really convincing argument that what your monkeys are doing is still random typing. For instance, if the monkeys' typing speed halves every minute, or the frequency of the letter Q being chosen halves every minute, I don't think we'd call what they're doing random typing anymore. Rp (talk) 20:05, 29 June 2015 (UTC)
Rp, you write I do not agree with Trovatore that the "infinite monkey theorem" really is a specific theorem. But I did not in fact say that it was. I actually think that name for the article is at least potentially problematic; that's one small point of common ground between me and Dave Apter. --Trovatore (talk) 21:02, 29 June 2015 (UTC)

I will briefly note that it is extremely important for readers to realize that this is in the limit of infinite keystrokes and that no finite number of keystrokes can achieve that probability. This already confused some readers before (see some of the previous sections on this talk page).--Jasper Deng (talk) 00:36, 30 June 2015 (UTC)

That's certainly true, but not obviously related to Dave Apter's points, unless I've completely misunderstood them. --Trovatore (talk) 00:48, 30 June 2015 (UTC)
Oh, just a quibble — it's true that the limit of the probability of getting whatever text, as the number of keystrokes approaches infinity, is one, but that's not what it means that the probability is one when you have a completed infinity of keystrokes. The latter statement is to be understood as a statement about probability measures on the space of infinite sequences of keystrokes, not about limits. --Trovatore (talk) 00:56, 30 June 2015 (UTC)

Thank you Rp and Jasper Deng for your contributions to the discussion. For the record, your remark definitely is related to my points. Almost surely is defined as having a probability of 1 (which is not the same as 'certain to occur'). All we can say is that the probability approaches a limit of 1 as the time increases without limit. This is uncontroversial but not particularly illuminating. Talking about what happens "in an infinite amount of time" is mathematically sloppy and would fail any maths exam. Furthermore it is counterfactual speculation since there is no such thing as an infinite amount of time. DaveApter (talk) 10:48, 2 July 2015 (UTC)

So first of all, there is nothing wrong with counterfactuals.
But supposing there were — I'm gonna have to call "citation needed" on "there is no such thing as an infinite amount of time". How do you know that, exactly? --Trovatore (talk) 01:48, 8 July 2015 (UTC)
It is irrelevant whether there is physically an infinite amount of time or not, since this is about an abstract Gedankenexperiment about probability, and not a discussion about real monkeys or typewriters. — Loadmaster (talk) 16:34, 18 July 2015 (UTC)

(a) I never said there was anything "wrong" with counterfactuals. (b) Of course I don't "know" whether there there is an infinite amount of time, and neither does anyone else, but it's my understanding that this is the current consensus among astrophysicists. But enough of this sophistry. I've amended the lead section to give a clearer and more accurate summary. Please feel free to discuss further improvements. There's still a great deal of cruft needing to be trimmed out of the rest of the article. DaveApter (talk) 10:41, 15 July 2015 (UTC)

Dave, I give you credit, that version is better than your earlier attempts. But I don't really think it's better than what was there. For one thing, it's too focused on Borel's initial entry, whereas the formulations have evolved. Also it doesn't cover the notion of "almost surely" up front, which is one of the most important ideas to get across. I have reverted to the version of 03:03, 15 July 05. --Trovatore (talk) 19:02, 16 July 2015 (UTC)

I think we are going to need to get some other opinions on this. The lead as it stood conveyed precisely the opposite impression of what Borel and Eddington were driving at. Rather than suggesting that such an outcome from such a process should bear consideration, they were attempting to illustrate the absurdity of the idea. I've kept your elision of the words 'popular' and 'misnomer', although I think that's an accurate description. Borel did not describe the idea as a "theorem", and so far as I can see it isn't one. I don't think the purported "proof" is actually a rigorous demonstration that the proposition is almost surely true. DaveApter (talk) 08:47, 18 July 2015 (UTC)

On the naming issue you may have a point. I think we could profitably discuss that separately.
Whatever Borel and Eddington were driving at, specifically, is not really the point. The idea is not specific to Borel and Eddington, not by any means. It's just a fact that, with infinitely many random independent identically distributed trials, for any event that has positive probability in a given trial, you get probability one for the infinitely many trials. (By the way, this is can be expressed with complete mathematical rigor in terms of completed infinite collections — you seem to think that it needs to be expressed in terms of limits, but that is not so. However that is a side angle.)
You seem to have a dislike of this fact, for some reason, but it doesn't change it, it is still a fact. --Trovatore (talk) 09:03, 18 July 2015 (UTC)

I've started a new section about renaming this article. — Loadmaster (talk) 16:27, 18 July 2015 (UTC)

Applications and criticism: every possible text[edit]

An obvious criticism that isn't mentioned: The theorem tells us that the monkey typing for an infinite time will not only type Hamlet, but every other finite text. Given that, how can there be anything notable about it typing Hamlet? What conclusion can be drawn from the theorem that has any possible application? Isn't it entirely meaningless?

It's amusing to think about, for instance it also types this Wikipedia article, and every previous version of this Wikipedia article... and the entirety of Wikipedia, and the entirety of the Internet and everything ever written and spoken, both in chronological order and reverse chronological order and alphabetical order and every other possible order... as well as literally every other thing you can possibly imagine, and every possible variation or misspelling of it, so long as it's not infinitely long. ··gracefool💬 11:25, 18 May 2015 (UTC)

I think that's dealt with to some extent in the quote by Borges. I would sort of hope it's obvious, anyway. But I'm not in principle against saying it somewhere more prominent.
Suggestions? What should we say, where do we put it, how do we source it? --Trovatore (talk) 18:39, 18 May 2015 (UTC)
That only deals with the question of authorship, not relevance to probability or the anthropic principle (evolution). So it's a new point worth saying, we just need to find a source. I'm sure there are some good ones out there but I don't know how to find them, it's not an easy thing to search for. ··gracefool💬 11:34, 21 May 2015 (UTC)
I'm not sure what you mean by "authorship". The Borges quote covers your point about "every other finite text" pretty well.
It's possible I hadn't read your first paragraph in your first comment carefully. I don't see how it makes it "entirely meaningless".
Consider: Suppose that the universe is infinite and basically homogeneous. That may well be the case, so this consideration is not vacuous. Then, on the shores of some unimaginably distant ocean, light and dark sand grains have arranged themselves into a sharp readable copy of the King James Bible, except with the book of Judges replaced by a subtly distorted Quechua translation of the third chapter of Atlas Shrugged. Not for sure, but almost for sure. How is that meaningless, just because there are other worlds with the same text, except that the name Adam is replaced by Alex? --Trovatore (talk) 14:30, 21 May 2015 (UTC)
I meant it's only dealing with the question of accidentally writing something, rather than the broader picture like evolution.
It would be meaningless because everything would be meaningless. Everything would exist almost infinite times, and there would be an almost infinite multiple of those in slight variations, and an almost infinite multiple of those in slightly larger variations, etc... So for starters, by definition no single thing in any particular world could be significant. Everything happening is equivalent in meaning to nothing happening, there's no basis for saying that any variation is more meaningful than any other.
But it's worse than that. Extremely improbable things become probable. For every world, there would be a near-infinite number of alternative, almost-identical worlds, but where a thought you had based on evidence is replaced with a thought based on hallucination, psychosis or other illusion. Although this is improbable, it still happens in a huge number of worlds. Thus it is unreasonable for anyone to trust their own thoughts, or anyone else's, and all rationality is unfounded. Thus the argument defeats itself.
There are huge numbers of worlds where *every* thought you have is unconnected to reality. If you take this all the way you end up with Boltzmann brain worlds (a thought experiment created for the purpose of arguments like this) - worst of all, Boltzmann brain worlds should be vastly more common than worlds like ours. ··gracefool💬 07:27, 23 May 2015 (UTC)
Oh, I misunderstood you. You weren't saying that the statement was meaningless (in the sense of not having an interpretation); you were saying that it makes existence or experience meaningless (in the sense of having no ultimate importance). I don't really see it that way, but the way that I do see it is probably not relevant to the article so I won't get into it here.
What might be relevant to the article is if there are notable thinkers that have interpreted the result that way and if we can find RSs for those thoughts. I don't know of any, but if there were such, it strikes me as at least plausible that we'd consider mentioning it here, even though it's maybe a bit of a tangent. --Trovatore (talk) 17:29, 23 May 2015 (UTC)
Yeah you get me. Though the statement is also meaningless in a fashion: if a corollary of a statement is that all statements are unreliable or meaningless, then it's self-refuting. ··gracefool💬 00:38, 24 May 2015 (UTC)

RfC: Which of these versions of the lead is the more accurate and informative?[edit]

1) Trovatore's version was preferred; 2) Reliable sources classify this as a theorem; 3) Reliable sources state accepted proofs of this theorem. DaveApter (talk) 16:31, 28 July 2015 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

  1. Which of these two versions of the lead is the more accurate and informative: [5]?
  2. Is there any reference anywhere in the literature that classes this proposition as a theorem?
  3. Is the purported "proof" given in this article a mathematically valid demonstration of the proposition stated in the lead (in the more recent version from the above diff) (With a reliable source)? DaveApter (talk) 10:37, 18 July 2015 (UTC)
  • Comment — A general phrasing of the idea is appropriate for the lede section. Details about its origin and the subsequent modifications to it belong in a separate "History" or "Background" section. — Loadmaster (talk) 16:36, 18 July 2015 (UTC)
  • Comment. Regarding (1), the first sentence of the original revision reads:

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.

The first sentence of the new revision, that was reverted, reads:

The infinite monkey theorem is the name often used to refer to an idea from Emile Borel's book on probability, published in 1909.

The first is a clear statement of the proposition. The second is not, and I do not think many sources place special emphasis on Borel's role, even if he was the first to formulate the theorem. (Also why is the year linked?) The next sentence is:

The book introduced the concept of "dactylographic1 monkeys" seated in front of typewriter keyboards and hitting keys at random.

This still lacks a clear statement, and brings in a word "dactylographic" which is not explained and does not appear in my dictionary. A reader lacking a knowledge of Greek might legitimately wonder if dactylographic was a synonym for "infinite", the subject of the article presumably being about "infinite monkeys". Whether there are infinitely many monkeys, a single monkey with infinite time (or neither) is never made clear, regardless of the intended meaning of the neologism. This is a grave omission, since it fails to articulate the conditions under which the "theorem" holds.
The third sentence is:

Borel exemplified a proposition in the theory of probability called Kolmogorov's zero-one law by saying that the probability is 1 that such a monkey will eventually type every book in France's National Library.

Even now, we lack a clear statement of the result. This version still does not mention infinite monkeys, just a single monkey, so the relation to the article remains obscure. Furthermore, the article does not discuss the relation to Kolmogorov's zero-one law, so I think discussing this in the lead (without a source) is ipso facto problematic. Moreover, given the statement presented in the new lead, it is not clear that Kolmogorov's law is even applicable because the event "The Bibliotheque nationale eventually appears" is not a tail event, so any invocation of the 0-1 law needs explanation (with a source). The actual tail event that might be intended here is "The monkey types the text infinitely often", but this is not discussed in the article, and the 0-1 law would only give probability 0 or 1 for this event (it has probability 1, but not by Kolmogorov). So, that the IMT "exemplifies" the Kolmogorov zero-one law is dubious at best.
The next sentence is

There need not be infinitely many monkeys; a single monkey who executes infinitely many keystrokes suffices.

Which is perfectly true, but not very helpful in the context where it is used, since this is the first time the reader is actually told that there were infinitely many monkeys at all: the previous sentence already appears to have been about the actions of a single monkey. The final sentence is a further remark on the red herring of Kolmogorov's 0-1 law, which borders on original research.
So, given that the proposed revision is not accurate and does not contain a clear statement of the subject of the article, I conclude in regards to (1) that: the original revision is more accurate and informative.
Now, regarding point (2), I have found the following sources that use the exact term "infinite monkey theorem":
  • Marc Paolella (2007) Intermediate Probability: A Computational Approach, Wiley.
  • Simon N. Chandler-Wilde, Marko Lindner (2011) Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, Memoirs of the American Mathematical Society.
  • Ian Stewart (2010) Professor Stewart's hoard of mathematical treasures, Profile Books.
  • Christopher R S Banerji, Toufik Mansour, and Simone Severini (2014) "A notion of graph likelihood and an infinite monkey theorem", Journal of Physics A, 47 035101 doi:10.1088/1751-8113/47/3/035101
  • Eric S. Raymond (1996) The New Hacker's dictionary, MIT Press.
  • Prakash Gorroochurn (2012) Classic problems of probability, Wiley.
  • Edward B. Burger, Michael P. Starbird (2005) The Heart of Mathematics: An Invitation to Effective Thinking, Springer.
  • G. Spencer-Brown (1957) Probability and scientific inference, Longmans-Green. (Refers to it as the "monkey theorem".)
So, clearly yes, this is regarded as a theorem by many references.
I feel that point (3) must be a trick question. I count two proofs. The first is an intuitive argument, which I don't think is intended to be completely rigorous, but it can be made so without too much effort. I would say that constitutes a "mathematically valid demonstration", modulo quibbles about standards of rigor. Here is a more rigorous version of the same argument. Let denote the event that the word "banana" fails to appear after n blocks of 6 letters have been typed. Then, assuming that each character is independent and uniformly distributed, we compute . Observe that the are a nested sequence of subsets of the σ-algebra: . Let . Then the event E is that the word banana never appears. Then E is a measurable set, and . Thus, the complement of the event E has probability 1. That is, the word "banana" must appear almost surely. These details are sufficiently routine that anyone with a passing familiarity with the mathematical foundations of probability can supply them, and I do not think the article would benefit from such added details.
One thing that could be made slightly clearer is that the same argument applies with the word "banana" replaced by any particular (finite) string of length k (e.g., the collected works of Shakespeare), but with instead. The simple word "banana" is just being used for illustration purposes.
The second proof shows that the theorem is a straightforward application of the second Borel-Cantelli lemma. I would also call this proof "mathematically valid", although I think it should be clarified that the infinite string is broken into non-overlapping blocks of length k (otherwise the events referred to there are not independent). Sławomir Biały (talk) 12:55, 20 July 2015 (UTC)
  • Comment on 1) Trovatores version is a better lede than DaveApter's in the diff posted in the RFC. On 2)A quick google for "infinite monkey" produces about 350 results of which about 290 contain "infinite monkey theorem" so clearly that is the common term. I have no comment on 3)SPACKlick (talk) 11:34, 21 July 2015 (UTC)
  • Comment. (1) Trovatore's version is better, because one can read it without reading anything from the rest of the article and still learn something useful about the subject. That is not true for the other version. (2) Yes, it can be classified as a theorem, but that's the wrong question: the right question is whether "infinite monkey theorem" is a common name of this subject, ignoring whether it's actually an accurate name. As SPACKlick's search results suggest, the answer to the right question is also yes. (3) There is more than one proof in the article, and yes, they are valid mathematical demonstrations of a mathematical abstraction of the claim. The scare quotes in your question are inappropriate editorialization. You might argue that because they prove an abstraction rather than the actual real-world claim, the answer should be no, but I think that would be a mistake: the claim itself is already an abstraction . The part in the claim about an "infinite amount of time" should have been a clue to that. —David Eppstein (talk) 06:05, 22 July 2015 (UTC)
  • Comment. 1) Trovatores's version is better, because it's less technical and more readily understandable. 2) I've always encountered it as "infinite monkey theorem", and a Web of Science search finds some results as well, so yes. 3) The theorem is intuitively obvious to me, and the "direct proof" more than suffices to 'prove' the theorem to my satisfaction. I'm not a mathematician, and they may have different and far more rigourous standards, but for the purposes of an encyclopedia I think it's sufficient. Banedon (talk) 08:54, 22 July 2015 (UTC)
    • OK, thanks for the support all, but just for the record, it isn't my version. I don't think I've even contributed much to it. It's just the longstanding version that I think is preferable to Dave's changes (which is certainly not to say that it can't be improved, or even that some full rewrite might not be indicated — just not one along Dave's current line of thinking). --Trovatore (talk) 16:26, 22 July 2015 (UTC)
  • Clarification. Thanks to everyone who has commented so far. It's clear that the consensus on point (1) is that my attempt at a boldly revised wording is not an overall improvement, and I accept that conclusion. However, I get the impression that there seems to be some misunderstanding of what I am driving at with the other two questions:
    • Regarding my second question, I'm not disputing that "infinite monkey theorem" is the common name for this proposition, or suggesting that the title of the article should be changed. My point is this (and it is interrelated with the third question): is there an authoritative source that states explicitly states that this is a theorem, rather than just acknowledging that this is the name colloquially used to refer to it? If merely the latter, I should have thought that it would be important to establish this clearly at the outset. An earlier draft of my lead said that it was "...a common misnomer for an idea...", but Trovatore objected to that and instantly reverted it, hence the weaker wording " the name often used to refer to an idea...".
    • My concern regarding the proof is as follows: the concluding line is that 'As n approaches infinity, the probability Xn approaches zero.' Entirely uncontroversial, and actually obvious; but this was not the proposition to be demonstrated. To prove that "a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text,", the conclusion of the proof would have to be 'At infinity the probability Xn is equal to 0.' Talking about limits as a value approaches infinity is precise and rigorous; talking about results "after an infinite amount of time" is sloppy and can lead to paradoxical conclusions (such as this one!). If the proof doesn't establish the proposition, then it isn't a theorem. DaveApter (talk) 10:20, 22 July 2015 (UTC)
  • Comment on the "clarification". Limits are not required to make the statement of the theorem rigorous. See the more rigorous version of the first proof that I articulated. If is the event "The word banana does not appear after n blocks of 6 characters have been typed", then the event is the event "The word "banana" never appears". The concept of a limit is not required; one simply can prove the equality of these two sets in the usual way, by establishing two inclusions. A fully expanded version of the first proof would not need the use of limits either, just the Archimedean property of the real numbers, and the monotonicity of the probability measure. One can show that in this way, although it is probably simplest (as the article does) just to show that . This is true, not because the statement of the theorem requires limits to be invoked, but because of the probability axioms (see also probability measure). In any case, fully rigorous formulations of two versions of the theorem appear in the section "Infinite strings". Neither of these statements requires limits. One simply has a probability measure on the space of all infinite strings.
I think the objection has more to do with a reluctance to consider the idea that an infinite string is meaningful. While there are schools of mathematics and philosophy (for example ultrafinitism) in which the concept of an infinite string would be rejected as meaningless, in the conventional axiomatization of mathematics, such infinite objects are indeed allowed and meaningful statements can be made about them independently of any notion of limit. In fact, the axiom of infinity is logically prior to the concept of a limit, so one can have infinite sets without a concept of limit, but not the other way around. So arguing that infinite sets or infinite sequences are meaningless in favor of infinite limits is just begging the question. Sławomir Biały (talk) 13:21, 22 July 2015 (UTC)
So is that your personal assessment of the validity, or are there reliable sources that could be cited? DaveApter (talk) 15:16, 22 July 2015 (UTC)
I've already given sources. The cited source by Gut shows in fact that the probability of the event "The monkey types 'banana'" infinitely often is equal to one. Limits do not come into it, although they are of course used in the proof. Clear formulations appear in several of the abovementioned works as well. The cited source by Isaacs, for example, contains a lengthy discussion (although it is written in a somewhat chatty style which someone lacking background in the mathematical formulations of probabilty might find fault with). The proof given there is almost identical with the expanded version I gave at length.
Not one of these sources points out that the correct formulation should be, as you hold, that "'As n approaches infinity, the probability Xn [sic] approaches zero.'" So, if you're going to challenge the validity of such statements of the theorem, it might be better if you showed some indication of having read and understood thso e cited in the article and RfC. Insisting that educated readings of probability sources by individuals with some familiarity with the fundamentals of probability theory is just a "personal assessment" is an unconstructive ad hominem dismissal. You brought up limits, because you reject the idea of infinite time as mathematically meaningful in this context. I was pointing out why this is a misapprehension. But you don't show any indication if having understood the reply, the sources, or the comments above. So I don't think further discussion is likely to be constructive. Consensus is clear. WP:STICK. Sławomir Biały (talk) 16:04, 22 July 2015 (UTC)
I don't see any reference to a source from 'Gut' on this page - where is that? Regarding my quote above, this is at the end of the third paragraph in the section 'Direct proof' in this article. DaveApter (talk) 16:24, 22 July 2015 (UTC)
It's referenced in the article: "The first theorem is proven by a similar if more indirect route in Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 97–100. ISBN 0-387-22833-0."
I don't see why you think that a statement in the article regarding limits means that the statement of the theorem should be changed. Indeed, one of the defining properties of a probability measure is that if is a nested family of measurable sets ("events") whose intersection of E. Notice that a limit appears on one side of this equation, but not the other. So if we had , we would be justified in saying that as well. The statement "" is then just as true as the equivalent statement "", but does not involve the use of the limit concept. That's generally what identities are good for. You will often see this property used without comment in sources on probability (see any textbook on the foundations of probability or measure theory, e.g., Gut, Theorem 1.3.1, first chapter of Rudin's "Read and complex analysis", etc).
I have already indicated that, just because a proof uses limits, does not mean that the theorem is about limits. Also, I have explained how the limits that appear in the proof are merely expedient; they are not actually essential to the proof anyway. An example is the theorem, that the area of the plane region bounded by the curves is equal to one half. This is a true theorem of plane geometry, and there are several ways to prove it. One is using elementary methods, and other is to calculate the integral . This is a kind of limit. But the statement of the theorem is not really about limits, it's about plane areas. It's true that the limit is also equal to one half, but one still has to prove that this limit gives the area of the plane region. The theorem is not really about that limit, even if one proof can be reduced to computing this limit at some level. Sławomir Biały (talk) 17:08, 22 July 2015 (UTC)
I was merely expressing a preference for a convention well established since the time of Euclid whereby the final line of the proof states the proposition to be proved - indicated by 'Quod erat Demonstrandum'. DaveApter (talk) 10:07, 27 July 2015 (UTC)
The thirteen books of Euclid were originally written in Greek, and would not have included such Latin text (he wrote ὅπερ ἔδει δεῖξαι at the end of propositions). Anyway, this practice is now deprecated in most modern mathematics writing. For example, Goursat's 1905 treatise Cours d'analyse mathématique does not use such a phrase or abbreviation to denote the end of a proof. The end of a proof environment in the standard AMSTeX maintained by American Mathematical Society is a small square box at the right margin of the page. Since Wikipedia is not a textbook, we don't usually include "proofs" as such. In this case, we are giving a mathematically valid argument that can be made into a rigorous proof without much effort. Sławomir Biały (talk) 12:23, 27 July 2015 (UTC)
I can't be sure whether you misunderstood my point, or are just deliberately being obtuse. Strange to say, I did realise that Euclid wrote in Greek and used a Greek phrase rather than a Latin one. Nor is my point whether the end of the proof is indicated by the annotation 'QED', by a square, or any other convention. My point was that the purported 'direct proof' given in the article stops short of deriving the proposition. Regardless of whether Wikipedia is a textbook or not, it would serve readers if sections identified as proofs are rigorous expositions. The purpose of this page is for discussion of how to improve the article (which is sorely needed in this case), not to indulge in displays of erudition. The questions I asked could readily be answered by giving a precise quotation and exact reference for (a) a reputable mathematician stating that the proposition is a theorem (strictly speaking); and (b) a mathematician endorsing proof of same. It would be preferable (though not of course essential) if the ref can be checked out online without having to track down a hard copy of the text. For instance, the online source I found for Gut returned 'not found' when I searched for Monkey; and I didn't see anything in the Contents listing that gave a clue where there might be some discussion of this. Once these two points have been settled, maybe we could move on to address some of the other numerous shortcomings in this article. DaveApter (talk) 14:18, 27 July 2015 (UTC)
@DaveApter: I'm puzzled by your claim that this does not appear in the cited (2005) work of Allan Gut, Probability: a graduate course. The article refers to pages 97-100 of that work, on which pages there is a discussion of "the monkey and the typewriter". The moniker "infinite monkey theorem" is sometimes used to refer to Theorem 18.4 of Gut, which is the second theorem stated in the "infinite strings" section, modulo some trivial details (see, for instance, Chandler-Wilde and Lindner, p. 94, where they refer to this principle in this way). Prakash Gorroochurn states on p. 209, "The result we have just proved is the so-called infinite monkey theorem", referring to Gut and Isaac for a fuller discussion. Isaac has the "direct proof" on pages 48-50 in his (1991) The pleasures of probability, where he says on page 50: "a theorem can be proved asserting that the monkey will type out the works of Shakespeare not only once but actually infinitely often with certainty." Bio-inspired computation in telecommunications (2015) by Xin-She Yang, Su Fong Chien, T.O. Ting has, on page 4, a statement of the infinite monkey theorem: "the probability of producing any given text will almost surely be 1 if an infinite number of monkeys randomly type for an infinitely long time". Ian Stewart (2012), in Professor Stewart's Hoard of Mathematical Treasures, has a heading "The infinite monkey theorem" wherein it is said on p. 225, "if a monkey sat at a typewriter and kept hitting keys at random, then eventually it would type the complete works of Shakespeare". So, yes, this is regarded as a "theorem" in the literature. Proofs aimed at various levels of rigour can be found there. Sławomir Biały (talk) 15:30, 27 July 2015 (UTC)
Thanks very much - that is helpful. Incidentally, I didn't 'claim that it doesn't appear' in Gut - just said that the search (on wherever I was looking, Google books or Amazon look inside etc) didn't return any search results; obviously some quirk of the indexing, or of the selection of the pages available. DaveApter (talk) 15:53, 27 July 2015 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Renaming this article[edit]

It has been mentioned that using the word "theorem" in the title is not entirely accurate. If we are serious about changing the title of this article, how about Infinite Monkey Conjecture or Infinite Monkey Principle? — Loadmaster (talk) 16:26, 18 July 2015 (UTC)

As to whether the subject is a real mathematical theorem or not, it is actually fairly easy to cast it as one: given a monkey (or the more general case of N monkeys) typing a truly random infinite sequence of characters, it is almost surely (some would argue a certainty) that within that sequence lies the complete corpus of Shakespeare, and indeed every other chosen finite sequence. — Loadmaster (talk) 16:29, 18 July 2015 (UTC)

The name "Infinite Monkey Theprem" is widely used (e.g. see [6]), and I would guess it is by far the most common way of referring to this topic. Whether it is actually a valid mathematical theorem or not (though I'm convinced it is) is, as regards the aricles name, irrelevant, it only matters that it is most commonly called a theorem. Paul August 16:49, 18 July 2015 (UTC)

I guess what I'm wondering is whether the article itself propagated the name. That's the bad case that we don't want to get into (it's a little like "citogenesis"). Even if it's so, though, I don't know whether it can be fixed now, and I also have no alternative naming suggestion. --Trovatore (talk) 17:29, 18 July 2015 (UTC)
I'm sure there is a certain amount of propagation. But not all of the sources I cited above are from the past 10 years or so, and one is from 1996. It's clear to me that we didn't invent the term. Moreover, "infinite monkey principle" only gets one hit on Google books, compared to over 500 for "infinite monkey theorem". Whatever the reason, "infinite monkey theorem" seems to be the standard term for this now and it's too late to try to legislate an alternative term. Sławomir Biały (talk) 13:01, 20 July 2015 (UTC)

Correspondence between strings and numbers...[edit]

This section is a mixture of original research and complete and utter garbage. I'm not sure which has the highest proportion. — Preceding unsigned comment added by (talk) 19:37, 5 September 2015 (UTC)

I don't actually see anything in the section that isn't true, and I'm sort of curious what you think is "garbage" and why. But my curiosity is not really on topic here, and I do have to say that I'm not sure what the point of the section is in this particular article. If kept, it would need to be sourced, but before going to that effort, someone should probably explain the rationale for having it at all. --Trovatore (talk) 20:10, 5 September 2015 (UTC)
I don't see the point of it either. Let's be bold and remove it. Otherwise we could wait a few weeks for possible objections. ··gracefool💬 10:04, 8 September 2015 (UTC)
I think the main point is to link the subject of the article explicitly with the concept of a normal number. It seems to me that this is worth doing. We can cite Borel's 1909 paper, if necessary, where he proves both the Borel-Cantelli lemma, and uses it to show that almost all numbers are normal. Sławomir
10:47, 8 September 2015 (UTC)
Why do we need to link it to normal numbers? ··gracefool💬 22:32, 8 September 2015 (UTC)
Because (a version of) the infinite monkey theorem is that almost every real number is normal. (See the cited work by Alan Gut, which we summarize with "Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.") Normality is a slight strengthening of this result that, with probability one, the frequency of any given word in a random infinite string is equal to the natural frequency of that word relative to all words of that size. Sławomir
22:56, 8 September 2015 (UTC)


I just added "It is a veridical paradox (a valid argument with a seemingly absurd conclusion) that demonstrates counterintuitive properties of infinity." to the opening and was immediately reverted. I took this from Hilbert's paradox of the Grand Hotel. This theorem is clearly similarly counterintuitive. Why am I wrong? ··gracefool💬 22:38, 8 September 2015 (UTC)

The article does not discuss the paradoxical nature of the theorem, and neither do any of the sources I have read. There is nothing especially paradoxical about the notion that certain enormously improbable events can happen with a small positive probability. In the context of the original examples by Borel and Eddington, our intuition is actually correct, that a monkey sitting at a typewriter typing out the play Hamlet is an enormously improbable event. So it isn't really a "paradox" at all construed in this way. The point of their metaphor is that, while we have a pretty good idea how unlikely it is to get quality output from monkeys, we don't have much intuition for what 10^23's of molecules are doing. Sławomir
23:13, 8 September 2015 (UTC)
The paradox is that anything, no matter how ridiculously improbable, becomes almost certain. There's nothing intuitive about a process creating every piece of literature ever written, as well as a myriad of slight misspellings of each, and a variation of each where people are replaced by carrots, etc for everything imaginable... ··gracefool💬 01:49, 9 September 2015 (UTC)
Fine. You're welcome to think of it as a paradox or not. I, for one, see nothing paradoxical in small probabilities getting larger with many repetitions. (And I do not think I am unique in that regard.) But anyway, to write an encyclopedia article claiming that it's a paradox requires sources. The references in the article do not use this metaphor as a paradox. On the contrary, they employ it as an intuitive result to illustrate some counterintuitive consequences of statistical mechanics. That's something like the opposite of a paradox. Sławomir
02:12, 9 September 2015 (UTC)
Fair enough. ··gracefool💬 22:51, 9 September 2015 (UTC)

Popular culture ... wtf?[edit]

The Popular Culture section bit about the Ricky Gervaise show ... is it essential, encyclopaedic, etc? Doesn't really read that way to me. In particular, is there some dialect of English in which this bit makes sense: "He then doubled down by using the acumen that ..." He what? The what?? Help meeeeeeeeee ... (talk) 20:23, 13 November 2015 (UTC)

Yes, that was over the top. Removed, thanks. Johnuniq (talk) 23:05, 13 November 2015 (UTC)

Actual immortal monkeys[edit]

Recently a dubious sentence in the lead, marked {{citation needed}}, was edited to an (in my opinion) even more dubious sentence:

It should also be noted that real monkeys do not produce actually random output, which means that an "actual" monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text - some letters or combinations of letters in Hamlet may have precisely zero probability of being typed.

How do "actual" monkeys differ from actual monkeys? An actual or "actual" monkey hitting keys for an infinite amount of time is apparently immortal. I pronounce with absolute certainty that no such actual monkeys exist. Even if they did, the typewriter would be total loss within 101010 seconds, a negligeable fraction of an infinite amount of time. Therefore the argument about what combinations of letters "actual" monkeys might or might not produce appears to be without merit. I also cannot think of any reasonable argument why "some letters or combinations of letters in Hamlet may have precisely zero probability of being typed". Granted the existence of immortal monkeys with everlasting typewriters, it may take eons before one hits the letter Q, but if it is physically possible, it is bound to happen eventually. I think this sentence is unconvincing original research; I have therefore removed it.  --Lambiam 11:22, 22 April 2016 (UTC)

"Ridiculous" probability[edit]

@Trovatore, Adrums63, and Slawekb:, Firstly my edit should not have been reverted because it contained two unrelated edits and the reverter didn't seem to be opposed to the other edit.

Secondly just because something seems subjective doesn't mean it is. The probabilities involved are objectively ridiculous. If three hundred and sixty thousand orders of magnitude longer than the estimated total age of the universe isn't ridiculous, nothing is. I can't think of a better word - if you can think of one, please use it, but at this stage, "ridiculous" is more accurate than weaker language like "far more". "Far" doesn't even begin to cover it. ··gracefool 💬 22:49, 3 May 2016 (UTC)

I agree that "ridiculous" is no good in an encyclopedia article. I think the article does a pretty good job of expressing the size of the numbers involved. And anyway "ridiculous" is subjective. There are numbers so large that the computational complexity needed to express those numbers exceeds the total number of particles in the observable universe. These are numbers that are useless, and can never even in principle be used in a mathematical argument, whose size can never be comprehended in any human terms. The numbers in the article are peanuts by comparison.* Sławomir
22:57, 3 May 2016 (UTC) *Actually, some infinitesimally small peanut, whose smallness can never be comprehended in human terms, by comparison.
In addition to the subjectivity, there's also the question of encyclopedic tone. In informal discourse, "ridiculous" is a fine word for "extreme beyond ordinary conception". But that's not what the word means, in formal writing.
In formal writing, something that is "ridiculous" is worthy of ridicule. It is not clear why one would ridicule a tiny probability or a very long time, or what good it would do to do so. --Trovatore (talk) 04:03, 4 May 2016 (UTC)

gracefool: I don't see any objection to the other part of your edit, that is true. So don't get upset; just restore that part of it. You can't expect people reviewing edits to sort through the whole thing, when there are multiple pieces to it. They just revert the whole thing; then you can restore the non-controversial parts. --Trovatore (talk) 23:32, 3 May 2016 (UTC)

Yes of course numbers can't be ridiculous in themselves. What's ridiculous - as in worthy of ridicule - is actually using that number to say anything about actual probabilities. But okay we'll leave that to the criticisms section.

@Trovatore: people certainly do expect others reviewing edits to sort through the whole thing. Unless it's very complicated and you intend to revert the vast majority of it (but that means you still had to sort through it to some degree to make the decision). In this case the "multiple pieces" were two pieces. You're supposed to explain your revert; it seems your explanation is "I couldn't be bothered". Why should other people have to do what you're too lazy to do? Also there are a lot of good reasons to keep reverting to a minimum. ··gracefool 💬 01:34, 6 May 2016 (UTC)

No, I'm sorry, I completely disagree with you. Reversion is ordinarily best done atomically. That minimizes the "version hell". Also, it's the new edit that has to justify itself, not the revert. --Trovatore (talk) 04:40, 6 May 2016 (UTC)
@Trovatore:: But my edit was a revert! By your logic, by restoring the original edit, you need to justify it. ··gracefool 💬 21:26, 12 May 2016 (UTC)

A Universe Full of Monkeys[edit]

The opening paragraph states the following: "However, the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero)". I'd love to see the rough math on this, or at least an indication of how many monkeys the author thinks would fit into the universe. Even if we consider a non-expanding universe fixed at it's current dimensions, that's a butt-load of monkeys! — Preceding unsigned comment added by (talk) 02:34, 4 January 2017 (UTC)

The section Infinite monkey theorem#Probabilities has the details you seek. Of course, notably the section excludes the fact that if you were to fill the universe with monkeys event horizons would form all over the place, so that the question of "did the collection of monkeys filling the entire universe type the text of Hamlet" arguably becomes meaningless anyway. In fact, based on my own calculation, any collection of about or so monkeys, if they are closely but comfortably gathered together into a roughly spherical region, will produce a horizon. Sławomir Biały (talk) 13:12, 4 January 2017 (UTC)
Hmm? The universe may be infinite in extent. Of course that's a problem with the text as it stands as well — the common conflation of "universe" with "observable universe" is common, but still sloppy. We should fix that in the article. --Trovatore (talk) 20:02, 22 April 2017 (UTC)
I understand it that the universe is filled with monkeys, but in the age of the universe (that is, in our particle horizon) the probability that a complete work could have been typed is very small. This amounts to the same thing as "observable universe". Sławomir Biały (talk) 10:44, 23 April 2017 (UTC)

Transfinite Typing[edit]

"So the probability of the word banana appearing at some point after an infinite number of keystrokes is equal to one."

I'm confused by this. It seems to assert that the probability of the word banana appearing after an infinite number of keystrokes might not be one initially but if we keep typing it eventually will be. I read the comment that an infinite (ordered) cadre of monkeys yields an infinite sequence of letters with each keystroke, so if they type together a countable infinity (i.e. ω) of keystrokes their output has order type ω². Here we have continued typing after an infinite number of keystrokes. Is this the idea? Would striking the words "at some point" change the meaning of the sentence? Lewis Goudy (talk) 19:59, 22 April 2017 (UTC)

Agreed - the entire article is somewhat confused. To say that it is 'almost certain' that a specified string would be typed in an infinite amount of time is meaningless, and also misleading - bearing in mind that the probability of its happening is negligible in the entire age of the universe. DaveApter (talk) 16:22, 14 September 2017 (UTC)
It's neither meaningless nor misleading. It is meaningful and true. --Trovatore (talk) 00:03, 15 September 2017 (UTC)
So what is the meaning? DaveApter (talk) 15:59, 18 September 2017 (UTC)
From the article: "Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position." Almost surely means with probability one. Sławomir Biały (talk) 16:59, 18 September 2017 (UTC)
So, given that there is no such thing as an infinite string, what is the meaning of that? DaveApter (talk) 18:38, 18 September 2017 (UTC)
There is in fact such a thing as an infinite string. --Trovatore (talk) 18:41, 18 September 2017 (UTC)
An infinite string is a mapping from the set of positive integers to the alphabet of characters. Sławomir Biały (talk) 23:19, 18 September 2017 (UTC)

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An editor is apparently confused about the appropriateness of including proofs in mathematics articles. From WP:MSM: "we often want to include proofs, as a way of really exposing the meaning of some theorem", and from Wikipedia:WikiProject Mathematics/Proofs: "Proofs are often discussed in Wikipedia's mathematics articles, just as axioms, definitions, theorems, and lemmas are. Because much of the published professional literature of mathematics consists of the details of proofs, it would be very difficult to write in any depth about mathematics without including at least some proofs or proof sketches." We quite regularly include proofs in articles, when the proof is important to a reader understanding a topic. In this case, it is very clear that a proof, written at a level and style appropriate to likely readers, is an essential part of the article. Maintenance templates should not be added without discussion of issues, and should certainly not be restored after they have been removed without first developing consensus on the discussion page. Sławomir Biały (talk) 12:25, 10 February 2018 (UTC)

(edit conflict)

Wikipedia is not a guidebook, manual or textbook. A general outline of a proof in prose, describing the approach a proof takes, is fine. But when you see fragments like this, it's a warning sign that the article is slipping into a textbook-like essay, which is not appropriate for an encyclopedia:
  • As an introduction, recall that... – this is not a description, this is standard verbiage from the set-up to a proof. Not appropriate language here.
  • Suppose... – don't suppose. Describe.
  • Let X...
  • Then, the chance that... – Don't follow the proof logic step by step; just describe it.
  • From the above... – Nope; just describe the proof, don't give it.
  • As n grows, Xn gets smaller... – this is not description, this is detail of the internals.
Just describe the proof (if needed), don't go into the actual details of it. Wikipedia is not a mathematical journal. Mathglot (talk) 12:37, 10 February 2018 (UTC)
The above suggests a lack of familiarity with how mathematical proofs are written in prose. Imperative verbs like "let", "suppose", and "then" are regularly used in proofs. Encyclopedias on mathematics regularly use these words too. See, for instance, the Springer series Encyclopedia of the mathematical sciences. In a Wikipedia article, we are expected to present content in a way that readers will find understandable. That often means describing aspects of the details. There is nothing in this article that is like a "mathematical journal". That also suggests unfamiliarity with how mathematical journals are written. Likewise, the proof would look very different from this if it were presented in a textbook on the subject. Instead, the proof is presented in a manner suitable for a general reader of an encyclopedia, not a student reader of a textbook or a researcher reading a journal article. Sławomir Biały (talk) 12:43, 10 February 2018 (UTC)
(edit conflict)
Reading WP:MSM#Proofs, it appears that the guideline allows for Proofs in some circumstances; whether this one helps is debatable.
What's not debatable, is WP:WNTRMT; and your comment above that Maintenance templates should not be added without discussion of issues, and should certainly not be restored after they have been removed.. has no basis in any guideline that I've seen, and is just your own, personal preference. I haven't removed the proof section itself, *that* would have been subject to WP:BRD, I agree. However, *you* removed the template; that *is* subject to WP:WNTRMT, and you should defend that here, or revert yourself. Cordially, Mathglot (talk) 12:48, 10 February 2018 (UTC)
Tags should generally be accompanied by talk page discussion, and are usually for obvious and uncontroversial problems (e.g., an article has no references or is an essay). Per Wikipedia:Template messages/Cleanup: Tags must be accompanied by a comment on the article's talk page explaining the problem and beginning a discussion on how to fix it, or for simpler and more obvious problems, a remark using the reason parameter as shown below. Tagging editors must be willing to follow through with substantive discussion. Furthermore, whether a tag is included in an article or not is still governed by WP:CON policy. (You've cited a help page, which is not a guideline or policy.) So far you've not convinced anyone that there is actually a problem with the section that warrants tagging. Sławomir Biały (talk) 13:12, 10 February 2018 (UTC)

Sloppy treatment of infinities[edit]

Loose talk about infinities and 'almost certain' causes more confusion than enlightenment. The mathematically correct way of dealing with these issues is to speak of the approach to a limit, as the variable expands indefinitely. DaveApter (talk) 17:18, 5 December 2018 (UTC)

No, that is incorrect. The modern treatment is to use measure theory, which does not rely on limits per se. --Trovatore (talk) 21:30, 5 December 2018 (UTC)

Possible mistake about probabilities[edit]

In the second paragraph of 'Almost Surely,' the article states that "Equally probable is any other string of four characters allowed by the typewriter, such as "GGGG", "mATh", or "q%8e"." However, this is not true as (for example) to type "GGGG," not only would four 'g's need to be pressed, but the shift key would need to be held down for all four of them. 2600:6C50:17F:F672:B9B3:58F6:4BF8:9E2D (talk) 18:25, 27 March 2020 (UTC)

The example is supposing that capital letters and lower case letters are considered distinct keys with the same probability for simplicity. Note also that it includes %, which would have similar rules given a normal keyboard. I'm not sure which would be better: adding a note explaining that factoring in shift makes the math more complicated, or just forgetting about it and removing shift-key-only letters from the examples. Integral Python click here to argue with me 21:39, 30 August 2020 (UTC)

Archiving request[edit]

The page WP:AATP suggests a talk page start to archive content once its over 70,000 bytes for several reasons, and this talk page is currently twice that at about 150,000. I would do it myself, but I lack the technical know-how, so could somebody more versed in this area look into archiving some of the much older sections on this talk page, and/or set up the auto-archive bot? Thanks! Integral Python click here to argue with me 23:32, 30 August 2020 (UTC)

Proof flawed[edit]

The flaw is here:

« Because each block is typed independently, the chance Xn of not typing banana in any of the first n blocks of 6 letters is… »

This is wrong precisely because each « block » is independent of all others. That argument requires that all « blocks » be different. Regardless of how many times you pull 6 letters, the probability remains 1 in 15 billions each time. 2605:B100:E034:D642:9DDC:E402:9D28:67F3 (talk) 01:51, 2 October 2020 (UTC)

No, it doesn't require that the blocks are different; it only requires that all of them are unequal to "banana". Rp (talk) 15:40, 5 October 2020 (UTC)