Of Monkeys and Typewriters

by Miles Mathis

I won’t claim that this is one of the greatest standing errors in mathematics, but it certainly still gets a lot of attention. It would appear that the standard and current answer to this question is that, given infinite time, any number of monkeys typing randomly would create the complete works of Shakespeare. Since this answer is in error, it may merit some attention from me.

The problem is that, once again, mathematicians have begun calculating before they have finished thinking through the given situation. We are told that this is a strictly statistical problem, and--assuming it is a professional mathematician who is trying to sell us the current answer--we are shown the strings of equations which prove it. Wikipedia has published this proof. But I will show that the proof is flawed, not in the equations, but in the assumptions or postulates. As we know, a true and correct series of equations grounded on a false postulate is still false.

The history of this problem is long. Aristotle and Cicero mention forms of it, and Cicero comes out decidedly against it (and therefore with me). Pascal and Swift also argued against it. But in the 20th century, that all changed. Thomas Huxley and Arthur Eddington both came out for the proposition, and this was enough for the writer Jorge Luis Borges, who then popularized the notion in The Library of Babel. Now it is a commonplace, advanced by everyone from professional mathematicians like Richard Duda to comedians like Ricky Gervais.

In the contemporary world, the argument usually devolves to one of religion against science, and anyone who questions the current answer is immediately suspected of Christian or humanist leanings. This dichotomy was solidified and perhaps created by the debates of Huxley and Wilberforce in the 19th century. The argument has never really advanced beyond the precedent set by that debate. But my argument here will be strictly logical. I have no intention of bringing gods or teleology into such a simple question.

To get right to it, the problem with the current answer is that it assumes, without proof or argument, that taking a random string of items to infinity will necessarily cover all possibilities. That is to say, those who assert the monkey theorem assume that in infinite time the monkeys will type everything: all combinations will be covered. That is a false assumption.

It is easy to propose an infinite number of combinations where all possible combinations are not covered. Let us say that we take Ricky Gervais to a basketball court and allow him to shoot baskets. No matter how long we leave him there, he will never be caught slam dunking the basketball. We could leave an infinite number of short guys there an infinite time, and we would never find any of them slam dunking the basketball. What we would find, instead, is an infinite number of similar non-dunking combinations. Increasing the number of shots taken does nothing to increase the likelihood that we will see a slam dunk. And taking the experiment to infinity certainly does not guarantee we will see one. This is simply because there is no reason to believe that all possible outcomes must be covered.

This was demonstrated, though of course not proved, by a real experiment with monkeys and typewriters. It was found that the monkeys typed various strings of the letter “S” or the letter “P”, for example. This should remind us that there is no reason to assume that all combinations would be covered in infinite time. According to the laws of probabilities, there is the possibility that in infinite time the monkeys would create an infinite number of strings that were not the complete works of Shakespeare. Purely by chance, they failed to create the complete works of Shakespeare an infinite number of times. Can you give me a reason why this sentence must be false? If not, then Huxley is wrong.

What Huxley should have said is that the monkeys may, purely by chance, create the complete works of Shakespeare. That is true. It is nearly infinitely unlikely, but it is possible. That is one of the possible outcomes, and it may be hit in infinite time. But there is certainly no guarantee that it will be hit. There is no “must” about it.

As I said, the only way you could claim that the monkeys must create the complete works of Shakespeare is if you can prove mathematically and logically that all possible combinations must be covered in infinite time. But you cannot prove that; therefore it is nearly infinitely unlikely that the monkeys will do it. Not “must”, but nearly “must not.”

Let us hit it one more time, for good measure. Using only the terminology and logic of modern statistics, let us notice that there is a possibility that the first string created by the first monkey is all S’s. If the complete works of Shakespeare is composed of 107 letters, say, it is possible the stupid monkey sat there for years and never typed anything but the letter S, 107 times. That is one of the potential outcomes, and no math or logic can deny it. If we add a second “toss”, either by adding another monkey or letting our first monkey have another go, we find, among all the other possibilities, the possibility that S is again typed 107 times. In every toss, this is one of the possibilities. Therefore, by extrapolation, we can easily see that in an infinite number of tosses, there must exist the possibility that S and only S is typed. QED, there is a possibility that the complete works of Shakespeare is not typed. It is not typed each time and it is not typed every time. In infinite time, it is never typed.

In fact, as I hope you can now see, there are an almost infinite number of combinations where the monkeys fail, even at infinity. This is enough, by itself, to disprove the current theorem.

Interestingly, one of the possible outcomes is that every monkey writes the complete works of Shakespeare every single time. Because that is one of the potential outcomes, does it mean it must happen? No, of course not. It is easy to see that going to infinity does nothing to increase the odds of it happening. In fact, going to infinity infinitely decreases the odds of it happening.

But otherwise intelligent people assume that because one monkey might succeed, he must succeed in infinite time. This assumption is false. In one attempt, he might succeed, but he might not. In a billion attempts, he might succeed, but he might not. Making the number larger does not change this. Going to infinity does not change it either. Unless there is a pre-established mechanism that guarantees that all possible combinations will be covered, there is no logical reason to assume they will be. As I have shown, there are an infinite number of combinations where the monkeys fail, even at infinity.

Huxley might say that my basketball example was not to the point, since I have proposed a task that Ricky Gervais cannot accomplish. This is not analogous to the monkeys, since we are assuming the monkeys can reach all the keys. But just because we assume they can reach all the keys does not in any way imply they will cover all possible combinations of keys. There is only one combination that we have defined as correct, and it is conceivable that in any number of tosses, even an infinite number, the monkeys would always be incorrect.

Let me give another example. Say you are blessed with a magnificent voice and can hit a high A. But let us say you don’t know this and so you never do hit a high A. Is this possible to imagine? Certainly. It has no doubt happened thousands of times in history to poor laborers who never had the opportunity to train their voices. It happens all the time to this day, even to wealthier people. Many (or most) opportunities--those that are innate as well as those provided by experience--never happen, due to various circumstances. Now, if you had a longer life, or multiple lives, would all of your possibilities be realized? Perhaps, but not necessarily. It is possible that in a longer life you would do more varied things; it is also possible you would only do more of what you already do. We can see this simply by looking at the lifespan we are actually given. Do older people commonly live a more varied life than younger people, or do they commonly just have more days like the days they already had? We all know the answer. Even in this short existence we are given, variety is not the rule. If we all begin to live to a thousand, it may be that we will live lives of more variety, exploring more of our talents. But this is not a necessary conclusion. It is also possible we would live the same sorts of narrow lives we now live, only ten times as long.

Notice that we do not have to decide the question. We do not have to make any judgments of human nature or decide any factual questions. I do not know whether longer lives will be more varied or not. It doesn’t matter here. What matters is that it is possible that longer lives will not be more varied, simply as a statistical statement. Because it is a possible outcome, we must include it in our probability math. Once we include it in our probability math, the monkey theorem immediately crashes.

So you see that increasing “tosses” does not necessarily lead to variety, much less to a covering of all potentialities. A talented person may eventually find his talent. But he may not. Given a thousand lives, you might live a thousand different ways; but you might live the same way a thousand times. Either possibility is mathematically and logically equal. The equations of probability have nothing to say about it one way or the other. They certainly provide no guarantee or proof of variety. Friedrich Nietzsche believed or proposed that we in fact live this same life over and over, down to the tiniest detail. I am not here to confirm or deny it, I only point out that this is mathematically a possible outcome. And because it is a possible outcome, the monkey theorem must be false.

The monkey theorem is falsifiable. What’s more, it is easily falsifiable. What’s more, I have falsified it with a specific set, above. The set of “All S” an infinite number of times falsifies it. If you claim that the monkeys must create the complete works of Shakespeare, given infinite time, you must show why my potential set is not a potential set. The very fact that it exists as a potential set or outcome is enough to doom the monkey theorem.

As has so often happened in the recent past, the thinking has gotten sloppy once the debate has begun. The original debate was science versus religion, and the monkey theorem was only a passing example. In the heat of the moment, science (Huxley) asserted something that wasn't true, and the truth or falsity of the assertion crossed over from science to polemics. That is to say, the question was no longer one of fact, it was one of rhetoric. Since all scientists were thought to be on one side of the argument, there was no need to back down in the face of this rather large error. All the religious people could be browbeaten with false equations and treated like idiots when they raised religious objections. Few scientists would be savvy enough themselves to see the error, and those that did would keep quiet--most from a sense of obligation. Many would feel that the monkey theorem, even if flawed, was miles closer to the truth than any religion, so why quibble? "May" or "Must"?: small difference when faced with having to admit an error to the enemy. Better to pretend to a mathematical infallibility than to expose science to the ignorant slings. In the social setting, all this is understandable, but it has had the rather large side effect of miseducating the vast majority of people. So that, for example, we now have Ricky Gervais--a very intelligent and well-read man and a vocal non-Christian--stating with absolute certainty that the monkey theorem is true. He does this not as a scientist who knows the social reasons for the muddy waters, but as a modern man who trusts science. He does not trust science to be "more correct" than religion: he expects that what they say is true is true. After reading my paper, Gervais might not be satisfied to continue asserting something that is false, simply because it is "less false" than Wilberforce's view of the matter. Mathematicians may know more about math than religious people, but that does not mean they have therefore earned the right to do bad math.

Science doesn't need bad math to debate religion. Good math would be better, in any possible outcome.

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