The infinite monkey theorem is just that—what happens when you put an infinite amount of monkeys into a room with typewriters? Well, the law of big numbers and probabilities dictate that eventually, they will produce a word for word script of any of Shakespeare’s plays. In truth, this is a nod to how anything is possible with enough inputs, and conversely, nothing is impossible with enough inputs. You can’t rule out anything that has a slight possibility or chance, and you must always account for everything that is possible. Something is possible from almost nothing. It doesn’t sound like a revelation, but it’s a more eloquent way of encouraging thoroughness and looking at the concept of randomness and serendipity.
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Shakespeare, I Presume?
A related but equally mind-boggling train of thought to follow is one you’ve probably heard of before: the thought experiment concerning an infinite number of monkeys sitting at an infinite number of typewriters, banging the keys randomly. This famous “infinite monkey theorem” helps stretch the bounds of how we think about infinity, but also large numbers and probability.
The theorem goes like this: an infinite number of monkeys, if sat down in front of an infinite number of typewriters for an infinite amount of time, will eventually write, by sheer accident, an exact copy of the complete works of Shakespeare.
In fact, it’s more than this—since the monkeys have infinite time, they would in fact produce the complete works of Shakespeare an infinite number of times over, along with every other great work, also an infinite number of times. Émile Borel outlined this idea way back in 1913, although we can understand that he used monkeys as a metaphorical stand-in for any random generator of letters.
The mathematical way of putting it is to say that random inputs will produce all possible outputs, given infinite time. The real-world translation is that any problem can be solved, or anything created, if time and resources are infinite.
However, the probability of a monkey actually producing Shakespeare is, although not technically or mathematically zero, very small. For those interested in the maths, there is a theorem proving that the likelihood of monkeys producing Shakespeare under these conditions is exactly 100 percent. However, it might also interest you to know that staff and students from the University of Plymouth conducted a (tongue-in-cheek) real-life experiment, giving a number of monkeys a number of typewriters and watching to see what happened. After a month, unsurprisingly, the monkeys had produced five pages of the letter “S.”
Dan Oliver also wrote a computer program that randomly generated letters, and after the equivalent of 42,162,500,000 billion billion monkey-years (stop to think how enormously large that number is), one of the “monkeys” wrote:
VALENTINE. Cease toIdor:eFLP0FRjWK78aXzVOwm)-‘;8.t
Hilariously, the first nineteen letters of this string can be found in Shakespeare’s Two Gentlemen of Verona. Other teams with similar programs also produced results from other Shakespeare plays. Computational speed is a natural hard limit to how much can be done practically speaking, but these real-life attempts don’t add much to the fundamental question at hand—we will never be in the position to conduct this experiment for real.
So, what can we glean from the monkeys with typewriters theorem? Surely it only has mathematical value? Anything is possible with enough inputs, and conversely, nothing is impossible with enough inputs. You can’t rule out anything that has a slight possibility or chance, and you must always account for everything that is possible. Something is possible from almost nothing. It doesn’t sound like a revelation, but it’s a more eloquent way of encouraging thoroughness and looking at the concept of randomness and serendipity.
As it happens, this “thought experiment” (more properly a proof or theorem) has inspired much debate and criticism. What can we do with the knowledge that although some things are mathematically possible, they aren’t realistically feasible given what we know about the world (i.e. no resources are infinite, etc.)?
Some thinkers have attempted to use this experiment to show how evolution could have occurred—the analogy is that simple elements, given enough time and resources, will spontaneously assemble themselves into more complex organisms, without the need for an agent to cause it.
A related analogy is designed to show the impossibility of the claim that life did not evolve, but emerged wholly formed. Imagine you were walking on a beach and saw a watch on the sand. How did it get there? The analogy claims that evolutionary theory (or life randomly emerging by itself) is akin to saying the watch elements washed up onto a beach and randomly assembled themselves into a perfectly functioning watch—possible, but enormously unlikely.
The proposed alternative is that a watchmaker made the device—an argument for intelligent design.
Others have commented on the validity of the thought experiment itself. R. G. Collingwood claimed, “Any reader who has nothing to do can amuse himself by calculating how long it would take for the probability to be worth betting on. But the interest of the suggestion lies in the revelation of the mental state of a person who can identify the ‘works’ of Shakespeare with the series of letters printed on the pages of a book.”
His point is that merely producing a string of symbols associated with a work is not the work itself—a classic example of how a thought experiment designed to engage with questions in one realm (random number generation and probability) can be used to inspire questions and dialogue in another (what it means to create art, and the relationship between symbol and its referent).
For many people, complex systems like living organisms are so far removed from abstract mathematical strings of data as to make any metaphor or analogy useless. Some have similarly criticised the analogy as incomplete, saying that for the product to be meaningful you would need to factor in the beliefs, semantic structures, morality, science, linguistic patterns and so on that came from Shakespeare’s time—i.e., already banked knowledge that Shakespeare himself would have worked with (and not, as the monkeys would have to, starting from scratch with merely the fifty symbols of a typewriter).
Using this theorem to talk about evolution is tricky for this same reason—selection of the fittest individuals will mean that strings of DNA that “work” will be retained and passed on; it is not as if every organism existing today had to arise from scratch in each instance.
Like most mathematics, the theory doesn’t map neatly onto life as we know it. Economists can predict the consumer behavior of “rational” humans only to find their models fail miserably to predict anything. Statistics can be used to make perfectly true claims (“the average person has one breast”) that are nevertheless inaccurate, and AI game theory can tell us about actions, choices and solutions that are most optimal, even though to the average human being they make no sense at all (consider the chess-playing AI, Alpha GO, whose style is no longer comprehensible to ordinary human players.)
Borel argued that although some events might seem mathematically possible, they are for all intents and purposes impossible. Just because we can think about outlandishly huge numbers, it doesn’t mean we can easily translate those concepts to our real, flesh-and-blood world.
Of course, there is a chance that the sun might rise in the west one day, but it is so unlikely as to be impossible. When we are comfortable understanding what mathematical probabilities actually mean for our lives and the choices we make, we can use thought experiments like this and the insights that come with them to live (and think) better.
Mathematics can allow us to plumb the very outer edges of what is theoretically the case, while common sense then enables us to fill in the concrete details using what we see in the real world. Likewise, though we can agree with Descartes in a theoretical sense that we can know nothing, we can still reasonably inhabit the world using our senses and do pretty well.
Though it might not seem like much, this is a monumental shift: knowing the difference between yes and no, luck and probability, chance and circumstances. It tells us the proper place for abstract theory in our lives: it’s worth learning the difference between “in theory” and “in practice”!
In Eastern philosophy, great teachers attempt to point to the fundamental limitations of our conception of reality indirectly, by using nonsense stories, jokes, unanswerable questions or statements like, “What is a buddha? Three pounds of flax.”
By their nature, these statements are invitations to step outside of the occupied realm of thought and look at life from a wholly different perspective—an elevated one.
Perhaps we can imagine that Western philosophy does the same, only a little more formally. In seemingly bizarre, paradoxical or nonsensical territory, we uncover not the strangeness of reality itself, but our own limits in being able to conceive of it, or represent it in symbols. It is as though the thinking mind, having looked around at the world, gets the idea to turn back and look on itself, and watch itself as it thinks.