You’ve probably heard this argument: Give enough monkeys enough typewriters and enough time and they will type out Shakespeare. The Infinite Monkeys Theorem is meant to show that chance rules the universe.

If you sense a problem, you are right. What does “enough” mean here?

Image Credit: ostap25 - Adobe Stock

University of Technology Sydney mathematicians Associate Professor Stephen Woodcock and Jay Falletta decided to find out. Let’s start with the fact that our universe is finite, so far as we know. It has a beginning and will have an end. That means we can put some numbers into the picture:

The Infinite Monkey Theorem only considers the infinite limit, with either an infinite number of monkeys or an infinite time period of monkey labour,” said Associate Professor Woodcock.

“We decided to look at the probability of a given string of letters being typed by a finite number of monkeys within a finite time period consistent with estimates for the lifespan of our universe.

University of Technology Sydney. “It’s not to be. Universe too short for Shakespeare typing monkeys.” ScienceDaily. 30 October 2024 The paper is open access.

They assumed that the estimated world population of 200,000 chimpanzees types one key per second until the universe ends in about 10¹⁰⁰ years (1 followed by 100 zeros).

How far would the monkeys get with typing so many words in a correct sequence?

The results reveal that it is possible (around a 5% chance) for a single chimp to type the word ‘bananas’ in its own lifetime. However, even with all chimps enlisted, the Bard’s entire works (with around 884,647 words) will almost certainly never be typed before the universe ends. “Too short”

Although Woodcock and Falletta’s approach is light-hearted, they have a serious point to make:

“This finding places the theorem among other probability puzzles and paradoxes — such as the St. Petersburg paradox, Zeno’s paradox, and the Ross-Littlewood paradox — where using the idea of infinite resources gives results that don’t match up with what we get when we consider the constraints of our universe,” said Associate Professor Woodcock. “Too short”

In other words, if we want to play around with infinity concepts, we can have fun. But we can’t really use them to make decisions about what might or might not happen in our finite universe.

And they think that argues against a reductive approach to life in general:

In the era of generative AI, the Infinite Monkey Theorem, and its finite version, perhaps also challenge readers to consider philosophical questions around the nature of creativity, meaning and consciousness, and how these qualities emerge. “Too short”

If the monkeys couldn’t type out Shakespeare within the life of the universe, human creativity is not just a matter of chance.

Quintillions of years?

As it happens, theoretical physicist Manon Bischoff offered similar thoughts earlier this year at Scientific American: “An infinite number of random events can produce just about anything if you have quintillions of years to wait.”

She offers a brief history of the idea of the infinite monkeys:

The infinite monkey theorem owes its name to mathematician Émile Borel, who used the animals metaphorically to illustrate his theory of probability in 1913. The ideas behind the theorem are much older, however. In antiquity, Roman philosopher and politician Marcus Tullius Cicero wrote that one might “believe that if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form [the epic poem] the Annals of Ennius. [But] I doubt whether fortune could make [even] a single verse of them.”

Since our universe is estimated to be 13.8 billion years old, we would have to wait nearly 7 x 1048 times as long as the time that has passed between the big bang and today. And all this just to produce a single sentence from Hamlet by chance. In this respect, Cicero was right: it is very unlikely that chance will produce even a single readable verse of a poem—or any other text—after a finite amount of time.

“The Mathematical Case for Monkeys Producing Shakespeare—Eventually,” May 7, 2024

That’s part of the reason why chatbots scarf things up off the internet rather than producing anything of their own by juggling random numbers. Creativity is not a product of randomness.

Can we work with infinity?

We can work with infinity in mathematics, as Jonathan Bartlett pointed out a couple of years ago:

The hyperreal number system extends the reals so as to handle infinity precisely. Note that there are actually multiple ways to handle infinitely large numbers. But I have found one particular system to be of more practical use than other systems, which is why I think it is worth discussing. Thinking about infinities is somewhat mind-bending, but it turns out that actually manipulating infinities with the hyperreal system is incredibly easy if you are familiar with basic algebra.

“Yes, You Can Manipulate Infinity — in Math, Mind Matters News, October 28, 2019

We can multiply infinities indefinitely — as abstractions. But we can’t assume that what we can do mathematically unfolds in the physical universe we live in.