The Infinite Monkey Theorem

KEVIN HARRIS: When I was a little boy, I was told by an older kid that given enough time, monkeys, and typewriters, eventually at least one of those monkeys will type Shakespeare’s Hamlet with no errors. In fact, the way I heard it, it involved an infinite amount of time and monkeys so an actual infinite is in the mix somehow. The New York Times just came out with an article on some of the latest research on this, and I'd like to see what you think about it and how it applies to the kalam, your other work on God, time, and probability theory, and so on. Did you grow up hearing about the typing monkeys like I did? Or did it come up later in your studies?

DR. CRAIG: I think this typically came up with regard to evolutionary biology. Even though human beings and other biological organisms are breathtakingly complex, given enough time even a battery of monkeys sitting at a typewriter would eventually type out the sonnets and plays of Shakespeare, and so we shouldn't be so surprised to see that by chance alone these complex organisms have evolved. Of course, the discovery that the universe had a beginning only around 14 billion years ago kind of puts the kibosh on that argument because there isn't infinite time to arrive at these life forms.

KEVIN HARRIS: The article^[1] is by Alexander Nazaryan, and he writes that the monkey math problem “was first described in a 1913 paper by the French mathematician Émile Borel, a pioneer of probability theory.” Nazaryan writes,

In 1979, The New York Times reported on a Yale professor who, using a computer program to try to prove this “venerable hypothesis,” managed to produce “startlingly intelligible, if not quite Shakespearean” strings of text. In 2003, British scientists put a computer into a monkey cage at the Paignton Zoo. The outcome was “five pages of text, primarily filled with the letter S,” according to news reports. In 2011, Jesse Anderson, an American programmer, ran a computer simulation with much better results . . .

A new paper by Stephen Woodcock, a mathematician at the University of Technology Sydney, suggests that those efforts may have been for naught: It concludes that there is simply not enough time until the universe expires for a defined number of hypothetical primates to produce a faithful reproduction of “Curious George,” let alone “King Lear.” Don’t worry, scientists believe that we still have googol years — 10¹⁰⁰, or 1 followed by 100 zeros — until the lights go out. But when the end does come, the typing monkeys will have made no more progress than their counterparts at the Paignton Zoo, according to Dr. Woodcock.

Woodcock's paper deals with finite time and resources. Woodcock says,

“The study we did was wholly a finite calculation on a finite problem,” he wrote in an email. “The main point made was just how constrained our universe’s resources are. Mathematicians can enjoy the luxury of infinity as a concept, but if we are to draw meaning from infinite-limit results, we need to know if they have any relevance in our finite universe.”

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The new paper has been mocked online because the authors purportedly fail to grapple with infinity. Even the paper’s title, “A numerical evaluation of the Finite Monkeys Theorem,” seems to be a mathematical bait-and-switch. Isn’t infinity a basic condition of the infinite monkey theorem?

DR. CRAIG: I think Woodcock is absolutely right here where he says that you can use the mathematical concept of infinity, but if we're to have meaningful results from these infinite limit speculations they need to be relevant in our finite universe. He's talking about a result into the future, not into the past as I was a moment ago, where in the past there has been only a finite amount of time in order for biological complexity to evolve on this planet. But now what he's experimenting with is if you project this into the future what will happen? There, you see, because the future is only potentially infinite (not actually infinite), you will always be finite in your result. The monkeys at every point in the future will have typed only a finite number of pages. They will never arrive at infinity. Infinity is just a mathematical limit which is endlessly approached but is never reached. So Woodcock is absolutely right. By 10 to the 100th years all the stars will burn out and the universe will enter what's called the dark era. That will be the end of all life and heat in the universe. It represents the heat death of the universe. Therefore it's impossible for these monkeys to ever type out even Curious George much less the sonnets and plays of Shakespeare. And even if this dark era were not to arrive – even if somehow new sources of energy were to prolong the life of the universe – you never reach infinity. You're always in the domain of the finite, and therefore the monkeys will never be able to produce this result.

KEVIN HARRIS: You can slap the hand of the monkey doing transfinite math. [laughter] By the way, this article confirms what you've said about the heat death of the universe. It's kind of a side issue, and it's a long way off, but eventually the lights go out. That has ramifications not only in astrophysics but the meaning of life.

DR. CRAIG: Exactly. With respect to cosmology, not only cosmogony (the origin of the universe as a domain of contemporary astrophysics) but also eschatology is now a domain of physics, not just theology. Scientists project in the future what the universe will be like in X number of years. As you mentioned and as Woodcock mentions, the universe will eventually use up all its energy and go black and all life will perish. This raises a profound question. If in a finite amount of time this will happen then why hasn't it happened already if the universe has existed for infinite past time. We should have already reached this state of darkness, dilution, and death. Indeed an infinite time ago if the universe is beginningless. So this represents a very powerful thermodynamic argument for the finitude of the past and the beginning of the universe and the need for a cosmic creator. The other thing that you mentioned is that it puts a question mark behind, I think, the meaning of life. I, as a non-Christian, was aware that we were all doomed ultimately in this universe – that someday the last human being would draw his last breath and mankind would be no more. Not only each of us individually, but our entire human race is doomed to extinction in the heat death of the universe. Everything will turn out the same no matter what we do, no matter how hard we try. It's hard to avoid the sense of futility and meaninglessness that that imparts. It means that no matter what you do, say, think, it ultimately is in vain. It makes no ultimate difference; everything will end up the same – namely in extinction and death.

KEVIN HARRIS: The paper continues,

The chances of a monkey spelling out “bananas” are “approximately 1 in 22 billion,” Dr. Woodcock said.

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The chances of a monkey replicating that book [Curious George] are 1 in 10¹⁵⁰⁰⁰ (a 1 followed by 15,000 zeros). And, at nearly 836,000 words, the collected plays of Shakespeare are about 464 times longer than “Curious George.”

Woodcock writes,

“If we replaced every atom in the universe with a universe the size of ours, it would still be orders of magnitude away from making the monkey typing likely to succeed.”

Sounds like even in a multiverse it's impossible for a monkey to randomly write a book.

DR. CRAIG: I think that's true if the multiverse had a beginning because then you've got this same problem of being unable by finite steps to reach an infinite limit. So eventually every universe in the multiverse will come to this kind of thermodynamic destruction.

KEVIN HARRIS: The article continues,

This conclusion circles back to the French mathematician Borel, who took an unlikely turn into politics, eventually fighting against the Nazis as part of the French Resistance. It was during the war that he introduced an elegant and intuitive law that now bears his name, and which states: “Events with a sufficiently small probability never occur.” That is where Dr. Woodcock lands, too. (Mathematicians who believe the infinite monkey theorem holds true cite two related, minor theorems known as the Borel-Cantelli lemmas, developed in the prewar years.)

“Borel’s Razor”, sounds like?

DR. CRAIG: What Borel, I think, is saying is that there is a kind of probability bound, and if some event exceeds that bound then effectively it never happens. This is applied by theorists like William Dembski to the evolution of biological complexity and the amount of time available in our past. Dembski has argued that the complexity manifest in life on Earth exceeds the acceptable probability bound. It would never have happened if it were by a blind, undirected chance.

KEVIN HARRIS: The article concludes with, you guessed it, the future of artificial intelligence. Everything I read today has got to bring up AI these days.

The new paper offers a subtle comment on the seemingly unbridled optimism of some proponents of artificial intelligence. Dr. Woodcock and Mr. Falletta note, without truly elaborating, that the monkey problem could be “very pertinent” to today’s debates about artificial intelligence.

For starters, just as the typing monkeys will never write “Twelfth Night” without superhuman editors looking over their shoulders, so increasingly powerful artificial intelligences will require increasingly intensive human input and oversight.

Is there an argument for God embedded in this?

DR. CRAIG: I take it that what they're saying here is that no matter how sophisticated AI becomes, no matter how powerful, it's still going to require increasingly intensive human involvement. That you can't just let it run by itself and expect it to get to meaningful results. I think the importance of this for the existence of God will be with respect to the probability of the origin of life and the evolution of biological complexity on Earth and whether or not this is too improbable to have occurred in the time available, and therefore is evidence for a creator and a designer of the world.

KEVIN HARRIS: One more thing. The article also cites Eric Werner of Oxford who has studied various forms of complexity. He wrote a paper in 1994 that applies to typing monkeys and AI. He says,

“Complex structures can only be generated by more complex structures.”

That kind of reminds me of what Dawkins was saying – God would be more complex than the universe in order to create the universe.

DR. CRAIG: I think this point is more akin to William Dembski’s point, although Dembski says it's not just complexity that requires an explanation. It's specified complexity. It is a complexity that conforms to an independently given pattern that cries out for some sort of explanation. The question then will be whether or not a chance hypothesis is sufficiently credible to account for that. What Dawkins was talking about was that a creator and designer of the universe would have to be composed of complex parts in order to cause our universe, and that simply doesn't follow. A mind is distinct from a mind's ideas and actions, and a mind can be partless. It doesn't have any parts as an immaterial entity, and yet the mind can be very powerful in thinking very complex thoughts and performing complex actions.

KEVIN HARRIS: As we wrap it up today, it seems to me that the current pages of The New York Times are very interested in a lot of what we've been talking about in these podcasts for a long time – actual infinities, probability theory, design, complexity, the works. Any concluding thoughts?

DR. CRAIG: I just hope that the interest that these sorts of articles generate on the popular level will make our arguments for God's existence resonate more with laypeople when they encounter them and make them more open to the conclusions that we draw.^[2]