Deiah veDibur - Information & Insight

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'25 Teves 5759 - Jan. 13, 1999 | Mordecai Plaut, director Published Weekly







Opinion & Comment
Monkeys and Typewriters: The Myth Goes On
by Joshua Josephson

Part II

Part I of this article ended with a quotation from a recent book which uses the parable of monkeys and typewriters. I challenged the readers to try to intuit its exact meaning. Here's the quote again:

"It is a bit like the well-known horde of monkeys hammering away on typewriters -- most of what they write will be garbage, but very occasionally by pure chance they will type out one of Shakespeare's sonnets."

This quote is from A Brief History in Time (p. 123), a book that was very popular in the secular world despite the fact that it deals with some of the most difficult topics of theoretical physics. The author of the book, Stephen Hawking, was considered by many to be the preeminent theoretical physicist of the past decade. He was certainly not unaccustomed to thinking about, or intimidated by, large numbers.

Readers were asked to try and pin down the meaning of the phrase "very occasionally" used by Hawking in the passage. Just how long would it take a "horde of monkeys" to type out one of Shakespeare's sonnets by chance?

We will try to estimate this more exactly. To do this we first have to determine about how long a Shakespearean sonnet is, and then how long it would take Hawking's monkeys to type one out.

It is commonly accepted that Shakespeare wrote about 154 sonnets. (The word sonnet is derived from the Italian sonetto, meaning "little song.") Each is a poem of exactly fourteen lines written with a special rhythm called iambic pentameter. An iamb consists of an unstressed syllable followed by one that is stressed and a poem written in iambic pentameter has lines which usually contain five groups of iambs, each of two syllables. Thus it follows that a typical sonnet should have about 140 syllables.

Considering that in the English language syllables are on the average three to four letters long, the typical sonnet should have around 500 letters. (I checked this by counting several lines and got results which were consistent with this estimate.)

Now let us make some assumptions. I would like to be as generous as possible to Hawking, to make it as likely as possible for his horde of monkeys to produce what he claims they will. Thus, I will make several assumptions that are extremely favorable to him and his monkeys.

First, we will offer the monkeys scaled down versions of typewriters. We will set the monkeys to work on typewriters that have keys with only the twenty-six letters of the alphabet. There is no space bar, no shift key, no period -- nothing more than the raw alphabet. We will even do away with capitalization. Imagine how miserable a monkey would feel after it typed virtually an entire sonnet but hit the "&" or "#" symbols just before the very end.

All we will ask these monkeys to do is to type away, to produce a long string of letters, all in lower case, without spaces anywhere. We will search these strings to try locate something that matches perfectly the sequence of the letters of one of Shakespeare's sonnets, or for that matter, any sequence that has meaning in English.

In addition, we will use a very large number of monkeys, give them the ability to type very quickly and give them a tremendous amount of time within which to work. Let's suppose that every single atom in the universe is a monkey at a typewriter. Let's suppose that every single monkey types one trillion letters per second. And let's further suppose that all these monkeys type nonstop for a trillion years.

All this is way beyond any normal physical scale. To put it into some perspective, let's try to get a handle on the size of the total output of these typing monkeys.

Assume that we had a technology which could fit ten trillion letters -- about the number of letters in all the books in the Library of Congress or the British Museum -- onto one CD- ROM. (Today's CD-ROMs can hold "only" about 600 million characters -- the equivalent of six hundred books, each five hundred pages long. This is an amount of writing way beyond what Shakespeare produced.) Assume also that a CD- ROM occupies only one cubic inch of space (today's are slightly bigger). Still the total output of these monkeys would be incredibly staggering. The CD-ROMs that would be needed to hold their entire output would fill up not merely every square inch of this universe but we would need trillions of similar sized universes all packed solidly with CD-ROMs to accommodate them all.

Tucked away somewhere in all this output, this mass of "literature," so to speak, we would find very little of value. We almost surely would not encounter even a single one of Shakespeare's sonnets. Indeed, we almost surely would not be able to find a single piece of error free, consecutive, meaningful text in the English language that would be anywhere near 500 letters long.

For there to be a any reasonable chance for monkeys typing randomly to produce 500 letters of the sort we are discussing, it would be necessary to convert every single atom in our universe into a universe and assume the same thing all over again for each new universe. In other words, we would have to blow up each atom in this universe to the size of our entire universe, have each new universe contain as many atoms as our existing universe, convert each of the new atoms into monkeys at typewriters, and so on. Only then would there be a chance -- though still a remote one -- that these monkeys would type, after a trillion years, consecutive strings of 500 error-free letters in the English language. It's beyond me how anyone would find it, however.

As difficult as it may be to believe, we almost surely still would not be able to find within this entire output what Hawking claimed his horde of monkeys could do. The chances of finding a single string of text that matches even one of Shakespeare's 154 sonnets would still be extremely remote.

One thus must wonder what Hawking meant when he said that a horde of monkeys would "very occasionally" type out a sonnet. Is it that he did not bother doing the calculation? Or was he simply speaking tongue in cheek, saying something that most normal people would take to mean one thing, but which he intended to mean something vastly different?

I am quite sure that many readers are having difficulty accepting what I am saying. Is it really true that monkeys typing randomly could not produce an appreciable length of meaningful text? How does one evaluate such a statement? A simple analogy should give the reader a tool which will enable him or her to grasp what is being said here.

Think of an old fashioned, non-digital car odometer. (The odometer is the part of the speedometer that gives the readout of the total miles the car has been driven.) An odometer is a set of drums that rotate. Each drum contains the numbers from 0 to 9. The drums are behind a screen that shows only one of the numbers on each of the drums at a time. Each time the car travels a mile, at least one of the drums moves one notch and a new number becomes visible.

Consider now the odometer of a new car that has never been driven. If the odometer consists of five drums, it would read 00000. As the miles go by, the drums in the odometer turn and they show all the numbers in succession -- from 00000 to 99999.

Since our odometer has only five drums, it can display only five numerals. Thus, it cannot display the number 100,000, which would require a sixth drum. As any savvy used car buyer ought to know, when mile 100,000 is reached, the odometer reads 00000 again, and as even more miles are added to the car, it displays the numbers from 00001 to 99999 all over again.

How many different numbers can an odometer with five drums display? The answer should be obvious: exactly 100,000 -- the number 00000 and the 99,999 numbers from 00001 to 99999.

What happens if we add one more drum to the odometer (instead of 5 drums, we now have 6)? How many different numbers or combinations can such an odometer display? The pretty obvious answer is one million: again, the number 000000 and all the numbers from 000001 to 999999. In general, the total of the numbers that an odometer can display is one more than the highest number that can be written with it.

What this implies is very important for understanding the calculations we need for monkeys and typewriters. Think about the implications. By adding just one drum with ten numerals to an odometer, the total numbers the odometer can display is multiplied by ten. With seven drums we get ten million different numbers. Add just one more, and the odometer can display one hundred million different numbers. The addition of one new drum has the effect of multiplying the total numbers by ten.

We can now convert our odometer into a "monkey and typewriter" calculating machine. Suppose we add a lever to the odometer to make it into something like a slot machine: instead of showing the sequence of numbers in succession as in a car odometer, the drums of this slot- odometer will rotate wildly and randomly every time the lever is pulled. Instead of placing the numbers 0 to 9 on each drum, we will place the twenty-six letters of the alphabet on them.

Suppose we now place a monkey in front of this modified odometer, and instead of asking it to type on a typewriter, we ask it to repeatedly pull the lever. We will record and collect the results. How often will the random letters form words? What are the chances that our slot machine will produce meaningful text, or a sequence of the letters that are part of one of Shakespeare's sonnets?

Consider the problem. An odometer consisting of one hundred drums containing the numbers 0 to 9 can be used to write the total number of atoms in the universe, with a lot of room to spare. In other words, there are more combinations of numbers possible on such an odometer than all the atoms of the universe. (No, I did not count the atoms. But scientists make estimates of their numbers. The highest estimate I have seen in print is 10^88 [a 1 followed by 88 zeros], but I have heard that more recent estimates are higher. Still the number 10^100, which is the number just after the highest number which can be written by an odometer with 100 drums, is far greater than the highest estimate of atoms in the universe.)

Because there are significantly more letters in the alphabet than numbers from 0 to 9, an odometer consisting of one hundred drums that has the letters of the alphabet on it has vastly more combinations than the odometer with numbers on it. Out of this huge amount of possible combinations, a precise sequence of letters found in any of the sonnets of Shakespeare form an extremely small part. The chances of a monkey pulling the lever and by chance hitting on a sequence that matches one found in Shakespeare's sonnets is virtually nil.

Even a horde of monkeys that is as numerous as all the atoms in the universe, all working at the same time, would take an inordinately long time to pull the levers often enough to create even a remote likelihood of getting a right sequence.

And this is only an extremely small fraction of the problem. To get a match with an entire sonnet of Shakespeare requires the production of about 500 matching letters. This essentially means dealing with an "odometer" that has 500 drums. The number of possible combinations for such an odometer is way beyond any number that is meaningful in our universe or indeed in any imaginable number of universes. In sum, it cannot be done unless we posit things which are way beyond our finite experiences.

It must be stressed that what is being done here is not simply a cute little exercise in arithmetic or Hawking- bashing. When biologists maintain that life with its incredible complexity arose, developed and flourished through random combinations of small molecules or random changes in existing molecules, they are positing a situation that in a limited sense is similar to the case of monkeys and typewriters. Evolution of life through random mutation and random combination alone is much more farfetched than monkeys on typewriters producing meaningful English text of some appreciable length.

It would be a gross oversimplification to maintain that the argument made in this article is sufficient to prove that evolutionary theory is wrong. Much more needs to be explained and said. However it is a start, and the sort of exercise we have just done in this article can help lay the groundwork for an examination of the claims of scientists. It provides a necessary tool for evaluating statements which posit that order arises from chaos or that pure and simple randomness can be the driving force behind the majestically beautiful world we see before us.

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