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.
