THE IMPROBABLE IT; SOME REFLECTIONS ON MEANING AND CHANCE
by
Roger E. Ball
Delivered to The Chicago Literary Club
February 4, 1991
Copyright 1991 Roger E. Ball
Achilles and the Tortoise were seated on the sea wall that
stretched in a curve along the margin of the bay. Behind them, the white houses
of the village clustered tightly together on the hill that rose steeply behind
the road. The sun was warm, and the air had that crystalline, super-transparent
quality that one still finds on Aegean islands when the wind blows the
pollution of
The two friends had lunched on figs, olives, and cheese, along with bread and
half a bottle of retsina. Now the lunch was over and they were relaxing,
watching the gulls and an occasional dolphin that surfaced briefly just inside
the line of foam that marked the entrance to the bay.
Before lunch they had walked along the shingle, and Achilles had amused himself
by finding a handful of symmetrical pebbles. Now he was tossing them in the air
and watching the patterns that formed as they fell on the blanket that they had
spread for their lunch. Suddenly the Tortoise gave an exclamation. "Look
Achilles. How extraordinary! The pebbles have fallen so as to spell out clearly
the name ZENO."
"Wasn't he that old fool that claimed that I could never catch you in a
foot race if I once gave you a head start?"
"He not only claimed it but proved it by irrefutable logic. He discovered
many other wonderful things, such as that an arrow in flight never actually
moved."
"If he had stood with me and the host on the plain of Illium he would have
learned quickly enough that an arrow moved. But never mind that. You are right
that something truly extraordinary has happened. The chances against it must be
overwhelming."
"Not at all."
"What do you mean, not at all?"
"I mean that each pattern that the pebbles can fall in is equally
unlikely. The ZENO pattern is no more unlikely than any other."
"Oh come on! The other patterns that I threw didn't have any meaning.
Surely, to throw a pattern that means something is much much more unlikely than
just getting a random arrangement."
"How do you know that the other patterns didn't mean something?"
"Because I was watching and saw them."
"You recognized the meaning in the ZENO pattern because you know Greek.
Are you acquainted with all of the other languages, alphabets, and systems of
writing that are used in different parts of the world? Surely you know that in
God-like Achilles, fleetest of mortals, snorted derisively. "Well, I'm
sure they didn't, although I'm not quite sure how I know that. And I'm also
sure that to throw pebbles and have them come up ZENO is really much more
remarkable than just throwing random patterns."
The Tortoise thought for a moment before replying. "Look Achilles, let's
try an experiment. Throw the pebbles again."
Achilles gathered them up and threw. "There, Tortoise. That's a random
pattern if I ever saw one. And I'll wager my sword and spear that the pattern
doesn't mean anything in any language."
You're probably right." said the Tortoise. "But you'll agree, won't
you, that I can define my own system of symbols if I choose?"
"Yes, of course." said Achilles.
"Well, I've just devised such a system, and in my system the pattern that
you just threw means `Zeno was right'. Now, by giving a meaning to your
pattern, have I radically changed the probability of its occurrence, which
would seem to be implied by your statement that a meaningful pattern is more
unusual than a random one?"
"Oh Tortoise, in the name of Zeus, stop this intellectualizing. My head is
beginning to ache. Let's watch the dolphins that are playing around the reef. A
minute ago, I could swear I saw a mermaid sporting with them."
In this relatively brief dialogue the two improbable friends succeeded in
touching on some of the deepest problems in philosophy, mathematics, and logic.
Although a whole library shelf would be needed to deal adequately with all of
the implications, we can at least sketch out some of the main problems,
beginning with the famous foot race that so disturbed Achilles.
The idea that Achilles would not be able to catch the Tortoise if the latter
had a head start was advanced by Zeno of Elea, a student and friend of
Parmenides, who lived about 500 B.C. It was included in a book that he wrote
describing several paradoxes of motion.
When the conclusion of an argument is absurd it is easy to think that the
reasoning behind it is equally absurd. This would be a great mistake when
thinking about Zeno's paradoxes. They raise issues that are immeasurably subtle
and profound, and which occupied some of the greatest minds for over 2000 years
without noticeable success until the late 19th century. These issues involve an
accurate understanding of the infinitesimal, the infinite, and continuity.
There are different ways of describing the paradox of Achilles and the
Tortoise, but the version by Bertrand Russell shows the difficulty in its
clearest form. The argument runs as follows:
Achilles and the Tortoise agree to race. Since Achilles is much faster, the
Tortoise is given a head start. At every moment Achilles is somewhere and the
Tortoise is some where, but neither is ever twice in the same place. Therefor
the Tortoise goes to as many places as Achilles because each is in one place
one moment and in another place at another moment. But Achilles can never catch
up, because to do so the places where the Tortoise would have been would be
only part of the places where Achilles would have been, and this is impossible
as has been shown.
This argument is strictly correct if we allow the axiom that the whole always
has more terms than the part. If you believe that no part of any collection is
ever as great as the whole, you are in agreement with common sense and with the
majority of people. The trouble is that this belief forces you to agree that
Achilles can never catch the Tortoise. This is the Catch 22 that stymied the
philosophers, and led to a great deal of ingenious sophistry in an effort to
sweep the whole thing under the rug.
Enlightenment finally came. That no part of a collection is ever as great as
the whole is only true of finite sets. The "places" in Zeno's paradox
are like the mathematical points on a line segment, they are infinite in
number. Late in the 19th century George Cantor proved that any sub-set of an
infinite set has as many terms as the set as a whole. In fact, this is the
defining characteristic of an infinite set. With this discovery, Zeno's paradox
was at last laid to rest.
Achilles and the Tortoise showed an intuitive sense of the probabilities
involved in throwing the pebbles, but had difficulty in making quantitative
comparisons. Assuming that their conversation took place sometime during the
Classic or Hellenistic eras, the mathematics of probability had not yet been
discovered. This mathematics began to be developed in
After Cardano, little further progress was made for over 100 years until the
problems were taken up again in
Progress has continued down to the present, and probability theory is now one
of the most widely used and practical branches of mathematics. Forming the
heart of statistical analysis and sampling theory, it is used to assess the
results of almost every scientific experiment, to control production processes,
determine the television programs that we are permitted to see, and report to
us on our opinions and activities from the political to the sexual. By allowing
us to make rational decisions under conditions of uncertainty it is at the core
of almost all strategic planning, both for corporations and governments. And
finally, it provides the measure of ultimate reality, because at the quantum
mechanical level all events are measured in terms of probability densities.
However, there is a worm at the core of this apple. Central to the idea of
probability is the idea of the random event that cannot be foreseen or predicted
in advance. Since our ability to predict the outcome of an event depends on our
knowledge, probability depends as much or more on conditions inside our heads
as it does on factors directly associated with the event itself. This
subjective aspect of probability can lead into a quagmire of philosophy and
even theology. As we shall see later, modern theory avoids this problem by
resolutely refusing to think about it.
For example, the standard example of a chance event is the flip of a coin.
However, it only takes a little reflection to realize that the result of the
coin flip should be fully predictable. Given information on the initial
conditions, the force vectors applied when the coin is flipped, and the density
of the air, the computer should be able to calculate the trajectory easily, and
tell us if heads or tails will come up. That we assign even odds to heads or
tails is due to lack of knowledge or inability to handle the mathematics
involved.
Bernouilli was well aware that the probabilities that he was measuring depended
on human uncertainty. He lived in a completely deterministic world under the
firm control of divine providence. God left nothing to chance, every event was
determined by Him, and therefor every thing was objectively certain. Bernouilli's
mathematics provided a way to make reasonable decisions in instances where the
outcome was uncertain because of lack of knowledge of God's will.
After
The advent of the 20th century saw the progressive dismantling of this entire
deterministic world machine. Einstein disposed of the absolute nature of time
and space. Heisenberg showed that it was impossible to simultaneously determine
the location and velocity of a particle. Schroedinger proved that the particle
itself has no objective reality, since it could just as easily be a wave. Godel
found a fatal flaw in the supposably seamless structure of mathematics. Chaos
theory revealed what had become a growing suspicion: Ultimate reality is not
just hard to understand, it is impossible to ever know.
Uncertainty had now moved into a new dimension. We not only have the
uncertainty deriving from our lack of knowledge, but also the uncertainty built
into the structure of reality. What then is the exact definition of
probability? This definition has proved so difficult to formulate that the
attempt to do so is now generally avoided. A leading modern textbook on
probability theory has this to say: "From a mathematical point of view . .
. it is not necessary or even desirable to define probability explicitly.
According to the usual axiomatic procedure, probability is an undefined notion,
a real number between zero and one, satisfying certain rules of operation, from
which the calculus of probability is developed by deduction."
This sort of thing inevitably reminds me of the old recipe for cooking a
porcupine. You put the porcupine in a pot of water with a stone and boil them
together until the stone is tender, after which you throw away the porcupine
and eat the stone.
Be that as it may, these philosophical problems are quibbles for people like me
who fond of quibbling. The mathematics works very well, and we can use it to
get a good approximation of the probabilities in Achilles' game with the
pebbles.
Assume that there were twenty pebbles, and that they were falling in a circle
40 centimeters in diameter. The total number of possible distinguishable
patterns works out to be about 3.998 x 10 to the 49th power. This is a enormous
number, more than equal to all of the seconds that have passed since the big
bang created the universe.
We can probably arrange the pebbles to spell ZENO in less than fifty different
ways. Each of these is a possible pattern, and the Tortoise was right in saying
that each is equally likely or unlikely. Clever sophist that he was, the
Tortoise tried to use this to trick Achilles, whose stubborn intuition told him
that a meaningful pattern was more unlikely than a random one. He was right,
but to see why we must descend again into the labyrinth of meaning.
The Tortoise challenged Achilles by pointing out that the latter could not be
sure that the other patterns did not mean anything, since he was unacquainted
with many of the world's languages and symbol systems. The Tortoise was trying
to plant the idea that, if a large number of the patterns that Achilles had
thrown actually did mean something, then the appearance of a pattern spelling
out ZENO was much less improbable and mysterious than it at first appeared.
This is flawed logic and actually irrelevant, as we shall see later. However,
it points to a stubborn problem in communication theory, namely that of
determining whether or not a seemingly meaningless set of symbols actually
contains a hidden message.
Spies and other people anxious to preserve the privacy of communications use
ciphers that employ a "key" to convert the readable "plain text"
of their communication into a presumably unreadable "ciphertext". The
privileged receiver of the communication uses the key to convert the ciphertext
back into plaintext.
Actually, all communications are sent in cipher. The sender uses a key consisting
of the vocabulary, grammatical rules, and orthography of his language to
convert the thought that is in his head into a set of symbols that can be
decoded and understood by the reverse application of the key. If the receiver
of the communication does not have the key, that is if he does not understand
the language and symbol system employed, he is not only unable to understand
the message, but in the absence of other information, usually cannot know
whether or not a message has been intended.
An instructive example is provided by the Voynich Manuscript. This is a
232-page illuminated book written entirely in a cipher that has never been
decoded. No one knows when, where, or by whom it was originally written,
although some scholars believe that it can be traced to Roger Bacon in the 13th
century. The manuscript was bought by the Holy Roman Emperor Rudolph II of
The margins of the manuscript contain drawings of nude women frolicking in
basins of water connected by a strange sort of plumbing that leads the water
from one basin to another. Other pages show very detailed drawings of plants
and flowers that cannot be identified by botanists, and constellations of stars
that do not appear in any sky.
The writing is an apparent cipher, using approximately twenty-one symbols that
bear no resemblance to those used in any known language. They are in groups,
resembling words, with spaces between them. No one knows in what language the
text would be in if it were deciphered. Some believe that there is no language,
and that the whole thing is a hoax, although this is unlikely.
The Voynich manuscript has been a challenge to cryptographers for over
seventy-five years, and the best of them have worked at it with absolutely no
success. This includes Herbert Yardley, the American code expert who broke the
German cipher in World War I and William Friedman, who defeated the Japanese
"purple code" of the 1940's. It recent years it has been subjected to
elaborate statistical analysis, using powerful computers. This analysis shows
regularities that make it unlikely that the whole thing is a hoax, although the
regularities are not those found in any European or Asiatic language. On the
other hand, it seems highly unlikely that anyone prior to the 16th century
could have created a cipher that would withstand modern analytical tools. The
whole thing is a complete mystery.
There is, of course, a significant difference between the Voynich Manuscript
and the patterns that Achilles formed by tossing the pebbles. Whether a message
or a hoax, the manuscript was the creation of a human being, whereas the
patterns formed by the pebbles were the result of a random process. This
difference is important to those who claim that meaningful messages must
originate in the mind of a sentient being. To those who hold this view, the
ZENO pattern was an accidental artifact, devoid of real meaning because it did
not originate in a mind.
The occurrence of repeated patterns of regularity in a set of symbols has
traditionally been taken as evidence of a mind at work, and therefor the
probability that the symbols contain a message. This idea has been badly shaken
by recent developments in chaos theory. It appears that pattern and regularity
can originate spontaneously out of chaotic data.
Last summer in
As might be expected, the immediate result was what we recognize as
"snow" on our television screens. But after about two hours a
completely unexpected thing happened. The snow cleared and a pattern appeared.
The pattern was quite complicated. If anything, it resembled an architect's
rendering for some futuristic city plan, with roads, plazas, and strange
looking buildings. After a few minutes this pattern disappeared and the screen
was again covered with snow. But in another hour another, different but equally
complicated pattern appeared.
There is something weird and disturbing about this sort of thing. It is hard to
believe that there is not some kind of meaning in these patterns, but what is
it and where does it come from? Is someone or something trying to communicate
with us? It is all a mystery, but the similarity to Achilles' pebbles and the
Voynich Manuscript seem obvious.
However, all of these difficulties with meaning are really beside the point.
The problem of Achilles' ability to recognize meaningful patterns was a
smokescreen thrown out by the Tortoise. Despite the ZENO throw, the probability
of throwing any such pattern again is so vanishingly small that Achilles was in
no danger whatever of losing his sword and spear.
The Tortoise pointed out that each pattern of the pebbles is equally likely or
unlikely. This is true but not relevant to the probability of throwing a
meaningful pattern. That probability is obtained by comparing the size of the
population of meaningful patterns with the total number of all possible
patterns, which we have seen to be 3.998 x 10 to the 49th power.
There are approximately 4000 spoken languages in the world, of which about 500
use some form of written communication. Two of these, Chinese and Japanese, use
a logographic system in which a single symbol stands for a word or part of a
word. There are about 7000 such symbols in the two languages that would be
recognized by an educated user.
The rest of the languages use alphabets of one kind or another. Using twenty
pebbles, it is difficult to form a word of more than five letters. English has
a very rich vocabulary and an educated user can recognize as many as 150,000
words. I have drawn a random sample of fifty pages from the American Heritage
Dictionary and counted the number of letters in each word. From this I estimate
that about 13% of the words in the language have five letters or less. On the
very generous assumption that we can use the same statistics on the other
languages, we come up with a total of 9,750,000 recognizable words.
To these words we should probably add about 1,500 specialized symbols used in
such professions as mathematics, astrology, pharmacy, and medicine. Adding it
all up indicates that there are probably no more than ten million patterns that
the pebbles could have fallen in that would be recognized as meaningful by
persons acquainted with all of the languages and symbol systems of the world.
Dividing this number by the total number of possible patterns shows that
Achilles is much more than 99.999999% sure of keeping his sword and spear. In
fact, the real probability is so large that, even if my estimates are wrong by
several orders of magnitude, it would not seriously change this conclusion.
It should be evident by now that probability is entangled with problems of
meaning and causality. The latter is, of course, a notoriously sticky part of
philosophy. Its intersection with probability opens up interesting problems and
many opportunities for error. One of these is in the interpretation of chains
of contingent causes.
A basic of probability theory is that the chance of a number of successful
occurrences is the product of their individual probabilities. Thus, the chance
of rolling an ace with a die is 1/6, but the chance of rolling two successive
aces is 1/36. The odds increase very rapidly as the sequence is extended. The
probability of rolling five successive aces is one out of 7,776.
No matter what "it" is that we are considering, its present existence
is the result of a chain of linked causal events going back far into the past,
even to the creation of the universe. Many of these events must have been
fortuitous, and some of them very unlikely. From this we might argue that the
existence of anything is extremely improbable. Consider, for example, the
improbability of my grandchildren. Start with events that occurred south of
Manila in 1899.
The Philippine Insurrection had begun with sporadic engagements on the
outskirts of the city. Two weeks later the American forces, led at this point
by the 22nd Kansas Volunteers, had forced Aguinaldo and his small band to fall
back into the rugged country near the small village of Calumpit. As they
advanced cautiously on a road that was little more than a trail, a well hidden
rifleman aimed his Mauser at one of the soldiers. He aimed for the head, but
the round struck the American on the neck.
That soldier was my father. When the field dressing was taken off the wound at
the hospital in Manila the surgeon said, "Young man, you have no right to
be alive." The bullet had missed the carotid artery by less than one
quarter inch. By that margin, the line of descent that led eventually to my
grandchildren had been preserved.
Invalided back to his home in the states, my father received a present from
relatives in New York. The package had been wrapped in a newspaper that
contained a picture of an attractive young woman. She had an unusual name and
was reported to live in a very small Hudson Valley town. On an impulse, my
father wrote her a letter which she received. This began a correspondence that
ended with their marriage. That young woman was my mother.
What was the probability that my father would see the picture and be impressed?
That he would take the trouble to write? That the letter would be received?
That it would be answered? That it would lead to marriage? Nevertheless, it was
this extremely unlikely conjunction of events that preserved the line of
descent that led eventually to my grandchildren.
Come now to 1939. I was walking the streets of New York, broke and out of a
job. I had decided to ride the freights out to the Northwest where I had heard
that they were hiring men to work on the Grand Coulee Dam. I met a young woman
that I knew who worked in the personnel department at Macy's. She told me that
the partner of a Chicago management consulting firm was in New York looking for
bright young men as junior staff trainees. I met him and was hired in spite of
having no visible qualifications. When I entered the Chicago office of the
consulting company, the receptionist was a strikingly beautiful young woman. A
year later, that young woman became my wife. Again, through an improbable
sequence of events, the line of descent to my grandchildren was preserved.
But why stop with this relatively short time frame? Behind each individual
stands the entire history of human matings going back to our emergence as a
species. Many of these matings must have occurred as a result of improbable
events, and each carried the possibility of changing the line of descent in
ways that would eliminate me and my family. And why stop with the human species?
In a recent book, Wonderful Life, Stephen Jay Gould points out that we are the
result of a series of extremely improbable evolutionary accidents. Time after
time in the course of evolution a precursor species to the human line survived,
not because it was "the fittest", but simply because that is the way
things happened.
And finally, what was the probability that any kind of life as we know it would
evolve on one of the nine planets circling a small sun at the edge of a minor
galaxy? Carbon based life in an aqueous environment is only possible within a
narrow range of temperatures and severely limited radiation exposure. No other
planet in our solar system meets these requirements, nor as far as we know does
any other planet anywhere, whether in our galaxy or any other. Life seems to be
an extremely improbable thing.
Let us now apply our multiplication rule to the probability that my
grandchildren really exist. Take the probability that life would evolve on our
earth, that it would evolve into the human species, that each of the millions
of individual matings that would lead to my family line would actually occur.
Multiply each of these small fractions together and the result is evident. The
probability has become so vanishingly small that my grandchildren do not exist.
Yet they do. Something is evidently wrong here. What is it?
The difficulty lies in a fallacy that has led to the ruin of many a gambler. In
every casino there are people who meticulously chart the frequency with which a
particular result occurs. For example, they may note that black has come up
twelve times in succession at the roulette table. They believe that something
called the "law of averages" means that red is now much more likely.
However, the roulette wheel has no memory. It is true that a run of twelve
blacks is very unlikely, but once it has occurred, the probability of red on
the next turn is exactly what it always was, namely ½, minus the house
percentage.
The world is what it is, regardless of the prior hazards and improbabilities
that have led to its present state. My father might never have written that
fateful letter, our primordial ancestor might never have crawled out of the
oozy swamp. But they did, and in the end I was left walking down that street in
New York. At this point the probability question should have been put in its
proper form. "What is the chance that this healthy and ambitious young man
will marry, and in the end have grandchildren?" It doesn't take any fancy
mathematics to see that this probability was very high.
Having saved my grandchildren from annihilation by logical fallacy, we need to
deal with the final slight of hand that The Tortoise attempted to pull on
Achilles. You will remember that after the latter had thrown the pebbles, The
Tortoise suggested that he could radically change the probability of what had
happened by devising a new symbol system that would give meaning to the
pattern. This is, of course, strictly illegal. Whatever the likelihood of an
event, we cannot change it by redefining the event after the fact.
Is your head beginning to ache? I suggest that we quit all this
intellectualizing, and go look for mermaids. They are very improbable, but I am
told that they are sexy and not too unapproachable.
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