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man but the sum of his thoughts?
- Mortimer Adler's
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Essays
Change:
Editor's
1-minute essay
The Greek Heraclitus claimed that all things change, that everything is in
continual flux. He is the one credited with the proverb, "No man can step into the
same river twice."
- But, among the ancients, only Zeno, a disciple of Parmenides,
denied that there is such a thing as change.
2500 years ago Zeno created an elaborate paradox to support his contention.
Amazingly, so skillfully constructed was his illusion, that, only in modern times, with a
better understanding of calculus, have mathematicians properly solved his riddle!
Let's allow Isaac Asimov to tell us about it:
- CLOSER AND CLOSER AND CLOSER . . .
Consider a series of fractions like this: 1/2, 1/4, 1/8, 1/16 ... and so on endlessly.
Notice that each fraction is one-half the size of the preceding fraction, since the
denominator doubles each time. (After all, if you take any of the fractions in the series,
say 1/128, and divide it by 2, that is the same as multiplying it by 1/2, and 1/128 x 1/2
= 256, the denominator doubling.)
Although the fractions get continually smaller, the series can be considered endless
because no matter how small the fractions get, it is always possible to multiply the
denominator by 2 and get a still smaller fraction and the next in the series. Furthermore,
the fractions never quite reach zero because the denominator can get larger endlessly and
it is only if an end could be reached (which it can't) that the fraction could reach zero.
The question is, What is the sum of all those fractions? It might seem that the sum of an
endless series of numbers must be endlessly large ("it stands to reason!') but let's
start adding, anyway.
First 1/2 plus 1/4 is 3/4. Add 1/8 and the sum is 7/8; add 1/16 and the sum is 15/16; add
1/32 and the sum is 31/32, and so on.
Notice that after the first two terms of the series are added, the sum is 3/4 which is
only 1/4 short of 1. Addition of the third term gives a sum that is only 1/8 short of 1.
The next term gives a sum that is only 1/16 short of 1 ... and so on.
-
- In other words, as you sum up more and more terms of that series of fractions, you
get closer and closer to 1, as close as you want, to within a millionth of one, a
trillionth of one, a trillionth of a trillionth of one. You get
closer and closer and closer and closer to 1, but you never quite reach 1.
Mathematicians express this by saying that the sum of the endless series of fractions 1/2,
1/4, 1/8 ... "approaches 1 as a limit."
This is an example of a "converging series," that is, a series with an endless
number of members but with a total sum that approaches an ordinary number (a
"finite" number) as a limit.
The Greeks discovered such converging series but were so impressed
with the endlessness of the terms of the series that they did not realize that the sum
might not be endless. -
- Consequently, a Greek named Zeno set up a number of problems called
"paradoxes" which seemed to disprove things that were obviously true. He
"disproved," for instance, that motion was possible. These paradoxes were famous
for thousands of years, but all vanished as soon as the truth about converging series was
realized.
Zeno's most famous paradox is called "Achilles and the Tortoise." Achilles was a
Homeric hero renowned for his swiftness, and a tortoise is an animal renowned for its
slowness. Nevertheless, Zeno set out to demonstrate that in a race in which the tortoise
is given a head start, Achilles could never overtake the tortoise.
Suppose, for instance, that Achilles can run ten times as fast as the tortoise and that
the tortoise is given a hundred-yard head start. In a few racing strides, Achilles wipes
out that hundred-yard handicap, but in that time, the tortoise, traveling at one-tenth
Achilles's speed (pretty darned fast for a tortoise), has moved on ten yards. Achilles
next makes up that ten yards, but in that time the tortoise has moved one yard further.
Achilles covers that one yard, and the tortoise has traveled an additional tenth of a
yard. Achilles...
But you see how it is. Achilles keeps advancing, but so does the tortoise, and Achilles
never catches up, Furthermore, since you could argue the same way, however small the
tortoise's head start---one foot or one inch--Achilles could never make up any head start,
however small. And this means that motion is impossible.
Of course, you know that Achilles could overtake the tortoise and motion is possible.
Zeno's "proof" is therefore a paradox.
Now, then, what's wrong with Zeno's proof? Let's see. Suppose Achilles could run ten yards
per second and the tortoise one yard per second. Achilles makes up the original
hundred-yard head start in 10 seconds during which time the tortoise travels ten yards.
Achilles makes up the ten yards in 1 second, during which time the tortoise travels one
yard. Achilles makes up the one yard in 0.1 second during which time the tortoise travels
a tenth of a yard.
In other words, the time taken for Achilles to cover each of the successive head starts of
the turtle forms a series that looks like this: 10, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001,
and so on.
How much time does it take for Achilles to make up all the head starts? Since there are an
endless number of terms in this Zeno series, Zeno assumed the total sum was infinite. He
did not realize that some series of endless numbers of terms "converge" and have
a finite sum.
For instance, the sum of the first two terms in the Zeno series above is 11; the sum of
the first three is 11.1; of the first four, 11.11; of the first five, 11.111 and so on. As
you see, if you add up all the endless series of terms, you get an endless decimal as the
sum: 11.111111111111111111 ... and so on forever.
But if you work out the decimal equivalent of the number 11 1/9, you find that it also is
the endlessly repeating decimal 11.111111111111111111111 ... and so on forever.
The sum of the Zeno series is therefore 11 1/9 seconds and that is the time in which
Achilles will overtake and pass the tortoise even though he has to work his way through an
endless series of continually smaller head starts that the tortoise maintains. He will
overtake the tortoise after all; motion is possible, and we can all relax.
If, after this, you still have trouble understanding Asimov's point, maybe this
will help.
Our common experience says that if we have, let's say, a pile of bricks, the pile
keeps getting larger and larger every time we add one more brick. So far so good.
But let's now reverse the process; let's start removing the bricks one-by-one. The
pile will get smaller and smaller -- but, of course, only to a point. Once the last brick
is removed, the pile is gone, and we've "hit the wall" -- the pile can't get any
smaller.
- And this is the slip in logic that stumped people for
2000 years!
Zeno was implying that if distances are continually halved, smallness continues to
regress indefinitely -- and Achilles never catches the tortoise. But, like the reducing
pile of bricks, diminution of distance has its limits. It's so simple -- once you see it.
Most thinkers, unlike Zeno, accept:
- that change is -- that it is evident; but what
change is, is neither evident nor easy to define.
That change to the ancients was a subject of great importance is evidenced by the
Greek word phusis from which "physics" comes to us. In its original
significance it refers to a sensible reality in motion and change -- the then-general view
of the nature of things!
Aristotle taught of different kinds of change:
(1) the change of local motion: spatial change
of a body from one place to another;
(2) the change of alteration: change in terms
of quality, for example, the ripening of an apple;
(3) the change of increase and decrease:
change in terms of quantity, for example, entities becoming larger or more numerous;
(4) the change of coming to be or passing away:
the change in living organisms: birth, growth, death.
Later, this view was challenged by Descartes, one who saw the world in terms of
only one kind of change, that of local motion.
This holding found its highest expression one hundred years later in Isaac
Newton's famous three laws of motion.
Newton seemed to be the final word on the matter as he conceived the universe as a
grand and complex machine, ticking away methodically and predictably. (So convincing was
this view that even theology was transformed by it -- giving rise to deism. God,
having created the world, was seen to have gone off somewhere, now allowing Newton's
sturdy laws of mechanics to do the rest.)
Classical mechanics ruled the intellectual scene for 200 years -- like the Roman
Empire at its zenith, Newton's clock-work universe seemed to be quite unassailable; that
is, until the coming of Einstein, Bohr, Heisenberg, and others. Their brave new scientific
world disallowed Newton's laws of motion -- at least, on the sub-atomic level.
Another most interesting issue related to change is that of time. Change, it is
said, exists only in a temporal world. Julian Barbour, British physicist, in his recent
book, The End of Time: The Next Revolution in Physics, posits that, along with
Parmenides of old, change is illusion -- but, to do this, he also suggests that time, too,
does not exist.
- A student of St. Augustine, concerned with issues of time and
creation, once asked of his master, "What did God do before he created the
world." Annoyed with this untidy query, the saint responded with: "He was
thinking up punishments in hell for fools who ask questions like that."
On this note, one encouraging a spirit of inquiry into the great idea of change,
and of all things, we close this introductory discussion.
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