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Achilles in the Quantum Universe
The Definitive History of Infinity
By Richard Morris Henry Holt and Company
Copyright © 1997 Richard Morris
All rights reserved.
ISBN: 978-1-62779-750-4
CHAPTER 1
THE PARADOXICAL NATURE OF INFINITY
A BASEBALL PLAYER DIES and goes to heaven. He likes nothing better than playing baseball, so God decrees that he shall be allowed to play every day for all eternity. In other words, he will be able to play an infinite number of games. Furthermore, it is ordained that every tenth hit shall be a home run. "George Herman," says God (who likes to address people by their first and middle names), "you will have an infinite number of hits and an infinite number of homers."
God doesn't mean that this player can expect to hit a home run after every set of nine singles, doubles, and triples. That would take all the excitement away. George Herman will sometimes get twenty or even thirty hits before he slams one out of the park. On other occasions, he will homer on consecutive times at bat. In the long run, as he accumulates hits, the ratio will get closer and closer to an exact one in ten.
Now, an infinite number of games is quite a lot. It is obvious that they are going to require an infinite supply of baseballs. Since God doesn't want to be bothered with having to constantly perform miracles to create them, He gives each team a somewhat nondescript wooden bin containing an infinite number of balls. Obviously, the supply can never be exhausted. However many times you subtract one from infinity, an infinite number remains.
George Herman is a very good hitter. But one of the pitchers on the team does not hit too well at all. In fact, it has been ordained that although he will get a great number of batters to strike out, he himself will finish the infinitely long season with a batting average of zero. One day, to everyone's surprise, the pitcher gets two hits. At first, no one can understand how this is possible. After all, it is inconceivable that God would lie to His players. However, one of the other players, a man named Yogi who is something of a philosopher, soon hits upon a solution. If the pitcher only gets a finite number of hits in an infinite series of games, he will indeed wind up with a .000 batting average. If you divide any number by infinity, Yogi says, the result is always the same. Two divided by infinity is zero. If you divide two thousand, or two million, or even two trillion by infinity, the result is still zero. "It ain't over till it's over," he adds. Not all of the players are convinced by this analysis. "How can you divide a number by infinity?" one mutters.
Naturally, George Herman's team plays half its games on the road. And of course when the players travel, they must have a hotel to stay in. They never bother to make reservations in advance — in heaven, all the hotels have an infinite number of rooms.
But one day the team arrives in another celestial town and discovers that the hotel is full. There are already an infinite number of guests. At first it appears it will be necessary to seek other accommodations, but then the archangel who manages the hotel quickly assures the team that this will be unnecessary. Yes, the hotel is full, yet there will be no problem assigning each player a room. The archangel asks how many rooms he must provide and is told that, counting players, coaches, and so on, a total of forty is required. The manager then does a little juggling. He moves the guest occupying room number 1 into number 41. He shifts the guest in number 2 into number 42, and so on. When he is finished, forty rooms are available. Furthermore, no one has been evicted. Everyone previously occupying a room has simply been moved into one with a higher number. The woman who was originally in room 41 is now in room 81, for example.
"That was easy," the archangel said to himself after each of the baseball players has been given a key to his room. "If I had to, I could have accommodated an infinite number of new guests." And indeed he could. He could have moved the guest in room 1 to room 2, while moving the guest in room 2 to room 4, the guest in room 3 to room 6, and so on. This would have emptied all the rooms with odd numbers — and of course there were an infinite number of them.
It appears that infinite numbers — if they can indeed be called numbers — are paradoxical entities. You might legitimately wonder if it is meaningful to imagine a baseball season of infinite length, or to talk of a hotel with an infinite number of rooms, as such things just don't exist in the real world. But before I attempt to address this concern, let's look at yet another paradoxical property of infinite numbers.
It can be easily shown that if George Herman gets one home run for every ten hits, then the number of home runs and the number of hits are equal. To show this, you need only to pair hits and home runs in a one-to-one manner. The first home run is matched with the first hit, the second home run is paired with the second hit, and the nine hundred and ninety-ninth home run corresponds to the nine hundred and ninety-ninth hit. For every hit, there is always a corresponding home run. Since the number of home runs is never exhausted, there are no hits "left over."
Some matters are easier to understand if expressed in a visual form. I will therefore give another example of this kind of pairing of sets of numbers, and show that the number of positive integers (the whole numbers 1, 2, 3, and so on) is equal to the number of integers that can be divided by two. I need only write down the positive numbers in one row, and the even numbers in the row below it, as follows:
[ILLUSTRATION OMITTED]
Here the double-edged arrows indicate that each number is paired with the one below it, and the three dots at the end of each row indicate that both series go on forever.
At first, this may look like a somewhat suspicious argument. But in fact the reasoning is perfectly valid. Pairing one collection of objects with another is the most fundamental form of counting. It is a procedure that can be employed even if one knows no arithmetic.
This can be illustrated by the following example: Imagine that a young girl wants to know if the number of cups and saucers that her mother keeps in the cupboard are equal. Although the little girl is precocious, she has not yet learned to count. However, this is no hindrance. She simply places each cup in a saucer. If there are no cups and no saucers left over when she finishes doing this, the numbers of each are the same.
Even though intuition would tell us that there are twice as many positive integers as there are even integers, we must conclude that the two infinities are equal. Indeed, it is possible to find even more extreme examples. In 1638, for instance, the great Italian scientist Galileo noticed that the number of positive integers was equal to the number of squares.
A square is a number formed by multiplying any number by itself. The first square is 1, the result obtained when one carries out the multiplication 1 × 1. The next squares are 4 (2 × 2), 9 (3 × 3), 16 (4 × 4), 25 (5 × 5), and 36 (6 × 6). They can be paired in the same manner that numbers were paired above:
[ILLUSTRATION OMITTED]
When Galileo obtained this result, he concluded that there was something very bizarre about infinite numbers, and that they were best avoided. Infinity, he said, was "inherently incomprehensible." He was not the first or only person to come to this conclusion. The fact that "unequal" infinite collections could be paired with one another in this way had been noticed in ancient times, and for a couple of centuries or so after Galileo's death, mathematicians generally denied that it was meaningful to speak of infinite numbers. They did make use of unending "infinite series" of numbers. A simple example of such a series would be the set 1, 2, 3, ... that we have already encountered. Another would be the series of fractions 1/2, 1/4, /. ... But here the basic idea was that such a series, like the Energizer bunny, kept going, and going, and going. Even in a steadily increasing series such as 1, 2, 3, ... you never actually got to infinity, only to progressively larger numbers.
TRANSFINITE NUMBERS
The concept of infinite numbers continued to baffle mathematicians until the latter part of the nineteenth century. Then, in a series of papers published between 1874 and 1884, the German mathematician George Cantor showed that infinity could indeed be treated in a mathematically rigorous way. He began by defining an infinite number to be one that could be put into a one-to-one correspondence with some part of itself. I have already given two examples of this by showing that the positive integers can be matched both with the even integers and with the set of square numbers. This gives us the not very surprising result that, according to Cantor's definition, the numbers 1, 2, 3, ... constitute an infinite collection.
But Cantor did much more than define infinity. He achieved a number of surprising results. For example, he was able to prove that the set of positive integers had the same number of members as the set of all proper and improper fractions. To do this, it was only necessary to set up the following one-to-one correspondence:
[ILLUSTRATION OMITTED]
Note that the lower series is ordered in such a manner that no fraction will be left out. Cantor begins by including all fractions in which the numerator and denominator add up to 2. There is exactly one of these, /. He then lists the fractions in which this sum is 3. This time there are two: / and 1/2. Next, there are three fractions for which the sum is four, four for which the sum is five, and so on.
Some of the results that Cantor obtained were surprising indeed. For example, in 1874 he set out to prove that the number of points on a line was less than the number of points on a plane or in a space of any number of dimensions. Instead he discovered a proof of the opposite. No matter how many dimensions there were, the number of points was always the same. "I see it, but I do not believe it," he said in a letter he wrote to the German mathematician Richard Dedekind in 1877.
But don't imagine that Cantor's work implied that all infinite numbers were equal. This was definitely not the case. For example, he was able to show that the positive integers 1, 2, 3, ... could not be put into one-to-one correspondence with the points on a line. This meant that the latter infinity had a greater magnitude. Even though both were infinite, the number of points on a line was larger than the number of positive integers.
Eventually, Cantor was able to show that there were many different infinite numbers — an infinite number of them. He assigned the symbol [??]0 to the smallest infinity, the one represented by the positive integers. Here is the first letter of the Hebrew alphabet, aleph, and [??]0 is said as "aleph- null." The next larger infinite number is [??]1, aleph-one, and is followed by an unending series of infinite numbers, all represented by the same Hebrew letter. Cantor called the alephs transfinite numbers, and they are still known by that name today.
As might be expected, Cantor's discoveries did not gain immediate acceptance among mathematicians. Many of them wanted to avoid the use of the concept of infinity altogether, and here was Cantor speaking of an infinite number of infinities. One of Cantor's former professors, the German mathematician Leopold Kronecker, was especially critical of his work. He attacked Cantor's ideas as being "mathematically insane," and later prevented his former student from obtaining a post at the University of Berlin. Another, even more eminent mathematician, the Frenchman Henri Poincaré, described Cantor's mathematical theory of infinity as something that later generations would regard "as a disease from which one has recovered."
Such attacks had an unfortunate emotional effect on Cantor. Somewhat paranoid to begin with, Cantor began imagining conspiracies. He refused to have anything to do with the only mathematical journal that had been receptive to his work, believing its editor was involved in a plot against him. He experienced a nervous breakdown in the spring of 1884. After recovering, he withdrew from mathematical work and began to publish essays in philosophical journals. During the latter part of his life, Cantor experienced severe depressions and several mental breakdowns. He was eventually relieved of his teaching duties at the University of Halle and ended his life in a mental hospital in 1918.
By this time, a younger generation of mathematicians and philosophers was beginning to understand the importance of the work Cantor had done. In 1926, the great German mathematician David Hilbert summed up the newfound regard for Cantor by saying, "No one shall expel us from the paradise that Cantor has created for us." But of course, by then Cantor had been dead for eight years.
ACHILLES AND THE TORTOISE
It is not my intention to discuss Cantor's theory of transfinite numbers in great detail. The theory is an example of what is called "pure" mathematics — that is, mathematics for its own sake — and has no applications in the natural sciences. Physics, for example, does not make use of transfinite numbers, nor does any other scientific field. My purpose in introducing the topic was simply to show that, however paradoxical the concept of "infinity" may seem, it is nevertheless one that can be put on a firm logical foundation. You cannot simply refuse to admit infinite numbers into mathematical discourse, as Leopold Kronecker wanted to do.
This makes the idea of infinity more, not less, baffling when it is met in a nonmathematical context. When infinite quantities are encountered, we cannot wish them away by saying that infinity is an illogical or self-contradictory concept. We can't dismiss infinite numbers as "inherently incomprehensible," as Galileo did. If we encounter some situation in which infinite quantities appear, it is necessary to examine matters carefully and to try to find a way to work with these numbers.
It is true that infinite quantities are not encountered in the everyday world. Nothing travels at an infinite velocity. There are not an infinite number of stars in the sky, or an infinite number of grains of sand on the beach. However, one encounters the concept of the infinite again and again in philosophy, in modern science, and occasionally in literature. In everyday speech, the word infinite is still often used as a synonym for "that which is beyond human comprehension." When the infinite is encountered in a scientific or philosophical context, however, it cannot be evaded that easily. Science and philosophy, after all, are attempts to understand the world.
One of the earliest and most famous uses of the idea of infinity is the paradox of Achilles and the Tortoise, which was conceived by the Greek philosopher Zeno of Elea sometime around the middle of the fifth century B.C. It can be stated as follows: Suppose that the swift Greek warrior Achilles is to run a race with a tortoise. Because the tortoise is much the slower of the two, he is allowed to begin at a point some distance ahead. But then, says Zeno, Achilles can never overtake his opponent. For to do so, he must first reach the point from which the tortoise began. By that time, the tortoise will have run to some point farther down the racecourse. And by the time Achilles reaches that point, the tortoise will have advanced farther yet. It is obvious, Zeno maintains, that the series is never-ending. There will always be some distance, however small, between the two contestants.
(Continues...)
Excerpted from Achilles in the Quantum Universe by Richard Morris. Copyright © 1997 Richard Morris. Excerpted by permission of Henry Holt and Company.
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