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Mathematical Fallacies and Paradoxes
By Bryan Bunch Dover Publications, Inc.
Copyright © 1982 Bryan H. Bunch
All rights reserved.
ISBN: 978-0-486-13793-3
CHAPTER 1
Thinking Wrong about Easy Ideas
One should not forget that the functions, like all mathematical constructions, are only our own creations, and that when the definition with which one begins ceases to make sense, one should not ask, What is, but what is convenient to assume in order that it remains significant.
Karl Friedrich Gauss
Everybody makes mistakes. In particular, everybody makes mistakes in mathematics. "Everybody" includes mathematicians, even some of the greatest of all time.
If you add two numbers and get a wrong sum, the mistake is just a mistake. If the wrong answer results from an argument that seems to make it correct, the mistake is a fallacy. Sometimes students give explanations for mistakes that sound very logical. But the sum is still wrong.
Here is a very simple example of such a mistake. This one is made deliberately. You may have seen the following trick played on a child (or even on an unsuspecting adult). You go to the child and say that you can prove that he or she has 11 fingers. "How?" says the child, who knows perfectly well that almost everyone has 10 fingers. "By counting."
You proceed to count the fingers in the ordinary way, laying one of your fingers on each one of the child's in turn.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
"Oh," you say, "I must have made a mistake. Let me do it the other way." Starting with the little finger you count backwards:
10, 9, 8, 7, 6
You are now at the thumb. You stop and say, "And 5 more from the other hand, which we counted before, makes 11."
An incorrect result coupled with an apparently logical explanation of why the result is correct is a fallacy. (The word fallacy is also used to refer to incorrect beliefs in general, but in mathematics the incorrect chain of reasoning is essential to the situation.) In fact, the word in mathematical usage could also refer to a correct result obtained by incorrect reasoning, as, for example, the student who used cancellation for reducing fractions as follows:
16/64 = 1/4
and
19/95 = 1/5
The student got the correct result in these cases, but the method has no logical basis and generally would fail.
Sometimes the mistakes in reasoning come because your experience with one situation causes you to assume that the same reasoning will hold true in a related but different situation. This mistake can happen at a very simple level or at a more complex one. At a simple level, the most common conclusion is that you know you have to reject the reasoning, although it may be difficult to say why. At the more complex level, you may conclude that the reasoning must be accepted even when the results seem to contradict your notion of how the real world works.
No one is quite sure why reasoning and mathematics so often seem to explain the real world. Experience has shown, however, that when the results of reasoning and mathematics conflict with experience in the real world, there is probably a fallacy of some sort involved. As long as you do not know what the fallacy is, the situation is a paradox. In some cases, as you will see, the paradox is entirely within mathematics. In others, it is in language or in events in the real world (with associated reasoning). For most paradoxes that are within mathematics, elimination of the fallacious reasoning produces a "purified" mathematics that is a better description of the real world than the "impure" mathematics was. So be it.
In looking at wrong thinking about easy ideas, you will find some cases in which reasoning about the real world is wrong because of a lack of experience with parts of the real world. In other cases, reasoning about mathematics is wrong because certain operations must be ruled out of mathematics. There are also situations in which reasoning is not the right approach to take to the real world.
SEEING IS BELIEVING
You think that the world you see before you is reasonable. Reason can play peculiar tricks on you, though. If you use the right reasons, you may get the wrong answers, at least from the point of view of common sense (or of your uncommon senses). Consider the following argument.
Which has the longer outside rim, a dime or a half-dollar? Going by the judgment of the eye and common sense alone, most persons would answer that the rim of the half- dollar is considerably longer than that of a dime. There is a reason, however, to believe that these lengths are the same.
Probably you have been taught that there is a relation between the distance across a circle and the distance around it. This relation is usually expressed as
C = πd
where C represents the number of units in the circumference, or distance around the circle; d represents the number of units in the diameter, or distance across the circle; and the Greek letter pi, pi, stands for a mysterious number slightly larger than 3. How did mathematicians come to this strange conclusion? After all, it is very hard to measure a curve, such as a circle, with any accuracy.
If this theory about the connection between the distance across a circle and the distance around it is true, then the length of the rim of a half-dollar is actually greater than the circumference of a dime, for the diameter of a half-dollar, which is easy to measure because it is straight, is longer than the diameter of a dime. You should not, however, accept a theory just because it was told to you in school. A simple experiment can settle the matter. In fact, if you just visualize the following experiment, you can probably see the answer to the question.
Suppose you bore a small hole through the center of each coin and then put an axle through the holes. The coins should touch, as shown in Fig. 1-1.
Furthermore, fix the two together so that one will not slip when the other is turned. As you roll the larger coin along a stretch of level tabletop, the small one also turns. If you mark a point on the rim of the half-dollar and, just above that point, another on the rim of the dime, you can keep track of how far you have rolled the coins. Mark the point on the half dollar A and the point above it B.
Start with the point A touching the table and roll the half-dollar along until A touches the table again. Naturally, the distance between the first and second places where A touched the table is equivalent to the distance around the rim, if you were careful not to let the half-dollar slip. Since one rotation of the dime would bring B around to the starting point again, measuring off the circumference of the dime, you can find whether or not the dime's circumference is greater than, equal to, or less than the length of the rim of the half-dollar. You can see the result in Fig. 1-2 without actually performing the experiment.
There is no question that the length labeled C is the circumference of the half-dollar. The dime has also rotated once only, and it laid off its circumference, which is also obviously equal to C. Contrary to what you were taught in school, it seems that two circles with different diameters have the same circumference. In fact, it is easy to see that the same method could be used to prove any pair of circles have the same circumference; hence, all circles are the same distance around.
This startling discovery was made by Aristotle more than 2,000 years ago. He did not believe it any more than you do. There must be something wrong with this reasoning (and, indeed, there is).
For one thing, if the experiment is. repeated along the edge of the table with the rolling done by the dime, the length of the "circumference of all circles" turns out to be different, as in Fig. 1-3.
If you have lost faith in rolling coins as a means of measuring the circumference of circles (which would not be surprising), do not bother to take the dime and half-dollar apart to test them along the same stretch of tabletop. Try wrapping a piece of string around the rim of each coin. When the string is unwrapped, it should equal the circumference. If you are quite careful, this method will yield a measure for the circumference that is just a little more than three times the diameter of each coin. Possibly the familiar formula is right after all.
Aristotle's proof, just given, is a classic example of a paradox. There are two conclusions. One can be obtained by direct measurement (with a string). The other involves reasoning. Both conclusions seem to be based upon fairly sound evidence. Most persons certainly have a hard time explaining why rolling the two coins with their centers fastened together does not provide the true answer. Although both conclusions are reasonable, they are in direct opposition.
I. The length of the circumference depends upon the diameter.
II. The length of the circumference does not depend upon the diameter.
Since it is reasonably certain that II is false, it is a fallacy. When one conclusion of a paradox is false, that conclusion is a fallacy. You cannot be certain that it is a fallacy, however, until you find what mistake was made in the reasoning process.
Exactly what are you measuring by the method of rolling coins? It appears that you are measuring circumferences, but this surely cannot be. If you were correctly measuring the distances around the coins, the result would not be a fallacy.
When you roll the half-dollar, the point A traces a peculiar curve before it comes to rest again on the table, as shown in Fig. 1-4.
This curve is called the cycloid. It has a number of interesting properties. For one thing, the area of a figure bounded by the cycloid and the line along which the coin rolls is exactly three times the area of the disk of the coin. Also, the cycloid in Fig. 1-5 is the curve along which an object would slide in the least time from one point to another point not directly below it.
(You may have thought that a straight line between S and F would be the fastest. In this case, common sense is in error. You can see, however, that sliding down a cycloid would produce the fastest acceleration at the beginning of the trip, thus raising the average velocity. Perhaps some enterprising toy manufacturer will someday make a slide for children in the shape of a cycloid.)
Since the straight line segment AA' between the starting and ending positions of the rolling coin is the shortest distance between the points, the cycloid must be longer. As the coin rotates, then, point A must be moving faster than the imaginary "point" where the coin touches the table. "Point" is in quotation marks because it is a different point (without quotation marks) at every moment of the coin's trip. Yet you can perceive this succession of points as a continuously existing entity --- a "point."
It may seem odd, but every point on the rim has a different speed than the "point" touching the table. This strange phenomenon can be accomplished because the speed at which point A moves forward is constantly varying. When A begins its trip on the back side of the coin, it is moving slower than the "point" touching the table; when it reaches the top of the coin, A is impelled by the rotation to move faster than the "point" on the table, and, after it has reached the last quarter of its journey, it begins to slow to the speed of the "point" on the table.
These differences are imparted by the rotation of the coin. You may think of A as a passenger in a bus. As the bus goes forward at a constant speed, A walks to the back (going slower than the bus), remembers to walk to the front to pay his fare (traveling faster than the bus), and walks back again (slower than the bus once more). Despite these complications, A and the bus reach the stop together.
In the connected pair of coins, point B also follows a peculiar path which is another kind of cycloid, as shown in Fig. 1-6.
The point B, like the point A, travels at different rates of speed during different parts of the journey. As you can see, the curve has apparently been pulled out of shape by something (in relation to the ordinary cycloid). That something is the forward motion of the coin as a whole. Notice particularly near the ends of the curve how the path has been flattened. Since B is moving slowest in this region, its motion is mainly created from the point being dragged along.
This is the solution, then, or the key to it. Although you are careful not to let the half- dollar slip on the tabletop, the "point" tracing the line segment at the foot of the dime is both rotating and slipping all the time. It is slipping with respect to the tabletop. Since the dime does not touch the tabletop, you do not notice the slipping. If you can roll the half-dollar along the table and at the same time roll the dime (or better yet the axle) along a block of wood, you can actually observe the slipping. If you have ever parked too close to the curb, you have noticed the screech made by your hubcap as it slips (and rolls) on the curb while your tire merely rolls on the pavement. The smaller the small circle relative to the large circle, the more the small one slips. Of course the center of the two circles does not rotate at all, so it slides the whole way. The center is the only point that travels the true circumference of the larger circle. To measure the circumference without taking rotation effects into account, you must measure from the point at the center of the circles before you start to roll the coins to where that point is when you stop. Since the "point" below the center on the tabletop must move at the same speed as the center, however, you are safe in using that "point" in marking off a line segment the length of a circumference. You cannot use the "point" below the center on the small circle to measure the circumference of the small circle, for that "point" also always travels the same distance as the center of both circles, no matter the size of the circle.
In considering the motion of a disk rolling down a hill, a scientist must take the different rates of speed of the various points into account. The point A moves in one continuously varying way, the point B in another, and the disk itself, considered as a whole, in still another. The problem is much more difficult than the consideration of an object sliding down a hill. The physicist uses the various relations discussed with regard to the circle paradox in studying the rolling disk.
When the dime and half-dollar are put together as suggested and rolled along the rim of the dime, the point A on the rim of the half-dollar must move much faster in relation to the "point" beneath the center. This is because it travels farther. Since the rotation speed is constant, point A is sometimes moving backward faster than it is going forward. The path is looped as a result of these short excursions into backward travel.
The same general situation occurs also with the rims of the flanges on a railroad train's wheel. The inner part rolls on the track, so the points on the rim describe loops as shown in Fig. 1-7. Even in the fastest train, there are always some points on the train that are moving backward.
The Aristotle circle paradox is connected to another paradox: two circles with the same diameter exist such that the circumference of one is twice that of the other. It is best appreciated if you perform it yourself as a parlor trick.
Take two Lincoln pennies. Place them flat on a table beside each other with the Lincoln heads up, as illustrated in Fig. 1-8.
What do you think will happen if you carefully roll the penny on the left along the half- circumference of the penny on the right, being careful not to let either penny slip? Try to predict the answer without actually performing the operation. Since the penny is rolling along half its circumference, Lincoln ought to turn upside down. (After all, this is what happens if you roll a penny along half its circumference on a tabletop.) Perhaps your experience with the Aristotle paradox will make you suspicious of this.
When the experiment is carefully performed, it appears that Lincoln is right side up at the end of a half turn. Sometimes a person viewing this experiment for the first time will try it over and over, insisting that the coins must be slipping.
One proposed explanation of this trick is that two circles may have the same diameter and different circumferences. The full circumference on one equals half the circumference of the other. Once again, the idea that the ratio of the diameter and circumference is constant is called in question.
(Continues...)
Excerpted from Mathematical Fallacies and Paradoxes by Bryan Bunch. Copyright © 1982 Bryan H. Bunch. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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