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Paradoxes and Math   


The Liar Paradox

We are caught in the notorious liar paradox. Here's it's simplest form:

This sentence is false.

Is it true? If so, it's false. Is it false? If so, it's true! Contradictory statements like this are more common than you think. Other liar paradoxes:

   A button saying "Ban Buttons!"

   The graffiti sign saying "Down with graffiti!"

   A bumper sticker that says "Eliminate bumper stickers."

   A sign that says "Don't read this."

   A bachelor declares that the only kind of girl he would marry is one smart enough not to marry him.

   A man refuses to join any club willing to have him as a member.

   A little girl says she is glad she hates broccili because if she liked it she would eat lots of it, and she just can't stand the stuff.

The Crocodile and the Baby

Greek philosephers liked to tell about a crocodile that snatched a baby from its mother. Said the crocodile:

    "Will I eat your baby? Answer correctly and I'll give your baby back to you unharmed."

    "Oh! Oh! Your'e going to eat my baby."

    "Hmmm. What shall I do? If I give you back your baby, you will have spoken falsely. I should have eaten it."

    The crocodile has a problem. He has to both eat the baby and give it back at the same time.

    "Okay, so I won't give it back."

    "But you must. If you eat my baby, I spoke correctly and you have to give it back."

    The poor crocodile was so freaked out that it let the baby go. The mother grabbed her child and ran.

    "Zounds! If only she'd said I'd give the baby back. I'd have had a juicy meal."

    Let's examine this famous paradox more carefully to make sure you understand how clever the mother is. She said to the crocodile: "You are going to eat my baby." Whatever the crocodile does is sure to contradict his promise. If he gives the baby back, the mother spoke falsely, which entitals him to eat the baby. And if he eats it, the mother spoke truly, which forces him to return the baby unharmed. The crocodile is caught in a logic paradox from which he can't excape without contradicting himself.

    Suppose, instead, the mother had said: "You"re going to give the baby back." Now the crocodile can return the baby or eat it, in both cases without contradiction. If he gives it back, the mother spoke truly, and the crocodile has kept his word. On the other hand, if he is mean enough, he can eat the baby. This makes the mother's statement false, which frees the crocodile from the obligation to give the baby back.

The Unexpected Tiger

    "You're the king, father. Can I marry Michael?" "My dear, you may if Mike kills the tiger behind one of these five doors. Mike must open the doors in order, starting at one. He won't know what room the tiger's in until he opens the right door. It will be a unexpected tiger."

    When Mike saw the doors he said to himself: "If I open four empty rooms I'll know the tiger's in room 5. But the king said I wouldn't know in advance. So the tiger can't be in room 5. Five is out, so the tiger must in one of the other four rooms. What happens after I open three empty rooms? The tiger will have to be in room 4. But then it won't be unexpected. So 4's out too."

    By the same reasoning, Mike proved the tiger couldn't be in room 3, or 2, or 1. Mike was overjoyed. "There's no tiger behind any door. If there were, it wouldn't be unexpected, like the king promised. And the king always keeps kis word.

    Having proved there was no tiger, Mike boldly started to open the doors. To his surprise, the tiger leaped from room 2. It was completely unexpected. The king kept his word after all. So far logicians have been unable to agree on what is wrong with Mike's reasoning.

Mathematical Stories

The Parable of Π

    Many, many years ago, there was a big party being given for the numbers of the time. 1 was there in all its glory. 2 showed up with all the other even numbers in tow. And as many prime numbers as could be found had come. There were even some fractions like 1/2, 1/4 and 2/3. A few of the radicals had showed up like sqrt of 2 and sqrt of 7 who had just arrived off the sides of a right triangle with 3. But when π rolled in everyone asked, "Who invited you?". "What do you mean, 'Who invited me?'", asked π. "I'm a number." "Yes you are, but do you know your location on the number line?" "What about sqrt of 2?" asked π. "Thanks to Pythagoras and the use of a compass, I know exactly where I belong on the number line," replied sqrt of 2.

    π felt embarrassed and hurt, but said, "I'm a little after the number 3."

    "But exactly where?," they all chimed in.

    Since 1 was a factor of every number, 1 felt π's pain and said, "Let's give π a chance to describe itself."

    So π began to tell his story. "As you all know the Babylonians probably first discovered me Some ancient scribe had drawn circles with different sized radii. The scribe took the diameter (by doubling the radius) of each circle. And just for kicks, decided to wrap each circle's diameter around it. To his surprise, he found that regardless of the size of the circle, its diameter always wrapped around it 3 and a little bit. This was an exciting discovery. The news spread quickly all over the world from Egypt to Greece to China. Everywhere people were learning about me. Because of my special connection to circles they were now devising new ways to find the area and distance around a circle by using me in their calculations. People were anxious to find my exact value. No offense, but they knew I was no ordinary number, especially since they had never come across a number quite like me. They were not able to derive me from any of their regular algebraic equations, so later on they also labeled me transcendental. You would have thought that people would have given up on finding my exact number name. I'm satisfied with π. It suits me just fine. But no, you know how stubborn some mathematicians are, they wanted to be more and more precise. So for the centuries that followed to the present, new means and methods have been developed to get more accurate approximation.

    The famous mathematician Archimedes found me to be between 3 10/71 and 3 1/7. In the Bible I appear twice and my value is given as 3. Egyptian mathematicians used 3.16 for me. And Ptomely estimated me as 3.1416 in 150 A.D.

    Mathematicians know they will never get my exact amount, but they keep on drawing me out to more and more decimal places. You can't imagine how much a burden it is carrying around all those decimal places. Once calculus and computers are used, I'll be out to millions of places.

    They say I am essential to computing various things, such as volumes, areas, circumferences, and anything that deals with circles, cylinders, cones, and spheres, I also play a role in probability. And with my millions of decimal approximation, modern day computers will rely on me to put them through their paces and test them for accuracy and speed.

    "Say no more", shouted 1. 1 continued, "I'm sure we all agree that such a renowned number as π should be counted among us. After all, we know we each have our own point on the number line. No number can ever have another number's point. π has its point. It is not the most important thing about a number to know the exact location of its point."

    "Agreed," shouted 3, one of the mystical numbers. "I think π lends the party a bit of a mystery, variety and intrigue," said sqrt of 2. "Welcome", chimed the rest of the numbers. "Let's get our party under way. Let's start counting", said π.

The Story of the Quaternion

    As usual at the numbers convention, the cliques were already beginning to form. It was a shame that numbers could not think of themselves as a happy family.

    Over the centuries when just the counting numbers were around, the odd and even numbers would argue about which was more useful. But, they united forces when the integers entered the picture with their negative numbers.

    Now sides were already beginning to form over the big issue of the convention- who would accept the newcomer, the quaternion? The counting numbers had always been very elitist- only open to whole numbers greater than or equal to 1. They were dressed in their natural garb- all in increasing order with only one unit between consecutive numbers. They had to study the newcomer, and decide if it fell into its set. The integers were both hot and cold about the quaternion, while zero, being neutral, never took sides, since it was neither negative or positive.

    Surely the rationals would consider it more seriously. But the fractions, as usual, were more interested in displaying their numerators and denominators than to even talk to the decimals. Over the years, the decimals had gotten used to the fractions. They knew how much easier they were to operate with than fractions, especially on calculators. .007 even called the fractions passè. But 1/7 jumped up to say "Although one has to find common denominators to add or subtract us, and we require some fancy footwork for multiplication and division, and while we prefer to be in lowest terms- some of your decimal representations of rational numbers are way out there. In fact, some calculator's memories can't hold your decimal representation." Thus the rationals, which included the counting numbers, the integers, fractions, and decimals, continued to fight among themselves.

    Having seen the bickering that went on with the "rational" numbers, quaternion was understandably fearful of a group of the radicals by the snack bar that included sqrt of 2, sqrt of 3, sqrt of 15 and sqrt of 6. Quaternion had heard irrational they could get. But to its surprise, they were interested in talking. "So I hear that you have a number of parts and are what they call 4-dimensional," sqrt of 2 said. "Well, don't feel bad about it, I'm never ending non-repeating when in decimal form. All those digits are a drag to carry around so I prefer wearing my square root suit instead. Perhaps you too can find a more abbreviated way of expressing yourself."

    Quaternion was a bit encouraged and felt more relaxed. Not wanting to get quaternion's hopes up, sqrt of 3 added, "You'll need to be cleared with complex numbers set. It keeps track of all of us- the counting, integers, rationals, irrationals, reals and imaginarys."

    "But I heard the complex set has a split personality, and fluctuates between the real and imaginary numbers," said quaternion.

    Suddenly the complex number 3-5i walked over saying " You got that right, but the complex number plane gives everyone of us our own single point on which to reside. When worse comes to worst, I can always take refuge there. I know it's my very own point, no one else has that location, so there I can be alone, and regroup, relax, and meditate. We each have our own spot we can call home."

    You seem to have a multiple personality, what with your vectors and scalar," 3-5i said to quaternion. "I'm sure the complex plane has no place for you."

    "I hope I can find my own point and home," quaternion sid. With a sad note to its voice it continued, " One doesn't know which way to turn, or should I say which set to seek out."

    "It certainly is difficult," said a rather deep voice. Quaternion turned around and saw π. "It was very hard for me to be accepted by the real numbers. Although I'm irrational as sqrt of 2 and others, they would not let me into the reals right away, saying there was no way to find my exact location on the real number line, unlike the sqrt of 2, sqrt of 3, sqrt of 5,... who use the Pythagorean theorem to find their locations. So what was I to do? I had to do some fast talking, and the reals finally realized how important an irrational number I am, especially since all the circles depended on me for their circumference and area, and I am transcendental to boot." "Well, π, you speak as though you're the only transcendental number," said the number e, who was known to be a braggart. "I also happen to be transcendental, the base for the natural logarithms, and found extensively in nature besides being used in calculus."

    Quaternion was beginning to get a headache and was tired of all the teasing and bickering. "Perhaps I don't belong here," quaternion said. "Perhaps you don't," all the complex numbers shouted. "But where do you belong?" they taunted quaternion.

    "I'm different from all of you. I have more depth, more dimensions. Perhaps I belong to my own set. Yes, That's it! I am a member of the quaternions. The set of 4-dimensional numbers, since my general form is q=a+xi+yj+zk." Having said this, quaternion began to rise from the convention floor, and suddenly disappeared, as if it had gone to another dimension.

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