Reasoning With Mathematics

Lesson Two ­ But Is this Mathematics?

OK, you've been through some starting exercises for this course and maybe you have a question something like "Well all right, but what does this have to do with mathematics?"

The short answer is you used mathematics to answer the questions or solve the problems in the starting exercises. Longer, more elaborated answers will emerge as you go ahead in this lesson and following ones.

Numbers and More

Most of us think of mathematics as numbers only. But mathematics is not only numbers, important as they are. As a demonstration of this in a moment you'll go through a problem which is very much like one of the starter exercises ­ the one about knights and knaves. (If you didn't look at the knights and knaves starter, don't worry. You won't need it to follow the example.) Incidentally, a MindQuest student submitted this example to the Student Center's Puzzlers folder early in 1997. Notice that there are no numbers at all in the problem.

A missionary visits an island where two tribes live.

One tribe always tells the truth. The other always lies.

The truth tellers live on the western side of the island and

the liars live on the eastern side of the island.

The missionary's problem is to find someone who tells the truth by

asking one native only one thing.

The missionary, seeing a native walking in the distance, asks a nearby native: "Go ask that native in the distance which side of the island he lives on."

When the messenger returns he answers: "He says he lives on the

western side of the of the island."

Is the messenger a truth teller or a liar? How can you be sure?

One Approach to Answering the Questions

For the first question, you could take a guess. And you'd have a fair chance of getting it right because there are only two possibilities: the messenger is either a truth teller or a liar. But you can't very well guess an answer to the second question. So you have to figure it out; you have to reason it out.

Now, a few people will get the answer immediately and they might think the problem is so obvious it needs no explanation. If you're one of those people, please be patient as you go along one path of reasoning out the answer:

If the person in the distance is a truth-teller, then he'll answer "western side" when asked where he lives because truth-tellers live on the western side and they always tell the truth.

If the person in the distance is a liar, then he'll also answer "western side" because liars live on the eastern side and they always lie.

So no matter whether the other person is a truth-teller or a liar, he will tell the messenger "I live on the western side of the island."

If the messenger is a truth-teller, then he'll tell the missionary "He says he lives on the western side of the island" because that's what he heard and truth-tellers always tell the truth.

If the messenger is a liar, then he'll tell the missionary "He says he lives on the eastern side of the island" because the liars always lie, in this case about what the other person said.

So because the messenger reported "He says he lives on the western side of the island," the messenger is telling the truth.

And as you have probably figured out already, this does not tell anything about the person who was asked the question. That person could be either a truth-teller or a liar.

Is This Approach Mathematical?

You could say that this is a reading problem and be entirely correct. But it is also a mathematical puzzle in at least two ways. First, mathematics is very concerned with proving statements. So the second question of the problem, how can you be sure of your answer, is characteristic of mathematics. Getting an explanation that other people can follow and use is crucial. A second way that the approach is mathematical comes out of seeing and using patterns. In this problem what you're told in advance sets up the pattern: truth-tellers always tell the truth and live on the western side of the island, liars always lie and live on the eastern side. You use that pattern to answer the question and prove your answer.

A Specialist's Comment About Mathematics and Patterns

Ian Stewart wrote a short but challenging book for readers who are not mathematicians but who are interested in understanding more about mathematics. It's called Nature's Numbers, but in the first chapter Stewart writes that the book " ought to have been called Nature's Numbers and Shapes." That title, he suggests, would have said more about the ways in which patterns are so significant in mathematics.

Stewart also gives two reasons for using the title he did. One reason is non-mathematical. He and his editor decided the shorter title is snappier and more attractive to readers. The second reason is more specialized. Stewart says that mathematics can always express a shape in numbers, for instance as a computer handles graphics through coding different dots on a monitor to create a given shape.

Patterns, Patterns Everywhere

Maybe you've taken a test that included a question such as this: What is the next number in this series: 3, 6, 9, 12, ?

You can give the answer 15 if you recognize the pattern "increasing by three."

Well, how about this series: 3, 5, 8, 13, 21, 34, ? This one probably takes more time and thought, but you might notice that if you add 3 and 5 you get 8, if you add 5 and 8 you get 13, and if you add 8 and 13 you get 21. There's a definite pattern: each number is the sum of the two number before it in the series. This is called the Fibonacci series and it has helped scientists understand patterns in plant flowers. In Nature's Numbers you can read that lilies have three petals; buttercups have five, many delphiniums have eight; marigolds, thirteen; asters twenty-one; most daisies have thirty-four, fifty-five, or eighty-nine. All of these are Fibonacci numbers. [If you want more details about why these plant patterns might have developed, see Nature's Numbers, Chapter 9, especially pages 135 through 139.] By the way, this number series was invented about 1200 by Leonardo Fibonacci while working on a problem about rabbit populations, not plants.

Other patterns are not so numerical. A full moon rises on a regular schedule; stars move in a regular path across the night sky; white pine trees have five needles in a bunch, red pines have two; rainbows seen from the ground look like an arc, seen from an airplane look like a circle; comet tails point away from the sun; birds won't come to your feeder if a squirrel is there; your daughter wants a snack as soon as she comes home from school.

Examples seem endless.

Mathematics can help us recognize, understand, and use patterns. Ian Stewart put it this way in his book:

Human mind and culture have developed a formal system of thought for recognizing, classifying, and exploiting patterns. We call it mathematics. By using mathematics to organize and systematize our ideas about patterns, we have discovered a great secret: nature's patterns are not just there to be admired, they are vital clues to the rules that govern natural processes. [Nature's Numbers, pg. 1]

Two Examples of Using Patterns to Make Discoveries

Finding Neptune ­ Before the days of powerful telescopes people knew of six planets in our solar system, the ones that could be seen in those days. In 1781 a seventh planet, Uranus, was discovered. A lot of work, including mathematical work, went into understanding the movements of those seven planets ­ the patterns of their movements, that is. This work was tested by seeing how exactly the movements could be predicted, but the predictions for Uranus kept coming in slightly off. This continued until two scientists, Jean Leverrier who lived in France and John Adams who lived in England, both worked out the same idea: that there was an eighth planet, not yet seen from earth, affecting the movement of Uranus. Working from the pattern of Uranus's movements, they started calculating where this unseen planet would most likely be found.

Adams finished his calculations in 1845 but other English scientists were so skeptical that he didn't publish his work until the next year. Working independently, Leverrier finished his calculations and sent them to astronomers in Berlin who, on September 13, 1846 found the planet within one degree of where Leverrier figured it would be on that night. Eventually, the planet got the name Neptune.

Eliminating Smallpox ­ Smallpox used to be the most destructive of human diseases. It covered more territory, affected more people, and lasted longer than other afflictions such as cholera, yellow fever, or bubonic plague. It was often fatal and even when not it could cause blindness or leave gruesome scars on victims.

An English physician named Edward Jenner thought there might be a pattern in what he saw among some of his patients. Around 1796 Jenner noticed that young women who worked as dairy maids would become ill with cow fever, a mild disease similar to smallpox, but then would never get smallpox. He inoculated an eight-year-old boy with matter from cow fever pustules, and later injected smallpox matter into the boy. The boy did not come down with smallpox.

This led to smallpox vaccine which became used throughout the world. In the late 1970s the World Health Organization reported there were no cases of smallpox known anywhere.