know as many of them as possible. Your bag of tricks is part of your mathematical toolbox. As Feynman says somewhere, don’t despise tricks; make them your own.
5. The following experience is very common among students of mathematics. You stew over a problem for hours or even days, unable to see how to do it. Then, later, you’re doing something else and not even thinking about mathematics. Suddenly you see the solution in your mind, and it’s obvious.
This happens to everyone, beginners and professionals alike. The time spent stewing over the problem was not really wasted. It helped you to find the “obvious” solution.
What is “obvious” depends on what you know. In mathematics, everything you already understand seems easy, and everything you don’t understand yet seems impossible.
A variant of this experience occurs as follows. A professional mathematician is reading a mathematical paper in a scholarly journal. One paragraph contains the sentence: “It is obvious that 0.”
The mathematician works for three days, and finally proves that x = 0.
“Oh, yes,” says the mathematician, without conscious irony. “That is obvious.”
You should not use the phrase “it is obvious” in your proofs, even though you may sometimes see this phrase in print.
6. Sometimes you, the student, may feel that everyone else in the class understands the mathematics in this course easily, and that you alone are confused. This impression is almost certainly false. In any case, if you are confused about some detail of a proof that is being presented, you should ask a question immediately. Since mathematics that one doesn’t understand seems just like meaningless gibberish, it’s always considered polite to ask a question during a mathematical lecture.
If a person at the blackboard keeps writing x where you think y should be, ask about this right away. A person writing on a black-board needs help from the audience to get the details right. And even if your intended correction is wrong, you need to know that it’s wrong in order to understand the rest of the argument.
Often the students who have the least confidence are actually the best students in verbal mathematics. So take heart when you feel that everyone else knows more than you do. It probably isn’t true.
7. When you do ask a question in class, you may be disconcerted to find that the professor does not understand what you are asking. This can be frustrating, but it is a perfectly natural occurrence. What we are studying in this class, even more than in verbal mathematics in general, lies right at the threshold of intelligibility. Until your question can be translated into formal mathematical language, its meaning is genuinely unclear. Pronouns such as “it” seem innocuous to the questioner but confuse the hearer. Be patient.
8. A professor who seems irritated by a question is not angry at the questioner. The professor does not understand the question, and is annoyed at not being able to figure it out. Be patient. Professional mathematicians have difficulties akin to those of beginners. Mathematics is not personal, and no one will ever be personally angry with you for asking a mathematical question. The gruffer-seeming mathematicians are not angry; they are just trying to think under pressure, concentrating on the mathematics and not on smiling reassuringly at the student. Gruff professors are much kinder than they seem, and often turn out to be good company.
9. Sometimes the professor makes mistakes, and sometimes mathematical texts contain mistakes. What’s great about mathematics is that it makes sense. If you and I make opposing mathematical claims, we can use mathematics to settle which of us (if either) is correct. Don’t believe a mathematical statement just because a book or a professor says that it’s true and claims to prove it. Is the proof correct? Is the statement true? Can you disprove the statement?
When a professor writing on the blackboard seems to be making a mistake, be sure to ask about it. Whether you’re wrong or right, you want to know the truth. One of the most refreshing features of mathematics is that mathematical disagreements are not personal. Any disagreements should be resolved as soon as possible.
10. No methods of thinking, figuring, or computation are beneath the professional mathematician. If we can add better by counting on our fingers, we will do so without shame. If we feel that a picture will help us think about a problem, we draw a picture. (If you would like a professor to draw a picture to illustrate a proof, just ask.)
Like a magician, a mathematician does plenty of work behind the scenes. When you write a proof for homework, you are performing a magic trick. Your first draft can be messy, with all sorts of false starts and side figuring. Your final draft contains nothing but the finished proof. Looking at your final draft, a reader would think that you had written the proof easily and smoothly, without effort, getting everything right the first time. But you, the magician, know better.
11. Finally, mathematics is one of the great pleasures of the human mind. Even calculating is fun for those who can do it well. The real fun in verbal mathematics comes in seeing something familiar in a new way. When you suddenly understand how an argument works, when you get the joke, that’s where the real fun lies.
Here in the U.S.A. at the start of the twenty-first century, most people are completely unacquainted with verbal mathematics. Thus the fun of verbal mathematics has become an unintentional secret. By learning the language of mathematics, you are putting yourself in a position to enjoy some little-known pleasures. Welcome to our secret society.
Credits
Grateful acknowledgment is made for permission to reprint excerpts from the following:
THE BEST OF MYLES by Flann O’Brien. Copyright © 1968. Reprinted by permission of The Dalkey Archive Press, Illinois State University.
THE BLUE FLOWER by Penelope Fitzgerald. Copyright © 1995 by Penelope Fitzgerald. Reprinted by permission of Houghton Mifflin Company. All rights reserved.
CHERRY by Mary Karr. Copyright © 2000 by Mary Karr. Reprinted by permission of Viking Penguin, a division of Penguin Group (USA) Inc.
JR by William Gaddis. Copyright © 1951 by William Gaddis. Reprinted by permission of the Wylie Agency.
“The Kid Who Learned About Math on the Street” from THE FOUR ELEMENTS by Roz Chast. Copyright © 1988 by Roz Chast. Reprinted by permission of the Wylie Agency.
LITTLEWOOD’S MISCELLANY by J. E. Littlewood. Edited by Béla Bollobás. Copyright © 1986 by B. Bollobás. Reprinted by permission of The Cambridge University Press.
MURPHY by Samuel Beckett. Copyright © 1938 by Samuel Beckett. Reprinted by permission of Grove/Atlantic Inc.
NIAGARA FALLS ALL OVER AGAIN by Elizabeth McCracken. Copyright © 2001 by Elizabeth McCracken. Reprinted by permission of Random House, Inc.
ON BEYOND ZEBRA! by Dr. Seuss. Copyright © and ™ by Dr. Seuss Enterprises, L.P., 1955, renewed 1983. Reprinted by permission of Random House Children’s Books, a division of Random House, Inc.
THE THIRD POLICEMAN by Flann O’Brien. Copyright © 1967. Reprinted by permission of The Dalkey Archive Press, Illinois State University.
Acknowledgments
We would like to thank Shwu-Yen T. Lin and You-Feng Lin for permission to use the first half of their book, Set Theory with Applications (1985, Book Publishers, Tampa, Florida) as the basic template for this book. We have taken our basic outline, many problems, and some examples and explanations from Lin and Lin.
The authors would also like thank the following people for their input and feedback: Adam, Cheri, Jo and Walter Babich, Vasily Cateforis, Orlando Canelones, Sandra Cox, Harold Ellingsen, Warren Hamill, Lynn Huang, Victoria Klawitter, Brook Landor, Sam Moczygemba, Vivek Narayanan, Jeff Reeder, Irene Schensted, Chris Shera, Kang-Ho Song, and Armand Spencer (author of Dr. Spencer’s Mantra).