thoughts and rejecting most of them. The computer is not so well set up for this. Also, it is by no means as easy to type mathematics as to type ordinary language. Furthermore, it can be better not to have your thinking tied to a machine that cannot be with you everywhere. It can be fun to carry an unsolved problem in your head. If you can work by hand on a scrap of paper, it’s easier to play with the problem in times of boredom or distress. Also, writing by hand seems somehow to make a stronger impression on the mind than typing.
Some students will find it hard to be patient enough to write things down. (This is particularly true of high school students, who like to rush through problems.) We would suggest that students cultivate a taste for the slow, deliberate writing of proofs. A person in too big a hurry will not be able to learn to prove theorems. It can actually be fun for a fast-paced person to slow down for a change and look at the scenery.
It may seem that, so long as the argument presented is correct, there is no difference between a badly-written proof and a well-written one. This is not true at all. A well-written proof is easy to follow. (This is an odd statement, because of course the argument presented may be inherently so difficult that no elucidation of it can really be “easy.” The often-stated ideal of mathematical lucidity is that any uneducated person who reads the proof should understand it immediately. This is an exaggeration, but it has a point.) A badly written proof creates unnecessary difficulties.
Beginning students often regard all their mathematical textbooks as badly written, because they find the proofs hard to read. Actually, some of these textbooks are written very well indeed, but there is an inherent difficulty in learning to use mathematical language. When you write proofs yourself, you will appreciate the difficulties of writing good mathematical English. The less mathematics your intended audience knows, the harder it is to write mathematics clearly. It’s easiest to write for experts.
For the student who hates writing but is truly interested in mathematics, we would suggest bearing in mind a remark attributed to Einstein, saying that he thought that his job at the patent office, which consisted of writing a short but complete and accurate description of each invention submitted for patenting, helped develop his mind in ways that were helpful for thinking of and writing down his physics. It’s not enough to think of something; you also need to be able to describe it clearly to other people. Thus, it is worth learning to write mathematically in order to present one’s own ideas in physics, engineering, or computer science. And it really doesn’t matter if you can’t spell.
In this course, the more “verbal” students will have an advantage. There are also people with deep mathematical insight who are “non-verbal.” These people will struggle, but will be helped greatly in communicating with others about their ideas. Writers and students of writing will find the precision of mathematical language useful and its peculiar syntax entertaining.
We hope that every student who uses this book will enjoy the experience, but perhaps this is not possible. Should you truly dislike it, we would suggest stopping, and finding another approach to the subject. Mathematics should bring pleasure, not pain.
Foreword D
Amazing Secrets of Professional Mathematicians REVEALED!!!
Throughout our years of compulsory education, we study a subject called mathematics. We learn to count and do calculations in arithmetic, algebra, geometry, and calculus. (If we’re lucky, we might get some proofs in geometry.) By the end of high school, students have some beliefs about mathematics.
Many high school students believe that talent in mathematics consists in quickness in calculating, preferably in one’s head. The smartest people, it is believed, never write anything down and never study. High school students also tend to believe that success in mathematics is a special gift, a savant’s magic facility for calculating, and that people who have this gift are essentially different from those who lack it.
In the world of verbal mathematics, these assumptions are stood on their heads. In verbal mathematics, in the realm of proof, we do not try to think as fast as possible. Instead, we take steps to down our thinking, so that we can more or less watch ourselves think. One way to do this is to write our thoughts down on paper. It is much easier to think mathematically if one writes things down.
When mathematicians speak of “reading” a mathematical book or paper, we do not mean “reading” in its ordinary sense. In fact, most mathematicians, when reading a mathematical document, actually copy out much of the text with pencil onto paper, word for word. As we do this, we frequently have to pause and prove to ourselves on paper that each claim made in the text is true. This is much harder work than the ordinary reading of a newspaper, textbook, or novel. It is akin to reading a novel in a foreign language which we have studied a little bit but don’t know very well.
Ordinary reading, in which one simply sits and stares at the page while mentally hearing the words, is woefully inadequate for learning mathematics. Reading while underlining or highlighting is likewise useless. Ordinary reading, with or without highlighting, simply does not make a strong enough impression on the mind. “Reading mathematics” requires more attention than ordinary reading. The act of writing focuses the mind’s attention. It helps to engage the part of the unconscious mind that is interested in mathematics.
Students who are new to verbal mathematics sometimes find it frustrating. Often such students are used to doing well in mathematics classes without effort. Suddenly effort is required, effort of an unaccustomed kind. And you, the good student who always makes 100, keep getting your homework back all marked up with corrections. It’s as if, suddenly, you aren’t good at math anymore.
Relax. Verbal mathematics is considerably harder than calculation. In this introductory course you will learn how to prove theorems using correct mathematical notation. To prove a theorem is to write a mathematical proof. A mathematical proof is written in the form of a paragraph or a few paragraphs. A proof is somewhat like a joke or a magic trick, in that it has a punchline. And, in general, a mathematical proof is written in formal mathematical language.
In this first course, you are learning the basic grammar of mathematical language. When you do your homework, you write proofs. Since you are writing essays in an unfamiliar foreign language, it’s not surprising that you make many mistakes. As with a foreign language, you must make mistakes in order to learn how to write mathematics properly. That you can’t instantly write the language perfectly does not mean that you are not good at math. Don’t be afraid to make some beginner’s mistakes. There’s no other way to learn a foreign language.
Unfamiliar notation seems repellent and hard to use, you say. Is it really necessary to use quantifiers? Yes, it is necessary. Experienced mathematicians do not enjoy learning unfamiliar notation any more than you do. And yet we must put up with learning unfamiliar notation all the time. We put up with this annoyance because it’s worth it to us. Unfamiliar notation is üke vocabulary in a foreign language: wonderful to know but a nuisance to learn.
In a way, nobody is good at verbal mathematics. People don’t learn to prove theorems instantly. It takes a few semesters of study to get good at it. Some people learn faster than others, but even the fastest don’t learn everything perfectly the first time. Most professional mathematicians are not lightning calculators or savants. Their minds are like yours. What makes them “good at mathematics” is a strong interest in mathematics and years of experience in working with it.
Here are a few secrets known to every professional mathematician and graduate student, and unknown to nearly every beginner in verbal mathematics. (That these are secrets is not intentional; it has simply never occurred to anyone to state these truths in print.)
1. You need to learn the definitions by heart. Memorize the definitions. Make
