It is usual in U.S. mathematics departments to classify calculus as a freshman course and all proof classes as “advanced.” This is primarily because the U.S. mathematics curriculum is designed for engineers and scientists. Calculus is taught early so that the student may start learning physics as soon as possible. Since many engineering and science students, in contrast to “verbal” students, are more comfortable with computations than with language, proof classes are relegated to junior or senior year. Moreover, successful completion of a calculus class is often a prerequisite for entry into a proof class. This requiement ignores the needs of the verbal student, because the verbal student is assumed to be uninterested in mathematics.
The authors of this book have taught this introductory proof class several times. We have found that the students who do best in the course are those who are most sensitive to language.
We would like to see an end to “the two cultures.” We would like the language and conventions of mathematics to be part of the intellectual birthright of all students. We would like to stop reading in the newspaper that “Americans can’t do math.” And we would like to see a return to the notion that amateurs — novelists, diplomats, classicists, historians, lawyers, librarians and so forth — not only can enjoy mathematics, but can contribute significantly to it.
Mathematics belongs essentially not only to the numerical, but also to the verbal culture. Mathematics is free, and should belong to us all, not just to a privileged elite. We who speak the language of mathematics have no interest in being a privileged elite. One thing we like about mathematical notation is that it makes communication possible between mathematicians who have no other common language. Mathematical language is for letting people in, not for keeping people out. To learn mathematics is to love it. We would like more people to share this experience.
For this reason, the authors of this book do not assume that the reader has studied calculus. We think that there are many people (especially verbal people) who can enjoy calculus only after completing an introductory class like this one and a first course in real analysis. In the U.S. at the present time, such people generally stop studying mathematics either before or just after calculus, and never see a proof class at all. We hope that tins situation will change. Currently, it often turns out that only a few undergraduate mathematics students like or do well in proof classes. It may be that some of the best proof students — the verbal students — are inadvertently being excluded. It might even turn out that girls and Americans are good at mathematics after all.
Foreword C
Mathematical Proof as a Form of Writing
Lucidity is nine-tenths of style. Elisha said to the boys: If you do that again I will tell a big bear to come and eat you up. And he did. And it did. (It could do with the old tenth.)
J.E. Littlewood, Littlewood’s Miscellany
This book is about the experience of writing and reading mathematical proofs. A mathematical proof is a peculiar and amusing sort of written document. The language is formal. It sounds solemn and grandiose and absurd. It can also sound very elegant.
Of course, the way the proof looks or sounds is not the point. The point of the proof is the theorem it proves and the way it proves the theorem. The shape of the logical argument is what makes a proof elegant.
Mathematical writing tries to be as clear as possible. This is much harder than it might seem. Our language of sets, quantifiers, relations and functions is a great help in writing proofs and in understanding one another’s proofs.
Since the goal of a mathematical proof is to show as clearly as possible how a piece of deductive reasoning works, it is not a flaw in mathematical writing to use the same sentence structure several times in a row. Variation in language just for its own sake should, in general, be avoided, especially by beginners. (Of course, beginners love to play with language. This isn’t really a bad thing, and, even if it were, could not be helped. But be warned that some professors may find it more annoying than amusing when you give all your variables funny names and salt your proofs with such phrases as mutatis mutandis or per impossible.) The focus should be on the argument.
In some ways, mathematical writing is like poetry. A mathematician, like a poet, gets stuck and requires inspiration. Of course, it does no good to wait around for inspiation to strike; the only thing to do is to attempt to write the proof or the poem, or at least go through the motions. Here the mathematician may have the advantage over the poet that there are several known strategies to try. (But then, the poet also has a bag of tricks.) Often known strategies don’t seem to work. Then the mathematician or poet goes out for a walk (or even just as far as the next room) and an idea starts to form. The person has an idea, but doesn’t yet know what the idea is. The hope is to put the idea into language that will clearly reveal its lineaments to the writer as well as to readers.
In other ways, mathematical proofs are like plays. They are rather formal plays, in which each character must be introduced before it (mathematical objects seem genderless) can play its role in the drama.
As readers of fiction and poetry we often wish that writers in general knew more about mathematics. For us, mathematics is a part of life. But characters in books (especially female characters) usually seem untouched by it. As women, we are grieved when female writers we like seem to feel that mathematics is masculine, or boring, or ‘‘linear.” Mathematics is none of these things.
While we think it inadvisable for the beginner to write proofs in the style, say, of S.J. Perelman, we would like to have read Perelman’s mathematical pastiches, had he written any. We think that if more writers knew more verbal mathematics, some entertaining books might be written.
Writers of novels, poems, histories, and so on may find that they enjoy writing mathematical proofs. Writers may find that their mathematics and their non-mathematical writing enrich each other. But there are some people who hate to write, in some cases because the native language is not English and they make mistakes, forgetting articles, misspelling words, mixing up singular and plural. For these people, we have good news.
In many ways, mathematical English is much easier to write than conversational, journalistic, or literary English. Mathematical English uses a limited vocabulary. We introduce variables and write formally and re-dundantly. This means that a mathematician who knows no English may very well understand a talk in mathematical English, and even learn a few English phrases in the course of it. (It would be hard, for example, to avoid learning the phrase “such that.”) It is very common for people to write mathematical papers in a language (often English) that is not the own. Therefore, nobody cares very much about small grammatical solecisms such as misuse of articles or omission of plural endings. All anybody really cares about is understanding the proof. If the specifically mathematical parts of the language are correct, the proof will be understood.
Mathematical language is redundant. We keep reminding ourselves what sets our characters belong to. This is useful, because written mathematical documents nearly always contain errors. Mathematicians like to get things right, and try hard to produce error-free documents. However, this is nearly impossible. Despite our efforts, there will be a missing subscript, or “=“ will be printed instead of “≠.” The redundancy of mathematical language helps the reader to catch and correct errors.
Occasionally, in our experience, a student shows up in class with a laptop computer and proposes to take notes and do the homework on the computer. This turned out well only in one case, where, anomalously, the student was blind. As far as we know, anyone who is able to write (or print) by hand will be most successful writing proofs by hand. Probably one reason for this is that, when writing proofs by hand, we don’t just write but also
