高级数学入门：Chartrand等人的《数学证明》第三版解析

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"《数学证明：迈向高级数学的过渡》第三版是由 Gary Chartrand、Albert D. Polimeni 和 Ping Zhang 联合编著的一本数学教材，专为帮助读者从基础数学向高级数学领域过渡而设计。该书是系列'过渡到高级数学'的特色作品，旨在通过系统的证明方法来培养学生的逻辑推理和抽象思维能力。此版本为第三版，于2012年9月27日由 Pearson 出版社发行，共424页。该书作者 Gary Chartrand 是来自 Western Michigan University 的数学专家，而 Albert D. Polimeni 来自 State University of New York at Fredonia，一同与 Ping Zhang 合作，共同构建了这本书的核心内容。书中涵盖了丰富的数学理论，从基本的证明技巧到复杂的论证方法，为学生在进一步学习高等数学时打下坚实的基础。本书的结构严谨，适合各个层次的学生使用，无论是在大学的入门课程中还是在自我学习过程中。它不仅提供了大量的例题和习题，还注重引导读者掌握如何清晰地表达思想、构造证明以及分析数学命题的真伪。每一章都有精心设计的问题，帮助读者逐步提升对数学证明的理解和实践能力。此外，本书的编辑团队包括 Deirdre Lynch 作为主编，William Hoffman 担任高级收购编辑，Brandon Rawnsley 是助理编辑，Jeff Weidenaar 负责执行营销管理，Caitlin Crain 作为市场助理，以及 Beth Houston 管理生产项目。封面设计由 Suzanne Behnke 完成，艺术元素则来自 Shutterstock.com。版权信息显示，本书享有 ©2013, 2008 的版权，并且书中引用和参考了其他来源的材料，在文本中得到适当的授权和致谢。全书由 Aptara, Inc. 提供全方位项目管理，印刷和装订工作分别由 Courier Westford 和 Lehigh/Phoenix 完成。《数学证明：迈向高级数学的过渡》第三版是一本不可或缺的教育资源，对于那些希望深入探索数学逻辑和证明方法的学生来说，它既实用又系统，能够有效地引导他们在学术道路上取得进步。"

Communicating Mathematics

uite likely, the mathematics you have already encountered consists of doing prob-

lems using a speciﬁc approach or procedure. These may include solving equations

in algebra, simplifying algebraic expressions, verifying trigonometric identities, using

certain rules to ﬁnd and simplify the derivatives of functions and setting up and evaluat-

ing a deﬁnite integral that give the area of a region or the volume of a solid. Accomplishing

all of these is often a matter of practice.

Many of the methods that one uses to solve problems in mathematics are based on

results in mathematics that were discovered by people and shown to be true. This kind

of mathematics may very well be new to you and, as with anything that’s new, there are

things to be learned. But learning something new can be (in fact should be) fun. There

are several steps involved here. The ﬁrst step is discovering something in mathematics

that we believe to be true. How does one discover new mathematics? This usually comes

about by considering examples and observing that a pattern seems to be occurring with

the examples. This may lead to a guess on our part as to what appears to be happening.

We then have to convince ourselves that our guess is correct. In mathematics this involves

constructing a proof of what we believe to be true is, in fact, true. But this is not enough.

We need to convince others that we are right. So we need to write a proof that is written

so clearly and so logically that people who know the methods of mathematics will be

convinced. Where mathematics differs from all other scholarly ﬁelds is that once a proof

has been given of a certain mathematical statement, there is no longer any doubt. This

statement is true. Period. There is no other alternative.

Our main emphasis here will be in learning how to construct mathematical proofs

and learning to write the proof in such a manner that it will be clear to and understood

by others. Even though learning to guess new mathematics is important and can be fun,

we will spend only a little time on this as it often requires an understanding of more

mathematics than can be discussed at this time. But why would we want to discover new

mathematics? While one possible answer is that it comes from the curiosity that most

mathematicians possess, a more common explanation is that we have a problem to solve

that requires knowing that some mathematical statement is true.

2 Chapter 0 Communicating Mathematics

Learning Mathematics

One of the major goals of this book is to assist you as you progress from an individual

who uses mathematics to an individual who understands mathematics. Perhaps this will

mark the beginning of you becoming someone who actually develops mathematics of

your own. This is an attainable goal if you have the desire.

The fact that you’ve gone this far in your study of mathematics suggests that you

have ability in mathematics. This is a real opportunity for you. Much of the mathematics

that you will encounter in the future is based on what you are about to learn here. The

better you learn the material and the mathematical thought process now, the more you

will understand later. To be sure, any area of study is considerably more enjoyable when

you understand it. But getting to that point will require effort on your part.

There are probably as many excuses for doing poorly in mathematics as there are

strategies for doing well in mathematics. We have all heard students say (sometimes,

remarkably, even with pride) that they are not good at mathematics. That’s only an alibi.

Mathematics can be learned like any other subject. Even some students who have done

well in mathematics say that they are not good with proofs. This, too, is unacceptable.

What is required is determination and effort. To have done well on an exam with little

or no studying is nothing to be proud of. Conﬁdence based on being well-prepared is

good, however.

Here is some advice that has worked for several students. First, it is important to

understand what goes on in class each day. This means being present and being prepared

for every class. After each class, recopy any lecture notes. When recopying the notes,

express sentences in your own words and add details so that everything is as clear as

possible. If you run into snags (and you will), talk them over with a classmate or your

instructor. In fact, it’s a good idea (at least in our opinion) to have someone with whom

to discuss the material on a regular basis. Not only does it often clarify ideas, it gets you

into the habit of using correct terminology and notation.

In addition to learning mathematics from your instructor, solidifying your under-

standing by careful note-taking and by talking with classmates, your text is (or at least

should be) an excellent source as well. Read your text carefully with pen (or pencil)

and paper in hand. Make a serious effort to do every homework problem assigned and,

eventually, be certain that you know how to solve them. If there are exercises in the text

that have not been assigned, you might even try to solve these as well. Another good idea

is to try to create your own problems. In fact, when studying for an exam, try creating

your own exam. If you start doing this for all of your classes, you might be surprised at

how good you become. Creativity is a major part of mathematics. Discovering mathe-

matics not only contributes to your understanding of the subject but has the potential to

contribute to mathematics itself. Creativity can come in all forms. The following quote

is due to the well-known writer J. K. Rowling (author of the Harry Potter novels).

Sometimes ideas just come to me. Other times I have to sweat and almost bleed

to make ideas come. It’s a mysterious process, but I hope I never ﬁnd out exactly

how it works.

Learning Mathematics 3

In the book Defying Gravity: The Creative Career of Stephen Schwartz from

Godspell to Wicked, the author Carol de Giere writes a biography of Stephen Schwartz,

one of the most successful composer-lyricists, in which she discusses not only creativ-

ity in music but how an idea can lead to something special and interesting and how

creative people may have to deal with disappointment. Indeed, de Giere dedicates her

book to the creative spirit within each of us. While he wrote the music for such famous

shows as Godspell and Wicked, Schwartz discusses creativity head-on in the song “The

Spark of Creation” he wrote for the musical Children of Eden. In her book, de Giere

writes:

In many ways, this song expresses the theme of Stephen Schwartz’s life—the

naturalness and importance of the creative urge within us. At the same time he

created an anthem for artists.

In mathematics our goal is to seek the truth. Finding answers to mathematical

questions is important, but we cannot be satisﬁed with this alone. We must be

certain that we are right and that our explanation for why we believe we are correct

is convincing to others. The reasoning we use as we proceed from what we know

to what we wish to show must be logical. It must make sense to others, not just to

ourselves.

There is joint responsibility here. As writers, it is our responsibility to give an

accurate clear argument with enough details provided to allow the reader to understand

what we have written and to be convinced. It is the reader’s responsibility to know the

basics of logic and to study the concepts involved so that a well-presented argument

will be understood. Consequently, in mathematics writing is important, very important.

Is it really important to write mathematics well? After all, isn’t mathematics mainly

equations and symbols? Not at all. It is not only important to write mathematics well,

it is important to write well. You will be writing the rest of your life, at least reports,

letters and e-mail. Many people who never meet you will know you only by what you

write and how you write.

Mathematics is a sufﬁciently complicated subject that we don’t need vague, hazy

and boring writing to add to it. A teacher has a very positive impression of a student

who hands in well-written and well-organized assignments and examinations. You want

people to enjoy reading what you’ve written. It is important to have a good reputation

as a writer. It’s part of being an educated person. Especially with the large number of

e-mail letters that so many of us write, it has become commonplace for writing to be

more casual. Although all people would probably subscribe to this (since it is more

efﬁcient), we should know how to write well formally and professionally when the

situation requires it.

You might think that considering how long you’ve been writing and that you’re set

in your ways, it will be very difﬁcult to improve your writing. Not really. If you want

to improve, you can and will. Even if you are a good writer, your writing can always be

improved. Ordinarily, people don’t think much about their writing. Often just thinking

about your writing is the ﬁrst step to writing better.

4 Chapter 0 Communicating Mathematics

What Others Have Said about Writing

Many people who are well known in their areas of expertise have expressed their thoughts

about writing. Here are quotes by some of these individuals.

Anything that helps communication is good. Anything that hurts it is bad.

I like words more than numbers, and I always did—conceptual more than com-

putational.

Paul Halmos, mathematician

Writing is easy. All you have to do is cross out all the wrong words.

Mark Twain, author (The Adventures of Huckleberry Finn)

You don’t write because you want to say something; you write because you’ve

got something to say.

F. Scott Fitzgerald, author (The Great Gatsby)

Writing comes more easily if you have something to say.

Scholem Asch, author

Either write something worth reading or do something worth writing.

Benjamin Franklin, statesman, writer, inventor

What is written without effort is in general read without pleasure.

Samuel Johnson, writer

Easy reading is damned hard writing.

Nathaniel Hawthorne, novelist (The Scarlet Letter)

Everything that is written merely to please the author is worthless.

The last thing one knows when writing a book is what to put ﬁrst.

I have made this letter longer because I lack the time to make it short.

Blaise Pascal, mathematician and physicist

The best way to become acquainted with a subject is to write a book about it.

Benjamin Disraeli, prime minister of England

In a very real sense, the writer writes in order to teach himself, to understand him-

self, to satisfy himself; the publishing of his ideas, though it brings gratiﬁcation,

is a curious anticlimax.

Alfred Kazin, literary critic

The skill of writing is to create a context in which other people can think.

Edwin Schlossberg, exhibit designer

A writer needs three things, experience, observation, and imagination, any two

of which, at times any one of which, can supply the lack of the other.

William Faulkner, writer (The Sound and the Fury)

If confusion runs rampant in the passage just read,

It may very well be that too much has been said.

Mathematical Writing 5

So that’s what he meant! Then why didn’t he say so?

Frank Harary, mathematician

A mathematical theory is not to be considered complete until you have made it

so clear that you can explain it to the ﬁrst man whom you meet on the street.

David Hilbert, mathematician

Everything should be made as simple as possible, but not simpler.

Albert Einstein, physicist

Never let anything you write be published without having had others critique it.

Donald E. Knuth, computer scientist and writer

Some books are to be tasted, others to be swallowed, and some few to be chewed

and digested.

Reading maketh a full man, conference a ready man, and writing an exact man.

Francis Bacon, writer and philosopher

Judge an article not by the quality of what is framed and hanging on the wall,

but by the quality of what’s in the wastebasket.

Anonymous (Quote by Leslie Lamport)

We are all apprentices in a craft where no-one ever becomes a master.

Ernest Hemingway, author (For Whom the Bell Tolls)

There are three rules for writing a novel. Unfortunately, no one knows what they

are.

W. Somerset Maugham, author (Of Human Bondage)

Mathematical Writing

Most of the quotes given above pertain to writing in general, not to mathematical writing

in particular. However these suggestions for writing apply as well to writing mathemat-

ics. For us, mathematical writing means writing assignments for a mathematics course

(particularly a course with proofs). Such an assignment might consist of writing a single

proof, writing solutions to a number of problems or perhaps writing a term paper which,

more than likely, includes deﬁnitions, examples, background and proofs. We’ll refer to

any of these as an assignment. Your goal should be to write correctly, clearly and in an

interesting manner.

Before you even begin to write, you should have already thought about a number

of things. First, you should know what examples and proofs you plan to include if this

is appropriate for your assignment. You should not be overly concerned about writing

good proofs on your ﬁrst attempt—but be certain that you do have proofs.

As you’re writing your assignment, you must be aware of your audience. What is the

target group for your assignment? Of course, it should be written for your instructor. But

it should be written so that a classmate would understand it. As you grow mathematically,

your audience will grow with you and you will adapt your writing to this new audience.

Give yourself enough time to write your assignment. Don’t try to put it together

just a few minutes before it’s due. The disappointing result will be obvious to your

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