数学阅读与证明：通往更高层次的桥梁

数学证明

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"《Reading, Writing, and Proving 2nd Edition》是基于Pólya的问题解决方法编写的一本教材，旨在帮助学生从微积分或预备微积分过渡到更高层次的数学学习。这本书首先提供了大量关于如何理解数学中的定义、例子和定理的指导，然后逐步引导学生进入更深入的数学思维，最终提供独立研究项目，以培养学生的自主学习能力。" 《阅读、写作与证明》第二版是数学教育领域的一本重要著作，由Ulrich Daepp和Pamela Gorkin共同编著。它属于 Springer 的“Undergraduate Texts in Mathematics”系列，旨在为大学生提供一个逐步深入数学世界的平台。该书特别关注数学证明这一核心概念，因为证明不仅是数学的基石，也是培养逻辑思考和批判性思维的关键。书中遵循Pólya的解决问题方法，这是一种系统性的解决问题和学习策略，包括理解问题、制定计划、实施计划和回顾过程四个步骤。在处理数学定义时，书中的指导强调了理解其精确性和重要性，因为定义是构建数学理论的基础。对于例子，书中的处理旨在展示概念的实际应用和可能的变体，以深化理解。对于定理，读者将学会如何分析、证明和应用这些定理，从而形成自己的数学推理能力。书中还注重写作在数学学习中的角色，因为清晰、准确地表达思想是有效沟通和证明的关键。通过写作练习，学生可以磨练他们的表达技巧，更好地理解和构造复杂的数学论证。此外，本书的独立研究项目鼓励学生自我探索，这不仅锻炼了他们的自主学习能力，也培养了他们独立解决问题和创新思考的能力。《Reading, Writing, and Proving》第二版对于那些希望深入理解数学证明及其在高级数学课程中的应用的学生来说是一份宝贵的资源。它不仅提供了学习工具，还为学生提供了从被动接受知识转向主动探索和创造的桥梁，是数学教育中不可或缺的一部分。书中的方法和练习有助于培养出具备强大数学思维能力的未来数学家和相关领域的专业人士。

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Chapter 1

The How, When, and Why of Mathematics

What is mathematics? Many people think of mathematics (incorrectly) as addition,

subtraction, multiplication, and division of numbers. Those with more mathematical

training may think of it as dealing with algorithms. But most professional mathe-

maticians think of it as much more than that. While we certainly hope that our

students will perform algorithms correctly, what we really want is for them to un-

derstand three things: how you do something, why it works, and when it works.

The problems we present to you in this book concentrate on these three goals. If

this is the ﬁrst time you have been asked to prove theorems, you may ﬁnd this to

be quite a challenge. Not only will you be learning how to solve the problem, you

will also be learning how to write up the solution. The necessary deﬁnitions and

background to understand a problem, as well as a general plan of attack, will always

be presented in the text. It’s up to you to spend the time reading, trying various ap-

proaches, rereading, and reapproaching. You will probably be spending more time

on fewer exercises than you ever have before. While you are now beyond the stage

of being given steps to follow and practice, there are general rules that can assist you

in your transition to doing higher mathematics. Many people have written about this

subject before. The classic text on how to approach a problem is a wonderful book

called How to Solve It by George P

olya, [84].

calls his suggestions “the list.” We have included the original in the Appendix on

and we suggest that you let it guide you as well. Here’s a closer look at “the list”

First. “Understanding the problem.” Easier said than done, of course. What

look something up in this book, or you may need to use another book. Look at the

ﬁgure out. If a picture will help, draw it. Will you be proving something? What?

Will you have to obtain an example? Of what? Check all conditions. Will you have

to show that something is false? Once you understand what you have to do, you can

move on to the next step.

should you do? Make sure you know what all the words mean. You may need to

page 366. This list has served as a guide for several generations of mathematicians,

statement to ﬁgure out carefully what you are given and what you are supposed to

with some 21st-century modiﬁcations.

U. Daepp and P. Gorkin, Reading, Writing, and Proving: A Closer Look at Mathematics,

In his text, Polya gives a list of guidelines for solving mathematical problems. He

Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4419-9479-0_1,

2 1 The How, When, and Why of Mathematics

Second. “Devising a plan.” How will you attack the problem? At this point, you

understand what must be done (because you have completed Step 1). Have you seen

something like it before? If you haven’t looked over class notes, haven’t read the

text, or haven’t done the previous homework assignments, the odds are slim that

you have seen anything that will be helpful. Do all that ﬁrst. Look over the text with

the problem in mind, read over your notes with the problem in your head, look at

previous exercises and theorems that sound similar. Maybe you can use some of

the ideas in the proof of a theorem, or maybe you can use a previous homework

problem. Mathematics builds on itself and the problems in the text will also. If you

are truly stuck, try to answer a simpler, similar question. Once you decide on a

method of approach, try it out.

Third. “Carrying out the plan.” Solve the problem. Look at your solution. Is each

sentence true? Sometimes it is difﬁcult to catch an error right after you have “found

a solution.” Put the problem down and come back to it a few hours later. Is each

sentence still true?

Fourth. “Looking back.” P

olya suggests checking the result and the argument, or

even looking for a different proof. If you are allowed (check with your teacher), one

really good way to check a proof is to give it to someone else. You can present it

to friends. Even if they don’t understand a word you are saying, sometimes saying

it out loud in a coherent manner will allow you to recognize an error you can’t spot

when you are reading. If you are permitted to work together, switch proofs and ask

your partner for criticism of your proof.

When you are convinced that your argument is correct, it is time to write up a

correct and neat solution to the problem.

Here is an example of the P

olya method at work in mathematics; we will decipher

a message. A cipher is a system that is used to hide the meaning of a message by

replacing the letters of the alphabet by other letters or symbols.

Exercise 1.1. The following message is encoded by a shift of the alphabet; that is,

every letter is replaced by another one that has been shifted n places further down

the alphabet. Once we reach the end of the alphabet, we start over. For instance, if n

were 7, we would make the replacements a → h, b → i, ... , s → z, t → a, .. .. Now

the exercise: What does the message below say?

PDEO AJYKZEJC WHCKNEPDI EO YWHHAZ W YWAOWN YELDAN.

EP EO RANU AWOU PK XNAWG, NECDP?

Let’s use the ideas from P

olya’s list to solve this. If you have solved problems

like this before, it might be a better exercise for you to try on your own to see how

this ﬁts P

olya’s method before you read on.

1. “Understanding the problem.” Each sequence of letters with no blank space be-

tween the letters represents one word. Each letter is shifted by the same number

of places: namely, n. So n is the unknown in this problem and it is what we need

to ﬁnd. Once we know the value of n, we can decipher the whole message. In

addition, once we know the meaning of one letter, we can ﬁnd the value for n.

1 The How, When, and Why of Mathematics 3

2. “Devising a plan.” A cipher text may have weak points. What are these? How

about the short words? Looking at the short words, in some sense, substitutes an

easier problem for the one we have.

3. “Carrying out the plan.” The short words are:

EO (which appears twice);

EP;

PK.

Try using the most common one- and two-letter words. For each guess, check

the beginning of the cipher text to see if it makes sense. It shouldn’t take long for

you to come up with the message.

4. “Looking back.” If your solution makes sense, then it is highly unlikely that a

different replacement is also possible. So the solution is (with high probability)

the only one.

Would there have been other solution methods? Sure. For instance, not all letters

have the same frequency in the English language. One analysis of English texts

showed the letter e occurring most frequently, followed by (in this order) t, a, o,

i, n, s, h, and r. (See [99, p. 19].) We could have used this information to guess

the assignment of letters.

We also could have simply tried one value of n after another until the message

made sense.

Have you now solved the problem? If you know what the message says, then

the answer to this question is yes. Are you done? Unless you solved the problem

and wrote up a clear, complete solution, the answer to this second question is no.

A solution consists of a report that tells the reader how you solved the problem and

what the answer is. This needs to be done in clear English sentences. As you write

up your solution, try to keep the reader in mind. You should explain things clearly

and logically, so that the reader doesn’t have to spend time ﬁlling in gaps.



We now move on to a very different kind of example. Consider the set of points in

three-space. In case you haven’t seen this before, these points are easily described.

We take the familiar xy-plane, and place it parallel to the ﬂoor. The z-axis is the

vertical line perpendicular to the xy-plane and passing through the origin of the xy-

plane (see Figure 1.1). We’ll review the important concepts before we begin our

example.

To locate a point, we will give three coordinates. The ﬁrst coordinate is the x-

coordinate and tells us the number of units to walk in the x-direction. The second

is the y-coordinate, telling us how to move in the y-direction, and the third is the

z-coordinate, telling us how far, up or down, to move. So a point in three-space is

denoted by (x,y,z). It is important to make sure you understand this. Try to think

of how you would plot points. The point (1,0,0) (plotted in Figure 1.2) would ap-

pear one unit in the positive direction on the x-axis (since it doesn’t move in the

y-direction or z-direction at all). The point (−1,1,0) would appear in the xy-plane,

1 The How, When, and Why of Mathematics 5

x

y

z

(x

0

,y

0

,z

0

)

(x

1

,y

1

,z

0

)

(x

1

,y

1

,z

1

)

Fig. 1.3

We follow P

olya’s method to ﬁnd the solution.

1. “Understanding the problem.” Before we begin, we make sure we really under-

stand the meaning of each word and symbol above. We spent the last few para-

graphs making sure we all understand the symbols, and all the words are familiar

ones that appear in a standard English dictionary. But, wait—has “distance be-

tween two points” really been deﬁned? We need to be sure that everyone means

the same thing by this. The distance between these two arbitrary points would

mean the length of the straight line segment joining the two points. That’s what

we need to ﬁnd. What were we given? Two points and their coordinates.

2. “Devising a plan.” How do we solve something like this? We haven’t covered

anything yet, so what can the authors be thinking? If you have no idea how to get

started, try thinking about ﬁnding the distance between two speciﬁc points. Of

course, (and this is very important) this won’t give us a general formula because

it is much too speciﬁc, but maybe we’ll get some ideas.

So what’s the distance between the two points (1,0,0) and (−1,0,0)? That ques-

tion is easier to answer—it’s two. What’s the distance between (1,1, 0) and

(−1,−2, 0)? This seems to be just the distance between two points in the fa-

miliar xy-plane. We saw a formula for that at some point. It was obtained using

the Pythagorean Theorem. What was it? If you can’t recall the formula, look it

up or (better, yet) try to derive it again.

Our reasoning now brings us to a simpler, similar question. As you recall, this is

precisely where P

olya suggested we look for a plan. So far, it seems we can ﬁnd

the distance between two points as long as they lie in a plane parallel to one of

the coordinate planes. But in this problem, if we look at the two points, they need

not lie in such a plane. We can try to insert a third point that helps us to reduce

the problem to one we can already solve. Which point? A picture will help here,

so we draw one in Figure 1.3.

We see that (x

) and (x

) lie in the plane z = z

, while (x

)

and (x

) lie on the same vertical line, in the intersection of the two planes,

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Pedersen (k, n) protocol

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