波利亚(Polya, George)的数学思维

2022-05-06

• 生词：sagacity睿智， 灵感。

Polya, George. How to solve it: A new aspect of mathematical method. No. 246. Princeton university press, 2004.(中译本：怎样解题：数学思维的新方法，上海科技教育出版社，2011.11)

• 这本书1944年就出版了，但国内译本2007年出现（存疑）。显示出滞后性。希望以后能更快地发现好书，并多传播好书。

  
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我买到了这本二手书，书内干净很。如果读者没怎么阅读的话，大为可惜。

这本书提到4条解决数学问题的步骤：
How to Solve It suggests the following steps when solving a mathematical problem:

1. First, you have to understand the problem.^[2]
2. After understanding, make a plan.^[3]
3. Carry out the plan.^[4]
4. Look back on your work.^[5] How could it be better?

If this technique fails, Pólya advises:^[6] "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

这本书的序里这样写道：

1. 老师怎么教的问题：用好教的机会。

a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.

以上的机器翻译：

数学老师有一个很好的机会。 如果他在分配的时间里训练他的学生进行常规操作，他就会扼杀他们的兴趣，阻碍他们的智力发展，并滥用他的机会。 但是，如果他通过设置与学生的知识相称的问题来挑战学生的好奇心，并通过激发性的问题帮助他们解决问题，那么他可能会让他们尝到独立思考的滋味和一些方法。

2. 学生怎么学的问题
   抓住课程能提供的独特的机会，而不是为了学分：(课程提供独特机会，同样适用于其他科目，而非只是数学。)

Also a student whose college curriculum includes some mathematics has a singular opportunity. This opportunity is lost, of course, if he regards mathematics as a subject in which he has to earn so and so much credit and which he should forget after the final examination as quickly as possible.
如果一个学生的大学课程包括了某些数学科目，那么他就有了难得的机会。 当然，如果他认为数学是一门他必须获得如此多学分的学科，并且在期末考试后他应该尽快忘记这门学科，那么这个机会就失去了。

怎么做题：不只是要答案，还要深入思考如下问题：

Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such
facts? And how could I invent or discover such things by myself?"
是的，该解答似乎有效，它似乎是正确的； 但是怎么可能发明这样的解答呢？ 是的，这个实验似乎奏效了，这似乎是事实； 但是人们怎么会发现这样的事实？ 我怎么能自己发明或发现这些东西呢？”

Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others.
不仅要了解这个或那个问题的解答，还要了解解答的动机和步骤，并试图向他人解释这些动机和步骤。

我的感觉是：在现实中，我们的教学中似乎过多地是老师在告诉学生这是什么。对于为什么提得太少。即使提了，学生愿意去主动深入思考得少。

这本书当有更高价值。
作者自己也认为书的价值不限于数学的师生，而是更广:

Although the present book pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of invention and discovery.

这比如作者对于出色的念头(bright idea)的描述。念头有好有坏，有深有浅。深刻的才是好的念头。

Most people will agree that conceiving a bright idea is an "act of sagacity." Aristotle defines ''sagacity" as follows: "Sagacity is a hitting by guess upon the essential connection in an inappreciable time. As for example, if you see a person talking with a rich man in a certain way, you may instantly guess that that person is trying to
borrow money. Or observing that the bright side of the moon is always toward the sun, you may suddenly perceive why this is; namely, because the moon shines by the light of the sun."
The first example is not bad but rather trivial: not much sagacity is needed to guess things of this sort about rich men and money, and the idea is not very bright. The second example, however, is quite impressive if we make a little effort of imagination to see it in its proper setting.

进一步的升华：

He conceives the sun and the moon not as flat discs but as round bodies, one giving and the other receiving the light. He understands the essential connection, he rearranges his former conceptions instantly, "in an inappreciable time": there is a sudden leap of the imagination, a bright idea, a flash of genius.

再如在讲到 倒着干Working backwards. 的困难时上升到如下：

There is a certain difficulty in turning round, in going away from the goal, in proceeding without looking continually at the aim, in not foIlowing the direct path to the desired end.

作者还有一节Wisdom of proverbs. 谚语的智慧
如：

Look around when you have got your first mushroom or made your first discovery; they grow in dusters.

这和万事开头难一脉相承。不过一旦开头后，就能有更多发现。所以做研究不用担心没问题可做。

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最后附上两张图，纪念伟人。

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