怎么学数学
——陶哲轩教我们数学学习方法
原文作者 Terence Tao 单位 UCLA
摘要： 大二下学期我闲着无聊的时候翻译了几篇著名数学家陶哲轩写的数学方法论文章.现在贴到这里来.让我们聆听大师教诲.
关键词：学习其它领域知识,写下来,学习其它数学家的工具,陶哲轩,学习方法
不要害怕学习其它领域的知识
尽可能地了解全局，同时精通局部.(赫胥黎)
数学恐惧症普遍存在于较广泛的群体中.不幸的是，它有时也存在于职业数学家们之间（伴随着它的远亲——数学势利眼）.
如果你不得不学些别的数学以使你的问题有所进展，这是一件好事——你自己的数学范围会扩大，你将会获得一些新的工具 ，而且，在与你同领域和不同领域的人眼里，你的工作都是变得更有意思了.
如果某个数学领域很活跃，去看一下它为什么会这么有意思，那里的人都在努力解决什么问题，那个领域产生了哪些“酷”的,或令人惊讶的想法(insights)，现象(phenomena)和结论.这通常都是值得的.(可以看看我在什么是好的数学 里的讨论).这样的话，一旦你在你的工作中碰到了一个类似的问题(problem)，障碍(obstruction)或现象(phenomena)，你就能知道该转向哪儿了.
学习别的领域的知识，参加一些其它领域的讲座和会议是个好方法.
也可看看我的博文 对你的领域学而时习之.
把你做过的写下来
每个作曲家都清楚，由于自己没有及时记录，而后遗忘想法会有多么的令人痛苦和绝望.（埃克托·柏辽兹）
在职业生涯的早期，我多次或者是读到，或者是听说，或者是发现一些干净的(neat)数学技巧 或论证(argument)，我觉得自己已经懂了，因此不必写下来，然后，比如说六个月之后，等到我真的需要回忆那个技巧时，却丝毫无法将其重建.最终我下定决心，要把我所遇到的任何有意思的论证都写下来(我更喜欢写电脑上)——不必达到能出版的标准，但是要详细到能让我安全地忘掉其中的细节，而且一旦在我需要的时候，能根据这些文字毫无困难地还原论证.
我建议你们也这样做，因为这有以下几个方面的用途：
1.这样能让论证永久地能被你所获得，而且，这些论证或许最终有助于你的研究论文，讲稿，和研究提案(research proposal).
2.这样能锻炼你的数学写作——同时在技术层面(比如，学习使用Tex)和阐述与教学法(pedagogical)的层面.
3.这样能测试你是否真的理解了论证，而不仅仅是表面的理解.
4.这样做能解放你的记忆空间，你不必再记住论证的准确细节，因此能把记忆用来学习新东西.
一旦你写下了这样的豆腐块，即使它没有达到能发表的原创性和深度要求,你也可以考虑使它能被人获得(比如，把它放到你的网站上).
基于类似的理由，如果你对你所研究的某个问题有不完整的(或不满意的)论证，而且你计划将其遗弃，你仍然可以为你自己写一份豆腐块(在豆腐块里给出勉强足够的细节，以使得你以后能毫无困难地还原整个事情)，然后把它存在电脑的某个地方，说不定在将来你会需要它.
学习其他数学家的工具的威力
如果你仅有的工具是一把锤子,你会倾向于把一切问题都看成钉子.(亚伯拉罕.马斯洛,科学心理学)
在听讲座或者读论文时，会有一些你感兴趣的问题被你所不熟悉的工具解决，而那个工具似乎并不为你所掌握.这时候，你要试试用自己已经掌握的工具能否解决类似的问题，也应当想想为什么那个新工具对于解决该问题如此有效——比如，去研究该问题的一个特例，看看在这个特例里，那个工具发挥了怎样的作用.
如果你比较了新工具相对于旧工具的优点和弱点，那么以后一旦需要，你就能想起它.经过足够多的练习后，这个新工具就会永久性地被你所掌握.因此花一些时间学习一下别的工具是值得的.即便那个工具不在你所从事的领域，也是值得的，这样做的一个方式是去阅读其它领域针对一般读者的研究文章.(四年一度的国际数学家大会是不错的资源.)
外文文献出处：terrytao.wordpress.com
附外文文献原文
Donrsquo;t be afraid to learn things outside your field
Try to learn something about everything and everything about something. (Thomas Huxley)
Maths phobia is a pervasive problem in the wider community. Unfortunately, it sometimes also exists among professional mathematicians (together with its distant cousin, maths snobbery).
If it turns out that in order to make progress on your problem, you have to learn some external piece of mathematics, this is a good thing – your own mathematical range will increase, you will have acquired some new tools, and your work will become more interesting, both to people in your field and also to people in the external field.
If an area of mathematics has a lot of activity in it, it is usually worth learning why it is so interesting, what kind of problems people try to work on there, and what are the “cool” or surprising insights, phenomena, results that that field has generated. (See also my discussion on what good mathematics is.) That way if you encounter a similar problem, obstruction, or phenomenon in your own work, you know where to turn for the resolution.
One good way to learn things outside your field is by attending talks and conferences outside your field.
See also “Learn and relearn your field“.
Write down what yoursquo;ve done
Every composer knows the anguish and despair occasioned by forgetting ideas which one had no time to write down. (Hector Berlioz)
There were many occasions early in my career when I read, heard about, or stumbled upon some neat mathematical trick or argument, and thought I understood it well enough that I didnrsquo;t need to write it down; and then, say six months later, when I actually needed to recall that trick, I couldnrsquo;t reconstruct it at all. Eventually I resolved to write down (preferably on a computer) a sketch of any interesting argument I came across – not necessarily at a publication level of quality, but detailed enough that I could then safely forget about the details, and readily recover the argument from the sketch whenever the need arises.
I recommend that you do this also, as it serves several useful purposes:
It makes the argument permanently available to you in the future, and may eventually be helpful in your later research papers, lecture notes, teaching, or research proposals.
It gives you practice in mathematical writing, both at the technical level (e.g. in learning how to use TeX) and at an expository or pedagogical level.
It tests whether you have really understood the argument on more than just a superficial level.
It frees up mental space; you no longer have to remember the exact details of the argument, and so can devote your memory to learning newer topics.
Once you have written up such a sketch, you might consider making it available (e.g. on your web site), even if it does not rise to the level of originality and depth required for a publishable paper.
For somewhat similar reasons, if you have an incomplete (or otherwise unsatisfactory) argument for a problem that you are working on, and you are planning to abandon it, you may still wish to write an informal sketch of it just for yourself (giving barely enough details to allow you to readily reconstruct the whole thing later on), and store it somewhere on your computer, just in case you find you have need for it some time in the future.
Learn the power of other mathematiciansrsquo; tools
I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail. (Abraham Maslow, “Psychology of Science”)
You will find, when listening to talks or reading papers, that there will be problems which interest you which were solved using an unfamiliar tool, but seem out of reach of your own personal “bag of tricks”. When this happens, you should try to see whether your own tools can in fact accomplish a similar task, but you should also try to work out what made the other tool so effective – for inst
剩余内容已隐藏，支付完成后下载完整资料
怎么学数学
——陶哲轩教我们数学学习方法
原文作者 Terence Tao 单位 UCLA
摘要： 大二下学期我闲着无聊的时候翻译了几篇著名数学家陶哲轩写的数学方法论文章.现在贴到这里来.让我们聆听大师教诲.
关键词：学习其它领域知识,写下来,学习其它数学家的工具,陶哲轩,学习方法
不要害怕学习其它领域的知识
尽可能地了解全局，同时精通局部.(赫胥黎)
数学恐惧症普遍存在于较广泛的群体中.不幸的是，它有时也存在于职业数学家们之间（伴随着它的远亲——数学势利眼）.
如果你不得不学些别的数学以使你的问题有所进展，这是一件好事——你自己的数学范围会扩大，你将会获得一些新的工具 ，而且，在与你同领域和不同领域的人眼里，你的工作都是变得更有意思了.
如果某个数学领域很活跃，去看一下它为什么会这么有意思，那里的人都在努力解决什么问题，那个领域产生了哪些“酷”的,或令人惊讶的想法(insights)，现象(phenomena)和结论.这通常都是值得的.(可以看看我在什么是好的数学 里的讨论).这样的话，一旦你在你的工作中碰到了一个类似的问题(problem)，障碍(obstruction)或现象(phenomena)，你就能知道该转向哪儿了.
学习别的领域的知识，参加一些其它领域的讲座和会议是个好方法.
也可看看我的博文 对你的领域学而时习之.
把你做过的写下来
每个作曲家都清楚，由于自己没有及时记录，而后遗忘想法会有多么的令人痛苦和绝望.（埃克托·柏辽兹）
在职业生涯的早期，我多次或者是读到，或者是听说，或者是发现一些干净的(neat)数学技巧 或论证(argument)，我觉得自己已经懂了，因此不必写下来，然后，比如说六个月之后，等到我真的需要回忆那个技巧时，却丝毫无法将其重建.最终我下定决心，要把我所遇到的任何有意思的论证都写下来(我更喜欢写电脑上)——不必达到能出版的标准，但是要详细到能让我安全地忘掉其中的细节，而且一旦在我需要的时候，能根据这些文字毫无困难地还原论证.
我建议你们也这样做，因为这有以下几个方面的用途：
1.这样能让论证永久地能被你所获得，而且，这些论证或许最终有助于你的研究论文，讲稿，和研究提案(research proposal).
2.这样能锻炼你的数学写作——同时在技术层面(比如，学习使用Tex)和阐述与教学法(pedagogical)的层面.
3.这样能测试你是否真的理解了论证，而不仅仅是表面的理解.
4.这样做能解放你的记忆空间，你不必再记住论证的准确细节，因此能把记忆用来学习新东西.
一旦你写下了这样的豆腐块，即使它没有达到能发表的原创性和深度要求,你也可以考虑使它能被人获得(比如，把它放到你的网站上).
基于类似的理由，如果你对你所研究的某个问题有不完整的(或不满意的)论证，而且你计划将其遗弃，你仍然可以为你自己写一份豆腐块(在豆腐块里给出勉强足够的细节，以使得你以后能毫无困难地还原整个事情)，然后把它存在电脑的某个地方，说不定在将来你会需要它.
学习其他数学家的工具的威力
如果你仅有的工具是一把锤子,你会倾向于把一切问题都看成钉子.(亚伯拉罕.马斯洛,科学心理学)
在听讲座或者读论文时，会有一些你感兴趣的问题被你所不熟悉的工具解决，而那个工具似乎并不为你所掌握.这时候，你要试试用自己已经掌握的工具能否解决类似的问题，也应当想想为什么那个新工具对于解决该问题如此有效——比如，去研究该问题的一个特例，看看在这个特例里，那个工具发挥了怎样的作用.
如果你比较了新工具相对于旧工具的优点和弱点，那么以后一旦需要，你就能想起它.经过足够多的练习后，这个新工具就会永久性地被你所掌握.因此花一些时间学习一下别的工具是值得的.即便那个工具不在你所从事的领域，也是值得的，这样做的一个方式是去阅读其它领域针对一般读者的研究文章.(四年一度的国际数学家大会是不错的资源.)
外文文献出处：terrytao.wordpress.com
附外文文献原文
Donrsquo;t be afraid to learn things outside your field
Try to learn something about everything and everything about something. (Thomas Huxley)
Maths phobia is a pervasive problem in the wider community. Unfortunately, it sometimes also exists among professional mathematicians (together with its distant cousin, maths snobbery).
If it turns out that in order to make progress on your problem, you have to learn some external piece of mathematics, this is a good thing – your own mathematical range will increase, you will have acquired some new tools, and your work will become more interesting, both to people in your field and also to people in the external field.
If an area of mathematics has a lot of activity in it, it is usually worth learning why it is so interesting, what kind of problems people try to work on there, and what are the “cool” or surprising insights, phenomena, results that that field has generated. (See also my discussion on what good mathematics is.) That way if you encounter a similar problem, obstruction, or phenomenon in your own work, you know where to turn for the resolution.
One good way to learn things outside your field is by attending talks and conferences outside your field.
See also “Learn and relearn your field“.
Write down what yoursquo;ve done
Every composer knows the anguish and despair occasioned by forgetting ideas which one had no time to write down. (Hector Berlioz)
There were many occasions early in my career when I read, heard about, or stumbled upon some neat mathematical trick or argument, and thought I understood it well enough that I didnrsquo;t need to write it down; and then, say six months later, when I actually needed to recall that trick, I couldnrsquo;t reconstruct it at all. Eventually I resolved to write down (preferably on a computer) a sketch of any interesting argument I came across – not necessarily at a publication level of quality, but detailed enough that I could then safely forget about the details, and readily recover the argument from the sketch whenever the need arises.
I recommend that you do this also, as it serves several useful purposes:
It makes the argument permanently available to you in the future, and may eventually be helpful in your later research papers, lecture notes, teaching, or research proposals.
It gives you practice in mathematical writing, both at the technical level (e.g. in learning how to use TeX) and at an expository or pedagogical level.
It tests whether you have really understood the argument on more than just a superficial level.
It frees up mental space; you no longer have to remember the exact details of the argument, and so can devote your memory to learning newer topics.
Once you have written up such a sketch, you might consider making it available (e.g. on your web site), even if it does not rise to the level of originality and depth required for a publishable paper.
For somewhat similar reasons, if you have an incomplete (or otherwise unsatisfactory) argument for a problem that you are working on, and you are planning to abandon it, you may still wish to write an informal sketch of it just for yourself (giving barely enough details to allow you to readily reconstruct the whole thing later on), and store it somewhere on your computer, just in case you find you have need for it some time in the future.
Learn the power of other mathematiciansrsquo; tools
I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail. (Abraham Maslow, “Psychology of Science”)
You will find, when listening to talks or reading papers, that there will be problems which interest you which were solved using an unfamiliar tool, but seem out of reach of your own personal “bag of tricks”. When this happens, you should try to see whether your own tools can in fact accomplish a similar task, but you should also try to work out what made the other tool so effective – for inst
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