分类目录归档:演讲访谈

做数学一定要是天才吗?(陶哲轩)[转载自刘小川WordPress]

做数学一定要是天才吗?(陶哲轩)

(原文:Does
one have to be a genius to do maths?

http://liuxiaochuan.wordpress.com/2008/03/30/%E5%81%9A%E6%95%B0%E5%AD%A6%E4%B8%80%E5%AE%9A%E8%A6%81%E6%98%AF%E5%A4%A9%E6%89%8D%E5%90%97%EF%BC%9F-%EF%BC%88%E8%AF%91%E8%87%AA-%E9%99%B6%E5%93%B2%E8%BD%A9-%E5%8D%9A%E5%AE%A2%EF%BC%89/

[转载自刘小川WordPress]

做数学一定要是天才吗?

这个问题的回答是一个大写的:!为了达到对数学有一个良好的,有意义的贡献的目的,人们必须要刻苦努力;学好自己的领域,掌握一些
其他领域的知识和工具;多问问题;多与其他数学工作者交流;要对数学有个宏观的把握。当然,一定水平的才智,耐心的要求,以及心智上的成熟性是必须的。但
是,数学工作者绝不需要什么神奇的“天才”的基因,什么天生的洞察能力;不需要什么超自然的能力使自己总有灵感去出人意料的解决难题。

大众对数学家的形象有一个错误的认识:这些人似乎都使孤单离群的(甚至有一点疯癫)天才。他
们不去关注其他同行的工作,不按常规的方式思考。他们总是能够获得无法解释的灵感(或者经过痛苦的挣扎之后突然获得),然后在所有的专家都一筹莫展的时
候,在某个重大的问题上取得了突破的进展。这样浪漫的形象真够吸引人的,可是至少在现代数学学科中,这样的人或事是基本没有的。在数学中,我们的确有很多
惊人的结论,深刻的定理,但是那都是经过几年,几十年,甚至几个世纪的积累,在很多优秀的或者伟大的数学家的努力之下一点一点得到的。每次从一个层次到另
一个层次的理解加深的确都很不平凡,有些甚至是非常的出人意料。但尽管如此,这些成就也无不例外的建立在前人工作的基础之上,并不是全新的。(例
如,Wiles 解决费马最后定理的工作,或者Perelman 解决庞加莱猜想的工作。)

今天的数学就是这样:一些直觉,大量文献,再加上一点点运气,在大量连续不断的刻苦的工作中慢慢的积累,缓缓的进展。事实上,我甚至觉得现实中的情
况比前述浪漫的假说更令我满足,尽管我当年做学生的时候,也曾经以为数学的发展主要是靠少数的天才和一些神秘的灵感。其实,这种“天才的神话”是有其缺陷
的,因为没有人能够定期的产生灵感,甚至都不能保证每次产生的这些个灵感的正确性(如果有人宣称能够做到这些,我建议要持怀疑态度)。相信灵感还会产生一
些问题:一些人会过度的把自己投入到大问题中;人们本应自己的工作和所用的工具有合理的怀疑,但是上述态度却使某些人对这种怀疑渐渐丧失;还有一些人在数
学上极端不自信,还有很多很多的问题。

当然了, 如果我们不使用“天才”这样极端的词汇,我们会发现在很多时候,一些数学家比其他人会反应更快一些,会更有经验,会更有效率,会更仔细
,甚至更有创造性。但是,并不是这些所谓的“最好”的数学家才应该做数学。这其实是一种关于绝对优势和相对优势的很普遍的错误观念。有意义的数学科研的领
域极其广大,决不是一些所谓的“最好”的数学家能够完成的任务,而且有的时候你所拥有的一些的想法和工具会弥补一些优秀的数学家的错误,而且这些个优秀的
数学家们也会在某些数学研究过程中暴露出弱点。只要你受过教育,拥有热情,再加上些许才智,一定会有某个数学的方面会等着你做出重要的,奠基性的工作。这
些也许不是数学里最光彩照人的地方,但是却是最健康的部分。往往一些现在看来枯燥无用的领域,在将来会比一些看上去很漂亮的方向更加有意义。而且,应该先
在一个领域中做一些不那么光彩照人的工作,直到有机会和能力之时,再去解决那些重大的难题。看看那些伟大的数学家们早期的论文,你就会明白我的意思了。

有的时候,大量的灵感和才智反而对长期的数学发展有害,试想如果在早期问题解决的太容易,一个人可能就不会刻苦努力,不会问一些“傻”的问题,不会
尝试去扩展自己的领域,这样迟早造成灵感的枯竭。而且,如果一个人习惯了不大费时费力的小聪明,他就不能拥有解决真正困难的大问题所需要耐心,和坚韧的性
格。聪明才智自然重要,但是如何发展和培养显然更加的重要。

要记着,专业做数学不是一项运动比赛。做数学的目的不是得多少的分数,获得多少个奖项。做数学其实是为了理解数学,为自己,也为学生和同事,最终要
为她的发展和应用做出贡献。为了这个任务,她真的需要所有人的共同拼搏!

Advertisements

我如何安排时间(译自陶哲轩博客)[转载自刘小川WordPress]

我如何安排时间(译自陶哲轩博
客)

http://liuxiaochuan.wordpress.com/2008/08/08/%E6%88%91%E5%A6%82%E4%BD%95%E5%AE%89%E6%8E%92%E6%97%B6%E9%97%B4%EF%BC%88%E8%AF%91%E8%87%AA%E9%99%B6%E5%93%B2%E8%BD%A9%E5%8D%9A%E5%AE%A2%EF%BC%89/
[转载自刘小川WordPress]

(原文:On
time management
by Terry Tao)

受到一些评论的鼓励,我最终决定在这里写一些关于如何安排时间的建议。其实,我有这个打算已经一段时间了,可是就我自己的情况而言,这方面也还在做
着探索(读者应该看看我等着写的论文排了多少!)而且很多想法未必成熟。(已经有一些经验写在advice on
writing papers
,比如page
on rapid prototyping
)而且,我的一些个人经验恐怕也不能对所有人通通适用,因为每个人都有不同的性格类型以及工作状态。
欢迎大家把自己的想法啊,经验啊,或者建议在评论中写出来。(其实,即使我自己的经验,我有时候也不能严格的遵照,挺遗憾的。)

这些经验并不系统,我慢慢的叙述如下。首先,我足够的幸运,自己的很多优秀的合作者都在我们合作的工作中付出了大量的心血。比如最近我的博客大家看
到的论文,很多都在很大的程度上是我的合作者们辛勤劳作的成果。一般来讲,我觉得几个人合作的时候,虽然常常要花费的时间要多一些,但是每个人实际花费的
力气却令人吃惊的少,而文章的质量却更高。我发现自己可以同时与很多人在不同的工作中合作(因为常常他们为主,或者该工作实际上在等待进一步发展。)可是
在我独自写论文的时候,我却只能同时只做一件工作。

由于一些学院时间的规定,在夏季很多的工作要做结,数量要比其他任何时候都多。这些工作都已经经历了相当长一段时间了(比如,很快就要有一篇文章完
成了。在这篇文章上,我们已经花费了三到四年;从2000年开始,我在关于波映射的全局正则问题(the global regularity
problem for wave maps
problem)上已经时断时续的花费了8年之久了。)所以说,当一篇论文一个星期就出现,这可不是说,从怀有这个论文的想法,到真正写出来,仅仅花费了
一个星期。其实往往整个漫长过程多是不被世人所知的。

另外,我解决严肃数学问题的能力常常上下变化,甚至每天都有区别。有时候我可以在一个问题上连续想一个小时之久;而有时我更适合去把我和合作者们的
草稿式的想法给具体到细节的写出来;另外一些时候,我觉得自己只能收收邮件,改改错误,甚至打个盹,散散步。我觉得非常重要的一点就是,我应该根据自己的
状态变化来调整自己的工作安排。如果我有一整个下午的时间,同时又有很好的状态,我可能就会关掉办公室的门,关掉网络,静下心来写这篇苦思已久的论文;而
状态不行的时候呢,我就看看这一周的e-mail,投几篇篇论文,写写blog。总之我要做些跟精力的热情的高低很相配的工作。做数学够幸运的一点就是,
你可以把大部分的工作在时间上做非常自由的调整(但是讲课是一个非常重要的例外,我们必须围绕讲课的固定时间来做安排)。能够准确的判断自己在某个时段的
工作能力以及对接下来的时间(比如这一天剩下的时间)做估计是很有帮助的。无论是太过自信,还是太不自信,在选择具体的工作内容的时候都会带来低效的后
果。(我在这两方面可以说都有反面的经验。)

类似的。我有时会有一大堆事情,在长度,复杂度,困难度都非常不同。这一堆问题写在我的“要做”清单之中,如果其中有某个需要很细致的思考的话,我
会完全排除掉其他干扰,只将注意力放在这一个问题之上,其他的能拖后的拖后,能放弃的放弃掉;我只有在各项工作都不会耗费我很多时间的情况下,才会同时在
各个方面工作。(而且,我还在这些工作中都没有什么灵感。)常常发生的情况是,这些任务要比我预想的难,需要更多的精力,时间或者是耐心才能够完成。这时
侯,你就必须要找到一个合适的“休息点”(比如,证明一篇论文中的关键命题;写下讨论中的一个想法,写出来黑板上的某个灵感,或者把一个论文草稿具体完成
到细节。)使得这件工作可以放下来不想一段时间,等到回来的时候依然能够很舒服从断开的地方直接继续原来的工作。应当避免在一件工作完成一半的时候就停下
来,没有找到合适的“休息点”。结果要么这件工作半途而废,要么留在脑袋中不能彻底忘掉,以至于影响其他的工作,而当你把这个问题捡起来的时候,常常要从
前面的什么地方重新开始思考,浪费了时间。但是也无必要拿到一个任务就一次完全的完成,只要找到合适的暂停的地方就好。举一个俗气一些的例子:我在写信的
时候(一般都是我工作状态较差,不能去做严肃的数学问题的时候),我会写完并打印好,装到信封之中,然后就把它们放在固定的地方,而一般不会马上就邮寄出
去(包括很多类似的东西都放在一起)。直到我“固定的地方”堆满了文件,而我有没有什么其他事情好做的时候,我会统一的把他们一起处理。(比如当我的电脑
出点问题的时候,就是个不错的时机。)

一般的讲,有些不需要很集中精力处理的问题最好能够成批的处理,而那些需要集中精力分别应对的任务,就不要被杂事分散了力量。

跟所谓的“休息点”的寻找相关的一点就是要会把又大又长的任务给切碎,让他们变成若干的小问题,而且每个问题又能够有很好的独立性以及自洽性。最好
不过的就是每个小问题都能有他自己的意义。举一个例子就是,我是完全不大可能一次性完整的写出关于庞加莱猜想的证明,而当我把它们分成了19个部分之后,
这些部分都相对可以很好的处理,而且又能够有其独立存在的价值。(而且,我还发现,把自己逼到悬崖边上也常常很有效。我提前宣布要讲庞加莱猜想的证明,这
给我带来很多动力,不至于半途而废。)

(译者注: lectures
on the Poincaré conjecture
是陶教授今年的一门博士课程,课程参考著名的Perelman的三篇论文,田
刚的500页的书,以及朱熹平,曹怀东的300页的论文为教材。由于课程非常艰深,因此上课的具体要求十分简单,只要坚持听课就好,没有任何的作业或者考
试要求。这门课的主页以及所有讲义在陶教授的博客上可以看到,
接在这里
。)

现代文字处理的优点就是,任何时候都可以间断下来,将草稿保存。又很容易找个时间继续。这个blog就是这样。我非常惊叹于在计算机时代之前的那些
数学家们,他们居然能够写出如此高质量的论文甚至是厚厚的一本书。而我即使有秘书的帮助,也会觉得这件事相当的困难。

有时候应当花费大段的时间来学习某种技术,因为这些技术将会在未来不断的被使用。这其中一个好的例子就是数学中的latex编辑软件。如果你打算写
很多的论文,那么就应该花费些时间仔细研究一下这个软件,给自己将来带来方便。好好的学习一下譬如怎么画图怎么做表等等。近来,我试图利用宏定义的方法将
标准的latex码(如\begin{theorem} … \end{theorem} \begin{proof} …
\end{proof}等等)简化,节约了击键次数。每次的时间节约当然很少,但是累计起来,效果就会不同。而且,在工作的时候如果有效率很高的感觉,人
也会精神抖擞,士气高涨。(写长论文的时候就能有体会。)

而在另外的一些情况下,却反而可以对一些任务进行推迟,延误,甚至放下去做些其他的工作。并不是所有的事情都同样的重要。面对一个给定的任务,如果
一个人等到自己的技能更强悍,或者是发生了某件事情使得这个任务变得不再那么重要,那么这个工作显然就变得简单了。比如,我目前关于波映射的论文( papers on wave maps
被延误了好些年,主要是因为我自己没能坚持。然而回想起来,我看到把论文放在那一段时间也有不错的方面。当初我计划中的方法在技术上简直是个噩梦。真的很
有必要等待合适的工具出现,等待对这个领域的理解的加深,然后在对问题有更深刻更有效的处理。  
[也许这篇文章本身就是一个很好的例子。在我的博客中,还有很多的文章草稿我觉得还不成熟,至少现在还不是露面的时候。它们还在等待进一步的修改。并不是
所有的想法或者话题都能够顺利完成,变成一个有意义的结果,参考我的另一篇“利用垃圾箱”。(use
the wastebasket
)]

我的最后一个建议就是要制定一个计划之后要尽最大努力坚持下去,一个不能全心投入的计划还不如干脆没有计划。我的计划包括我自己的PDA和笔记本,
我的e-mail同步。我的各种计划和办公室中其他的设计好的东西保持一致。我还有一个“保留的”黑板,上面写得东西也许只有我自己能完全明白。我并不想
很详细的在这里把这些写的很详细。总之,我已经很习惯于我的这些计划,而且到目前为止一切都非常好(尽管我可不希望有人把我的黑板擦干净!).选择怎样的
计划显然是一个非常隐私的事情,我当然也不大可能对每个人给出最好的建议,只能讲讲自己正在实施的方法.我认为这些方法给我赢得了很多的时间;我不用花费
精力去考虑自己在周二下午3点钟该做些什么;为了目的A,B,C,在X,Y,Z方面都需要做些什么也不用再操心,这样我可以投入更多的精力于理解数学本
身,抑或证明一个有难度的命题,或者什么其他的的工作. 
[I我还发现,当划掉计划中的任务时所带来的心理上的舒适感也会带来动力,不然,有些工作可能会由于没有兴致而最终搁置。]

哦,最后还有一条:有时又需要及时放弃自己的规则而容许有效地调整。比如说,当我在午饭时(随便抓些东西吃吃)为下午的工作做计划时,有时会被同事
或某个访问者所打断,结果要出去吃饭。结果常常发生的情况是,在这顿饭上我得到的比在办公室中更多更好(在数学上或者在其他方面),尽管不是按我事先所预
料的方式。而且这个过程常常更令人愉快 (有时候,脱离会议讲座甚至脱离会议本身去做自己的论文也会有相同效果)

Steve Jobs 在 Stanford 2005毕业典礼上的演讲

This is the text of the Commencement address by Steve Jobs, CEO of Apple Computer and of Pixar Animation Studios, delivered on June 12, 2005.

 I am honored to be with you today at your commencement from one of the finest universities in the world. I never graduated from college. Truth be told, this is the closest I’ve ever gotten to a college graduation. Today I want to tell you three stories from my life. That’s it. No big deal. Just three stories.

 The first story is about connecting the dots.

 I dropped out of Reed College after the first 6 months, but then stayed around as a drop-in for another 18 months or so before I really quit. So why did I drop out?

 It started before I was born. My biological mother was a young, unwed college graduate student, and she decided to put me up for adoption. She felt very strongly that I should be adopted by college graduates, so everything was all set for me to be adopted at birth by a lawyer and his wife. Except that when I popped out they decided at the last minute that they really wanted a girl. So my parents, who were on a waiting list, got a call in the middle of the night asking: "We have an unexpected baby boy; do you want him?" They said: "Of course." My biological mother later found out that my mother had never graduated from college and that my father had never graduated from high school. She refused to sign the final adoption papers. She only relented a few months later when my parents promised that I would someday go to college.

 And 17 years later I did go to college. But I naively chose a college that was almost as expensive as Stanford, and all of my working-class parents’ savings were being spent on my college tuition. After six months, I couldn’t see the value in it. I had no idea what I wanted to do with my life and no idea how college was going to help me figure it out. And here I was spending all of the money my parents had saved their entire life. So I decided to drop out and trust that it would all work out OK. It was pretty scary at the time, but looking back it was one of the best decisions I ever made. The minute I dropped out I could stop taking the required classes that didn’t interest me, and begin dropping in on the ones that looked interesting.

 It wasn’t all romantic. I didn’t have a dorm room, so I slept on the floor in friends’ rooms, I returned coke bottles for the 5¢ deposits to buy food with, and I would walk the 7 miles across town every Sunday night to get one good meal a week at the Hare Krishna temple. I loved it. And much of what I stumbled into by following my curiosity and intuition turned out to be priceless later on. Let me give you one example:Reed College at that time offered perhaps the best calligraphy instruction in the country. Throughout the campus every poster, every label on every drawer, was beautifully hand calligraphed. Because I had dropped out and didn’t have to take the normal classes, I decided to take a calligraphy class to learn how to do this. I learned about serif and san serif typefaces, about varying the amount of space between different letter combinations, about what makes great typography great. It was beautiful, historical, artistically subtle in a way that science can’t capture, and I found it fascinating.

 None of this had even a hope of any practical application in my life. But ten years later, when we were designing the first Macintosh computer, it all came back to me. And we designed it all into the Mac. It was the first computer with beautiful typography. If I had never dropped in on that single course in college, the Mac would have never had multiple typefaces or proportionally spaced fonts. And since Windows just copied the Mac, its likely that no personal computer would have them. If I had never dropped out, I would have never dropped in on this calligraphy class, and personal computers might not have the wonderful typography that they do. Of course it was impossible to connect the dots looking forward when I was in college. But it was very, very clear looking backwards ten years later.

 Again, you can’t connect the dots looking forward; you can only connect them looking backwards. So you have to trust that the dots will somehow connect in your future. You have to trust in something — your gut, destiny, life, karma, whatever. This approach has never let me down, and it has made all the difference in my life.

 My second story is about love and loss.

 I was lucky — I found what I loved to do early in life. Woz and I started Apple in my parents garage when I was 20. We worked hard, and in 10 years Apple had grown from just the two of us in a garage into a $2 billion company with over 4000 employees. We had just released our finest creation — the Macintosh — a year earlier, and I had just turned 30. And then I got fired. How can you get fired from a company you started? Well, as Apple grew we hired someone who I thought was very talented to run the company with me, and for the first year or so things went well. But then our visions of the future began to diverge and eventually we had a falling out. When we did, our Board of Directors sided with him. So at 30 I was out. And very publicly out. What had been the focus of my entire adult life was gone, and it was devastating.

 I really didn’t know what to do for a few months. I felt that I had let the previous generation of entrepreneurs down – that I had dropped the baton as it was being passed to me. I met with David Packard and Bob Noyce and tried to apologize for screwing up so badly. I was a very public failure, and I even thought about running away from the valley. But something slowly began to dawn on me — I still loved what I did. The turn of events at Apple had not changed that one bit. I had been rejected, but I was still in love. And so I decided to start over.

 I didn’t see it then, but it turned out that getting fired from Apple was the best thing that could have ever happened to me. The heaviness of being successful was replaced by the lightness of being a beginner again, less sure about everything. It freed me to enter one of the most creative periods of my life.

 During the next five years, I started a company named NeXT, another company named Pixar, and fell in love with an amazing woman who would become my wife. Pixar went on to create the worlds first computer animated feature film, Toy Story, and is now the most successful animation studio in the world. In a remarkable turn of events, Apple bought NeXT, I returned to Apple, and the technology we developed at NeXT is at the heart of Apple’s current renaissance. And Laurene and I have a wonderful family together.

 I’m pretty sure none of this would have happened if I hadn’t been fired from Apple. It was awful tasting medicine, but I guess the patient needed it. Sometimes life hits you in the head with a brick. Don’t lose faith. I’m convinced that the only thing that kept me going was that I loved what I did. You’ve got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven’t found it yet, keep looking. Don’t settle. As with all matters of the heart, you’ll know when you find it. And, like any great relationship, it just gets better and better as the years roll on. So keep looking until you find it. Don’t settle.

 My third story is about death.

 When I was 17, I read a quote that went something like: "If you live each day as if it was your last, someday you’ll most certainly be right." It made an impression on me, and since then, for the past 33 years, I have looked in the mirror every morning and asked myself: "If today were the last day of my life, would I want to do what I am about to do today?" And whenever the answer has been "No" for too many days in a row, I know I need to change something.

 Remembering that I’ll be dead soon is the most important tool I’ve ever encountered to help me make the big choices in life. Because almost everything — all external expectations, all pride, all fear of embarrassment or failure – these things just fall away in the face of death, leaving only what is truly important. Remembering that you are going to die is the best way I know to avoid the trap of thinking you have something to lose. You are already naked. There is no reason not to follow your heart.

 About a year ago I was diagnosed with cancer. I had a scan at 7:30 in the morning, and it clearly showed a tumor on my pancreas. I didn’t even know what a pancreas was. The doctors told me this was almost certainly a type of cancer that is incurable, and that I should expect to live no longer than three to six months. My doctor advised me to go home and get my affairs in order, which is doctor’s code for prepare to die. It means to try to tell your kids everything you thought you’d have the next 10 years to tell them in just a few months. It means to make sure everything is buttoned up so that it will be as easy as possible for your family. It means to say your goodbyes.

 I lived with that diagnosis all day. Later that evening I had a biopsy, where they stuck an endoscope down my throat, through my stomach and into my intestines, put a needle into my pancreas and got a few cells from the tumor. I was sedated, but my wife, who was there, told me that when they viewed the cells under a microscope the doctors started crying because it turned out to be a very rare form of pancreatic cancer that is curable with surgery. I had the surgery and I’m fine now.

 This was the closest I’ve been to facing death, and I hope its the closest I get for a few more decades. Having lived through it, I can now say this to you with a bit more certainty than when death was a useful but purely intellectual concept:

 No one wants to die. Even people who want to go to heaven don’t want to die to get there. And yet death is the destination we all share. No one has ever escaped it. And that is as it should be, because Death is very likely the single best invention of Life. It is Life’s change agent. It clears out the old to make way for the new. Right now the new is you, but someday not too long from now, you will gradually become the old and be cleared away. Sorry to be so dramatic, but it is quite true.

 Your time is limited, so don’t waste it living someone else’s life. Don’t be trapped by dogma — which is living with the results of other people’s thinking. Don’t let the noise of others’ opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. They somehow already know what you truly want to become. Everything else is secondary.

 When I was young, there was an amazing publication called The Whole Earth Catalog, which was one of the bibles of my generation. It was created by a fellow named Stewart Brand not far from here in Menlo Park, and he brought it to life with his poetic touch. This was in the late 1960’s, before personal computers and desktop publishing, so it was all made with typewriters, scissors, and polaroid cameras. It was sort of like Google in paperback form, 35 years before Google came along: it was idealistic, and overflowing with neat tools and great notions.

 Stewart and his team put out several issues of The Whole Earth Catalog, and then when it had run its course, they put out a final issue. It was the mid-1970s, and I was your age. On the back cover of their final issue was a photograph of an early morning country road, the kind you might find yourself hitchhiking on if you were so adventurous. Beneath it were the words: "Stay Hungry. Stay Foolish." It was their farewell message as they signed off. Stay Hungry. Stay Foolish. And I have always wished that for myself. And now, as you graduate to begin anew, I wish that for you.

 Stay Hungry. Stay Foolish.

 Thank you all very much

http://news-service.stanford.edu/news/2005/june15/jobs-061505.html 

Stay Hungry. Stay Foolish.——我的白话翻译:求知若渴,虚心好学