文章目录

Lesson One
Lesson Two
Lesson Three
Lesson Four
Lesson Five
Lesson Six
Lesson Seven
Lesson Eight
Lesson Nine
Lesson Ten

今天想为大家带来由 MIT 至今唯一同时担任数学教席与哲学教席的Gian-Carlo Rota（吉安-卡洛·罗塔）教授带来的 10 条人生谏言。

Lesson One

You can and will work at a desk for seven hours straight, routinely.

你可以且将会长期进行 7 小时的伏案工作

多年以来，我一直在教 18.30，一门微分课程，同时也是 MIT 数学系学生最多的一门课。多到什么程度呢？这门课每次都有超过 300 个学生同时在上。

这门课可以锻炼教师在面对很多学生教学时，显得更游刃有余。这门课上，他们说的每一句话都必须发音清晰、洪亮——最好是能说两遍，让台下的每个人都能听得见。黑板上的例子就算不够引人入胜，也要能最大程度上帮助学生理解。

不仅如此，老师们最好每 15 分钟左右能抖个包袱，可以是题外话、笑话、历史上的奇闻逸事，或是对当前在讲解的概念的一种不同寻常的应用。而当一名老师不能达到这些要求时，学生则会以带上书并离开教室的行为作为回应，以表达他们的不满。

然而，尽管老师们已经对课堂效果做了最大的努力，当一学期的时间悄然流逝，要留住学生们在课堂上的注意力越发成了一件难事，更多人开始在课上睡着。

不过从另一个角度来说，这或许也会成为老师们一部分成就感的来源，因为这证明了他们作为老师的的确确没有偷懒——学生们正在为解决繁重的课业以及应对期中考试而整天整夜地学习呢（不是开玩笑）。

每学期上四门科学及工程方面的课对于任何人来说都不是小工作量。正因为如此，几乎所有 MIT 的学生都学到了这一点，同时也是最重要的一点，那就是——拥有进行高强度和持续性工作的能力。

Lesson Two

You learn what you don’t know you are learning.

你在不知不觉中学到了东西。

我从 MIT 学到的第二件事可以用我所教授的另一门课的例子来说明。

这门课是 18.313，一门高等概率论课。这门课很难，难在它不仅把通常一年才能教完的内容浓缩在了一个学期里，还难在它要求学生每周写出一个即便在职业数学家看来也很难的习题作业——困难到每几年这门课的学生做题发现一个新解法时都能发表一篇期刊论文的程度。

这门课的学生们同心协力完成这个习题作业，而其中一些学生比其他人从中受益更多。那些最聪明的学生始终可以完成所有的习题，然后让其他学生抄他们的。

我总是假装为此感到苦恼，但实际上，我知道，相比于去解一个不那么难的问题，当学生努力从同学那里去弄懂一个真正有难度的问题解法时，他们能学到更多。

Lesson Three

By and large, “knowing how” matters more than “knowing what.”

授人以鱼不如授人以渔

半个世纪之前，哲学家吉尔伯特·莱尔论述了“知道如何做”的那些课程与“事实知识”的课程之间的区别。

“知道如何做”的课程包括数学、工程等精确科学、乐器演奏，甚至运动；而“事实知识”的课程则包括社会科学、创造性艺术、人文学科等，以及那些被形容为具有“社会价值”的学科。

在每学期的一开始，学生们会与他们的学术顾问见面并决定自己将要学习的课程，而许多时候，讨论的重心将会被放在学生是否应该把一些“知道如何做”的课程替换为“事实知识”的课，通过这种方法，他们通常可以减轻一些课业负担。

诚然，“事实知识”的课程有时会更加令人记忆深刻。相比一门热力学课程，一次对于美国宪法或《李尔王》的严肃研究很可能在学生的个性与品行上留下更深刻的影响。尽管如此，在 MIT，不管是老师还是学生通常都更看重“知道如何做”的课。

为什么呢？在我看来，这是因为“知道如何做”的课程更易于测试学生的掌握水平。我们可以测试出一个学生是否可以应用量子力学、用法语交流或克隆一个基因。然而，要评估学生对于诗歌的解读、在一个复杂的技术性方案中的谈判，或是对于一个小型且多样的工作群体里社会动力学的研究，则难多了。

在那些易于测试的学科领域，我们可以建立一个大家都认可的高标准，以此评估所有人的掌握能力；而对于在那些不易于精准测试的学科领域，一个人是否掌握、精通了这些知识则只取决于个人的主观判断。

在一些文理学院里，室外运动比一些教室内的学术科目更重要，这背后也有相应的原因。在这些学校，室外运动也许是唯一对于学生在“知道如何做”方面的训练了——只有在这个项目上，他们才能给予精准的掌握程度方面的测试。

在 MIT，运动则只是“爱好”（尽管许多人疯狂地热爱它），因为我们学校提供给了学生许许多多训练他们“知道如何做”方面的活动。

Lesson Four

In science and engineering, you can fool very little of the time.

在科学及工程领域，你很难浑水摸鱼。

人们听到的大多数关于 MIT 毕业生的传言都不那么可靠。然而，我意识到有其中一个传言是真的，那就是，MIT 的学生很天真——至少在统计学意义上是这样的。

举个例子，去年的时候，我们一位被华尔街企业录取的数学系毕业生打电话来和我们抱怨他公司的办公室“政治”像是肥皂剧。而毕业后在与职场首次接触后感到震惊的远远不止他一个，而是许许多多的 MIT 学生。

MIT 的环境——一个以科学客观与理论建构为中心的理想世界，不得不说，它与商业、医疗、法律或应用工程等行业现实之间有一条相当大的鸿沟。

科学或工程方面的教育是一种智性诚实的教育。其中，学生们不可避免要面对的是了解他们到底有没有学会的这一事实。特别是在经历了小测之后，所有 MIT 的本科生都知道，如果他们欺骗自己学会了没有学会的东西，他们一定会付出代价。

在校园里，他们习惯了对自己的或他人的错处、缺陷不加掩饰、直言不讳。不幸的是，这种智性诚实很多时候被认为是天真。

Lesson Five

You don’t have to be a genius to do creative work.

不只有天才才可以做创造性工作。

在浪漫主义时代（十八世纪末至十九世纪），天才是一个被过度强调的概念，这给我们今天的教育带来了不少害处。

那个时代将贝多芬、爱因斯坦、费曼等人塑造成圣人，他们在一次又一次对世界的洞察中不断成功、从未失误，而今天的年轻人又以他们为楷模，这如何不让人泄气呢？那些科学传记总是不能够现实地描述科学家们的个性，以至于让许多人对科学工作产生了误解。

然而，在来到 MIT 之后，年轻人会很快地打破他们自己对于天才的幻想并纠正这些误解。而当他们和其他 MIT 学生一样，开始跟着教授做研究之后，他们又会学到新的积极的一课，那就是，他们的教授自己也可能表现得像是个笨手笨脚的白痴。

在 MIT，人们对卓越和成就的追求随处可见，这样的氛围带来了一种民主的影响，它将教师和学生放在同一水平线上，使能力本身受到赞赏，而不考虑其来源。学生们会发现，许多卓越的想法来自于那些一起工作的科学家和工程师团队，而很少归结于特定的个人。

如果要形容，MIT 的科学工作模式更接近文艺复兴时期大工坊里的艺术家之间的交流融合，而不是孤独的浪漫主义天才形象。

Lesson Six

You must measure up to a very high level of performance.

你一定要为自己立下高标准。

我能想象到一些学生与家长会问：“我（我的孩子）为什么非得在 MIT 学微积分，威斯康星大学不香吗？不论在哪个学校里学习微积分，所学到的知识概念不都是一样的吗？但上 MIT 却要承担更高昂的费用。”

我见过一些对这个问题的回答，但感觉都没回答到点子上。他们可能会认为：

从致力于研究数学分析的人那里学来的微积分知识，会比从没有在这个领域中发表过著作的人学得多。
一些致力于研究数学分析的教师，教学水平会比从没有在这个领域中发表过著作的人要教的好。
有些老师从未对此学科有过更深入的研究，但在传达微积分这门学科的思想上比最杰出的数学家要做得好。
我认为，以上观点都有所偏颇。为什么有条件就一定要在 MIT 读书呢？我认为最关键的一点在于，学科的教学氛围是不一样的。一群有天赋的学生在一起学习是会相辅相成、互相成就的。

因为有了从 MIT 毕业的光环在，毕业生都将受到不一样审视与期待。对高标准的期待在不知不觉中被学生吸收与接纳，并伴随着他们的一生。

Lesson Seven

The world and your career are unpredictable, so you are better off learning subjects of permanent value.

世界和你的职业生涯难以预料，所以最好学习一些能终生受益的学科

有些学生刚踏入 MIT 的校园时就有着清晰的职业规划，有些学生则没有，但实际上，无论是哪种都不是很重要。这个时代，我们一些最前端的计算机学家曾是数学逻辑学的博士，这个专业曾被人认为最不可能应用、但现在成为了软件开发关键的数学分支。与此同时，许多实验分子生物学的领军人物原本是物理学的博士。

仅仅几年间，人们的职业方向就发生了巨变，这样的事清变得越来越常见。

相比我找工作的上个世纪 50 年代的时候，会比现如今更好找到有意义的工作。市场所需要的技能，不论是在研究还是产业方面，都变幻莫测。新的职位被创造，旧的被淘汰。这样想来，现如今的大学生确实有理由对未来感到不安。

因此我建议，大多数 MIT 的本科学生在做专业选择的时候不应该选择那些更容易被技术改变的当下热门职业技能，应选择科学或工程这样的基础领域学科。

Lesson Eight

You are never going to catch up, and neither is anyone else.

你永远也赶不上其他人，而其他人亦赶不上你。

MIT 的学生时常抱怨起课业太重，平心而论，他们没说错。每次学期开始前，我作为指导老师看到学生制定的课表时，总会感叹他们究竟是如何完成如此繁重的课业的。想当年我还是本科生工作量完全没这么大。许多人说，这个时代，我们生活中的闲暇一去不复返了。令人遗憾的是，这是真的，MIT 的教职工也与上面的学生们一样有着沉重的负担。但或许也有一些好事，比如最近一位教职工遇见了进入医学院或法学院的 MIT 毕业生，他们说，相比起过去四年在 MIT 的艰辛，医学院和法学院的工作量少得让他们感到轻松多了。

Lesson Nine

The future belongs to the computer-literate-squared.

未来属于“平方计算机能力者”。

在此之前，已经有很多人说过计算机能力的问题，我猜你不想再听更多了。因此，我在这里想要提出的是“平方计算机能力”这个概念，换句话说就是，更高阶的计算机能力。众所周知，MIT 有许多本科生就读于计算机科学专业，如果不读这个专业，他们中的许多人也掌握了不少可应用于其他领域的计算机能力。而当这些人升上大二时，他们会发现，学校计算机科学的必修课没能提供这个学科的所有内容。——这不是在说课纲有缺漏和不足，相反，MIT 的计算机科学课程大概是这个世界上最前端和先进的了。这是在说，学生们在学习必修课的同时也会发现他们自己的“隐藏课程”。这些课程里包含刚刚被运用的新想法、新技巧，这些想法和技巧就像是野火一般迅速蔓延、传播，很快又带来了新的令人意想不到的应用，而这些最终又会被收录到学校的官方课程中。如果一个计算机科学家想在这个领域保持领先地位，ta 就必须不断更新这些“隐藏课程”、与时俱进。而那些不拥有“平方计算机能力”、没能够成为计算机科学家的人则最终只能成为应用他人想法的程序员。

Lesson Ten

Mathematics is still the queen of the sciences.

数学仍然凌驾于所有科学之上。

在前九个道理中，我尝试从一个客观公正的角度来看待 MIT 这个整体，而最后，我决定为我自己的领域——数学，打个广告作为总结。每当有本科生问我，他们是否应当选择数学专业而不是一个其他任意专业 X 时，我总是会以下面这句话作答：“如果你主修数学，你可以在任何时候换别的专业学，但反之，就行不通了。”

附原文：

10 Lessons of an MIT Education

by Gian-Carlo Rota

Lesson One: You can and will work at a desk for seven hours straight, routinely. For several years, I have been teaching 18.30, differential equation, the largest mathematics course at MIT, with more than 300 students. The lectures have been good training in dealing with mass behavior. Every sentence must be perfectly enunciated, preferably twice. Examples on the board must be relevant, if not downright fascinating. Every 15 minutes or so, the lecturer is expected to come up with an interesting aside, joke, historical anecdote, or unusual application of the concept at hand. When a lecturer fails to conform to these inexorable requirements, the students will signify their displeasure by picking by their books and leaving the classroom.

Despite the lecturer’s best efforts, however, it becomes more difficult to hold the attention of the students as the term wears on, and they start falling asleep in class under those circumstances should be a source of satisfaction for a teacher, since it confirms that they have been doing their jobs. There students have been up half the night-maybe all night-finishing problem sets and preparing for their midterm exams.

Four courses in science and engineering each term is a heavy workload for anyone; very few students fail to learn, first and foremost, the discipline of intensive and constant work.

Lesson Two: You learn what you don’t know you are learning. The second lesson is demonstrated, among other places, in 18.313, a course I teach in advanced probability theory. It is a difficult course, one that compresses the material typically taught in a year into one term, and it includes weekly problem sets that are hard, even by the standards of professional mathematicians. (How hard is that? Well, every few years a student taking the course discovers a new solution to a probability problem that merits publication as a research paper in a refereed journal.)

Students join forces on the problem sets, and some students benefit more than others from these weekly collective efforts. The most brilliant students will invariably work out all the problems and let other students copy, and I pretend to be annoyed when I learn that this has happened. But I know that by making the effort to understand the solution of a truly difficult problem discovered by one of their peers, students learn more than they would by working out some less demanding exercise.

Lesson Three: By and large, “knowing how” matters more than “knowing what.” Half a century ago, the philosopher Gilbert Ryle discussed the difference between “knowing how” courses are those in mathematics, the exact sciences, engineering, playing a musical instrument, even sports. “Knowing what” courses are those in the social sciences, the creative arts, the humanities, and those aspects of a discipline that are described as having social value.

At the beginning of each term, students meet with their advisors to decide on the courses each will study, and much of the discussion is likely to resolve around whether a student should lighten a heavy load by substituting one or two “knowing what” courses in place of some stiff “knowing how” courses.

To be sure, the content of “knowing what” courses if often the most memorable. A serious study of the history of United States Constitution or King Lear may well leave a stronger imprint on a student’s character than a course in thermodynamics. Nevertheless, at MIT, “knowing how” is held in higher esteem than “knowing what” by faculty and students alike. Why?

It is my theory that “knowing how” is revered because it can be tested. One can test whether a student can apply quantum mechanics, communicate in French, or clone a gene. It is much more difficult to asses an interpretation of a poem, the negotiation of a complex technical compromise, or grasp of the social dynamics of a small, diverse working group. Where you can test, you can set a high standard of proficiency on which everyone is agreed; where you cannot test precisely, proficiency becomes something of a judgment call.

At certain liberal arts colleges, sports appear to be more important than classroom subjects, and with good reason. A sport may be the only training in “knowing how”-in demonstrating certifiable proficiency-that a student undertakes at those colleges. At MIT, sports are a hobby (however passionately pursued) rather than a central focus because we offer a wide range of absorbing “knowing how” activities.

Lesson Four: In science and engineering, you can fool very little of the time. Most of the sweeping generalizations one hears about MIT undergraduates are too outrageous to be taken seriously. The claim that MIT students are naive, however, has struck me as being true, at least in a statistical sense.

Last year, for example, one of our mathematics majors, who had accepted a lucrative offer of employment from a Wall Street firm, telephoned to complain that the politics in his office was “like a soap opera.” More than a few MIT graduates are shocked by their first contact with the professional world after graduation. There is a wide gap between the realities of business, medicine, law, or applied enginering, for example, and the universe of scientific objectivity and theoretical constructs that is MIT.

An education in engineering and science is an education in intellectual honesty. Students cannot avoid learning to acknowledge whether or not they have really learned. Once they have taken their first quiz, all MIT undergraduates know dearly they will pay if they fool themselves into believing they know more than is the case.

On campus, they have been accustomed to people being blunt to a fault about their own limitations-or skills-and those of others. Unfortunately, this intellectual honesty is sometimes interpreted as naivete.

Lesson Five: You don’t have to be a genius to do creative work. The idea of genius elaborated during the Romantic Age (late 18th and 19th centuries) has done harm to education. It is demoralizing to give a young person role models of Beethoven, Einstein, and Feynman, presented as saintly figures who moved from insight to insight without a misstep. Scientific biographies often fail to give a realistic description of personality, and thereby create a false idea of scientific work.

Young people will correct any fantasies they have about genius, however, after they come to MIT. As they start doing research with their professors, as many MIT undergraduates do, they learn another healthy lesson, namely, a professor may well behave like a fumbling idiot.

The drive for excellence and achievement that one finds everywhere at MIT has the democratic effect of placing teachers and students on the same level, where competence is appreciated irrespective of its provenance, Students learn that some of the best ideas arise in groups of scientists and engineers working together, and the source of these ideas can seldom be pinned on specific individuals. The MIT model of scientific work is closer to the communion of artists that was found in the large shops of the Renaissance than to the image of the lonely Romantic genius.

Lesson Six: You must measure up to a very high level of performance. I can imagine a propective student or parent asking, “Why should I (or my child) take calculus at MIT rather than at Oshkosh College? Isn’t the material practically identical, no matter where it is taught, while the cost varies a great deal?”

One answer to this question would be following: One learns a lot more when taking calculus from someone who is doing research in mathematical analysis than from someone who has never published a word on the subject. But this is not the answer; some teachers who is doing research in mathematical analysis than from someone who has never published a word on the subject. But this is not the answer; some teachers who have never done any research are much better at conveying the ideas of calculus than the most brilliant mathematicians.

What matters most is the ambiance in which the course is taught; a gifted student will thrive in the company of other gifted students. An MIT undergraduate will be challenged by the level of proficiency that is expected of everyone at MIT, students and faculty. The expectation of high standards is unconsciously absorbed and adopted by the students, and they carry it with them for life.

Lesson seven: The world and your career are unpredictable, so you are better off learning subjects of permanent value. Some students arrive at MIT with a career plan, many don’t, but it actually doesn’t matter very much either way. Some of the foremost computer scientists of our day received their doctorates in mathematical logic, a branch of mathematics that was once considered farthest removed from applications but that turned out instead to be the key to the development of present-day software. A number of the leading figures in experimental molecular biology received their doctorates in physics. Dramatic career shifts that only a few years ago were the exception are becoming common.

Our students will have a harder time finding rewarding jobs than I had when I graduated in the fifties. The skills the market demands, both in research and industry, are subject to capricious shifts. New professions will be created, and old professions will become obsolete with the span of a few years. Today’s college students have good cause to be apprehensive about future.

The curriculum that most undergraduates at MIT choose to follow focuses less on current occupational skills than on those fundamental areas of science and engineering that at least likely to be affected by technological changes.

Lesson Eight: You are never going to catch up, and neither is anyone else. MIT students often complain of being overworked, and they are right. When I look at the schedules of courses my advisees propose at the beginning of each term, I wonder how they can contemplate that much work. My workload was nothing like that when I was an undergraduate.

The platitudes about the disappearance of leisure are, unfortunately, true, and faculty members at MIT are as heavily burdened as students. There is some satisfaction, however, for a faculty member in encountering a recent graduate who marvels at the light work load they carry in medical school or law school relative to the grueling schedule they had to maintain during their four years at MIT.

Lesson Nine: The future belongs to the computer-literate-squared. Much has been said about computer literacy, and I suspect you would prefer not to hear more on the subject. Instead, I would like to propose the concept computer-literacy-squared, in other words computer literacy to second degree.

A large fraction of MIT undergraduates major in computer science or at least acquire extensive computer skills that are applicable in other fields. In their second year, they catch on to the fact that their required courses in computer science do not provide the whole story. Not because of deficiencies in the syllabus; quite the opposite. The undergraduate curriculum in computer science at MIT is probably the most progressive and advanced such curriculum anywhere. Rather, the students learn that side by side with required courses there is another, hidden curriculum consisting of new ideas just coming into use, new techniques and that spread like wildfire, opening up unsuspected applications that will eventually be adopted into the official curriculum.

Keeping up with this hidden curriculum is what will enable a computer scientist to stay ahead in the field. Those who do not become computer scientists to the second degree risk turning into programmers who will only implement the ideas of others.

Lesson Ten: Mathematics is still the queen of the sciences. Having tried in lessons one through nine to take an unbiased look at the big MIT picture, I’d like to conclude with a plug for my own field, mathematics.

When an undergraduate asks me whether he or she should major in mathematics rather than in another field that I will simply call X, my answer is the following: “If you major in mathematics, you can switch to X anytime you want to, but not the other way around.”

Alumni who return to visit invariably complain of not having taken enough math courses while they were undergraduates. It is a fact, confirmed by the history of science since Galileo and Newton, that the more theoretical and removed from immediate applications a scientific topic appears to be, the more likely it is to eventually find the most striking practical applications. Consider number theory, which only 20 years ago was believed to be the most useless chapter of mathematics and is today the core of computer security. The efficient factorization of integers into prime numbers, a topic of seemingly breathtaking obscurity, is now cultivated with equal passion by software desigers and code breakers.

I am often asked why there are so few applied mathematicians in the department at MIT. The reason is that all of MIT is one huge applied mathematics department; you can find applied mathematicians in practicially every department at MIT except mathematics.

From the Association of Alumni and Alumnae of MIT April 1997

参考资料：

10 Lessons of an MIT Education by Gian-Carlo Rota

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