作者： 陶哲轩 The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Where phases. For instance, the first phase is characterized by the question 'How can we eat?', the second by the question 'Why do we eat?' and the third by the question, 'Where shall we have lunch?' ( Douglas Adams , The Hitchhiker's Guide to the Galaxy ) One can roughly divide mathematical education into three stages: The pre-rigorous stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years. The rigorous stage, in which one is now taught that in order to do maths properly, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually mean. This stage usually occupies the later undergraduate and early graduate years. The post-rigorous stage, in which one has grown comfortable with all the rigorous foundations of one's chosen field, and is now ready to revisit and refine one's pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the big picture. This stage usually occupies the late graduate years and beyond. The transition from the first stage to the second is well known to be rather traumatic, with the dreaded proof-type questions being the bane of many a maths undergraduate. (See also There's more to maths than grades and exams and methods .) But the transition from the second to the third is equally important, and should not be forgotten. It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that fuzzier or intuitive thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as non-rigorous. All too often, one ends up discarding one's initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one's mathematical education. (Among other things, this can impact one's ability to read mathematical papers; an overly literal mindset can lead to compilation errors when one encounters even a single typo or ambiguity in such a paper.) The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions ; another is to relearn your field . The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. Then you will be able to tackle maths problems by using both halves of your brain at once - i.e. the same way you already tackle problems in real life. See also: Bill Thurston's article On proof and progress in mathematics ; Henri Poincare's Intuition and logic in mathematics ; this speech by Stephen Fry on the analogous phenomenon that there is more to language than grammar and spelling; and Kohlberg's stages of moral development (which indicate (among other things) that there is more to morality than customs and social approval). Added later: It is perhaps worth noting that mathematicians at all three of the above stages of mathematical development can still make formal mistakes in their mathematical writing. However, the nature of these mistakes tends to be rather different, depending on what stage one is at: Mathematicians at the pre-rigorous stage of development often make formal errors because they are unable to understand how the rigorous mathematical formalism actually works, and are instead applying formal rules or heuristics blindly. It can often be quite difficult for such mathematicians to appreciate and correct these errors even when those errors are explicitly pointed out to them. Mathematicians at the rigorous stage of development can still make formal errors because they have not yet perfected their formal understanding, or are unable to perform enough sanity checks against intuition or other rules of thumb to catch, say, a sign error, or a failure to correctly verify a crucial hypothesis in a tool. However, such errors can usually be detected (and often repaired) once they are pointed out to them. Mathematicians at the post-rigorous stage of development are not infallible, and are still capable of making formal errors in their writing. But this is often because they no longer need the formalism in order to perform high-level mathematical reasoning, and are actually proceeding largely through intuition, which is then translated (possibly incorrectly) into formal mathematical language. The distinction between the three types of errors can lead to the phenomenon (which can often be quite puzzling to readers at earlier stages of mathematical development) of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors. (In contrast, when unchecked by a solid intuition, once an error is introduced in an argument by a pre-rigorous or rigorous mathematician, it is possible for the error to propagate out of control until one is left with complete nonsense at the end of the argument.) See this post for some further discussion of such errors, and how to read papers to compensate for them. One can roughly divide mathematical education into three stages: The pre-rigorous stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years. The rigorous stage, in which one is now taught that in order to do maths properly, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually mean. This stage usually occupies the later undergraduate and early graduate years. The post-rigorous stage, in which one has grown comfortable with all the rigorous foundations of one's chosen field, and is now ready to revisit and refine one's pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the big picture. This stage usually occupies the late graduate years and beyond. The transition from the first stage to the second is well known to be rather traumatic, with the dreaded proof-type questions being the bane of many a maths undergraduate. (See also There's more to maths than grades and exams and methods .) But the transition from the second to the third is equally important, and should not be forgotten. It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that fuzzier or intuitive thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as non-rigorous. All too often, one ends up discarding one's initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one's mathematical education. (Among other things, this can impact one's ability to read mathematical papers; an overly literal mindset can lead to compilation errors when one encounters even a single typo or ambiguity in such a paper.) The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions ; another is to relearn your field . The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. Then you will be able to tackle maths problems by using both halves of your brain at once - i.e. the same way you already tackle problems in real life. See also: Bill Thurston's article On proof and progress in mathematics ; Henri Poincare's Intuition and logic in mathematics ; this speech by Stephen Fry on the analogous phenomenon that there is more to language than grammar and spelling; and Kohlberg's stages of moral development (which indicate (among other things) that there is more to morality than customs and social approval). Added later: It is perhaps worth noting that mathematicians at all three of the above stages of mathematical development can still make formal mistakes in their mathematical writing. However, the nature of these mistakes tends to be rather different, depending on what stage one is at: Mathematicians at the pre-rigorous stage of development often make formal errors because they are unable to understand how the rigorous mathematical formalism actually works, and are instead applying formal rules or heuristics blindly. It can often be quite difficult for such mathematicians to appreciate and correct these errors even when those errors are explicitly pointed out to them. Mathematicians at the rigorous stage of development can still make formal errors because they have not yet perfected their formal understanding, or are unable to perform enough sanity checks against intuition or other rules of thumb to catch, say, a sign error, or a failure to correctly verify a crucial hypothesis in a tool. However, such errors can usually be detected (and often repaired) once they are pointed out to them. Mathematicians at the post-rigorous stage of development are not infallible, and are still capable of making formal errors in their writing. But this is often because they no longer need the formalism in order to perform high-level mathematical reasoning, and are actually proceeding largely through intuition, which is then translated (possibly incorrectly) into formal mathematical language. The distinction between the three types of errors can lead to the phenomenon (which can often be quite puzzling to readers at earlier stages of mathematical development) of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors. (In contrast, when unchecked by a solid intuition, once an error is introduced in an argument by a pre-rigorous or rigorous mathematician, it is possible for the error to propagate out of control until one is left with complete nonsense at the end of the argument.) See this post for some further discussion of such errors, and how to read papers to compensate for them. Links:
在我知道的数学博客里， 陶哲轩 ( Terence Tao ) 的博客 What's new 是最好的。它好在其更新之勤快，内容之丰富，和问答之即时。 大家都知道陶哲轩是一位华裔大数学家。他在2000年获颁塞勒姆奖(Salem Prize)，2002年获颁博谢纪念奖(Bocher Prize)，和在2003年获颁克雷研究奖(Clay Research Award)，以表扬他对分析学的贡献，当中包括挂谷猜想 (Kakeya conjecture)和wave map。在2005年，他获得美国数学会的利瓦伊·L·科南特奖 (Levi L. Conant Prize)，澳大利亚数学会奖 (Australian Mathematical Society Medal)，和奥斯特洛斯基奖 (Ostrowski Prize)。2006年，他获得印度拉马努金奖 (SASTRA Ramanujan Prize) 和菲尔兹奖 (Fields Medal)并在国际数学家大会做了一小时报告。2007年，他被选为澳大利亚2007年名人(Australian of the Year)并获得麦克阿瑟奖 (MacArthur Award)。2008年，他获得美国奖励科学家的最高奖艾仑·T·沃特曼奖 (Alan T. Waterman Award)。这样一位大数学家能够如此辛勤地维持一个博客，使我们能近距离看到他思想的火花。这是我们网友们的荣幸。 陶哲轩虽然是华人，但他除了能说一些广东话以外，并不会中文。我看到有人留言希望他写中文，对这样无聊的留言他都不予理睬。但是如果你是正经跟他讨论学术问题，他回答的都很及时。 王昆扬 老师在翻译《 陶哲轩实分析 》过程中就和他有过通讯。陶哲轩表现得非常谦虚。 陶哲轩的博客建在了 WordPress.com ，这是一个很自然的选择，因为 WordPress.com 支持LaTeX。所以写起数学表达式就特别方便。我在自己的英文博客里介绍了 WordPress.com的LaTeX 。当然了，我的英文博客也是建在了那里。不过对於大陆的数学家来说，大家可能有时觉得看他的博客会遇到麻烦，因为有时候，政府的长城-火墙会把 WordPress.com 整个封掉。从这个意义上说，如果陶哲轩能有一个独立博客就好了。 陶哲轩的博客上的有些内容，比如他的新的研究结果，新的领域介绍，新的猜想，他的演讲等，特别高深，不是我们平常人能看的懂的。但也有许多博文很吸引我们。这些包括： 如何在网页上写数学表达式的讨论 ( Displaying mathematics on the Web ， Displaying maths online, II )； 化学元素表 ( Applications-oriented periodic table )； 对新网站“数学溢出”的介绍 ( Math Overflow )； 描写盖尔范德 ( Israel Gelfand )； 数学/统计博客和维基 ( Mathematics/Statistics blogs wiki page )； 数学家需要对博客知道什么 ( What do mathematicians need to know about blogging? )； 一个新的数学博客 ( New polymath blog, and comment ratings )； Google Wave ； WordPress被大陆封杀 ( WordPress blocked again by great firewall of China? )，后面的评论几乎都是大陆中国数学家写的； WordPress对LaTeX的支持 ( WordPress LaTeX bug collection drive )，其中实分析考试模拟试题 ( Sample midterm questions )，还有很多关于他的泛函分析课的注解对於学实分析的同学一定很有益； 一个与飞机场有关的智力测验 ( An airport-inspired puzzle )； 更新Java小应用程序 ( Upgrading old Java applets? )，瞧大数学家对技术也很通； 谈以因特网为载体的科技对学术界的影响 ( A speech for the American Academy of Arts and Sciences )，这篇应该翻译成中文。 陶哲轩博客上“友情连接”也非常丰富。这里就不再多罗嗦了。建议读者自己去点击一遍。我想你是不会发现浪费了时间的。 转自科学网蒋讯老师的博客。 附注: 数学文化 杂志网址主页有一个专栏数学人博客，介绍了不少个人数学博客站点。
作者： 蒋迅 在我知道的数学博客里， 陶哲轩 ( Terence Tao ) 的博客 What's new 是最好的。它好在其更新之勤快，内容之丰富，和问答之即时。 大家都知道陶哲轩是一位华裔大数学家。他在2000年获颁塞勒姆奖(Salem Prize)，2002年获颁博谢纪念奖(Bocher Prize)，和在2003年获颁克雷研究奖(Clay Research Award)，以表扬他对分析学的贡献，当中包括挂谷猜想 (Kakeya conjecture)和wave map。在2005年，他获得美国数学会的利瓦伊L科南特奖 (Levi L. Conant Prize)，澳大利亚数学会奖 (Australian Mathematical Society Medal)，和奥斯特洛斯基奖 (Ostrowski Prize)。2006年，他获得印度拉马努金奖 (SASTRA Ramanujan Prize) 和菲尔兹奖 (Fields Medal)并在国际数学家大会做了一小时报告。2007年，他被选为澳大利亚2007年名人(Australian of the Year)并获得麦克阿瑟奖 (MacArthur Award)。2008年，他获得美国奖励科学家的最高奖艾仑T沃特曼奖 (Alan T. Waterman Award)。这样一位大数学家能够如此辛勤地维持一个博客，使我们能近距离看到他思想的火花。这是我们网友们的荣幸。 陶哲轩虽然是华人，但他除了能说一些广东话以外，并不会中文。我看到有人留言希望他写中文，对这样无聊的留言他都不予理睬。但是如果你是正经跟他讨论学术问题，他回答的都很及时。 王昆扬 老师在翻译《 陶哲轩实分析 》过程中就和他有过通讯。陶哲轩表现得非常谦虚。 陶哲轩的博客建在了 WordPress.com ，这是一个很自然的选择，因为 WordPress.com 支持LaTeX。所以写起数学表达式就特别方便。我在自己的英文博客里介绍了 WordPress.com的LaTeX 。当然了，我的英文博客也是建在了那里。不过对於大陆的数学家来说，大家可能有时觉得看他的博客会遇到麻烦，因为有时候，政府的长城-火墙会把 WordPress.com 整个封掉。从这个意义上说，如果陶哲轩能有一个独立博客就好了。 陶哲轩的博客上的有些内容，比如他的新的研究结果，新的领域介绍，新的猜想，他的演讲等，特别高深，不是我们平常人能看的懂的。但也有许多博文很吸引我们。这些包括： 如何在网页上写数学表达式的讨论 ( Displaying mathematics on the Web ， Displaying maths online, II )； 化学元素表 ( Applications-oriented periodic table )； 对新网站数学溢出的介绍 ( Math Overflow )； 描写盖尔范德 ( Israel Gelfand )； 数学/统计博客和维基 ( Mathematics/Statistics blogs wiki page )； 数学家需要对博客知道什么 ( What do mathematicians need to know about blogging? )； 一个新的数学博客 ( New polymath blog, and comment ratings )； Google Wave ； WordPress被大陆封杀 ( WordPress blocked again by great firewall of China? )，后面的评论几乎都是大陆中国数学家写的； WordPress对LaTeX的支持 ( WordPress LaTeX bug collection drive )，其中实分析考试模拟试题 ( Sample midterm questions )，还有很多关于他的泛函分析课的注解对於学实分析的同学一定很有益； 一个与飞机场有关的智力测验 ( An airport-inspired puzzle )； 更新Java小应用程序 ( Upgrading old Java applets? )，瞧大数学家对技术也很通； 谈以因特网为载体的科技对学术界的影响 ( A speech for the American Academy of Arts and Sciences )，这篇应该翻译成中文。 陶哲轩博客上友情连接也非常丰富。这里就不再多罗嗦了。建议读者自己去点击一遍。我想你是不会发现浪费了时间的。
标签: 陶哲轩
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