博文
应用数学与理论工程学（转摘)
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本文是转摘著名的科学家何毓琦先生的博客中的两篇文章，适合数学和物理，工程等专业的工作者阅读。感谢何先生为我们提供了精美的博文。
应用数学与理论工程学(原文及译文）
Applied Mathematics and Theoretical Engineering
Many years ago (at least ten or more) a world renowned mathematician said to me “the kind of mathematics you do don’t even qualify to be called Mathematics”. I did not feel insulted since I learned shortly after I started my career that I’ll never be a good mathematician in his sense even though my ph.d degree is in applied mathematics. Just look at the example of the Fermat’s last theorem. Even though the statement of the theorem is easy to understand, the proof consumed almost the entire life of one person and used exoteric mathematical topics that most of us never heard of. World class (pure) mathematics departments live and converse in subjects so different from what popular conceptions of mathematics are and applied mathematicians do. Fields medal are awarded on topics we have not heard about and do not understand.
On the other hand, the usefulness of “mathematics” to science and technology is unquestionable. Modern civilization will not be where it is today without ‘mathematics”. This is the realm of applied mathematics. Back in late 19th century and the early part of 20th century, applied mathematics usually means mathematics applied to physical phenomena. The famed Cambridge University in England has a Department of Applied Mathematics and Theoretical Physics (DAMTP) where the Lucasian chair once occupied by Isaac Newton (current holder Stephen Hawkings) resides. However, since 1950 and the rise of computers, mathematics increasingly becomes applied to manmade phenomena (electric power grid, air traffic systems, manufacturing automation, etc). Here the nature of the problem undergoes a subtle change. In physical phenomena, you live with whatever God given existing laws whether it is fluid mechanics, electro-magnetic waves, thermodynamics, etc. Mathematics and mathematical models are primarily used to explain and understand such physics. It is a descriptive science used to explain “how thing are?” In mathematics applied to manmade phenomena, we are dealing often not with existing objects but objects we have only imagined. These objects behave according to manmade rules of operation which can themselves be changed to suit our desires. Thus we are creating as well as controlling such systems. Mathematics is used to design such systems. It is a prescriptive science more concerned with “how things should be?”. Here the importance of “rigor” in mathematics comes in, i.e., you better be absolutely sure what you are talking about or creating is only based on “truthful axioms” and “logical deductions”. Mere “Intuition” can be dangerous. The rationale here is that without “rigor” you may be building “空中楼阁” or a “house of cards”. Otherwise, how can you trust the outputs of a computer? On the other hand in mathematics of physical phenomenon you always have reality to face or to guide you. This dichotomy in the study of applied mathematics (physical vs. manmade) often comes out in the form of arguments for or against the requirement of “calculus” for computer science students. One faction takes the viewpoint that calculus is the basis of all applied mathematics while the other much prefer utilizing the students’ time to study combinatorics and abstract algebra (sometimes call discrete mathematics).
In the operations research, control and system field however, we enter the “twilight zone”. We are both designing (creating) systems as well as controlling existing physical systems with equal emphasis on both tasks. Here often the arguments and disputes for and against “rigor” become heated and confused. In my opinion, the need for “rigor” and the utility of “intuition” depend on what you do. If you are proving the convergence of a newly untested computing algorithm on a mathematical problem you created, you’d better be sure that the proof is rigorous. On the other hand, many well known and applicable results, such as the Draper prize winning Kalman filter, have been successfully used in situations where we know the assumptions and conditions are only approximately true or not even true. Here intuition and experience are as necessary if not more so than “rigor”. The difference here is not between absolute right or wrong but between applied mathematics with a big “A’; versus big “M”. Mutual respect and understanding will go a long way towards resolving conflicts.
Putting this in personal terms, I do not feel bad when I am accused of being not “rigorous” in my papers. I deal with real world problems and/or problem-driven methodology research. Reality always provides sanity checks for me and prevents me from being dead wrong. Ideas, intuition, and practical usefulness are important to me in my work. But I always insist that my engineering students be conceptually rigorous and understand what is meant by a “proof” (easily said than done). A working knowledge of the language of mathematics so that one is capable of accessing mathematical literature is useful regardless whether or not you are application or theory oriented (just like English is useful for S&T work regardless of your mother tongue) Instead of letting applied mathematics be synonymous with theoretical physics as in olden days, I’d like to suggest it can also be equated to “theoretical engineering”. The former notion of applied mathematics is more concerned with “descriptive science” while the latter (mathematics applied to manmade systems) principally deals with “prescriptive science”. But I cannot agree with some of my colleagues in the system discipline who worship “rigor” above everything else and denigrate perfectly good idea and applications as unworthy. Have they ever thought about letting the leading mathematics department judge the mathematical quality of their own works? The application side has already ignored them as impractical. The metaphor of an average or even stupid two-eye person can crown himself as king in the world of one eyes is applicable here. But he'd be just an ordinary person if he goes back to the two-eye world. Thus, to young scholars starting out in academic engineering departments, don’t become seduced by the beauty of mathematics alone and forget the ultimate customer you serve. You may end up not being appreciated by either group.
Of course nothing is strictly black or white. I deliberately exaggerated the contrasts above to clarify the distinction. Pure mathematicians of world renowned status can still give popular science talks accessible to average scientists. One of the best and recent examples is the 2006 Clay Mathematical Public Lecture http://www.claymath.org/public_lectures/sipser.php
It is well worth a hour of your time to view the lecture.
应用数学与理论工程学
许多年前（至少十年或者更久以前），一位世界闻名的数学家对我说：“你做的那类数学甚至不够资格称作‘数学’。”我不觉得受到了侮辱，我在职业生涯开始后不久，就意识到尽管我的博士学位是应用数学，但是我永远也不会成为一名他所认可的那种数学家。只要看看费马大定理就知道了。尽管该定理的表述本身很容易理解，但它的证明几乎耗尽了一个人毕生的心血，而且要用到绝大多数人闻所未闻的罕见的数学问题。世界级的（纯）数学系探讨的问题与人们熟悉的数学概念以及应用数学家所做的事情相去甚远。菲尔茨奖奖励的正是这些我们闻所未闻也一窍不通的数学问题。
另一方面，“数学”对于科学技术的重要性是毋庸置疑的。没有“数学”就没有今天的现代文明。这就是应用数学的领域。19世纪末20世纪初，当时的应用数学通常指的是应用于物理现象的数学。著名的英国剑桥大学有一个应用数学与理论物理系（DAMTP），牛顿曾经担任该系的卢卡斯数学教授席位（现在是霍金）。然而，自从20世纪50年代计算机兴起以来，数学日益被应用于人造现象（比如电力网、空中交通系统、制造业自动化等等）。如此一来，问题的本质发生了微妙的变化。在物理现象中，不论是流体力学、电磁波还是热力学等，人们面对的都是上帝设定的已有规律。人们主要用数学和数学模型来解释和理解这些物理问题。这是一种用来解释“事物的本来面目”的描述性科学。而数学一旦应用于人造现象，我们要面对的就往往不是业已存在的事物，而是人们想像出来的事物。它们依照人类制定的规则来运行，而这些规则本身可以按照我们的要求做出改变。因此我们不但创造了这些系统，而且控制了这些系统。数学则被用来设计这些系统。这是一种指令性科学，更关注“事情应该是怎样的”。在这里，数学“严密性”的重要性就体现了出来，也就是说，你最好完全确信你讨论的或者创造的事物是基于“真实的公理”和“合乎逻辑的演绎”。仅仅靠“直觉”是很危险的。这其中的逻辑是这样的：如果缺乏“严密性”，你在建造的或许就仅仅是“空中楼阁”。否则的话，如何能相信计算机的计算结果？另一方面，在针对物理现象的数学中，人们常常有实体可以用来参考或者指导自己。而对计算机科学领域的学生而言，应用数学研究中的二分法（自然的vs人造的）通常体现在是否需要“微积分”介入的争执上。其中一派人认为，微积分是所有应用数学的基础，而其他人则更倾向于利用学生时期来研究组合数学和抽象代数（有时称为离散数学）。
然而，在运筹学、控制与系统领域，我们进入了所谓的“朦胧地带”。我们同时设计（创造）系统和控制已有的物理系统，而且，把这两项工作放在同等重要的地位。这里，关于数学“严密性”的争论常常变得白热化，让人困惑。在我看来，对“严密性”的需求和对“直觉”的利用取决于你做的是什么。如果你试图在一个你创造的数学问题上证明一个未经检验的计算法则的收敛性，你最好保证该证明是无懈可击的。另一方面，许多具有应用前景的的著名的研究结果，比如获得德雷珀奖的卡尔曼滤波器，已经被成功用到很多情况中，其中的假设和条件只是大概真实甚至是不怎么真实的。在这里，直觉和经验即使不比“严密性”更重要，也至少同样必不可少。这其中的差异不在于谁绝对正确或者绝对错误，而是applied mathematics两个单词的首字母是A大写了还是M大写了（译者注：也就是我们强调的是“应用”还是强调的是“数学”）。只有相互尊重和理解才能让我们在解决这些争论的道路上走得更远。
就我个人而言，人家批评我的论文中不够“严密”我也不会生气。我处理的是现实世界中的问题和/或者问题驱动的方法学研究。我总是能够通过现实来检测我的研究的合理性，让我不至于铸成大错。在我的工作中，想法、直觉和实效性对我都很重要。但我一直强调，我的工程学学生要在概念上严密，并且明白所谓“证据”到底指的是什么（说着容易做着难）。不管你的研究是更偏应用还是更偏理论，熟练地掌握数学语言都是很有用的，这样你才能看懂数学文献（这就好比不论你的母语是什么，英语对于科学技术研究都是很有用的）。我觉得应用数学不必像过去那样只是理论物理的同义词，我建议把它等同于“理论工程”。应用数学的前一个概念更关注于“描述性科学”，而后者（应用于人造系统的数学）主要涉及“指令性科学”。不过，我无法认同系统领域的一些同事，他们认为“严密性”高于一切，同时把一些很好的想法和应用贬得一钱不值。他们考虑过让世界上最好的数学系来评价一下他们自己工作的数学性有多好吗？搞应用的人已经不搭理他们了，因为他们太不切实际。我们不妨打个合适的比喻，一个普通甚至是愚蠢的双眼人，在独眼人的王国里可以自冕为王，但是一旦他回到正常人的世界，他不过是个普通人罢了。因此，对于刚刚在大学的工程系里起步的年轻学者而言，不要被数学的美丽诱惑，而忘记了你所要服务的最终客户。你的下场可能是不被任何一方赏识。
当然，没有事情是绝对对或者错的。我有意夸大了这个对比，是为了澄清二者之间的区别。世界闻名的纯粹的数学家仍然可以做出一般科学家都懂的通俗科学报告，最近的也是最好的例子之一就是2006年克雷数学公共演讲：http://www.claymath.org/public_lectures/sipser.php
花一个小时的时间看看这些演讲内容是很值得的。
（科学网 任霄鹏译 何姣校）
数学、分析与工程学（2）（原文及译文）
Mathematics, Analysis, and Engineering (2)
Below is a guest Blog by Professor Xiren Cao of the Science and Technology University of Hong Kong. This article further clarifies and articulates the relationship between mathematics and engineering. I thank him for this timely piece.
Prof. Ho’s article “Applied Mathematics and Theoretical Engineering” brought up an important question about what is the relation between mathematics and engineering.
This question was asked quite often by young researchers, and there is a considerable amount of confusion in the research community. While Prof. Ho’s article gave a comprehensive answer to this question, I may add some words from my own experience. (Over the years, my research style has been influenced by Prof. Ho greatly.)
First, I always distinguish between “mathematics” and “analysis”. You may do research in engineering without deep mathematics, but you cannot do research in any subject without analysis. Analysis is a logical derivation/interpretation of the problem you are trying to solve (at a level Prof. Ho calls “conceptually rigorous”). Mathematics is a tool that helps you to express your analysis clearly, logically, and concisely. In most engineering problems, you are not developing mathematics (in a sense of pure mathematics); your work may be of a great value in engineering (such as the Kalman filter), but of no significant contribution from a pure mathematician’s point of view. Mathematics is important because the process of expressing your ideas at a proper level of mathematics helps you to clarify your ideas and to find logical flaws, if any, in your analysis. Because of this, one cannot live without mathematics (children study math from grade 1).
Second, in engineering, innovative ideas, intuitions, and motivations are always the most important thing. Purely working on mathematical expressions may extend the existing results and/or lead to more beautiful forms (those are important jobs!), but without conceptual breakthrough, it may hardly leads to new methodologies in engineering.
Third, do not over use mathematics. That is, if an engineering problem can be described clear with a lower level mathematics, do not use more complicated ones.
As Prof. Ho said, ``A working knowledge of the language of mathematics so that one is capable of accessing mathematical literature is useful regardless whether or not you are application or theory oriented (just like English is useful for S&T work regardless of your mother tongue).” Nowadays, many engineering papers are written in mathematical languages and without some fundamental understanding of it one can even hardly read these engineering papers. However, when you read such papers, try to translate back to intuitions and engineering ideas behind the heavy mathematics. Otherwise, you may not be able to remember the content of such a paper afterwards. Another reason that you may need to learn some (never enough) mathematics is because one day you may need to use some tools in mathematics to analyze your engineering problems but it may not be in your brain. I always read some new (to me) mathematics books or chapters every few years to enhance my knowledge. I suggest young researchers do the same if possible.
数学、分析与工程学（2）
下面的博文是香港科技大学曹希仁教授所写，它进一步澄清和阐明了数学与工程学之间的关系。我十分感谢他这篇及时的文章。
何教授的博文“应用数学与理论工程学”提出了一个重要的问题，即数学与工程学之间有怎样的关系。
这个问题经常被年轻的研究人员问及，科研群体中也存在相当多的混淆。而何教授的博文就该问题给出了一个全面的答案，我可以根据自身的经历来补充一些内容。（这些年来，我的研究风格已经深受何教授影响。）
首先，我常常将“数学”和“分析”区别开来。进行工程学研究可以不用深奥的数学，但无论研究什么主题都不能离开分析。分析就是对试图解决的某一问题的逻辑推导/解释（在何教授所谓的“概念上严密”的层面上）。数学作为一种工具，有助于清晰、逻辑、简明地表达一个人的分析。在大多数工程学问题中，一个人不是在发展数学（就纯数学意义而言），他的工作可能在工程学上具有重要价值（比如卡尔曼滤波器），但在纯粹的数学家眼里并没有什么太大贡献。数学很重要，因为在合适的数学层面上表达自己想法的过程，有助于一个人明晰自己的思路，并找出分析中可能存在的逻辑缺陷。正因为如此，一个人无法离开数学（小孩通常从一年级开始学数学）。
其次，在工程学领域，创新性的想法、直觉以及动机往往是最重要的。纯粹地在数学表达上下功夫或许可以拓展现有的结果或者产生更漂亮的表达形式（这些是重要的工作），但没有概念上的突破，这几乎不可能导致工程学领域新的方法学诞生。
再次，不要过度地使用数学。也就是说，如果一个工程学问题能够以较低水平的数学描述清楚的话，就不要使用那些更为复杂的数学。
正如何教授所言，“不管你的研究是更偏应用还是更偏理论，熟练地掌握数学语言都是很有用的，这样你才能看懂数学文献（这就好比不论你的母语是什么，英语对于科学技术研究都是很有用的）。”眼下，许多工程学论文都是用数学语言写就，没有一些基础性的了解，一个人甚至无法读懂这些论文。然而，在读这样的论文时，应该试着将繁重的数学背后的思想翻译回直觉或者工程学概念。否则，他可能无法在日后记得这样一篇论文的内容。而需要学习一些（永无止境）数学的另一个原因就是，终有一天，你要用一些数学工具来分析自己的工程学问题，而你可能还没有意识到这一点。我常常每隔几年就看一些新的（对我来说）数学书籍或者章节，来增加自己的知识。我建议年轻的研究人员如果可能的话，也一起这样做。（科学网 任霄鹏译 何姣校）
https://blog.sciencenet.cn/blog-81613-40960.html
上一篇：２０张真实的照片,看完大家都明白的（非原创，但希望大家能转摘）
下一篇：关于男生追女生的数学模型
应用数学与理论工程学(原文及译文）
Applied Mathematics and Theoretical Engineering
Many years ago (at least ten or more) a world renowned mathematician said to me “the kind of mathematics you do don’t even qualify to be called Mathematics”. I did not feel insulted since I learned shortly after I started my career that I’ll never be a good mathematician in his sense even though my ph.d degree is in applied mathematics. Just look at the example of the Fermat’s last theorem. Even though the statement of the theorem is easy to understand, the proof consumed almost the entire life of one person and used exoteric mathematical topics that most of us never heard of. World class (pure) mathematics departments live and converse in subjects so different from what popular conceptions of mathematics are and applied mathematicians do. Fields medal are awarded on topics we have not heard about and do not understand.
On the other hand, the usefulness of “mathematics” to science and technology is unquestionable. Modern civilization will not be where it is today without ‘mathematics”. This is the realm of applied mathematics. Back in late 19th century and the early part of 20th century, applied mathematics usually means mathematics applied to physical phenomena. The famed Cambridge University in England has a Department of Applied Mathematics and Theoretical Physics (DAMTP) where the Lucasian chair once occupied by Isaac Newton (current holder Stephen Hawkings) resides. However, since 1950 and the rise of computers, mathematics increasingly becomes applied to manmade phenomena (electric power grid, air traffic systems, manufacturing automation, etc). Here the nature of the problem undergoes a subtle change. In physical phenomena, you live with whatever God given existing laws whether it is fluid mechanics, electro-magnetic waves, thermodynamics, etc. Mathematics and mathematical models are primarily used to explain and understand such physics. It is a descriptive science used to explain “how thing are?” In mathematics applied to manmade phenomena, we are dealing often not with existing objects but objects we have only imagined. These objects behave according to manmade rules of operation which can themselves be changed to suit our desires. Thus we are creating as well as controlling such systems. Mathematics is used to design such systems. It is a prescriptive science more concerned with “how things should be?”. Here the importance of “rigor” in mathematics comes in, i.e., you better be absolutely sure what you are talking about or creating is only based on “truthful axioms” and “logical deductions”. Mere “Intuition” can be dangerous. The rationale here is that without “rigor” you may be building “空中楼阁” or a “house of cards”. Otherwise, how can you trust the outputs of a computer? On the other hand in mathematics of physical phenomenon you always have reality to face or to guide you. This dichotomy in the study of applied mathematics (physical vs. manmade) often comes out in the form of arguments for or against the requirement of “calculus” for computer science students. One faction takes the viewpoint that calculus is the basis of all applied mathematics while the other much prefer utilizing the students’ time to study combinatorics and abstract algebra (sometimes call discrete mathematics).
In the operations research, control and system field however, we enter the “twilight zone”. We are both designing (creating) systems as well as controlling existing physical systems with equal emphasis on both tasks. Here often the arguments and disputes for and against “rigor” become heated and confused. In my opinion, the need for “rigor” and the utility of “intuition” depend on what you do. If you are proving the convergence of a newly untested computing algorithm on a mathematical problem you created, you’d better be sure that the proof is rigorous. On the other hand, many well known and applicable results, such as the Draper prize winning Kalman filter, have been successfully used in situations where we know the assumptions and conditions are only approximately true or not even true. Here intuition and experience are as necessary if not more so than “rigor”. The difference here is not between absolute right or wrong but between applied mathematics with a big “A’; versus big “M”. Mutual respect and understanding will go a long way towards resolving conflicts.
Putting this in personal terms, I do not feel bad when I am accused of being not “rigorous” in my papers. I deal with real world problems and/or problem-driven methodology research. Reality always provides sanity checks for me and prevents me from being dead wrong. Ideas, intuition, and practical usefulness are important to me in my work. But I always insist that my engineering students be conceptually rigorous and understand what is meant by a “proof” (easily said than done). A working knowledge of the language of mathematics so that one is capable of accessing mathematical literature is useful regardless whether or not you are application or theory oriented (just like English is useful for S&T work regardless of your mother tongue) Instead of letting applied mathematics be synonymous with theoretical physics as in olden days, I’d like to suggest it can also be equated to “theoretical engineering”. The former notion of applied mathematics is more concerned with “descriptive science” while the latter (mathematics applied to manmade systems) principally deals with “prescriptive science”. But I cannot agree with some of my colleagues in the system discipline who worship “rigor” above everything else and denigrate perfectly good idea and applications as unworthy. Have they ever thought about letting the leading mathematics department judge the mathematical quality of their own works? The application side has already ignored them as impractical. The metaphor of an average or even stupid two-eye person can crown himself as king in the world of one eyes is applicable here. But he'd be just an ordinary person if he goes back to the two-eye world. Thus, to young scholars starting out in academic engineering departments, don’t become seduced by the beauty of mathematics alone and forget the ultimate customer you serve. You may end up not being appreciated by either group.
Of course nothing is strictly black or white. I deliberately exaggerated the contrasts above to clarify the distinction. Pure mathematicians of world renowned status can still give popular science talks accessible to average scientists. One of the best and recent examples is the 2006 Clay Mathematical Public Lecture http://www.claymath.org/public_lectures/sipser.php
It is well worth a hour of your time to view the lecture.
应用数学与理论工程学
许多年前（至少十年或者更久以前），一位世界闻名的数学家对我说：“你做的那类数学甚至不够资格称作‘数学’。”我不觉得受到了侮辱，我在职业生涯开始后不久，就意识到尽管我的博士学位是应用数学，但是我永远也不会成为一名他所认可的那种数学家。只要看看费马大定理就知道了。尽管该定理的表述本身很容易理解，但它的证明几乎耗尽了一个人毕生的心血，而且要用到绝大多数人闻所未闻的罕见的数学问题。世界级的（纯）数学系探讨的问题与人们熟悉的数学概念以及应用数学家所做的事情相去甚远。菲尔茨奖奖励的正是这些我们闻所未闻也一窍不通的数学问题。
另一方面，“数学”对于科学技术的重要性是毋庸置疑的。没有“数学”就没有今天的现代文明。这就是应用数学的领域。19世纪末20世纪初，当时的应用数学通常指的是应用于物理现象的数学。著名的英国剑桥大学有一个应用数学与理论物理系（DAMTP），牛顿曾经担任该系的卢卡斯数学教授席位（现在是霍金）。然而，自从20世纪50年代计算机兴起以来，数学日益被应用于人造现象（比如电力网、空中交通系统、制造业自动化等等）。如此一来，问题的本质发生了微妙的变化。在物理现象中，不论是流体力学、电磁波还是热力学等，人们面对的都是上帝设定的已有规律。人们主要用数学和数学模型来解释和理解这些物理问题。这是一种用来解释“事物的本来面目”的描述性科学。而数学一旦应用于人造现象，我们要面对的就往往不是业已存在的事物，而是人们想像出来的事物。它们依照人类制定的规则来运行，而这些规则本身可以按照我们的要求做出改变。因此我们不但创造了这些系统，而且控制了这些系统。数学则被用来设计这些系统。这是一种指令性科学，更关注“事情应该是怎样的”。在这里，数学“严密性”的重要性就体现了出来，也就是说，你最好完全确信你讨论的或者创造的事物是基于“真实的公理”和“合乎逻辑的演绎”。仅仅靠“直觉”是很危险的。这其中的逻辑是这样的：如果缺乏“严密性”，你在建造的或许就仅仅是“空中楼阁”。否则的话，如何能相信计算机的计算结果？另一方面，在针对物理现象的数学中，人们常常有实体可以用来参考或者指导自己。而对计算机科学领域的学生而言，应用数学研究中的二分法（自然的vs人造的）通常体现在是否需要“微积分”介入的争执上。其中一派人认为，微积分是所有应用数学的基础，而其他人则更倾向于利用学生时期来研究组合数学和抽象代数（有时称为离散数学）。
然而，在运筹学、控制与系统领域，我们进入了所谓的“朦胧地带”。我们同时设计（创造）系统和控制已有的物理系统，而且，把这两项工作放在同等重要的地位。这里，关于数学“严密性”的争论常常变得白热化，让人困惑。在我看来，对“严密性”的需求和对“直觉”的利用取决于你做的是什么。如果你试图在一个你创造的数学问题上证明一个未经检验的计算法则的收敛性，你最好保证该证明是无懈可击的。另一方面，许多具有应用前景的的著名的研究结果，比如获得德雷珀奖的卡尔曼滤波器，已经被成功用到很多情况中，其中的假设和条件只是大概真实甚至是不怎么真实的。在这里，直觉和经验即使不比“严密性”更重要，也至少同样必不可少。这其中的差异不在于谁绝对正确或者绝对错误，而是applied mathematics两个单词的首字母是A大写了还是M大写了（译者注：也就是我们强调的是“应用”还是强调的是“数学”）。只有相互尊重和理解才能让我们在解决这些争论的道路上走得更远。
就我个人而言，人家批评我的论文中不够“严密”我也不会生气。我处理的是现实世界中的问题和/或者问题驱动的方法学研究。我总是能够通过现实来检测我的研究的合理性，让我不至于铸成大错。在我的工作中，想法、直觉和实效性对我都很重要。但我一直强调，我的工程学学生要在概念上严密，并且明白所谓“证据”到底指的是什么（说着容易做着难）。不管你的研究是更偏应用还是更偏理论，熟练地掌握数学语言都是很有用的，这样你才能看懂数学文献（这就好比不论你的母语是什么，英语对于科学技术研究都是很有用的）。我觉得应用数学不必像过去那样只是理论物理的同义词，我建议把它等同于“理论工程”。应用数学的前一个概念更关注于“描述性科学”，而后者（应用于人造系统的数学）主要涉及“指令性科学”。不过，我无法认同系统领域的一些同事，他们认为“严密性”高于一切，同时把一些很好的想法和应用贬得一钱不值。他们考虑过让世界上最好的数学系来评价一下他们自己工作的数学性有多好吗？搞应用的人已经不搭理他们了，因为他们太不切实际。我们不妨打个合适的比喻，一个普通甚至是愚蠢的双眼人，在独眼人的王国里可以自冕为王，但是一旦他回到正常人的世界，他不过是个普通人罢了。因此，对于刚刚在大学的工程系里起步的年轻学者而言，不要被数学的美丽诱惑，而忘记了你所要服务的最终客户。你的下场可能是不被任何一方赏识。
当然，没有事情是绝对对或者错的。我有意夸大了这个对比，是为了澄清二者之间的区别。世界闻名的纯粹的数学家仍然可以做出一般科学家都懂的通俗科学报告，最近的也是最好的例子之一就是2006年克雷数学公共演讲：http://www.claymath.org/public_lectures/sipser.php
花一个小时的时间看看这些演讲内容是很值得的。
（科学网 任霄鹏译 何姣校）
数学、分析与工程学（2）（原文及译文）
Mathematics, Analysis, and Engineering (2)
Below is a guest Blog by Professor Xiren Cao of the Science and Technology University of Hong Kong. This article further clarifies and articulates the relationship between mathematics and engineering. I thank him for this timely piece.
Prof. Ho’s article “Applied Mathematics and Theoretical Engineering” brought up an important question about what is the relation between mathematics and engineering.
This question was asked quite often by young researchers, and there is a considerable amount of confusion in the research community. While Prof. Ho’s article gave a comprehensive answer to this question, I may add some words from my own experience. (Over the years, my research style has been influenced by Prof. Ho greatly.)
First, I always distinguish between “mathematics” and “analysis”. You may do research in engineering without deep mathematics, but you cannot do research in any subject without analysis. Analysis is a logical derivation/interpretation of the problem you are trying to solve (at a level Prof. Ho calls “conceptually rigorous”). Mathematics is a tool that helps you to express your analysis clearly, logically, and concisely. In most engineering problems, you are not developing mathematics (in a sense of pure mathematics); your work may be of a great value in engineering (such as the Kalman filter), but of no significant contribution from a pure mathematician’s point of view. Mathematics is important because the process of expressing your ideas at a proper level of mathematics helps you to clarify your ideas and to find logical flaws, if any, in your analysis. Because of this, one cannot live without mathematics (children study math from grade 1).
Second, in engineering, innovative ideas, intuitions, and motivations are always the most important thing. Purely working on mathematical expressions may extend the existing results and/or lead to more beautiful forms (those are important jobs!), but without conceptual breakthrough, it may hardly leads to new methodologies in engineering.
Third, do not over use mathematics. That is, if an engineering problem can be described clear with a lower level mathematics, do not use more complicated ones.
As Prof. Ho said, ``A working knowledge of the language of mathematics so that one is capable of accessing mathematical literature is useful regardless whether or not you are application or theory oriented (just like English is useful for S&T work regardless of your mother tongue).” Nowadays, many engineering papers are written in mathematical languages and without some fundamental understanding of it one can even hardly read these engineering papers. However, when you read such papers, try to translate back to intuitions and engineering ideas behind the heavy mathematics. Otherwise, you may not be able to remember the content of such a paper afterwards. Another reason that you may need to learn some (never enough) mathematics is because one day you may need to use some tools in mathematics to analyze your engineering problems but it may not be in your brain. I always read some new (to me) mathematics books or chapters every few years to enhance my knowledge. I suggest young researchers do the same if possible.
数学、分析与工程学（2）
下面的博文是香港科技大学曹希仁教授所写，它进一步澄清和阐明了数学与工程学之间的关系。我十分感谢他这篇及时的文章。
何教授的博文“应用数学与理论工程学”提出了一个重要的问题，即数学与工程学之间有怎样的关系。
这个问题经常被年轻的研究人员问及，科研群体中也存在相当多的混淆。而何教授的博文就该问题给出了一个全面的答案，我可以根据自身的经历来补充一些内容。（这些年来，我的研究风格已经深受何教授影响。）
首先，我常常将“数学”和“分析”区别开来。进行工程学研究可以不用深奥的数学，但无论研究什么主题都不能离开分析。分析就是对试图解决的某一问题的逻辑推导/解释（在何教授所谓的“概念上严密”的层面上）。数学作为一种工具，有助于清晰、逻辑、简明地表达一个人的分析。在大多数工程学问题中，一个人不是在发展数学（就纯数学意义而言），他的工作可能在工程学上具有重要价值（比如卡尔曼滤波器），但在纯粹的数学家眼里并没有什么太大贡献。数学很重要，因为在合适的数学层面上表达自己想法的过程，有助于一个人明晰自己的思路，并找出分析中可能存在的逻辑缺陷。正因为如此，一个人无法离开数学（小孩通常从一年级开始学数学）。
其次，在工程学领域，创新性的想法、直觉以及动机往往是最重要的。纯粹地在数学表达上下功夫或许可以拓展现有的结果或者产生更漂亮的表达形式（这些是重要的工作），但没有概念上的突破，这几乎不可能导致工程学领域新的方法学诞生。
再次，不要过度地使用数学。也就是说，如果一个工程学问题能够以较低水平的数学描述清楚的话，就不要使用那些更为复杂的数学。
正如何教授所言，“不管你的研究是更偏应用还是更偏理论，熟练地掌握数学语言都是很有用的，这样你才能看懂数学文献（这就好比不论你的母语是什么，英语对于科学技术研究都是很有用的）。”眼下，许多工程学论文都是用数学语言写就，没有一些基础性的了解，一个人甚至无法读懂这些论文。然而，在读这样的论文时，应该试着将繁重的数学背后的思想翻译回直觉或者工程学概念。否则，他可能无法在日后记得这样一篇论文的内容。而需要学习一些（永无止境）数学的另一个原因就是，终有一天，你要用一些数学工具来分析自己的工程学问题，而你可能还没有意识到这一点。我常常每隔几年就看一些新的（对我来说）数学书籍或者章节，来增加自己的知识。我建议年轻的研究人员如果可能的话，也一起这样做。（科学网 任霄鹏译 何姣校）
https://blog.sciencenet.cn/blog-81613-40960.html
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