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）