Alpha Chiang -- the author of Fundamental Methods of Mathematical Economics, the all-time best-selling math textbook for economics -- has shared many stories with us in his biography My First 90 Years. These stories may not be interesting to many students of the present generation, who are too young to know much of the persons or events involved. But I think one thing is surely motivating from this book, to me and to today's economics students: This most widely acknowledged math teacher for economics students (via his textbook) indeed didn't have a good math background (before he planned to write the book), didn't get too much training in math even when he took courses at PhD level, and so his math was virtually almost completely got by self-learning.
In this biography, his description about his math knowledge is not detailed. But certainly we can get from the book that his math training was inadequate before his self-learning. Both his master and PhD education did not teach him much about math. In 1950s, Prof Chiang studied for a PhD degree at the economics department of Columbia University. Here is how he described the math education there:
"The other source of frustration at Columbia, a more grave one, was the lack of mathematics course(s) to prepare the graduate students in economics to tackle the professional economics journal articles that had become increasingly more mathematical. I often found myself in the unenviable position of being able to read only the 'introduction' and the 'conclusion' of the article, but not the mathematical analysis in the body of the paper. And that was not only true for one or two articles, but was true for most articles assigned to us. And I suspect that I was not the only one facing this frustration. The option of taking courses in the mathematics department was not open to me; I could not afford the time and expenses. I became more and more resolved to launch a serious study of mathematics on my own." (p.169)
Perhaps some undergraduate students may have the same experience: being able to read only the "introduction" and "conclusion" of an economics paper. But notice that Prof Chiang was describing his experience for his PhD education, and what he read were journal articles published by 1950s (supposedly not too much math involved at that time). If even a PhD student did not get sufficient training in math to read the papers of only low-level math at that time, we can imagine there was even less training for undergraduate students at that time.
Then, how he learned the math later? Well, only when he started to teach economics at Denison University, he self-learned it. Let's also see how he described the difficulty and frustration from self-learning:
"Following the good advice of Professor Leland Gordon, my Chairman at Denison University on my first full-time teaching job, I spent all the summer months on self-study. Learning from mathematics books on my own proved to be much more difficult than I anticipated. Mathematicians as a group seem to me to be overly stingy with words. If a clear explanation of an idea should require 11 words, they would try to use 7. The saving of a few words often introduces a measure of ambiguity into the writing, making it necessary for the reader to wonder and guess what the author meant to say. On many occasions, when encountering a brevity-induced ambiguity, I would first try to figure out the exact meaning of the book statement, and then, to satisfy myself, rewrite the statement for greater clarity. Usually, this involved not much more than adding a few key words. Why save those words?
"The phrase in mathematics books that annoyed me the most is: 'It is obvious that...' Such a phrase is usually only a lame justification for the author to skip a few steps in the explanation. In one story (or joke) I heard, a well-known mathematician paused at one point of his lecture after saying, 'It is obvious that...,' then turned to the blackboard, and mumbled, '...just a minute.' Then, ignoring the class, he did a series of calculations by himself for 20 minutes, then turned around and continued with a smile of satisfaction: 'Yes, it IS obvious!' (p.170)
Prof Chiang later went to Yale University for a further study plan. Yale at that time has a Cowles Foundation that was a pioneer in integrating math and econ. It was also especially well known for its expertise in econometrics (a statistical discipline for economics). In those early days, the importance of math in economics had been growing but there might not be sufficient number of economists with the most adequate training. By visiting Cowles, these economists had the opportunity to learn the most fashionable math usable in economics.
At Yale, Prof Chiang originally intended to study econometrics. But he soon find math more interesting.
"But soon I found that econometrics did not excite me. As a result, I redirected my time and energy to my long-conceived plan to write a book on mathematical economics, tying various mathematical
methods to different types of economic analysis." (p.139)
So, we now have Fundamental Methods of Mathematical Economics. I think Prof Chiang's story of self-learning math is motivating. He is now the math teacher of most economics undergraduates (via the textbook). Of course, we appreciate that Prof Chiang is special: he mentioned in his biography that he was a perfectionist, having a drive to complete things perfectly. And of course, he was an excellent student from the very beginning. But yet he also faced a greater difficulty in math than us today: there was no good math textbook for economics students at that time and he had to learn math as late as he was already a teacher (normally the younger a person, the easier one can learn). Given this, I think students today can have much to learn from his story. In particular, if Prof Chiang can self-learn math without a good background, students can also learn math with his good textbook and teacher's guidance.
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