It may seem obvious or second nature, but I often have to remind myself when working on something to make sure that I’ve taken a look at what was done before. I am going to provide a story to illustrate what I mean and how this has helped me in my career.
I still remember “the dark days” of my years in graduate school. It was 1989-1990, my 4th-5th year in the mathematics Ph.D. program at UC Berkeley. I had passed my written and oral exams, had Henry Helson (a world-famous analyst) as my advisor and was now “doing research” in functional analysis. I use the quotes for emphasis (perhaps sarcasm is a better word), because what I was doing was trying to work on a problem that I’d found myself and was literally getting nowhere. I was learning new mathematics, but was really not making any headway on the unsolved problem that I hoped would bring me the coveted three letters after my name. Yup, each day I would bang my head against the wall for hour after hour, and at the end of the day I would be nowhere closer to solving the problem. I would talk to my advisor on a regular basis, but he also didn’t know how to approach the problem that I was trying to tackle using ideas/techniques that I felt might lead to a solution.
Eventually, Prof. Helson noticed that I had learned some things that could be useful in a different context. He said, “Why don’t you take a look at the thesis of my last doctoral student, G. Choe.” At first, I was somewhat annoyed with his advice. Why should I look at what his last student had done? I wanted to do something original and not look at something that had already been done. But I took his advice, and started looking at Choe’s thesis.
I quickly realized that during the time that I had been cracking my head on the other problem, I had learned techniques that could get a significant improvement on one of the theorems in Choe’s thesis. That was my first breakthrough! I finally had something new that I could call my own. Yes, even though it was a generalization of something already done, I had done it! I had a piece of original mathematics research. This first, very modest result, gave me confidence to look at other problems that I was able to solve. Some of these were different proofs or improvements on recently-proved results, but they were new, i.e., my own.
The lesson of looking at what has been done before, whether on a problem or some other context, is one that I have grown to very much appreciate. For example, now as a faculty member, whenever I teach a class that I haven’t taught before, I ask the colleague who taught it before me to share her/his material with me. I may end up only glancing at it, but it ends up informing my work. Whenever I become interested in a partially solved problem, I’ll try to understand the technique that was used to make headway on it, even though I understand that new techniques likely will be needed to further the work. I guess my point is that originality will most of the time mean variations on what’s been done before, and not a whole “new color.”
When I think about it, this is the norm and not the exception. For example, (in another setting) many of us Angelinos still remember Magic Johnson’s “baby hook” in the 1987 NBA Finals against the Celtics. Was it new? Not totally. It was really a variant of what he’d seen Kareem Abdul Jabbar do on a daily basis for years! What about when we read about the new gotta-eat-there restaurant. We most likely will read a review that says that the “YYY” is a “must-order.” That “YYY” is a dish whose name we will recognize so it really is an improvement on something that already exists. Both of these are examples of improvements, or new variations, on what has been done before.
I’ll end with a story that I believe is relevant to paying attention to what has been done before. I, like many graduate students, wanted to write a thesis that was groundbreaking and would set the mathematical world “on fire.” I didn’t. I wrote a thesis that “got me out.” I wrote a thesis that furthered solutions to problems that had been studied before. I didn’t prove the Riemann hypothesis nor even solved the problem that I racked my brains on for about two years. (BTW, that problem is still unsolved.)
I’m not quite sure if I ever came to grips with this until a few years ago when I had the privilege of meeting Elwyn Berlekamp, a world famous mathematician/computer scientist. In a conversation about being practical about one’s work, especially during graduate school, he said to me (something like this), “A lot of graduate students want to write a thesis that will ‘set the world on fire.’ They fail to realize that the world is a difficult place to burn.”