Anyone who has a passion for solving math problems or learning math should find the interview with the mathematician highlighted here interesting. The interview is an old one (posted in year 2000). But the lessons that can be drawn from it are timeless and it can inspire math students and enthusiasts at all levels. You probably know who this mathematician is. If not, here’s some hints. He spent 7 years working in isolation to tackle a problem that no one could solve for almost 350 years! The problem was a famous one that bears the name of the person who posed it. The poser gave a tease that he had a proof of the problem but the margin in the page of the book in which he was scribbling notes was too small to contain the proof. The mathematician is of course Andrew Wiles. The problem was Fermat’s Last Theorem. The statement of the problem can be readily found online. Here is an explanation of it (as a preface to the explanation of a generalization of Fermat’s Last Theorem called Beal’s conjecture).
Interesting reads about Wiles are also readily available in book forms and in web sites. This interview is a good start for anyone who is not familiar with Wiles. The interview was posted in the official website of Nova, a science program of PBS. Of course the 7 years he spent working on the problem did not include the times when he worked in it in his youth. He discovered Fermat’s Last Theorem when he was 10 years in reading the book The Last Problem by Eric Temple Bell. He was completely captivated by the conjecture that he was obsessed with it for a while until he realized that he was limited by his knowledge. So he gave up the dream of being the first person to solve the problem and instead focused on his education in mathematics.
Then in 1986 there was a breakthrough in that Fermat’s Last Theorem was shown to be equivalent to another unsolved math problem called Taniyama-Shimura conjecture, which is a modern math problem that provided a new angle for attacking the intractable original problem.
One thing that struck me the most was the question posed by the interviewer, “On a day-to-day basis, how did you go about constructing your proof?” This is something that anyone who learns math or work on math problems have to face. What do you do to advance your goal on a daily basis? If there is insight or if there are ideas that seem promising, no problem. You would just focus on those ideas. But if you are stuck, or if you have no ideas on what to do, what happens then? Of course, taking a break is a good idea. Let your mind rest. Eventually you may come back to the same place where you got stuck. What do you do? The following is his answer to the question.
I used to come up to my study, and start trying to find patterns. I tried doing calculations which explain some little piece of mathematics. I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about. Sometimes that would involve going and looking it up in a book to see how it’s done there. Sometimes it was a question of modifying things a bit, doing a little extra calculation. And sometimes I realized that nothing that had ever been done before was any use at all. Then I just had to find something completely new; it’s a mystery where that comes from. I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction, I would have the same thing going round and round in my mind. The only way I could relax was when I was with my children. Young children simply aren’t interested in Fermat. They just want to hear a story and they’re not going to let you do anything else.
One element of his day-to-day approach is about going small. Work on a little piece of a bigger problem and try to see how it fits in a broader framework. Another element is about tweaking something that had been done before and then trying to extend it. Then he also realized that known approach may not be useful at all for the problem at hand. So he had to find something new. But finding new idea or new approach is a mysterious process. There is not much point for him or anyone to describe how that mysterious process works. Here’s probably the most important point in the above paragraph. He carried the problem with him everywhere (in his head of course). When Wiles had no distraction, the problems and ideas in him just sloshed around in his head. Then when new ideas or good ideas came around, he was able to take advantage of them.
Fortunate for Wiles and for the world of mathematics, after years of digging and pushing the frontiers a little at a time (on many days no progress at all), certain right ideas finally emerged.
Is the experience of Wiles relevant to someone who is a mere mortal (mathematically speaking)? Obviously not many people are trying to solve problems that eluded the efforts of the brightest minds for centuries. I can see that the same ideas from Wiles’ epic conquest hold true in my own situation (in a much much smaller scale for sure). Going small definitely resonates with me. If I cannot solve a bigger problem, I try to pare down the problem, perhaps looking at a special case. Tweaking a known approach can also be useful. If doing so does not produce results, at least I know that a new idea is probably needed. I am a topologist by training. I certain use similar approaches when I learn something new or when I try to solve a problem.
Spending several years in isolation on a problem is certainly not easy to do for most of us. But we can be persistent in our own ways. We can be dogged in a way that fits our circumstances (not many of us has a professorship or a job that allows such unfettered access to free time). In other words, the example of Andrew Wiles can inspire us to chase our own dreams, solving problems that are meaningful to us.
Another thing that resonates with me is that he advises “heart” when choosing a problem – “always try the problem that matters most to you.” Otherwise it may be hard to stay with the problem through thick and thin.
The success meant that it was a melancholy moment for Wiles. It was a melancholy moment for many other people in mathematics too. Wiles said, “we’ve lost something that’s been with us for so long, and something that drew a lot of us into mathematics. But perhaps that’s always the way with math problems, and we just have to find new ones to capture our attention.”
A good popular science book about the search for a proof of Fermat’s Last Theorem is Simon Singh’s book. Singh also produced a documentary on the proof. Here is a video from PBS. For anyone who really wants to dig deep into the math, here is a two-volume book on the basics and the proof of Fermat’s Last Theorem by Takeshi Saito.
Here’s an article from The Guardian about Wiles’ being awarded the Abel Prize.