Shuvomoy Das Gupta
April 13, 2020
Recently, I came across a few old and new interviews of Donald Knuth, where he sheds light on his work habits, how he approaches problems, and his philosophy towards happiness. I really enjoyed reading the interviews. In this blog, I am recording his thoughts on approaching a problem, organizing daily activities, and the pursuit of happiness.
Seeing both the forest and the trees in research. “I’ve seen many graduate students working on their theses, over the years, and their research often follows a pattern that supports what I’m trying to explain. Suppose you want to solve a complicated problem whose solution is unknown; in essence you’re an explorer entering into a new world. At first your brain is learning the territory, and you’re making tiny steps, baby steps in the world of the problem. But after you’ve immersed yourself in that problem for awhile then you can start to make giant steps, bigger steps, and you can see many things at once, so your brain is getting ready for a new kind of work. You begin to see both the forest and the trees.”
How Knuth works on a project. “When I start to investigate some topic, during the first days I fill up scratch paper like mad. I mean, I have a huge pile of paper at home, paper that’s half-used, used on only one side; I’ve kept a lot of partially printed sheets instead of throwing them away, so that I can write on the back sides. And I’ll use up 20 sheets or more per hour when I’m exploring a problem, especially at the beginning. For the first hour I’m trying all kinds of stuff and looking for patterns. Later, after internalizing those calculations or drawings or whatever they are, I don’t have to write quite so much down, and I’m getting closer to a solution. The best test of when I’m about ready to solve a problem is whether or not I can think about it sensibly while swimming, without any paper or notes to help out. Because my mind is getting accustomed to the territory, and finally I can see what might possibly lead to the end. That’s oversimplifying the truth a little bit, but the main idea is that, with all my students, I’ve noticed that they get into a mental state where they’ve become more familiar with a certain problem area than anybody else in the world.”
Visualizers vs Symbolizers. Well, you know, I’m visualizing the symbols. To me, the symbols are reality, in a way. I take a mathematical problem, I translate it into formulas, and then the formulas are the reality. I know how to transform one formula into another. That should be the subtitle of my book Concrete Mathematics: How to Manipulate Formulas. I’d like to talk about that a little.
I have a feeling that a lot of the brightest students don’t go into mathematics because–-curious thing–-they don’t need algebra at the level I did. I don’t think I was smarter than the other people in my class, but I learned algebra first. A lot of very bright students today don’t see any need for algebra. They see a problem, say, the sum of two numbers is 100 and the difference is 20, they just sort of say, “Oh, 60 and 40.” They’re so smart they don’t need algebra. They go on seeing lots of problems and they can just do them, without knowing how they do it, particularly. Then finally they get to a harder problem, where the only way to solve it is with algebra. But by that time, they haven’t learned the fundamental ideas of algebra. The fact that they were so smart prevented them from learning this important crutch that I think turned out to be important for the way I approach a problem. Then they say, “Oh, I can’t do math.” They do very well as biologists, doctors and lawyers.
What graduate students should do when they have expertise in a certain area. “When they [the students] reach this point [expertise in a certain area] I always tell them that now they have a responsibility to the rest of us. Namely, after they have solved their thesis problem and trained their brain for this problem area, they should look around for other, similar problems that require the same expertise. They should use their expertise now, while they have this unique ability, because they’re going to lose it in a month. I emphasize that they shouldn’t be satisfied with solving only one problem; they should also be thinking about other interesting problems that could be handled with the same methods.”
On the importance of anthropomorphizing a problem. “Another aspect of role playing is considerably more important: We can often make advances by anthropomorphizing a problem, by saying that certain of its aspects are “bad guys” and others are “good guys,” or that parts of a system are “talking to each other.” This approach is helpful because our language has lots of words for human relationships, so we can bring more machinery to bear on what we’re thinking about.”
Why putting the discovery of a solution on paper is important. “Well, I have no sympathy with people who never write up an answer; it’s selfish to keep beautiful discoveries a secret. But I can understand a reluctance to write something up when another problem has already grabbed your attention. I used to have three or four papers al
