Sunday, October 28, 2007

Tao on Problem Solving

The following excerpt on problem solving by Terence Tao (winner of last year's Fields medal), is worthwhile reading a couple of times:

How does Tao describe his success?

"I don't have any magical ability," he said. "I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out; then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what's going on.

"Most mathematicians faced with a problem, will try to solve the problem directly. Even if they get it, they might not understand exactly what they did. Before I work out any details, I work on the strategy. Once I have a strategy, a very complicated problem can split up into a lot of mini-problems. I've never really been satisfied with just solving the problem; I want to see what happens if I make some changes.

"If I experiment enough, I get a deeper understanding," said Tao, whose work is supported by the David and Lucille Packard Foundation. "After a while, when something similar comes along, I get an idea of what works and what doesn't work.

"It's not about being smart or even fast," Tao added. "It's like climbing a cliff; if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools; you still need a plan – that's the hard part – and you have to see the bigger picture."

His views about mathematics have changed over the years.

"When I was a kid, I had a romanticized notion of mathematics -- that hard problems were solved in Eureka moments of inspiration," he said. "With me, it's always, ‘let's try this that gets me part of the way. Or, that doesn't work, so now let's try this. Oh, there's a little shortcut here.'

"You work on it long enough and you happen to make progress towards a hard problem by a back door at some point. At the end, it's usually, 'oh, I've solved the problem.'"

Tao concentrates on one math problem at a time, but keeps a couple of dozen others in the back of his mind, "hoping one day I'll figure out a way to solve them. If there's a problem that looks like I should be able to solve it but I can't, that gnaws at me."

Does theoretical mathematics have applications beyond the theory?

"Mathematicians often work on pure problems that may not have applications for 20 years -- and then a physicist or computer scientist or engineer has a real-life problem that requires the solution of a mathematical problem, and finds that someone already solved it 20 years ago," Tao said.

"When Einstein developed his theory of relativity, he needed a theory of curved space. Einstein found that a mathematician devised exactly the theory he needed more than 30 years earlier."

Will Tao become an even better mathematician in another decade or so?

"Experience helps a lot," he said. "I may get a little slower, but I'll have access to a larger database of tricks; I'll know better what will work and what won't. I'll get déjà vu more often, seeing a problem that reminds me of something."

What does Tao think of his success?

"I'm very happy," he said. "Maybe when I'm in my 60s, I'll look back at what I've done, but now I would rather work on the problems."

The above is an excerpt from a longer article, aptly titled Terence Tao: "The Mozart of Math". Terence Tao is not just the winner of Fields medal. He has won a whole lot of prizes. Lance's post on the same subject is worthwhile reading. My earlier short post on this subject.

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