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Interview with Andrew Wiles
Nova Website
by Simon Singh

Andrew Wiles has devoted his entire career to solving Fermat’s Last Theorem, the world’s most notorious mathematical problem. In 1993, he made front-page headlines when he announced a proof of the problem, but this was not the end of the story. An error in his calculation jeopardized his life’s work. Andrew Wiles spoke to NOVA and described how he came to terms with the mistake, and fought back to eventually achieve his life’s ambition.

NOVA : Many great scientific discoveries are the result of obsession, but in your case that obsession has held you since you were a child.

ANDREW WILES : I grew up in Cambridge in England, and my love of mathematics dates from those early childhood days. I loved doing problems in school, I’d take them home and make up knew ones of my own. But the best problem I ever found I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem - Fermat’s Last Theorem. This problem had been unsolved by mathematicians for 300 years. It looked so simple, and yet all the great mathematicians in history couldn’t solve it. Here was a problem, that I a ten year old could understand and I knew from that moment that I would never let it go. I had to solve it.

NOVA : And who was Fermat and what was his Last Theorem?

AW : Fermat was a seventeenth century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it. His proposition was about an equation which is closely related to Pythagoras’s equation. Pythagoras’s equation gives you: x2 + y2 = z2. You can ask what are the whole number solutions to this equation and you can see that: 32 + 42 = 52 and 52 + 122 = 132. And if you go on looking then you find more and more such solutions. Fermat then considered the cubed version of this equation: x3 + y3 = z3. He raised the question, can you find solutions to the cubed equation? He claimed that there were none. In fact, he claimed that for the general family of equations: xn + yn = zn, where n is bigger than 2, it is impossible to find a solution. That’s Fermat’s Last Theorem.

NOVA: So Fermat said because he could not find any solutions to this equation, then there were no solutions?

AW: He did more than that. Just because we can’t find a solution it doesn’t mean that there isn’t one. Mathematicians aren’t satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity. And to do that we need a proof - Fermat said he had a proof. Unfortunately, all he ever wrote down was: “I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.”

NOVA: What do you mean by a proof?

AW: In a mathematical proof you have a line of reasoning consisting of a many, many steps, what are almost self-evident. If the proof we write down is really rigorous then nobody can ever prove it wrong. There are proofs that date back to the Greeks that are still valid today.

NOVA: So the challenge was to rediscover Fermat’s proof of the Last Theorem. Why did it become so famous?

AW: Well, some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they’re extremely hard to solve. There’s no reason why these problems shouldn’t be easy, and yet they turn out to be extremely intricate. The Last Theorem is the most beautiful example of this.

NOVA: But finding a proof has no applications in the real world - it is a purely abstract question. So have people put so much effort into finding a proof?

AW: Pure mathematicians just love to try unsolved problems - they love a challenge. And as time passed and no proof was found, it became a real challenge. I’ve read letters in the early 19th century which said that it was an embarrassment to mathematics that Last Theorem had not been solved. And of course, it’s very special because Fermat said that he had a proof.

NOVA: How did you begin looking for the proof?

AW: In my early teens I tried to tackle the problem as I thought Fermat might have tried it. I reckoned that he wouldn’t have known much more math than I knew as a teenager. Then when I reached college I realized that many people had thought about the problem during the 18th and 19th centuries and so I studied those methods. But I still wasn’t getting anywhere. Then when I became a researcher I decided that I should put the problem aside. It’s not that I forgot about it - it was always there - but I realized that the only techniques we had to tackle it had been around for 130 years. It didn’t seem that these techniques were really getting to the root of the problem. The problem with working on Fermat was that you could spend years getting nowhere. It’s fine to work on any problem, so long as it generates interesting mathematics along the way - even if you don’t solve it at the end of the day. The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.

NOVA : It seems that the Last Theorem was considered impossible, and that mathematicians could not risk wasting getting nowhere. But then in 1986 everything changed. A breakthrough by Ken Ribet at the University of California at Berkeley linked Fermat’s Last Theorem to another unsolved problem, the Taniyama-Shimura conjecture. Can you remember how you reacted to this news?

AW : It was one evening at the end of the summer of 1986 when I was sipping iced tea at the house of a friend. Casually in the middle of a conversation this friend told me that Ken Ribet had proved a link between Taniyama-Shimura and Fermat’s Last Theorem. I was electrified. I knew that moment that the course of my life was changing because this meant that to prove Fermat’s Last Theorem all I had to do was to prove the Taniyama-Shimura conjecture. It meant that my childhood dream was now a respectable thing to work on. I just knew that I could never let that go.

NOVA : So, because Taniyama-Shimura was a modern problem, this meant that working on it, and by implication trying to prove Fermat’s Last Theorem, was respectable.

AW : Yes. Nobody had any idea how to approach Taniyama-Shimura but at least it was mainstream mathematics. I could try and prove results, which, even if they didn’t get the whole thing, would be worthwhile mathematics. So the romance of Fermat, which had held me all my life, was now combined with a problem that was professionally acceptable.

NOVA : At this point you decided to work in complete isolation. You told nobody that you were embarking on a proof of Fermat’s Last Theorem - why was that?

AW : I realised that anything to do with Fermat’s Last Theorem generates too much interest. You can’t really focus yourself for years unless you have undivided concentration, which too many spectators would have destroyed.

NOVA : But presumably you told your wife what you were doing?

AW : My wife’s only known me while I’ve been working on Fermat. I told her on our honeymoon, just a few days after we got married. My wife had heard of Fermat’s Last Theorem, but at that time she had no idea of the romantic significance it had for mathematicians, that it had been such a thorn in our flesh for so many years.

NOVA : On a day to day basis, how did you go about constructing your proof?

AW : 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 realised 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.

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