Solving the Unsolvable: Andrew Wiles
By: Zoë WesthofFermat’s Last Theorem: An Impossible Puzzle?
In the 17th century, it is à la mode for scholars to attempt to outdo the ancient masters by scrawling brilliant commentaries in the margins of ancient works. The French lawyer Pierre de Fermat happily adopts this habit, writing mathematical theorems and proofs in the margins of ancient texts or on lone scraps of paper. Fermat describes these theorems in letters to his friends, without bothering to include the proof he claimed to have worked out. Fermat thus earns the reputation of an amateur mathematician who gives even Descartes a run for his money. Most of his theorems are eventually proven — or disproven — when his friends manage to pull together proofs Fermat left scribbled on scraps of paper and book margins.
Fermat becomes most famous, however, for a certain scribble left in the margin of a book that none of his friends could find the proof for. In this scribble, Fermat claims to have proven that a certain equation referred to by Greek philosopher Diophantus in the book Arithmetica had no solutions, and that he has found “a truly marvelous demonstration of this proposition which this margin is too narrow to contain.”
The Curiosity and Audacity of Being Ten Years Old
Over 300 years later, 10-year-old Andrew Wiles is combing the stacks at his local Cambridge library in search of new math problems to solve. The problems he tackles at school aren’t difficult enough to quench his thirst. Stumbling upon Eric Temple Bell’s The Last Problem, Andrew discovers the centuries-old enigma known as Fermat’s Last Theorem. Undeterred by the failure of many before him to prove this great mystery, Andrew nurtures an obsession with the theorem throughout his teenage years, and into college.
Forgetting About It
Wiles finishes school and became a researcher. He decides it’s time to take a break on the theorum. The tools he had at hand were 130 years old; what if the tools he needed were not even invented yet? Fermat’s Last Theorem remaines a fixture in the back of Wiles’s mind, but he decides to wait until the time is right.
Serendiptity and Determination
Eleven years later, the right time comes on a seemingly ordinary evening over a glass of iced tea. Wiles’s friend mentions that another mathematician has proved a connection between the modern Taniyama-Shimura problem and Fermat’s Last Theorem. Armed with a contemporary problem that held the key to his ancient one, Wiles goes back to work. For seven years, he works in secrecy and isolation, telling no one but his wife of the work that consumes him.
He describes his experience:
“You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them.”
A Final Solution
One morning in 1993, Wiles is alone in the house, leafing through a research paper. A single sentence abruptly seizes his attention, a sentence that refers to a 19th-century construction. Wiles spends the rest of the morning and most of the afternoon entwining this construction into the last stage of his proof. At tea time, he goes downstairs and explains to his wife Nada why he missed lunch: he has finally solved Fermat’s Last Theorem.
Wiles presents the lengthy proof to a level of fanfare and celebration quite unusual for a mathematical problem. However, a slight error in a pivotal part of the argument is soon revealed, and Wiles goes back to work. This time, he enlists help from Cambridge mathematician Richard Taylor to untangle and rectify the error.
The Final Solution
A year later, Wiles presents an error-free, 150-page proof.
Many have speculated whether this could be the proof that Fermat was “unable” to fit in the margin of his book. Alas, no one will ever know for sure whether Fermat had really devised a proof — but we can be certain that he did not take the same route as Andrew Wiles. Wiles’s techniques had simply not been invented in the 17th century.
Action Summary:
- Trying to outdo the old masters is all well and good, but it will take curiosity, audacity, serendipity, and determination.
- Dedication and perseverance are crucial to accomplishing impressive feats, but patience is valuable too when the time is not yet right.
- The final solution is not always the Final Solution. When you’re almost there, don’t give up.
- Sometimes we can’t go it alone; enlisting the help of others may be the move that takes you to the next level.
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Picture of Andrew Wiles is copyright C. J. Mozzochi, Princeton N.J. Used by permission.
This post is by creative writer Zoë Westhof. Explore writing, innovation, choice, and change at Zoë’s blog, Essential Prose
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Wow. That’s some serious dedication there.
Great story! Loved the action tips too.
@ Jenny – Yes, it really is an incredible example. Dedication is all relative, so I would say different levels and times are right for different endeavors…
@ Leah – Storytelling always seems like the best way to show, but adding little summary points doesn’t hurt
I don’t think this is one of your better posts, just a standard motivational story which I could have gotten on any other blog. Another consideration is your target audience:
1) Consider those put off by maths. When they see the word ‘theorem’, they’ll probably immediately decide not to read your story as they’ll thnk its about complicated mathematical guff.
2) People like me who enjoy maths are disappointed with how you didn’t go into greater detail on the specifics of this problem, and how it was eventually solved – although I do understand how you can’t have the full 150 pages in your account.
But don’t worry, I won’t be unsubscribing any-time soon!
Thanks Shane for your valuable feedback.
We are experimenting with a “what-makes-geniuses-tick” column behind the scenes. These posts are a bit harder to write because we don’t want to sound like wikipedia. And so – thanks for bearing with our goof-ups till we perfect the genius column and make them better.
Hopefully – our next genius posts will hit the mark!
Thank You again for your very valuable comment and for not unsubscribing
Shane,
I’m sorry you didn’t enjoy the post. I wrote about Wiles because I thought it was a quirky story that I haven’t seen explored on other blogs.
The balance I struck while describing the math problem was precisely due to the points you listed: I didn’t get too in-depth so that “non-math people” wouldn’t be turned off, and I would hope that someone interested in math would have their curiosity piqued enough to look into the (lengthy) explanation further
“When you’re almost there, don’t give up.” This is my favorite line in the post.
Many people quit without realizing that when they quit, it is the time when they are almost about to succeed.
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