Monday, May 02, 2005

Fermat's Achievements

Pierre de Fermat published very little in his lifetime, and yet by most accounts, he is considered to be one of the greatest mathematicians of all time.

I was very surprised when I learned that Fermat's Last Theorem was published after Fermat's death. It is pretty amazing that the mathematical world took a note in the margin so seriously.

Fermat was very precise about his mathematical pronouncements. When he was not certain about a finding, he labeled it a conjecture. Indeed, shortly after Fermat's death, all but one of his theorems had a proof just as Fermat had claimed.

Fermat was born in Toulouse, France on August 17, 1601. He was born into a wealthy family. He studied law at the university in Orleans and became a councillor at the parliament in 1631. He quickly moved up the ranks and in 1652, he became the chief magistrate of the criminal court.

In 17th century France, magistrates were required to spend large amounts of time in isolation. It was during this time, that he worked on mathematics.

His achievements in mathematics are incredible.

Sir Isaac Newton said that his invention of calculus was based a large part on Fermat's method of tangents.

In 1654, Blaise Pascal wrote a letter asking about Fermat's views on probability. Their series of correspondences became the foundation of probability theory.

Rene Descartes is perhaps most famous for his invention of Cartesian coordinates and his classic work La Geometrie. Fermat independently came up with a three-dimensional geometry. While Fermat's version of analytic geometry was more complicated and advanced, Descarte's classic work became more popular because its notation was more convenient. Today, both men are seen as the fathers of analytic geometry.

He made advances in optics. He came up with Fermat's Principle, the idea that light always traverses the path that takes the least time.

With Wiles' amazing proof, we are still left with a question. Did Fermat really have a proof?

Wiles' proof rests on twentieth century mathematics including the theories of elliptic curves, modular forms, and Galois Representations.

Wile's proof goes way beyond the mathematics that Fermat helped to create. It is very possible that Fermat's proposed proof made an incorrect assumption about cyclotomic factorization. This was a mistake that got Gabriel Lame into falsely believing that he had found a proof of Fermat's famous problem (I will be talking more about this in a later blog).

It is also possible that Fermat, in his lifetime, figured out that he had made a mistake. If he had truly found such a marvelous proof, why did he never tell anyone about it while he was alive? Perhaps, he had figured that he had made a mistake but forgot to correct his notes.

Still, the possibility that Fermat had a proof raises an interesting question: is there a way to simplify Wiles' result? Is it possible to derive a result from Wiles' proof that corresponds to the mathematics of Fermat?

The information in this blog was taken from the following sources: