History of Math

How is Pythagoras (500 BC) linked to Galois (1800’s) or Wiles(1900’s)?  I recently watched a terrific documentary about Sir Andrew Wiles.  The video follows Wiles’ quest to prove Fermat’s Last Theorem, which states that there is no whole number solution to x^n+y^n=z^n greater than 2 (a cousin to the famous Pythagorean Theorem).   Fermat claims that he did not have room to write the full proof in the margin of his copy of the Arithmetica.  Over the centuries no one, not even brilliant mathematicians like Gauss, Kummer, Euler, or Germain, has been able to prove what Fermat claimed as true. Up until 1993, even with the advantage of computer assisted algorithms, thousands of people had tried to prove it but no one had succeeded.

Wiles’ dream began when he was only ten years old, after reading a math book in his local library and understanding that Fermat’s 300 year old theorem could be proven.  He worked tirelessly for many years -  researching and contemplating the works of many mathematicians, bridging seemingly unconnected areas of math, and handwriting all of his thoughts on paper and chalkboard.  Using ideas, theories, and proofs from mathematicians including Taniyama, Galois, Iwasawa, and Flach, in order to formulate his own conjectures and theories, Wiles finally revealed his proof of Fermat’s Last Theorem in 1993.  However, several months later, there was a problem discovered with the last part of the proof.  Wiles went back and started picking his proof apart to see where he went wrong, only to find that the dots could be connected using a path he had previously abandoned. So in the end, Wiles really did uncover a proof to Fermat’s Last Theorem.  One of Wiles’ great accomplishments on this journey was the webbing together of so many areas, ideas, and branches of math.  I made the word cloud (below) which contains some of the theories and mathematicians whose ideas Wiles used to uncover his proof to Fermat’s Last Theorem.

Quote from Documentary –

ANDREW WILES: “Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room, and it's dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and 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. At the beginning of September, I was sitting here at this desk, when suddenly, totally unexpectedly, I had this incredible revelation. It was the most—the most important moment of my working life.”

Modular Elliptic Curves and Fermat's Last Theorem