Alfred Adler: "The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then, little will ever be accomplished".
G. H. Hardy: "average age of election to Royal Society is lowest in mathematics". Ramanujan was elected FRS at 31; Abel did his best work at 19 and died in poverty at 27.
"I do not know of a major mathematical advance initiated by a man past fifty."
The book recounts the life of Pythagoras of Samos because the Pythagorean theorem is a precursor to Fermat's Last Theorem (FLT), and his life helps explain mathematical thinking.
The book describes Pythagoras (6th century BC) as an influential mathematician who created the term "philosopher" and also created a secret brotherhood that viewed numbers as divine.
Distinction between scientific theories (based on empirical evidence, at best approximately correct) versus mathematical logic (axiomatic, guaranteed to be true).
Mathematical patterns, though abstract, often underlie real-world phenomena, such as how musical notes are tuned.
Prior to Pythagoras, other cultures (Babylonians, Chinese, Indians) had already discovered the relationship that a^2 + b^2 = c^2.
Andrew Wiles first encountered FLT when he was a child (~10 years old) in a library through the book "The Last Problem" by Eric Temple Bell. 30 years later at the age of 40, Wiles gave a presentation at the Isaac Newton Institute at Cambridge announcing his proof.
Chapter 2 - "The Riddler" - Notes (Completed )
Pierre de Fermat was born in 1601 in France. At thirty, he was working in the civil service. He experienced great hardships such as the plague and the political dangers of 17th-century France.
Mathematics was not a prestigious subject in his time, and Fermat was a hobbyist isolated from the small math community.
Fermat was very secretive: "Whatever of my work is judged worth of publication, I do not want my name to appear there".
Fermat and Pascal exchanged letters on math, leading to the creation of probability theory as they studied gambling.
Fermat's work was instrumental to calculus — Newton wrote that he used "Monsieur Fermat's method of drawing tangents."
The book summarizes the history of mathematics after the death of Pythagoras, covering the Library of Alexandria, Euclid and Diophantus. Unfortunately, many of the ancient texts held in Alexandria were lost forever.
Mathematics was preserved by Indian and Arab scholars, who added the concept of zero and used a base-10 system.
Fermat encountered a copy of Diophantus' Arithmetica. He studied math as a hobby, writing down only what he felt was necessary and often leaving incomplete proofs.
Among Fermat's results in number theory were the discovery of the friendly pair 17,296 and 18,416, and a proof that 26 is the only number between a square and a cube.
Fermat liked to taunt his peers by announcing results and seeing if they could find the answer.
Fermat published little during his life, but after his death, his son compiled and published his father's works.
"I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain." For centuries, FLT would remain unsolved...
Chapter 3 - A Mathematical Disgrace (Completed )
FLT was Wiles' greatest passion since he was a child. After his initial attempts failed, he began to study the work of other mathematicians who worked on the problem.
Leonhard Euler, born in 1707, was a brilliant Swiss mathematician. His father, a pastor, wanted Euler to study theology. Thanks to the appeals of the Bernoulli family, Euler gained permission to study math.
Euler worked for the Russian tsars, Frederick the Great, and other European royals. He worked on problems in a multitude of domains, and was incredibly prolific.
Euler announced a proof of the n = 3 case of FLT in 1753, leveraging imaginary numbers and the method of infinite descent.
At 60, Euler went almost completely blind. Remarkably, he continued to do mathematics for seventeen years. While blind, Euler notably worked on the calculations of lunar positions - a problem that challenged Newton.
The book summarizes some elementary mathematical concepts and outlines the historical discrimination against women in mathematics.
Sophie Germain was born in 1776, and fell in love with mathematics. In 1794, the École Polytechnique - an elite French academy of science and math - was opened. Using the identity of a former male student "Monsieur Le Blanc", Germain retrieved lecture notes and submitted work to the academy. Eventually, Joseph-Louis Lagrange, the supervisor, noticed her work and met her, becoming her mentor and friend.
Germain wrote to Carl Friedrich Gauss, one of the most brilliant mathematicians who has ever lived. She shared her ideas on number theory and progress on FLT.
Germain's ideas for FLT were used by other mathematicians to prove the n = 5 and n = 7 cases.
During the Napoleonic Wars, Germain asked a French general to protect Gauss. Upon learning of her true identity, Gauss wrote a beautiful letter in admiration and support of her work.
Sadly, Germain died of breast cancer and did not receive the full recognition she deserved for her work, in part because she was a woman.
The French Academy offered a bounty of 3,000 francs for a solution to FLT. Two mathematicians - Gabriel Lamé and Augustin-Louis Cauchy - believed they were close, and announced their progress in an 1847 Academy meeting.
Unfortunately, the German mathematician Ernst Kummer identified key flaws in their approaches and wrote that both Lamé and Cauchy were wrong.
Wiles reviewed the progress and failed attempts to solve FLT, and resolved to apply 20th century mathematics to the problem.
Chapter 4 - Into Abstraction - Notes (Completed )
The German industrialist Paul Wolfskehl became interested in Kummer's work on FLT. His will established a 100,000-mark prize for a proof of FLT.
The University of Göttingen received many proof attempts for the Wolfskehl Prize - all failed.
At the time, mathematicians were interested in proving the foundations of mathematics from its most basic axioms.
This effort was championed by David Hilbert. In 1900, Hilbert posed 23 problems that he believed were important to mathematics. Some of these problems are unresolved to this day.
In 1931, 25-year-old mathematician Kurt Gödel published a groundbreaking paper: "On Formally Undecidable Propositions in Principia Mathematica and Related Systems".
The book describes the theorems as: "First Theorem of Undecidability: If axiomatic set theory is consistent, there exist theorems that can neither be proved or disproved. Second Theorem of Undecidability: There is no constructive procedure that will prove axiomatic theory to be consistent."
In 1963, 29-year-old Stanford mathematician Paul Cohen developed a technique for testing if certain questions are undecidable. Cohen proved that Hilbert's continuum hypothesis was undecidable.
The book recounts the cracking of the Enigma machine by Alan Turing and his team, and the rise of computers.
Leveraging computers, mathematicians and scientists were able to prove FLT for increasingly large n (such as n ≤ 10,000).
While interesting, these results did not constitute a proof. Euler famously conjectured there were no solutions to x^4 + y^4 + z^4 = w^4 (a case of the sum-of-powers conjecture). In 1988, Noam Elkies found an infinite number of solutions, including a very large counterexample. Another example of the dangers of using numerical evidence is the overestimated prime conjecture.
In 1975, Wiles started as a grad student at Cambridge. John Coates, Wiles' faculty supervisor, tasked Wiles with studying elliptic curves (y^2 = x^3 + ax^2 + bx + c; a, b, c are whole numbers).
Wiles' work on understanding elliptic equations would later help him prove FLT.
Chapter 5 - Proof by Contradiction - Notes (Completed on )
Goro Shimura was born in 1930 and Yutaka Taniyama was born in 1927. Both endured hardships in their early lives — Taniyama had tuberculosis and missed two years of high school, while Shimura worked in a factory during WWII and studied math at night.
The book describes modular forms, a topic of interest for both Taniyama and Shimura.
Taniyama believed there was a close relationship between modular forms and elliptic equations.
Sadly, Taniyama committed suicide in 1958: "...I am in the frame of mind that I lost confidence in my future."
Harvard professor Barry Mazur described the Taniyama-Shimura Conjecture (TSC) as "the surmise that every elliptic equation is associated with a modular form".
The book outlines a link created by mathematician Gerhard Frey between TSC and FLT.
Mathematician Ken Ribet, a professor at UC Berkeley, proved that TSC implies FLT.
Ribet: "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove this conjecture."
Chapter 6 - The Secret Calculation - Notes (Completed on )
After completing his PhD at Cambridge, Wiles moved to Princeton University.
To escape distractions, Wiles would work at home in his attic study.
"You have to really think about nothing but that problem - just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over, and it's during that time that some new insight comes."
Wiles was very secretive and didn't share his work on FLT with anyone in the math community.
Évariste Galois was born in France in 1811. He fell in love with mathematics, publishing his first paper at seventeen.
Galois was interested in solving quintic equations and was influential in the creation of group theory.
He was deeply involved in politics and died after a duel in 1832 at the age of twenty-one.
After two years of work, Wiles made progress using group theory.
"You might ask how could I devote an unlimited amount of time to a problem that might simply not be soluble. The answer is that I just loved working on this problem and I was obsessed."
In 1988, 38-year-old Yoichi Miyaoka of Tokyo Metropolitan University claimed a proof of FLT using differential geometry. After months of analysis, critical flaws were revealed in the proof.
Wiles believed he made critical progress using the Kolyvagin-Flach method, and asked his peer, Professor Nick Katz, for assistance and review.
After seven years of effort, Wiles believed he had completed a proof of FLT.
While Wiles waited for the referees, his work received major international news coverage in 1993.
Chapter 7 - A Slight Problem and Epilogue - Grand Unified Mathematics - Notes (Completed on )
Mazur appointed an unprecedented six referees to review the 200-page proof.
Nick Katz identified issues with the portion of the proof he was assigned to review. He himself missed the error in his earlier reviews with Wiles.
Wiles sent an email to the math community describing the status of the proof and explaining that he was working hard to solve the issues.
Even after six months of working on it, Wiles chose not to release the manuscript, fueling the mathematics community's gossip and doubt.
Wiles enlisted Richard Taylor, his former student and one of the referees, for help in completing the proof.
Eventually, Wiles found the fix, leveraging Iwasawa theory, and he submitted the final proof for review. "It was so indescribably beautiful; it was so simple and elegant... It was the most important moment of my working life. Nothing I ever do again will mean as much."
Wiles shared the Wolf Prize with Robert Langlands in 1996 and won the Wolfskehl Prize in 1997.
"I was so obsessed by this problem that for eight years I was thinking about it all the time - when I woke up in the morning to when I went to sleep at night."