Fermat’s Engima, by Simon Singh, is an engaging and readable history of the search for a proof of Fermat’s Last Theorem. Pierre Fermat was a 17th-century mathematician who wrote in the margins of a book that the equation (x^n + y^n = z^n) has no solutions for n greater than 2. He further stated that he had a proof for this statement, but the margin was too small to contain it. For the next 350 years, mathematicians struggled to find the elusive proof, until at long last Andrew Wiles completed a proof of the theorem in 1994. Singh expertly weaves together stories of the mathematician’s lives and explanations of some of the underlying mathematics related to the theorem, creating a book that is hard to put down.
The pursuit of pure mathematics intrigues me. This field strives for truth and proof in a way that no other discipline does. Every theorem must be proved using undeniable logic. The people Singh describes spent many hours simply thinking, as well as of course scribbling equations and notes on scraps of paper. It seems that what drives these people is an immense love and admiration for the beauty and elegance of numbers and mathematical logic and a strong desire to discover absolute truths. For example, Singh quotes Wiles as having said “it was so indescribably beautiful; it was so simple and so elegant” about the last link that finally completed his proof.
In some ways, I can understand this admiration of beauty and desire for proof. One of the many reasons I studied computer science as an undergraduate was because I had always particularly enjoyed logic and math. I appreciate the elegance of certain concepts, such as induction and the halting problem (a proof that there exist problems which computers cannot solve), and enjoyed learning the mathematical fundamentals of computer science. Although I am less familiar with mathematical proofs, I can see the appeal in constructing impeccable logical arguments that result in showing that something is irrevocably true. However, in order to construct these proves, mathematicians work for years on end to solve a single problem or set of problems. I am not able to relate so well to this single-minded long-term pursuit of solutions to mathematical problems. It is hard for me to see how one can work for so long in pursuit of solving a problem which does not necessarily have any practical applications. Granted, many parts of mathematics do have pratical applications. However, many mathematicians do not work on the problems because of those applications, but simply for the goal of solving the problem itself. While I appreciate the elegance of logic and proof, I do not think the sole goal of obtaining such proofs could ever be an effective motivation for myself.
One question remains at the end of Fermat’s Enigma: what proof did Fermat himself have of his theorem, if any? Wile’s proof was over 200 pages and relied on many discoveries made in mathematics after Fermat lived. It is highly improbable that Fermat had come up with this same proof. Some mathematicians therefore believe that there must be another proof that relies only on 17th-century mathematics. However, I find it hard to believe that is the case, given that many brilliant mathematicians have worked on the problem since Fermat and none have come up with such a proof. Furthermore, my impression from the book was that many, if not most, proofs initially have some small hole that is caught by other mathematicians when the proof is reviewed. Thus, I am of the opinion that Fermat probably thought he had a proof of the theorem, but it was one that contained some unpatchable logical error.