Andrew Wiles and Fermat’s Last Theorem

Here is one of my all time favorite documentaries, the 45 minute Fermat's Last Theorem made by Simon Singh and John Lynch for the BBC in 1996.  I've watched it many times and every time I am moved by unforgettable moments.

The plainspoken Goro Shimura talking of his friend Yutaka Taniyama, "he was not a very careful person as a mathematician, he made a lot of mistakes but he made mistakes in a good direction." "I tried to imitate him," he says sadly, "but I found out that it is very difficult to make good mistakes." Shimura continues to be troubled by his friend's suicide in 1958.

And then there is Andrew Wiles, the frail knight who for seven lonely years pursues the proof that has ensorcelled him since childhood. He announces the proof to the world, is featured on the front page of the New York Times and in People Magazine, he has the respect and admiration of his colleagues and then he discovers the proof is wrong.  He works another year trying to fix it but every time he patches one area another fault line opens up. Even speaking of it now you can see and hear his utter despair.  It is not too much to imagine that he was on the verge of a breakdown.  Unforgettable.

Hat tip to Kottke.



"Mathematicians and natural philosophers ... have little temptation to form themselves into factions and cabals, either for the support of their own reputation, or for the depression of that of their rivals."


Excellent; thanks for the pointer Alex.

@darren: I'm a math graduate student studying number theory. We ran a year-long seminar with the goal of understanding the main thing that went into Wiles' proof of the Taniyama-Shimura conjecture (which, as I suspect is explained in the video, is what implies Fermat via work of Ribet and Frey); this crucial ingredient is what's known as a "modularity lifting theorem". Suffice it to say that explaining the actual math that goes into the proof of such a theorem would be incredibly ambitious for a 1-hour popular documentary. Nonetheless, there is actually a wonderful popular book -- namely, "Fearless Symmetry", by A. Ash and R. Gross -- that explains the ideas behind the central mathematical objects involved, which are called Galois representations. (Named for Evariste Galois, a very smart man who became famous among non-mathematicians perhaps not so much for his seminal mathematical contributions as for his biography, for which see Wikipedia, or the book review .)

As far as histories of modern mathematics in the large, that's a rather tall order, and of course depends on your definition of the word "modern". But one book I like is "The Honors Class" by Yandell, which is biographical in nature and confined to the 20th century. It sketches the lives and contributions of all those who solved one of Hilbert's famous problems, which he presented at the 1900 International Congress of Mathematics. Since these involve a wide range of disciplines, they give a nice picture of modern mathematics as a whole. But I would stress that Hilbert's problems, from the point of view of 2010, don't strike me as particularly indicative of the trajectory of 20th century math. (To give an example, a big chunk of the problems, and hence of the book, are concerned with what's known as the "foundations" of mathematics or particular areas of it. This refers to the most basic logical underpinnings of the subject, such as set theory. While certainly people still study these questions, a casual reader of Yandell's book might come away thinking they are more important than they are.)

A comprehensive intellectual history of mathematics in the 20th century, informed by the viewpoint of contemporary mathematicians, would be fascinating to read. All I can point to is a couple of truly excellent histories of particular subdisciplines, namely J. Dieudonne's "History of Algebraic Geometry" and "History of Functional Analysis", as well as the volume "History of Topology" edited by I.M. James. All of these works, I think, require a bit of background in mathematics to read, so be warned. For some insightful reflections on such scholarship, see Andre Weil's 1978 ICM address "History of Mathematics: Why and How" at

To Tyler Fan: I think that Wiles spent the first 7 years of work in secrecy, doing also other, less important work publicly. That less important work enabled him to publish enough papers to be considered productive. So I do not think that tenure helped him in the most important period of work.

Gives me gooesbumps! Thank you for posting the video.

The current parallel, currently quieted down now, has been the fuss over the proposed
P != NP proof of a month ago or so. Discussion of that dragged in two Fields Medal winners
and a host of leading computer scientists, mathematicians of various stripes, including some
of the top logicians, as well as physicists and other riff raff, including even the stray
economist or two. There was discussion about if this was a waste of time by the top elite
of this bunch of people. Remains not clear, and not the same as a two month referee report.
However, when the question is viewed as being really big, and Fermat's Last Theorem was
certainly one of the biggest, and N != NP is also pretty big, then people are more willing
to devote extra time to figuring it out.

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I can't believe how much of this I just wasn't aware of. Thank you for bringing more information to this topic for me. I'm truly grateful and really impressed.

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