Gary Cornell on Andrew Wiles
With Gary Cornell’s permission, I reprint here an email he sent me with a correction to the wikipedia quotation in my post “Trillions and Trillions: Getting Used to Balance Sheet Monetary Policy.”
By the way, Gary was alluding to Alan Blinder’s book Hard Heads, Soft Hearts: Tough-Minded Economics for a Just Society, which in fact has been a book that influenced my thinking about economic policy. Because Alan Blinder was Greg Mankiw’s professor when Greg was an undergraduate at Princeton doing graduate-level work, I think of Alan as my grand-advisor.
Dear Miles,
Ihave very much enjoyed your recent blog entries which I was turned on to by the continuously interesting posts of your students Noah Smith. (Having such a brilliant student must be quite a lot of fun!) Anyway, I am glad you have started blogging because we definitely need more economists with “soft hearts and hard heads” like you seem to have.
That having been said, in one of your recent posts you quoted the Wikipedia article on Wileswhich said this:
“Starting in the summer of 1986, based on successive progress of the previous few years of Gerhard Frey, Jean-Pierre Serre and Ken Ribet, Wiles realized that a proof of a limited form of the modularity theorem might then be in reach.”
Since I actually helped organize the big conference on the proof that Wiles for example spoke at, I remember the time very well! And unfortunately this sentence is, I believe subtly but still quite wrong and so gives the wrong idea of what was going on at the time and I dare say of what Wiles himself was thinking when he took the leap into the unknown. As I read it, it says Wiles thought the that the work done on proving the amazing implication (what we call Ribet’s theorem now) that: “Modularity (Shimura-Taniyama-Weil) for semi stable elliptic curves implies Fermat” led Wiles to think that modularity was in reach. My understanding, based on what Wiles has said numerous times and also because of what was being done in the field of modular forms at that time (such as the then relatively recent proof of the main conjecture etc.) was rather that Wiles was willing to try so hard to prove modularity, even though nobody thought it was possible at the time *because* of Ribet’s theorem. In other words, he was willing to make the effort because Ribet’s theorem told him Fermat would fall *if* he could prove modularity. And since nobody really thought modularity was in reach at the time, it was truly amazing that Wiles believed he could prove even a special case of it and his willingness to basically risk all on the attempt remains so amazing. (Wiles himself is often quoted as saying that proving Fermat was a lifelong dream of his from the time he was a young student and thus provided the motivation once Ribet had shown the implication.)
Anyway if I was going to rewrite the Wikipedia article, here’s what I think it should say:
“Starting in the summer of 1986, based on the work of the previous few years including Gerhard Frey’s on the importance of what is now called the Frey curve, Jean-Pierre Serre who showed that the properties of the Frey curve plus what became known as the epsilon conjecture would contradict a weak form of the modularity conjecture and then finally Ken Ribet with his deep and difficult proof of the epsilon conjecture, everyone now knew that a proof of a limited form of the modularity theorem meantFermat would fall. Of course, at that point even a weakened form of the modularity conjecture was thought of as being totally beyond reach, so Ribet’s theorem was thought of as simply being a really cool result and hey maybe in a hundred years or so we would finally knew enough to prove modularity, and therefore Fermat itself would be a corollary! Somehow, Wiles started to believe that by combining many of the techniques developed in the 70’s and 80’s in Modular Forms specifically and Arithmetic Geometry more generally, along with techniques that he had only glimmerings of at that point, maybe, just maybe he could prove modularity and thus his lifelong dream of proving Fermat might actually be possible. So he was willing to devote all his efforts for so many years to the proof of the modularity conjecture.”
Finally, for those who want to go further into the details of the proof we turned that conference into what has become one of the more practical ways to “get into the proof” – well “practical” if you have two or three years of graduate math courses of course.
Otherwise, I strongly recommend
as the best possible treatment of the circle of ideas around Wiles work for a mathematically literate reader-any grad student in economics should have enough background to read this I think.
Best regards,
Gary Cornell