Students were asked to write about the life and work of a mathematician of their choice. Mozzochi, Princeton N. There is a problem that not even the collective mathematical genius of almost years could solve. When the ten-year-old Andrew Wiles read about it in his local Cambridge library, he dreamt of solving the problem that had haunted so many great mathematicians. Little did he or the rest of the world know that he would succeed I had to solve it.
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The methods introduced by Wiles and Taylor are now part of the toolkit of number theorists, who consider the FLT story closed. But number theorists were not the only ones electrified by this story. I was reminded of this unexpectedly in when, in the space of a few days, two logicians, speaking on two continents, alluded to ways of enhancing the proof of FLT — and reported how surprised some of their colleagues were that number theorists showed no interest in their ideas.
The logicians spoke the languages of their respective specialties — set theory and theoretical computer science — in expressing these ideas. Over the last few centuries, mathematicians repeatedly tried to explain this contrast, failing each time but leaving entire branches of mathematics in their wake. These branches include large areas of the modern number theory that Wiles drew on for his successful solution, as well as many of the fundamental ideas in every part of science touched by mathematics.
The computer scientist had recently been excited to learn about progress in automated proof verification , an ambitious attempt to implement the formalist approach to mathematics in practice. For formalists, a mathematical proof is a list of statements that meet strict requirements: The statements at the top of the list must involve only notions that are universally accepted. Each statement must be obtained by applying the rules of logical deduction to the preceding statements.
Finally, the proved theorem should show up as the last statement on the list. Mathematical logic was developed with the hope of placing mathematics on firm foundations — as an axiomatic system, free of contradiction, that could keep reasoning from slipping into incoherence. Mathematicians never write proofs this way, however.
Automated proof verification seems to offer a solution. It entails reformulating the proof as a series of discrete statements, each written in an unambiguous language that can be read by a computer, which then confirms that every step deserves to be stamped with a constitutional certification.
This painstaking method has been applied with success to many long and difficult proofs, most famously by Thomas Hales and his collaborators to the proof of the Kepler conjecture on the densest way to pack spheres.
More precisely, it had long been known how to leverage such an elliptic curve into C a Galois representation, which is an infinite collection of equations that are related to the elliptic curve, and to each other, by precise rules. The links between these three steps were all well-understood in By that year, most number theorists were convinced — though proof would have to wait — that every Galois representation could be assigned, again by a precise rule, D a modular form, which is a kind of two-dimensional generalization of the familiar sine and cosine functions from trigonometry.
But there are no such forms. Therefore there is no modular form D , no Galois representation C , no equation B , and no solution A. The only thing left to do was to establish the missing link between C and D , which mathematicians call the modularity conjecture. Twenty years after Yutaka Taniyama and Goro Shimura, in the s, first intimated the link between B and D , via C , mathematicians had grown convinced that this must be right.
The connection was simply too good not to be true. But the modularity conjecture itself looked completely out of reach. Objects of type C and D were just too different. I will not try to untangle this ambiguity. But if what the logicians had in mind was to formally verify the published proof of the relation between C and D , then they were setting their sights too low. For one thing, Wiles only proved a bit more than enough of the modularity conjecture to complete the A to E deduction. More recently, in the fall of , for example, 10 mathematicians gathered at the Institute for Advanced Study in Princeton, New Jersey, in a successful effort to prove a connection between elliptic curves and modular forms in a new setting.
If asked to reproduce the proof as a sequence of logical deductions, they would undoubtedly have come up with 10 different versions. Each one would resemble the A to E outline above, but would be much more finely grained. They would refer in a similar way to the proofs they studied in the expository articles or in the graduate courses they taught or attended.
And though each of the 10 would have left out some details, they would all be right. Nothing outside the formal system is allowed to contaminate the ideal proof — as if every law had to carry a watermark confirming its constitutional justification.
But this attitude runs deeply counter to what mathematicians themselves say about their proofs. But — unlike many who follow number theory at a distance — they were certainly aware that a proof like the one Wiles published is not meant to be treated as a self-contained artifact.
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Fermat's Last Theorem
The methods introduced by Wiles and Taylor are now part of the toolkit of number theorists, who consider the FLT story closed. But number theorists were not the only ones electrified by this story. I was reminded of this unexpectedly in when, in the space of a few days, two logicians, speaking on two continents, alluded to ways of enhancing the proof of FLT — and reported how surprised some of their colleagues were that number theorists showed no interest in their ideas. The logicians spoke the languages of their respective specialties — set theory and theoretical computer science — in expressing these ideas.
Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced
In —, Gerhard Frey called attention to the unusual properties of this same curve, now called a Frey curve. Mathematically, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x and y of the points on it. In , Jean-Pierre Serre provided a partial proof that a Frey curve could not be modular. However his partial proof came close to confirming the link between Fermat and Taniyama. His article was published in