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Posts Tagged ‘Fermat

“I have had my results for a long time, but I do not yet know how to arrive at them”*…



Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into the third talk, he came to the end. Then, to punctuate the result, he added:

=> FLT

“Implies Fermat’s Last Theorem.” The most famous unverified conjecture in the history of mathematics. First proposed by the 17th-century French jurist and spare-time mathematician Pierre de Fermat, it had remained unproven for more than 350 years. Wiles, a professor at Princeton University, had worked on the problem, alone and in secret in the attic of his home, for seven years. Now he was unveiling his proof.

His announcement electrified his audience—and the world. The story appeared the next day on the front page of The New York Times. Gap, the clothing retailer, asked him to model a new line of jeans, though he demurred. People Weekly named him one of “The 25 Most Intriguing People of the Year,” along with Princess Diana, Michael Jackson, and Bill Clinton. Barbara Walters’ producers reached out to him for an interview, to which Wiles responded, “Who’s Barbara Walters?”

But the celebration didn’t last. Once a proof is proposed, it must be checked and verified before it is accepted as valid. When Wiles submitted his 200-page proof to the prestigious journal Inventiones Mathematicae, its editor divvied up the manuscript among six reviewers. One of them was Nick Katz, a fellow Princeton mathematician.

For two months, Katz and a French colleague, Luc Illusie, scrutinized every logical step in Katz’s section of the proof. From time to time, they would come across a line of reasoning they couldn’t follow. Katz would email Wiles, who would provide a fix. But in late August, Wiles offered an explanation that didn’t satisfy the two reviewers. And when Wiles took a closer look, he saw that Katz had found a crack in the mathematical scaffolding. At first, a repair seemed straightforward. But as Wiles picked at the crack, pieces of the structure began falling away…

How mistakes– first Fermat’s, then Wiles’– reinvigorated a field, then led to fundamental insight: “How Math’s Most Famous Proof Nearly Broke.”

* Karl Friedrich Gauss


As we ponder proof, we might we might spare a thought for Josiah Wedgwood; he died on this date in 1795. An English potter and businessman (he founded the Wedgwood company), he is credited, via his technique of “division of labor,” with the industrialization of the manufacture of pottery– and via his example, much of British (and thus American) manufacturing.

Wedgwood was a member of the Lunar Society, the Royal Society, and was an ardent abolitionist.  His daughter, Susannah, was the mother of Charles Darwin.



What’s (the) matter?…


On the heels of yesterday’s film recommendation, another… albeit somewhat different:  Stanford physics professor, Leonard Susskind, one of the fathers of string theory, articulator of the Holographic Principle,  and explainer of the Megaverse, has a gift for making science accessible… a gift that is on display in this lecture, “Demystifying the Higgs Boson“:

(email readers, click here)


As we say “ahh,” we might spare a thought for Pierre de Fermat; he died on this date in 1665.  With Descartes, one of the two great mathematicians of the first half of the Seventeenth Century, Fermat made a wide range of contributions (that advanced, among other fronts, the development of Calculus) and is regarded as the Father of Number Theory.  But he is best remembered as the author of Fermat’s Last Theorem.* Fermat had written the theorem, in 1637, in the margin of a copy of Diophantus’ Arithmetica– but went on to say that, while he had a proof, it was too large to fit in the margin.  He never got around to committing his proof to writing; so mathematicians started, from the time of his death, to try to derive one.  While the the theorem was demonstrated for a small number of cases early on, a complete proof became the “white whale” of math, eluding its pursuers until 1995, when Andrew Wiles finally published a proof.

* the assertion that no three positive integers ab, and c can satisfy the equation an + bn = cn for any integer value of n greater than two


Written by LW

January 12, 2013 at 1:01 am

This is cool, but I’m holding out for a disease…


Hankering for a little immortality?  New Scientist has the answer:

While most mathematical theorems result from weeks of hard work and possibly a few broken pencils, mine comes courtesy of TheoryMine, a company selling personalised theorems as novelty gifts for £15 a pop.

Its automated theorem-proving software can churn out a theoretically infinite number of theorems for customers wishing to join the ranks of Pythagoras and Fermat. “We generate new theorems and let people name them after themselves, a friend, a loved one, or whoever they want to name it after,” explains Flaminia Cavallo, managing director of TheoryMine, based in Edinburgh, UK…

“We’re inventing totally novel theorems, and the tradition is you have the right to name these theorems,” explains Alan Bundy, professor of automated reasoning at the University of Edinburgh and another member of the TheoryMine team. “There are 10 star companies out there, and none of them have any affiliation to the International Astronomical Union.”

He’s got a point. Automated theorem proving is a well-respected mathematical field, used by manufacturers to guarantee that the algorithms in computer processors will work correctly. Bundy and his colleagues have worked in this area for a number of years, and Cavallo came up with the idea for TheoryMine during her final year of an undergraduate degree in artificial intelligence and mathematics at the University of Edinburgh, where she wrote a program to generate novel theorems for her dissertation.

From its library of mathematical knowledge, the program generates a set of mathematical axioms, then combines them in different ways to produce a series of conjectures. It then uses the library to discard a portion of these on the basis that there are already counter-examples, showing they can’t be true. Overly complex conjectures are also ignored. Then it applies a technique known as “rippling”, in which it tries out various sequences of logical statements until one of these sequences turns out to be a proof of the theorem…

“It’s a clever idea,” says Lawrence Paulson, a computational logician at the University of Cambridge and the creator of Isabelle, a theorem prover that Cavallo’s program uses. He is more interested in the theory behind the new program though, adding that “some of the technology here is quite impressive, and I would hope that it finds other applications apart from selling certificates”.

It may well do. Lucas Dixon, another TheoryMiner, is investigating the possibility of using the same techniques to elucidate the rules of algebra in quantum computing systems, which follow different mathematical rules to classical systems.

Don’t prepare your Fields medal acceptance speech just yet though, as TheoryMine’s theorems are unlikely to break drastically new ground. “We can’t say that we’ll never do that, but having looked at the things that come out, they’re not typically things that are going to change the world,” says Dixon.

Your correspondent just purchased “Eleanor’s Equation” for his daughter; reader’s can score their own mathematical monument at TheoryMine.

As we search for the “rum” in theorum,” we might wish a Buon Compleanno to Count Francesco Algarotti, the philosopher, critic, and popularizer of complex scientific ideas; he was born in Venice on this date in 1712– and wrote Neutonianismo per le dame (Newtonism for Ladies) when he was 21.



Oh, the places we’ll go…

The Atlas Obscura, “A Compendium of the World’s Wonders, Curiosities, and Esoterica”…  Consider, if you will:

The Cockroach Hall of Fame Museum

Featuring dead bugs dressed as celebrities and historical figures, this just might be the one time in your life that a cockroach puts a smile on your face.

On your visit, you’ll see cockroach displays featuring “Liberoachi,” “The Combates Motel,” and “David Letteroach,” among dozens of others.

See the Fremont Troll, the Wunderkammer, the Harmonic bridge and dozens of others, here.


As we re-plot our itineraries, we might offer a tip of the birthday beret to Blaise Pascal, born on this date in 1623.  Pascal was an extraordinary polymath: a mathematician, physicist, theologian, inventor of arguably the first digital calculator (the “Pascaline”), the barometer, the hydraulic press, and the syringe.  His principle of empiricism (“Experiments are the true teachers which one must follow in physics”) pitted him against Descartes (whose dualism was rooted in his ultimate trust of reason).  Pascal also attacked from the other flank; his intuitionism (Pensées) helped kick-start Romanticism, influencing Rousseau (and his notion of what Dryden called the “noble savage”), and later Edmund Husserl and Henri Bergson.  But perhaps most impactfully, his correspondence with Pierre de Fermat (the result of a query from a gambling-addicted nobleman) led to development of probability theory.


Written by LW

June 19, 2009 at 12:01 am

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