August 29, 2008
Proof Positive
From Harvard Magazine:
As academics work to understand the architecture of the universe, they sometimes uncover connections in mysterious places. So it is with Smith professor of mathematics Richard L. Taylor, whose work connects two discrete domains of mathematics: curved spaces, from geometry, and modular arithmetic, which has to do with counting. Taylor has spent his career studying this nexus, and recently proved it is possible to use one domain to solve complex problems in the other. “It just astounded me,” he says, “that there should be a connection between these two things, when nobody could see any real reason why there should be.”
This is not the first instance of finding in geometry an elegant explanation for a seem- ingly unrelated phenomenon. Scholars during the Renaissance, seeking a mathematical basis for our conceptions of beauty, fingered the so-called Golden Ratio (approximately 1.6 to 1). Some analyses find the ratio in structures—most famously the Parthenon—built centuries before its first written formulation. More recently, scientists have found that the faces people find most beautiful are those in which the proportions conform most closely to the ratio. The geometry-arithmetic connection explored by Taylor solves another puzzle that has enticed mathematicians across centuries. In 1637, French mathematician Pierre de Fermat scrawled in a book’s margin a theorem involving equations like the one in the Pythagorean theorem (a2 + b2 = c2), but with powers higher than two. Fermat’s theorem said such equations have no solutions that are whole numbers, either positive or negative. Go ahead, try—it is impossible to find three integers, other than zero, that work in the equation a3 + b3 = c3.
The French mathematician also wrote that he had discovered a way to prove this—but he never wrote the proof down, or if he did, it was lost. For more than 350 years, mathematicians tried in vain to prove what became known as Fermat’s Last Theorem. They could find lots of examples that fit the pattern, and no counterexamples, but could not erase all doubt until Princeton University mathematician Andrew Wiles presented a proof in 1993.
His discovery made the front page of the New York Times, but six months later, the Times reported that another mathematician had found a mistake in the new proof.
More here.
Posted by Azra Raza at 06:04 AM | Permalink
Comments
A great post.
For a superb read: The Golden Ratio by Mario Livio.
Posted by: Felix E F Larocca MD | Aug 29, 2008 7:10:26 AM
Anyone interested in Fermat's Last Theorem must read the works of the late Eric T. Bell, the distinguished mathematician at the California Institute of Technology. He referred to this theorem as "The Last Problem". He apparently believed it would never truly be "solved" and in a sense he was right after all. All the very best mathematicians worked on this problem but none was able to produce a proof involving known knowledge at the time which was great indeed and most working on the problem were genius level intellects. Many think Fermat did not have a proof at all. I disagree. I believe he did, in fact have a proof, but it did not resemble the now accepted one (1995) in any way, shape or form. What is so perplexing about this problem is that none of the known, usual, methods of proof work. The beautiful proof by contradiction that the square root of 2 is irrational fails. All efforts to arrive at contradictions between evens and odds fail. Efforts to use complex number theory fail, etc.
One way to think about this problem which provides insight available to ordinary mortals is to rewrite the equation by replacing x with z-N and y with z-M. N and M are obviously positive integers. The statement then becomes an nth order polynomial with real, rational coefficients. We know from general principles that this type of equation has n roots, complex in general. The imaginary parts of the roots may be zero in some cases. Closed form solutions are available only up to n=4 as proved by Abel. But even they are of little value as they become very cumbersome to work with even at n=3. The problem of course then is to prove that no solutions to this polynomial exist with z a positive integer and n greater than 2. I believe this would be the route of Fermat's proof if he had one.
Suppose all the printed proofs claimed to solve this problem in 1995 were destroyed and suppose its authors somehow developed amnesia on the proof.
How many people in the world could reproduce the proof? Indeed, could any of them reproduce it? Moreover, how many could explain it to others?
Richard Feynman is quoted with the remark: "You don't truly understand something until you can explain it to school children." In this sense, the great theorem of Fermat still remains unproved to this day as I suspect professor Bell thought it would be; hence, "The Last Problem".
And then there is the underlying problem with all proof as professor Hoffman has elegantly stated at www.contrarianisms.com: "Consider that the processes of reason involve dealing either in tautologies or else in error. That is, we state the exact same thing-albeit in a different way, or else we do not, and if we do not then we are in error. And who vouches for the original proposition in the first place?"
An interesting analogy with Fermat's Last Theorem to another theory for General Relativity involving Clifford Algebra is available at http://www.donaldgburkhard.com/manuscript.html. This work is by professor Donald G. Burkhard, a physicist.
Posted by: Winfield J. Abbe | Aug 30, 2008 9:24:57 PM
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