Thursday, May 19, 2005

Manila Times - Escultura, Wiles, and Fermat

Here's another update in the saga of Dr. Edgar Escultura, Dr. Andrew Wiles, and Fermat's Last Theorem (FLT). The Manila Times published a story asserting that Dr. Escultura had "proved that Wile's proof of Fermat's Last Theorem is false." This was a peculiar report, as we noted at the time and in a follow up a few days later. The only evidence presented for the claim that Wiles had "admitted" his error was an unverified letter, supposedly from Wiles, that was laced with sarcasm.

Mr. Roy Choco, of Manila, also smelled a rat, and he wrote a letter to the editor, which questioned the Times on the veracity of the story. To their credit, The Manila Times published it, and we reviewed the letter and the paper's response. They seemed to delight in pointing out some grammar issues with the letter rather than addressing the important points raised by Mr. Choco. It is clear they made no attempt to check the facts of the story with anyone other than Dr. Escultura.

Mr. Choco has now written The Times a second letter and also posted it on his blog: Random Thoughts. He addresses the paper's response and again takes them to task for not verifying the letter or the claim of refuting Wiles' proof. We will see if The Times will print the second letter, but our guess (and Mr. Choco's) is they are eager to put this behind them. It is clear the original story is incorrect.

In the original article, The Manila Times Internet Edition - "UP Math prof proves Princeton man wrong," they say:
"He [Dr. Escultura] took the position that the failure to resolve the problem for over 360 years reveals the inadequacy and defects of foundations, number theory and the real number system. He undertook a thorough critique-rectification of these fields and found, among others, that the real number system in basic algebra, the foundation of mathematics, is defective. Specifically, two of its axioms (the trichotomy and completeness axioms, for those who took basic algebra in high school and college) are false."
Axioms are by definition simple principles that are accepted without proof and used to prove other propositions. As such an axiom (or two of them) cannot be proven to be false. A set of axioms may be found to be in conflict with each other, in which case they are said to be "inconsistent," not "false."

The axiom of trichotomy is that for any two real numbers, "a" and "b," one and only one of the following is true: a = b; a > b; or a
"Escultura went on to overhaul the real number system and reconstructed it without these false axioms using only three simple axioms instead of 12. The result is a new real number system that is free from defects and contradictions, finite and enriched with new numbers that have important applications for physics.

Using the new real number system Escultura constructed many counterexamples to FLT showing that it is false."
Again, there may be some value in an alternative number system, but a proof under the standard system of axioms simply cannot be "refuted" by changing to a different system. It's perfectly possible for FLT to be true in one system and false in another. Furthermore, the whole basis of FLT is "positive integers greater than 2" not the real numbers (which include fractions, decimal numbers, and irrationals). Counter examples to FLT based on non-integers are not relevant.

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Update 5/19: Roy has posted what he believes is a letter from the UP math department to the Manila Times. It's unsigned and not published on the Times site yet, but it's written well by someone with a good math background. It also includes this link to the not be be missed story (mentioned in the original Manila Times article) of mathematicians going wild in the streets of Chicago after Wiles' proof was announced.

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