Thursday, May 05, 2005

Math Wars

A raging controversy in the world of mathematics has been "settled" according to The Manila Times (Hat tip: Lyle). The winner? The local hero:
UP Math prof proves Princeton man wrong: "Edgar Escultura, a professor of mathematics at the University of the Philippines, proved that Andrew Wiles' proof of Fermat's last theorem is false.

In 1993 Andrew Wiles of Princeton University announced at a lecture in London that he had proved Fermat's last theorem (FLT). This is a conjecture by the French mathematician Pierre de Fermat in 1637 that for any integer n greater than 2, Fermat's equation that claimed xn + yn = zn has no solution in integers x, y z except 0 which satisfies the equation.
N.B. The "n's"in Fermat's equation should be exponents (e.g. "x to the n power").

The Manila Times then undermines its math credibility by mis-defining "integers." Whole numbers are 0,1,2,3,4... While all whole numbers are integers, the integers also include negative numbers, which are not part of the set of whole numbers. But we digress...
Integers are whole numbers like 8, 73, 1,257, etc. Since that time mathematicians and amateurs had been trying to find a proof but failed.

When Wiles made the announcement it was celebrated around the world. In Chicago, for instance, mathematicians marched on the streets in euphoric celebration."
Yeah, we remember that. What a night it was! You know how wild those math parties can get.

Since those heady days, Filipino mathematician, Edgar Escultura, has been pushing his "refutation" of Wiles' proof. Escultura, who is also a former Manila Times columnist and math teacher at The Manila Times School of Journalism (We are not making this up.) has finally forced Wiles to "throw in the towel," as the Manila Times continues:
On April 26, Andrew Wiles conceded an error in his proof. His letter and Escultura’s reply are below.

Tuesday 04/26/2005 6:57:33am

Dear Sir,

Your work is incredible, I read all of it just yesterday and let me tell you I respect you. I am going to review all my ‘proof’ which I am sure is wrong (thanks to you!).

Would you like to collaborate with me in this work? I have noticed some imperfections in your perfect proof (that sounds like you), and I’d like to create a perfect proof with you, great professor.

Also I’d like to have the address of the guy who let you get a PhD 30 years ago. I’d like to discuss few things with him...

Very respectfully,
A. Wiles [elipses in the original]
Call it a hunch, but we don't think Prof. Wiles is convinced just yet. The last sentence is the tipoff, of course, but there is also the statement about having read all of Escultura's work of 30 years "just yesterday."

Escultura's reply plays it straight:
Dear Prof. Wiles,

I welcome and appreciate your comments and I hope we can have a continuing dialogue. Regarding your invitation to collaborate with you, I would be glad to. But here is the situation:

My critique of mathematics is focused not on your work but mainly on the underlying fields of FLT which are foundations, number theory and the real number system. Here is what I found:

1) Two of the axioms of the real number system are false, namely, the trichotomy and completeness axioms (the latter is a variant of the axiom of choice), counter examples to them were constructed by Brouwer and Banach-Tarski, respectively.

2) I also noted a flaw in the use of the universal or existential quantifiers on infinite set.

3) To avoid contradictions, it is necessary to well define a mathematical space and its concepts by a consistent set of axioms. A concept is well defined if its existence, properties and relationship with other concepts are specified by the axioms. Most of the concepts of mathematics today are ill-defined.

Based on these findings I constructed the new real number system on three simple axioms. Its most updated version appears in the Journal of Nonlinear Analysis and Phenomena. An extended abstract of it appears in my updated website in June.

Yes, indeed, there are imperfections in my work but not on principles. And if there are major ones, I would like to know.

Regarding my academic advisor, he was the late L. C. Young, distinguished research professor and professor emeritus at the University of Wisconsin.

Again, thank you for your interest in my work and for inviting me to collaborate with you.

With my very best wishes.

E. E. Escultura
We assume Prof. Escultura understood that Wiles's letter was being sarcastic, but he decided to take the high road with his reply. Still the overall context of the article makes us wonder. And how could The Manila Times represent Wiles's letter as conceding "an error in his proof?" Is this a joke by Escultura? Could he be clueless ennough to have mistaken Wiles's sarcastic letter for sincerity?

Update 5/9: Corrected spelling of "Filipino."
Update 5/13: See also "Wiles Escultura and Fermat" and "Escultura and Wiles III."

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