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Thursday, May 05, 2005


UP Math prof proves Princeton man wrong

By Rony V. Diaz 

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.

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.

Escultura, who had been working on the problem since 1992, disputed Wiles’ claim and inserted his refutation in the appendix to his book, Diophan­tus: Introduction to Mathematical Philosophy. He went on to present his results at the Second International Conference on Dynamic Systems and Applications in Atlanta in 1995.

In 1998 he published his formal refutation in “Exact solutions of Fermat’s equations (A definitive resolution of Fermat’s last theorem)” in the Journal of Nonlinear Studies, Vol. 5, No. 2, pp. 227–254. Since then Escultura has published over two dozen papers on the subject, and its applications to physics, in international scientific journals.

Escultura’s refutation sparked much discussion on the Internet that has spilled over to other fields such as physics, astronomy, cosmology, intelligence, learning, chaos, turbulence, gravity and nonlinear analysis.

He 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.

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.

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

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), counter­examples 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

Escultura was a former math and science editor and columnist of The Manila Times. He also taught math at The Manila Times School of Journalism.

He is currently working with Bernard Ziegler of the University of Texas at Houston on the new calculus based on Escul­tura’s real number system.

Ziegler and Escultura will also collaborate on a new nonstandard analysis, a subject on which Escultura has published many papers in peer-reviewed international journals.



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