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
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
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,
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. . .
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
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
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.