Thursday, May 19, 2005

Letter from the UP Math Department

A friend sent this letter from faculty members of the University of the Philippines' Math Department. Mr. Escultura taught at the University of the Philippines.
Letter to the Editor of the Manila Times

We find the Manila Times’ front page story (“UP Math Prof proves Princeton Man Wrong”, 5 May) disturbing as it has given many readers false and unfounded information, and hence, a false sense of pride about our nation's scientists. To say that the celebrated proof of Fermat’s Last Theorem (FLT) by Princeton mathematician Andrew Wiles is wrong is a tall claim and would put into question one of the 20th century’s greatest intellectual achievements.

In 1637, Pierre de Fermat conjectured that the equation x ^n + y^n= z^n has no non-zero integer solutions whenever the exponent n is greater than 2. Several generations of mathematicians searched for a proof of FLT, which became the Holy Grail of mathematics. By discovering astonishing connections between different areas of mathematics, Wiles finally proved FLT in 1995. His proof was carefully studied by many mathematicians all over the world before it was published.

Edgar Escultura now says that Wiles was wrong. A refutation of this importance would have sent out ripples in the scientific community and landed in pages of newspapers around the world. Instead, it has fed discussions on the Internet on cranks and scientific discovery.

Your reader Roy Choco was correct: a newspaper of your stature should have done more sleuthing and checking of facts. Certainly, you need to do more than echo the assertions of Escultura.

The Chicago event after the announcement of Wiles' proof of FLT (Manila Times: "In Chicago, for instance, mathematicians marched on the streets in euphoric celebration.") never took place. This must have been based on a June 1993 Chicago Tribune article (see for example http://www.langston.com/Fun_People/1993/1993AGD.html) which parodied the excitement of math people whenever a difficult math problem is solved. A fruitless google search on Escultura’s collaborator “Bernard Ziegler of the University of Texas in Houston” leads us to suspect that a mathematician of that name does not exist.

Ed Escultura first raised the issue of foundational problems of mathematics at least 15 years ago, asserting as shaky the grounds which concepts/definitions—including all the mathematics we were taught in school-- stand on. Then in 1993, he trained his guns on Fermat's Last Theorem, followed by the gravitational n-body problem. This last problem eluded Einstein until his death.

In his papers, Escultura does not give a critique of Wiles' work per se; FLT is simply the straw man he takes jabs at to question the real number system and its foundations. The foundations are wrong, he says, and therefore statements built on them, like FLT, are 'wrongly-formulated.' By rebuilding the real number system on new ground, he proceeded to construct counter-examples to Fermat's statement.

Counter-examples to Fermat are not new. If one changes the setting of the equations, then solutions to the equation of Fermat can be found. For example, it has been known since the early 1900s that solutions to Fermat’s equation exist in the world of “p-adic numbers”. But this does not refute Fermat and Wiles’ theorem: that there are no solutions in the world of integers.

What is meant by “foundations” in mathematics are the axioms (or postulates, as many of us have learned in high school geometry), from which mathematical statements are proved through a chain of logical reasoning. Axioms are very much like the rules of a game of chess. A configuration of pieces on the board is possible if it can be arrived at after a sequence of legal moves. In the same way, a mathematical statement is valid (or “correct”) if it can be proved from the axioms in a finite number of steps.

Escultura says that some axioms of the real number system are false. But axioms, just like the rules of a game, are just axioms—neither false nor true. There can be different systems of mathematics, each with its own set of axioms and semantics.

Escultura saw "errors" in FLT because he 1) insists that axioms such as trichotomy and completeness are 'wrong' and 2) he judges FLT from the standpoint of another axiom set. Allowed these liberties, the logician Bertrand Russell proved that he was the Pope. If Escultura's axiom set is consistent, then the mathematics that follows from it is neither more valid nor less valid than the mathematics we learn in school. We can have as many axiomatic systems as we want, each as logically valid as the rest.

Many years ago, New York University physicist Alan Sokal conducted an unusual experiment. In order to parody postmodernism and question the standards academia uses to judge what is fit for publication, he submitted to the prestigious journal Social Text a paper containing absolute nonsense written in academic gobbledygook. His hypothesis was that a journal accepts a paper if it "sounds good," and if it "flatters the editors' ideological preconceptions." Well the journal accepted the paper and how embarrassing it was a year later when Sokal revealed the hoax.

We wish Escultura writes the Manila Times and the “peer-reviewed journals” to say he was just kidding.

It does not speak well for the Manila Times that, without checking facts or consulting the scientific community, it has given front page coverage to a tiring issue that has made the rounds of math and science Internet forums many years ago. The only thing new here is Escultura's exchange of letters with Andrew Wiles. Wiles' letter itself drips with sarcasm, and it is amazing that someone could take it seriously.

EE has taken the Times for a ride, along with the rest of us, if we fall for this feel-good story of a Pinoy professor outwitting intellectual giants of the West.

The Faculty of the Department of Mathematics
University of the Philippines
Diliman, Quezon City

6 comments:

Abe said...

Roy,

Well, that should settle it.

For some reason the text in the message does not display properly in Safari (Mac browser), although Firefox handles it OK.

I suspect the problem is in this tag:
< pre wrap="" >Letter to the Editor of the ..
and
< /pre > at the end

If you just take those out of the post, it will probably fix things. (I put spaces in the tags to try to make them show here.)

Best regards,
Abe

roy choco said...

fixed it, thanks again

Sef said...

Idol!!!

Abe said...

By the way, if you haven't checked out that link to the "story" of the mathematicians going wild in Chicago, you should. It's a satire of the Chicago sports fans who went nuts after a championship.

roy choco said...

Yup, I've read it, but to be fair, if I did not know that it was satire, It would probably take me a while to "grok" the joke. Mathematicians are alien to me, I am not sure they are not capable of rioting in the streets :-)

E. E. Escultura said...

TOTAL IGNORANCE OF FUNDAMENTALS.

I just focus on a single paragraph of the poisoned letter from the Math Department that says: Axioms are like the rules of a game - neither true nor false.

This is total ignorance of the funamentals of mathematics. David Hilbert spent almost his entire mathematical career to elminate contradictions in mathematics a century ago and tried to prove the real number system consistent which he couldn't because it isn't. Even the rules of a game cannot be contradictory, otherwises, no one would judge it. A single false axiom makes a mathematical space nonsense. The trichotomy axiom of the real number system is false because a counterexample to it AMONG THE REAL NUMBERS exists. What this means is that the real number system (including the integers) is nonsense and FLT, being formulated in it, is also nonsense. Therefore, the first crucial step in resolving it (which Wiles did not do) is to make a critique of the underlying fields, namely, foundations, number theory, and the real number system, which I did, and provide the right remedy. Among the remedy is to reconstruct the real numbers as the contradiction-free new real number system R* consisting of decimals on these simple axioms:

1) R* contains the basic integers 0, 1, ..., 9.

2) The additon and
3) multiplication tables of primary school.

Then this new mathematical space yields the countably infinite counterexamples to FLT. They are as follow:

Let

x = (0.99...)10^T, T = 1, 2, ...

y = d* = 1 - 0.99...

x = 10^T.

Then, for n > n,

x^n + y^n = z^n.

Moreover, for k = 1, 2, ...,

(kx)^n + (ky)^n = (kz)^n.

For details see my websites, FLT and the Gravitational n-Body Problem, and http://home.iprimus.com.au.pidro/ .

Of course, those who do not publish in this field will never understand this issue and the counterexamples and I'm aftraid the writer of the article from the Math Department is one of them.

E. E. Escultura