Revisiting Andrew Wiles' Proof of Fermat's Last Theorem: A Critical Analysis

[Rick21383]

What do you guys think of this? I haven't been able to find any sort of verification from reputable sites.

"Edgar Escultura, a professor of mathematics at the University of the Philippines, proved that Andrew Wiles’ proof of Fermat’s last theorem is false."

http://www.manilatimes.net/national/2005/may/05/yehey/top_stories/20050505top4.html

 

[joeboo]

Haha. The letter "written" by Andrew Wiles is classic:

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

Yep. That sounds like a letter an intelligent person would write: "And let me tell you" ... come on. Further, I can't tell if the second paragraph is sarcasm or not.

The article also states:

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.

Note the bolded text ( by me ). It's richer, but it's also finite? Nice. So are the natural numbers finite? Certainly, this will have far reaching applications towards the Frivolous Theorem of Arithmetic ! Good stuff I say!

However, the letter written by Escultura is quite disturbing, namely these 2 parts:

2) I also noted a flaw in the use of the universal or existential quantifiers on infinite set.
[...]
Based on these findings I constructed the new real number system on three simple axioms.

This brings back some memories of futile arguments, and smacking my head against my keyboard.

 

[snoble]

Is manilatimes.net another onion? It looks like this Escultura guy has written other articles for them including some junk about 0.999...=1 and how that's a topic of real discussion. Is it possible that this guy is real and he's actually convinced a legitimate news paper that he is to be taken seriously? He seems to show up on the Math Genealogy but that doesn't mean much. I don't have access to mathscinet from home. Is anybody else interested in checking if he shows up there?

Even if he does I guess there is no guarantee that this isn't some guy capitalizing on the fact that he shares a name with a serious mathematician.

In case I'm not being clear I don't believe anything in the article including that letter which is supposed to be attributed to Wiles.

Steven

 

[matt grime]

E E is infamous on usenet and is a total crackpot, crank etc. You could, for instance even try locating him at the University of the Philipines, which is apparently quite tricky to do. He has in the past claimed to find a counter example to FLT, though he's never been able to produce the numbers of this counter example.

 

[Zurtex]

The whole thing is unbelievably stupid. What more is there to say?

 

[Icebreaker]

This thing sounds like something straight from TheOnion.com. Especially "Wiles's" letter.

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. . .

Haha.

Maybe it's not written by the Andrew Wiles, just a Andrew Wiles.

 

[bombadillo]

People here should read the Manila Times article: it will have them in hysterics. For example:

When Wiles made the announcement it was celebrated around the world. In Chicago, for instance, mathematicians marched on the streets in euphoric celebration.

I know the Escultura type. They avoid -- and are avoided by -- real professionals. Their language of discourse is gobbledygook, reminiscent of much of "postmodernist" writing. They publish their rubbish in "journals" that aren't peer-reviewed; if they are "peer-reviewed", the peers are just more crackpots. Take a look at an abstract from one of Escultura's "papers":

The main contribution of the paper is the critique of J.M. Henle's Nonnonstandard analysis [The Mathematical Intelligencer 21 (1) (1999) 67]. The principal criticism of Henle's work is the failure to do away with the axiom of choice as he claims he would. Therefore, it suffers from the same flaw that Robinson's Nonstandard Analysis has [Nonstandard Analysis, North-Holland, Amsterdam, 1966]. However, the paper also summarizes the new mathematics and physics generated by the resolution of Fermat's last theorem [Nonlinear Studies 5 (2) (1998) 227] and the solution of the gravitational n- body problem [Nonlinear Analysis 30 (8) (1997) 5021], respectively, including the resolution of the problems, paradoxes, contradictions and unanswered questions of mathematics and physics except the Bieberbach's conjecture and the Riemann hypothesis. Highlighted in the paper are astonishing results coming from the new real line (i.e., the reconstructed real line) as well as the proof of Goldbach's conjecture and the natural ordering of the new real line (which does not exist in the real line).

(This can be found at: http://portal.acm.org/citation.cfm?id=639152)

 

[theCandyman]

I thought everyone already knew that FLT had already been disproved. I posted about it in the General Discussion long ago, here is the link:

http://home.mindspring.com/~jbshand/ferm.html

And, to ask the question again, is that a real and creditable newspaper?

 

[Zurtex]

I thought everyone already knew that FLT had already been disproved. I posted about it in the General Discussion long ago, here is the link:

http://home.mindspring.com/~jbshand/ferm.html

And, to ask the question again, is that a real and creditable newspaper?

What a load of rubbish, all that says is sometimes |aⁿ + bⁿ - cⁿ| = 1 or 2. I don't trust any website that abuses the equal sign like so:

"54 cubed + 161 cubed = 163 cubed"

I love the line " Holy Grail of Mathematics: Fermat's Last Theorem" I think that is somewhat abusing the context of the theorem, difficult to prove maybe but in terms of usefulness to mathematics, not so much I think you'll find.

 

[shmoe]

I'm guessing that it's a joke page, the author is an I. Savant


The Manilla times appears real, but I question their journalistic integrity with that letter supposedly from Wiles (does Wiles have a cruel sense of humour? I dunno). Also notice that Escultura is a former employee of this paper.

 

[Icebreaker]

I guess news editors can't differentiate between good and bad credibility when it comes to a topic as esoteric as elliptic curves and modular forms.

 

[arildno]

I'm guessing that it's a joke page, the author is an I. Savant

Also, he calls his result an "idiotheorem"..

 

[HallsofIvy]

And this (from the I. Savant page) is lovely:

Savant says that the problem is unsolvable under the constraints of traditional thinking. Without his own expertise in Number Addition and Subtraction Theory, advanced superscripting, and home brewing techniques, it would have been impossible. Using an innovative combination of the theory of general relativity, quantum mechanics, and his own findings in the field of Jello Mold Topology, Savant first showed that any given number has a high probability of being equal to itself. This leads to the observation that there is actually a small probability that the number is not equal to itself, but rather a different number entirely.

(especially the "home brewing techniques" part!)

And you are still asking whether that is "real and "credible"?

 

[arildno]

I.Savant sounds cool..

 

[funkstar]

That Escultura guy looks like a real crackpot. I was meserised at his http://www.manilatimes.net/national/2004/aug/28/yehey/opinion/20040828opi4.html about $$1\neq0.99\ldots$$ Even if one takes his babble at face value, and take the reals to be constructable infinite sequences of naturals, ordered lexicographically, how does he avoid the standard reasoning of $$0.99\ldots = 1$$? You know the drill...

Let $$x=0.99\ldots$$.
$$10x = 9.99\ldots$$
$$9x = 10x - x = 9.99\ldots - 0.99\ldots = 9$$
and therefore $$x=1$$.

Certainly, this reasoning hold true in his constructivist number system...

He then says that lexicographical ordering gives $$0.99\ldots <_{lex}1$$ but that is tautological - it doesn't mean that $$0.99\ldots < 1$$ for the normal ordering of the reals. The above gives that then $$<_{lex}$$ must be reflexive. If he changes the equality sign to a lexicographical version $$=_{lex}$$, the above reasoning just shows that the equivalence classes defined by $$=_{lex}$$ are not singletons. The irony is that his new infinitesimal, the so called dark number $$d*=1-0.99\ldots$$ truly is well-defined, but he fails to appreciate that it absolutely must be, even lexicographically, equal to 0.

In other words, he has discovered that lexicographical ordering is not quite the same order type as normal ordering for infinitely long decimal expansions of reals, because - shocker! - some numbers have more that one decimal expansion. Give that man the Nobel prize.

 

[houserichichi]

MATH RIOTS PROVE FUN INCALCULABLE

by Eric Zorn

News Item (June 23) -- Mathematicians worldwide were excited and pleased today by the announcement that Princeton University professor Andrew Wiles had finally proved Fermat's Last Theorem, a 365-year-old problem said to be the most famous in the field.

Yes, admittedly, there was rioting and vandalism last week during the celebration. A few bookstores had windows smashed and shelves stripped, and vacant lots glowed with burning piles of old dissertations. But overall we can feel relief that it was nothing -- nothing -- compared to the outbreak of exuberant thuggery that occurred in 1984 after Louis DeBranges finally proved the Bieberbach Conjecture.

"Math hooligans are the worst," said a Chicago Police Department spokesman. "But the city learned from the Bieberbach riots. We were ready for them this time."

When word hit Wednesday that Fermat's Last Theorem had fallen, a massive show of force from law enforcement at universities all around the country headed off a repeat of the festive looting sprees that have become the traditional accompaniment to triumphant breakthroughs in higher mathematics.

Mounted police throughout Hyde Park kept crowds of delirious wizards at the University of Chicago from tipping over cars on the midway as they first did in 1976 when Wolfgang Haken and Kenneth Appel cracked the long-vexing Four-Color Problem. Incidents of textbook-throwing and citizens being pulled from their cars and humiliated with difficult story problems last week were described by the university's math department chairman Bob Zimmer as "isolated."

Zimmer said, "Most of the celebrations were orderly and peaceful. But there will always be a few -- usually graduate students -- who use any excuse to cause trouble and steal. These are not true fans of Andrew Wiles."

Wiles himself pleaded for calm even as he offered up the proof that there is no solution to the equation x^n + y^n = z^n when n is a whole number greater than two, as Pierre de Fermat first proposed in the 17th Century. "Party hard but party safe," he said, echoing the phrase he had repeated often in interviews with scholarly journals as he came closer and closer to completing his proof.

Some authorities tried to blame the disorder on the provocative taunting of Japanese mathematician Yoichi Miyaoka. Miyaoka thought he had proved Fermat's Last Theorem in 1988, but his claims did not bear up under the scrutiny of professional referees, leading some to suspect that the fix was in. And ever since, as Wiles chipped away steadily at the Fermat problem, Miyaoka scoffed that there would be no reason to board up windows near universities any time soon; that God wanted Miyaoka to prove it.

In a peculiar sidelight, Miyaoka recently took the trouble to secure a U.S. trademark on the equation "x^n + y^n = z^n " as well as the now-ubiquitous expression "Take that, Fermat!" Ironically, in defeat, he stands to make a good deal of money on cap and T-shirt sales.

This was no walk-in-the-park proof for Wiles. He was dogged, in the early going, by sniping publicity that claimed he was seen puttering late one night doing set theory in a New Jersey library when he either should have been sleeping, critics said, or focusing on arithmetic algebraic geometry for the proving work ahead.

"Set theory is my hobby, it helps me relax," was his angry explanation. The next night, he channeled his fury and came up with five critical steps in his proof. Not a record, but close.

There was talk that he thought he could do it all by himself, especially when he candidly referred to University of California mathematician Kenneth Ribet as part of his "supporting cast," when most people in the field knew that without Ribet's 1986 proof definitively linking the Taniyama Conjecture to Fermat's Last Theorem, Wiles would be just another frustrated guy in a tweed jacket teaching calculus to freshmen.

His travails made the ultimate victory that much more explosive for math buffs. When the news arrived, many were already wired from caffeine consumed at daily colloquial teas, and the took to the streets en masse shouting, "Obvious! Yessss! It was obvious!"

The law cannot hope to stop such enthusiasm, only to control it. Still, one has to wonder what the connection is between wanton pillaging and a mathematical proof, no matter how long-awaited and subtle.

The Victory Over Fermat rally, held on a cloudless day in front of a crowd of 30,000 (police estimate: 150,000) was pleasantly peaceful. Signs unfurled in the audience proclaimed Wiles the greatest mathematician of all time, though partisans of Euclid, Descartes, Newton, and C.F. Gauss and others argued the point vehemently.

A warmup act, The Supertheorists, delighted the crowd with a ragged song, "It Was Never Less Than Probable, My Friend," which included such gloating, barbed verses as --- "I had a proof all ready / But then I did a choke-a / Made liberal assumptions / Hi! I'm Yoichi Miyaoka."

In the speeches from the stage, there was talk of a dynasty, specifically that next year Wiles will crack the great unproven Riemann Hypothesis ("Rie-peat! Rie-peat!" the crowd cried), and that after the Prime-Pair Problem, the Goldbach Conjecture ("Minimum Goldbach," said one T-shirt) and so on.

They couldn't just let him enjoy his proof. Not even for one day. Math people. Go figure 'em.

http://xenia.media.mit.edu/~cahn/humor/math-riots.html

 

[theCandyman]

You misunderstood, HallsOfIvy. I was asking if the orginal posted link to The Manila Times, is a real newspaper. The link to the I. Savant page was something I thought would be funny to share.

A Nobel prize for that well known math puzzle?

 

[TenaliRaman]

You misunderstood, HallsOfIvy.

If i am not wrong, that comment was not aimed at you. It was aimed at general public who took your link as a real and credible link and fell for it Not to mention few of them even made comments at it

-- AI

 

[merab]

Dear All,
I think I find the original proof of Fermat’s Last Theorem.
Everyone can understand it who knows little about Numbers theory!
I need just 3 lines to right to proof it.
Whom and were I have to send the proof of Fermat’s Last Theorem. ?
He was right!
I need my right to be protected.
Regards,
Lelianeli

 

[3trQN]

Maybe the margin was too narrow to contain Edgar Escultura's proof also. ;)

Does anyone have a link to Wiles's paper?

 

[arildno]

Dear All,
I think I find the original proof of Fermat’s Last Theorem.
Everyone can understand it who knows little about Numbers theory!
I need just 3 lines to right to proof it.
Whom and were I have to send the proof of Fermat’s Last Theorem. ?
He was right!
I need my right to be protected.
Regards,
Lelianeli

Don't be an idiot!
Your proof is wrong.