Check out my post on Inside PCIJ about this. Did my own investigation, talked to both Dr. Escultura and Prof. Wiles (via email) and by all indications, the report isn't true.
The real number system has a number of defects among which are the following:
1) most of its concepts are ill-defined and are, therefore, ambiguous, nonsense.
2) The trichotomy axiom is false (i.e., it does not qualify as an axiom) because the real number system has no natural ordering.
It follows that FLT as formulated is ambiguous, nonesense, the reason it could not be solved. The first crucial step in resolving FLT would have been to fix the real number system first which Andrew did not do. I did by constructing the real numbers (the decimals, specifically) on three simple consistent axioms yielding the new real number system without ambiguity and contradiction. Moreover, this new mathematical space yields countable counterexamples to FLT proving that it is false. For details of this resolution, see my thread, Contradiction-Free Mathematics, in the math forum Sci Math that started last Jan. 5. All of these is accomplished in over half a dozen papers also listed in the same thread. Together with applications including physics there are now over three dozen papers on the subject in refereed international journals listed in the same thread.
I encourage viewers to take their comments from the top not from the flat of their foot.
E. E. Esccultura
All I have seen here so far are comments that miss the issue and focus on rumors and hearsay when the Manila Times headline of the April 1, 2005 article was quite categorical and unambiguous:
UP Prof Proves Princeton Man Wrong.
I explained in my previous post what the issue was and what I did and cited the publications where I accomplished this feat. No one is interested in those rumors anymore and this website will not thrive on them.
E. E. Escultura
All the comments I have seen here so fark miss the issue and focus on the rumors and hearsay when the headline of the April 1, 2005 was quite unambiguouus and categorical:
UP Prof. Proves Princeton Man Wrong.
I have explained what the issue was and what I did and cited the publications where I accomplished this feat. No one is interested on those rumors and issues anymore and this website will not thrive on them.
Cheers.
E. E. Escultura
For those interested in the countably infinite counterexamples to FLT, here they are:
Let x = (0.99...)10^T, where T is
ordinary integer.
y = d*, where d* = 1 - 0.99...
called dark number.
z = 10^T.
Then, for n > 2,
x^n + y^n = z^n.
The numbers x, y, z, are called new integers. They consist of d* and numbers of the form N.99..., N = 0, 1, ... They are isomorphic to the ordinary integers under the mapping
0 -> d*, N -> (N-1).99...,
N = 1, 2, ... This isomorphic embedding of the oridnary integers into the contradiction-free new real number system as integral parts of the decimals resolves the fundamental flaw of number theory, that the integers have no valid axiomatization at the present time. In other words, until this embedding they were nonsense.
The new integers x, y, z are among the countably infinite counterexamples to FLT that prove this conjecture false.
The rest of the countably infinite counter examples to FLT are also countable, the triples (kx,ky,kz), where k = 1, 2, ..., is an ordinary integer, since
(kx)^n + (ky)^n = (kz)^n.
These counterexamples and their applications are published in over two dozen scientific papers in renowned refereed internatitonal scientific journals and proceedings of international conferences.
For details see my websites.
E. E. Escultura
For details
More about this here.
For the benefit of the experts I'd like to share my successful strategy for capturing FLT.
In 1992 I pondered why FLT could not be resolved for centuries and I concluded that its underlying fields, namely, foundations, number theory and the real number system were defective or, at least inadequate. Therefore, I embarked on their critique rectification. Here's what I found on foundations:
1) I agree with David Hilbert that the concepts of individual thought cannot be the subject matter of mathematics since they are elusive being inaccessible to others and can neither be studied collectively nor axiomatized. Therefore, its subject matter must consist of symbols (I also call them concepts) in the real world whose behavior is specified by consistent axioms. I require more to avoid contradiction, as follows:
2) A mathematical space must be well-defined by a set of consistent axioms. A concept is well-defined if its EXISTENCE, properties and relations with other concepts are specified by its axioms. I stress existence because vacuous statement is ambiguous and a source of contradiction. The decimals have additional requirements to avoid contradictiton: a decimal is well-defined if every digit is known or computable, i.e., there is some algorithm or rule or scheme for determining it uniquely.
3) Since distinct mathematical spaces are independent, each being well-defined only by its own consistent set of axioms, they are independent and any proposition involving concepts form disctinct spaces is ambiguous, nonsense. In particular, Goedel's incompleteness theorems are nonsense because their statement and proofs involve concepts from the propositional calculus and the integers which are distinct spaces.
4) A statement about an infinite set involving the universal or existential quantifier is unverifiable and cannot be used as an axiom since it does not endow certainty to the conclusion of a theorm.
5)The real number system does not meet these requirements and, therefore, has the following defects:
a) Most of its concepts are nonsense; only periodic and nonterminating
decimals every digit of which is computable are well-defined. Even then, well-defined nonterminating decimals have inherent uncertainty because not all its digit can be computed being infinite (this is a situation where "every" is not necessarily the same as "all). This is acceptable because this ambiguity can be approximated by certainty. For example, a nonterminating decimal is approximated by its segment at the the nth decimal digit and the margin of error is 10^-n.
b) The trichotomy axiom is false because the real number system has no natural ordering since most of its concepts are nonsense. Morever, a pair of normal numbers is a counterexample to it. A decimal is normal if every digit is chosen at randome from the basic digits or integers, 0. 1, ..., 9.
c) Among the ambiguous concepts and propositions (which are sources of contradiction) are: vacu
Since my post does not fit, I continue:
c) Ambiguous concepts and propositions include: vacuous concept and proposition, infinite set and large and small numbers (depending on context).
d) Statement on infinite set involving the universal or existential quantifier is unveriafiable and cannot be used as an axiom because it does not endow the conclusion of a theorem with certainty. In particular, the completeness axiom does not qualify as an axiom.
6) The remedy for the real number system is its contruction on finite set into the new real number system R* consisting of decimals. Its axioms are: (1) R* contains the basic integers, 0, 1, ..., 9, (2) and (3) the addition and multiplication table of primary school.
7) Then R* is finite but unbounded (coutably infinite), discrete, has natural ordering, free from contradiction and enriched by the dark number d*, where d* = 1 - 0.99..., and
the new integers, N.99..., where N = 0, 1, ..., is ordinary integer. This is the appropriate mathematics for computer sciences which is digital and the natural sciences since nature is discrete consisting of the basic constituent of matter, the superstring.
The ordinary integers are now well-defined by its isomorphic embedding in R* as the integral parts of the decimal. This resolves the funamental flaw of number theory that the integers are presently ill-defined, nonsense.
9) Previously nonsense, FLT is now well-defined in R* which paves the way for its resolution.
For details visit my websites.
E. E. Escultura
It's getting pretty "deep" in here. What a bunch of psuedo-scientific gibberish.
"3) Since distinct mathematical spaces are independent, each being well-defined only by its own consistent set of axioms, they are independent ..."
Oh, they are independent because they are independent. I see.
"6) The remedy for the real number system is its contruction on finite set into the new real number system R* consisting of decimals. Its axioms are: (1) R* contains the basic integers, 0, 1, ..., 9, (2) and (3) the addition and multiplication table of primary school."
Talk about "ambiguous, ill-defined nonsense."
FLT is cast in the integer number system, not Real Numbers, and certainly not the Escultura Real Numbers. "Solving" FLT by changing the rules is meaningless. If changing the number system of FLT is allowed, it would be sufficient to change it to the Real Numbers to generate counter examples. Again, so what? That's not the problem that was posed by Fermat.
These crazy rantings have made me doubt one aspect of my position on this "controversy." I've said several times that there may be some uses of Dr. Escultura's number system. I'm beginning to see that is quite unlikely.
Just for the record (H/T Tycho):
Let x = 0.111...
so 9x = 0.999...
and 10x = 1.111...
10x - x = (1.111...) - (0.111...) = 1
but 10x - x is also 9x, which is 0.999...
So 0.999... = 1
Test
1) Your first remark is valid; it was my error. The sentence shoud read,
Since distinct mathematical spaces are well-defined only by their respective axioms they are indpendent.
2) The real numbers includding the integers are ill-defined. Are they decimals? p-adics? triadics? base 12?
They are distinct numerals and cannot be well-defined by the same set of axioms.
3) The trichotomy axiom is false because the real numbers have no natural ordering. If you well-define the real numbers I'll show you counterexamples to the trichotomy axiom. If you accept my definition of a real number as decimal every digit of which is known or computable I'll also show you counterexamples.
4) Infinite set is ambiguous because not all its elements can be identified. Therefore, any proposition about the elements of an infinite set involving the the universal (or existential quatifier) is unverifiable and cannot be used as axiom because it cnnot insure certainty to a theorem. To avoid amiguity we build a mathematical space on finite set first and introduce ambiguity when needed and still avoid contradiction provided it is approximable by certainty. For example, a well-defined nonterminating decimal can be approximated by its initial segment at the nth digit and the margin of error will be 10^n.
5) Your claim that FLT can be resolved on the integers does not hold because as subset of the real numbers they are also ill-defined. Of course, you can dissociate them from the real numbers and call them natural numbers. But then you have the problem of well-defining them as a mathematical space. There is none at this time.
6) There is no ambiguity about the basic integers and the addition and multiplication tables.
eee
You cannot add, subtract or multiplicy nonterminating decimals because that requires the last digits on the right of the numbers involved. You can divide any decimal by terminating decimal because that operation starts on the right and there is a first element there for both the divisor and the dividendl Thus
(1.99...)/2 = 0.99... Incidnetally, this is a counterexample to the theorem of elementary arithmetic that the average of two unequal numbers lies strictly between them.
E. E. Escultura
Correction: You can divide any decimal by terminating decimal because that operation starts on the left and there is a first element there.
So no one noticed that the operations + and x are valid only on terminating decimals? We need to be observant. Otherwise, we cannot go very far.
E. E. Escultura
I don't bother with typos and so I go straight to the point. I welcome your claim that you have an alternative approach towards the resolution of FLT as a problem on the integers. Then you need to build the integers alone as a contradiction-free mathematical space, not as a subspace of the new real number system. This alone would be a major contribution to mathematics even without actually resolving FLT. I would be glad to publish your result in my journal or referee and endorse it for publication in a suitable journal. I look forward to more development on this matter.
E. E. Escultura
Obviously, it's pointless to "discuss" this with Dr. E. His circular logic is:
a) 1 is not equal to 0.9999
b) anything that proves otherwise is a "contradiction" in conventional math
c) therefore conventional math is riddled with contradictions and must be discarded.
Without wading through all the bs he's piled in this thread, I'll just take this one as representative: "5) Your claim that FLT can be resolved on the integers does not hold because as subset of the real numbers they are also ill-defined."
I claimed nothing about whether or not "FLT can be resolved on the integers." I merely noted that's the system in which the problem is posed. If the underlying rules are changed, it's no longer the same problem. Furthermore, even if his claim that the real numbers are "ill-defined" were true, that would say nothing about the integers. It's as if problems with the complex numbers (real + imaginary) were cited to claim the real numbers are invalid.
In fact Dr. Wiles has solved FLT in the integers (as everyone except Dr. E defines them) and published his work in Annals of Mathematics, one of the oldest and most pretigious journals in the field. If Dr. E's ideas were as persuasive as he believes they are, he could publish them there.
Note that he says above, " I would be glad to publish your result in my journal or referee and endorse it for publication in a suitable journal." [emphasis added] Reading between the lines, we can see that his own results are published in his own journal.
Summarizing:
a) Wiles' results were published in a top mathematics journal, reviewed, critiqued, revised, and are holding up.
b) Escultura's results were published in a fourth tier journal of his own.
c) Wiles' results were reported in news publications around the world
d) Escultura's "refutation" was hailed in an article in the Manila Times, where he writes a column, and he was the sole source for the article.
a) You still missed a very very important point. The decimals are specific well-defined symbols. 1 and 0.99... are distinct symbols. How can they be equal?
b) Where is your proof that 1 = 0.99... (specify the mathematical space and the axioms you are using)?
c) You have not well-defined + and x on your nonterminating decimals. That, alone, is nonsense. Moreover, you have not proved that the real numbers have
natural ordering and so you don't even know if the trichotomy axiom, which is equivalent to natural ordering, is true or false. Therefore, you cannot use it as an axiom because that would not make sense of your theorems.
The implication of your statement is that if I want to resolve FLT I should do that among the integers. The integers are not well-defined as a mathematical space and, therefore, FLT does not make sense there. If you take the integers as real numbers, as everybody does, then they are also ill-defined, nonsense, and FLT formulated on the integers is nonsense. The only way to make sense out of FLT is to well-define a number system, which I did with the decimals, formulate FLT there and resolve it. All of this I have done. Now, find a contradiction in my axiomatization of the decimals.
Dr. Wiles solved FLT on the integers which are nonsense. Old, local and classical journals are not necessarily correct. In science the new, the international, supersede the old, the local and the classical. That's why, except for a small cult that publishes its own journals, nobody now believes that the Earth is flat. At any rate, if a piece of work is criticized it must be defended; otherwise, the criticism stands.
My results are published in half a dozen renowned, refereed, international journals. In other words, you don't know how to read between the lines or, more precisely, how to read, period.
Summary
a) I would hesitate to call The Annals of Mathematics a top mathematical journal. It's an in-house publication by Johns Hopkins University. The only mathematicians who reviewed Wiles' work are his friends, Ribet, Mazur and someone from Princeton (Sarnak?). And like Wiles, they are all quiet now.
b) It's not true that my results were published in my own journal (see above). This is a lie unbefitting a mathematician.
c) Newpaper reports about a scientific or mathematical result have nothing to do with its scientific or mathematical merit. The official arbiter of scientific merit is the network of refereed international scientific journals. My journal is one of them.
d) Even the Manila Times, ABS-CBN, Philipine Institute for Integrative Journalism, reports are nothing, nothing at all. Only the refereed journal articles matter in this business.
Isn't it a bit curious that Wiles, Ribet, Mazur and Sarnak are quiet despite the noise about FLT in your blog and many other blogs and website? Only his supporters in the Philippines are making a lot of noise about it. Even Sci Math has not had any comment on FLT
As in-house publication of Johns Hopkins University The Annals of Mathematics is in the same class as the Natural and Applied Sciences Bulletin, an in-house publication of the UP. It does not even belong to the same class as the AMS publications.
E. E. Escultura
I did not look up which journals Dr. E is referring to as "his." Therefore I could be wrong in inferring that his work has been only or mainly published there.
As I wrote previously, quoting Dr. E.:
So Nonlinear Studies is a "renowned journal," while Annals of Mathematics is just "a local US publication," according to Dr. Escultura. We note that NS has been published since 1993 and is located in Daytona Beach, Florida. On the other hand AoM has been published since 1884, giving it a 109-year headstart on earning "renown" among "local US publications." The online archiving and indexing service for scholarly journals, JSTOR, selects the best journals in a wide variety of scientific disciplines. They don't cover NS, but JSTOR says about AoM, "It is widely regarded as one of the main mathematics journals in the world."This page lists the top 10 math journals in 2001 based on their "impact factors" (a measure of how often they are cited by other journals). Annals of Mathematics is 6th. Nonlinear Studies is not listed, nor could I find an entry for it in this list of journals and their impact factors.
Now Dr. E argues that the news reports mean nothing, but earlier in this same thread he trumpeted the Manila Times headline as "proof" that he had "proved" Wiles is wrong.
As to why Wiles and the rest of the mathematics world are silent on Dr. E's "feat," I'd say the reason is obvious.
Even the archiving and indexing service, JSTOR, that Roy mentioned is probably old and relatively unknown. I am not even aware of it and I doubt if it made it to the UP's approved list of databases. Being from the UP Roy knows at least some of the major databases such as: ISI, Zentralblatt, CAS, Science Citation Index, Web of Science, Sci Search, CompuMath, Current Contents, Physical, Chemical and Earth Sciences, Compendex, Math Reviews and the Russian Academy of Sciences. These are top of the line. Another criterion is publisher. Among the top-of-the-line journal publishers are: Elsevier Science, Ltd. Academic Press, International Federation of Nonlinear Analysts and the Russian Academy of Sciences (both database and publisher). My publications are not in NS alone (in fact I only have one here). They are also in Nonlinear Analysis, Problems of Nonlinear Analysis in Engineering Systems and Applied Mathematics and Computation which are published by those I have just mentioned. Still another criterion for the greatness of the journal is the composition of the Editorial Board. It must be star-studied and international. An in-house publication does not meet any of these critieria.
Old journals are not necessarily updated since science advances quite rapidly. In this situation the old criterion of impact factor means nothing. Naturally, the old but unupdated journals have more citations that the new and updated ones.
A newspaper report does not prove anything and I stand by what I said in this thread: the the Manila Times article reported the fact that I have proved Wiles wrong and proceeded to list the publications where I accomplished this feat. I think Roy has difficulty in reading or understanding what he reads.
My criticisms of Wiles' work are published in several renowned journals and Wiles and his supporters are silent about them. I agree, the reason for the silence is quite obvious: he and his supporters can't refute them. Therefore, yy criticisms stand.
E. E. Escultura
Obviously, it is Dr. E who "has difficulty reading or understanding what he reads."
The site listing impact factors for journals doesn't mention Problems of Nonlinear Analysis in Engineering Systems. This is what they have for the other two:
0,314 NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
0.436 APPLIED MATHEMATICS AND COMPUTATION
In contrast:
1.905 ANNALS OF MATHEMATICS
Much higher.
There is no mention of Dr. E in the current Wikipedia article on FLT, however this older version calls out the Manila Times article as a hoax. The article was revised a few hours later, deleting all references to Dr. E's claims.
Problems of Nonlinear Analysis is a new publication (in English and Russian) by the Russian Academy of Sciences and may not be covered yet by the standard databases. Out of my over two-dozen publication two are in this journal. Most of my publications are published by Elsevier Science including Nonlinear Analysis and Appled Mathematics and Computation
(for physics articles)and Indian Academy of Sciences. Moreover, impact factor is not a standard database like ISI, etc., and is certainly an ureliable gauge of the stature of journals. Did you check if JSTOR and impact factor made it to the UP-approved listing of databases?
To give you an example of how important this updating is, all standard physics journals uphold Relativity as valid although in private conversation physicists admit its errors. For example, it contradicts quantum mechanic, meaning, one of them must be wrong (and the latter cannot be wrong being the mother of high technology). In the 80s physicists discovered that jet outflow of hot gas from the core of a nascent star or galaxy travels at 75 times the speed of light which disputes Relativity's Lorentz's transformation that sets the upper limit of speed of matter at the speed of light of 10^5 km/sec. For a long time physicists were buffled by the tremendous rate of expansion of our universe, meaning that the galaxies are flying at staggering speed far beyond the speed of light. I calculated that speed from Hubble's law and found it to be 10^20 km/sec and still increasing at acceleration of 10^-10 km/secsec. Moreover, the discovery of the formation of stars at the rate of one star/minute and just six months ago the discvery of a forming galaxy emerging from the supposedly empty Cosmos, disputes the main premise of Relativity, upon which Einstein's postulate of relativity is based. Thus, Relativity now belongs to the museum and all departments of Relativity abroad have folded up.
These results are part of my critique-rectification of physics that I published in several Elsevier Science journals and other refereed international journals and conference procedings. As in the case of FLT there is no challenge from the physicists.
Like this one, Wikepeda, Digital Mercenary, Sassy Lawyer, etc.,are blogs that dwell on claims and counterclaims, rumors, hearsay, and, in the case of the posts by Alecks, the writing coach of the Math Department, hallucinations as well. They will not even make to the footnote of scientific history.
If you want to make it in science and its history, PUBLISH.
You misread the crystal clear headline of the Manila Times, "UP Prof proved Princeton man wrong", which listed the published articles where I accomplished this feat. It reported a fact, not a claim.
E. E. Escultura
PS. I noticed that you are unfamiliar with isomorphic embedding, a topological technique for well-defining a set by making it a subspace of a mathematical space, in this case, the decimals now axiomatized as the new real number system. It maps the natural numbers into the integral parts of the decimals represented by N.00..., for N = 0, 1, ... BTW, your writing 0.99999 for 0.99..., is wrong. 0.99... is a nonterminaing periodic decimal.
E. E. Escultura
The impact factor is the number of citations a journal's articles receive divided by the number of articles. It is not a database.
The primary evidence cited in the Manila Times article was the "letter" from Dr. Wiles, supposedly renouncing his own work. Ultimately, it was revealed that:
1) the "letter" was an anonymous entry on Dr. E's website and a hoax
2) Wiles denied having even heard of the entry, let alone writing it
3) no attempt had been made to check the autheticity of the "letter" before publishing it in the Manila Times.
I have deleted Dr. E's latest rambling nonsense.
I saw that Abe. I was going to post a comment telling E.E. that he is not helping his cause.
Another one bites the dust.
I find it pretty funny that he kept calling me "Roy," as in "... Roy has difficulty reading or understanding what he reads." Obviously, Dr. E doesn't read very well. 
His lastest ramble cried "censorship." Censorship is when the government prevents you from speaking or writing about your ideas, and it has nothing to do with private actions. Dr. E has his own site, where he can put up whatever tripe he wants. Site owners always have the right to remove things and are never under any obligation to host comments of others. In this case his claim is even weaker, since I have left plenty of his trash on display.
That you have.
I thought all this time that he was talking about fellow blogger Roy Choco. I think that is his name. I kept wondering why he was bringing that to your blog.
I also thought all along that I was responding to Roy Choco's mathematical posts because I saw his post here sometime ago and Abe says he is a chemist. Since the posts attack my mathematical works don't I have the right to respond just as I allow viewers to post anything on my website?
E. E. Escultura
I'll leave the last one from Dr. E, but that's about it, as far as I'm concerned. He certainly has been permitted to defend his work here, but that has already been accomplished to the extent possible.
I was just a curious onlooker on this flt thing and about this grand unified theory and all these remarks and sarcasms being published on blogs and what not. I wanna ask Dr. E here on his perception as to why if Dr. Wiles and his minions are now silent on the refuted flt claim, then why is everyone else also silent about his claim or better yet what is he doing to convince the scientific and mathematics world to side with him and do more analysis of his work to give him recognition and certainly the nobel prize in the future?
B,
At first I thought your post might just be another one from Dr. E under a new name. Since I see your IP address is in Canada, that's evidently not the case.
Don't expect an answer here from Dr. E to your questions. You may wish to visit his site if you want a response from him on that.
I would say the reasons his work is being ignored by serious mathematicians around the world are quite simple, really. You can read the many posts on the topic on this blog. In a nutshell, his claim to have refuted FLT is bogus. It relies on recasting the problem into a different number system, which conveniently provides another form of zero.
Don't hold your breath waiting for that Nobel Prize for Dr. E. He's no closer to winning it than you and I are.
Questions raised here are answered in my blog, http://fltnewmathphysics.blogspot.com/, and, http://home.iprimus.com.au/pidro/ , and in many threads, forums, blogs and websites across the internet that I have noticed and responded to recently.
E. E. Escultura
Here's an antidote to the forever skeptic, an article in the Science and Technology website,
http://www.deccanherald.com/
decc...34020051212.asp,
which is online. E. E. Escultura
Tuesday, December 13, 2005
Deccan Herald » Science & Technology » Detailed Story
Nobility of the Nobel prize
By B M Hegde
[Full text of copyrighted article deleted. Follow the link above and/or read my post on the main blog about it. -- Abe]
Edited By Siteowner
I can see that someone cried uncle again and deleted some posts. Abe claimed earlier that he had as much chance winning the Nobel Prize in chemistry as I have in physics to which I responded:
I don't know about chemists but physicists don't expect a nobel unless they have significant contributions to their fields (and, may I add, they are the most outstanding ones for the year).
E. E. Escultura
Well, Dr. E, that post still stands in another comment thread.
I don't disagree with your statement about not expecting a Nobel without significant contributions in the field. I still maintain that our chances are equal: right about zero, or, if you prefer:
1 - 0.99...
I can see that someone cried Uncle again and deleted both the questions raised by Abe and my response to them. It's not in HaloScan either.
E. E. Escultura
Dr. E,
You have been given ample opportunity to make your case, such as it is, here. Most of your earlier posts in this thread and others have been allowed to stand.
However, consider your posting rights at this blog terminated.
Commenting by HaloScan
