Fermat’s Last Theorem
June 4th, 2007 31 comments link to (permalink) posted by david
If I were to pick my favorite documentary of all time – it would be this. Mathematics, the most rational of all subjects may seem to be an unemotional topic. For this very reason, seeing the tears in Wiles’ eyes when he recounts the inspirations that enabled him to finally solve a centuries old enigma make this a dramatic piece of the most subtle kind. Simultaneously gentle and powerful.
Running time: 43 mins.
31 responses so far »
Chris : Jun 5, 2007 at 1:27 am
Thanks for posting that, it was wondrous. I gave up smoking last week, and the past 45 minutes watching that documentary have been the first I’ve not consciously craved a fag.
adam : Jun 9, 2007 at 11:58 am
thanks for that, was really great! bless the nerds, they probably will inherit the earth
ET : Oct 14, 2008 at 4:37 am
Nice. You know there’s a random and undefined check-box above the embedded vid? Kinda random.
E. E. Escultura : Aug 11, 2009 at 3:04 pm
Two Fatal Defects in Andrew Wiles’ Proof of FLT
1) The field axioms of the real number system are inconsistent; Felix Brouwer and this blogger provided counterexamples to the trichotomy axiom and Banach-Tarski to the completeness axiom, a variant of the axiom of choice. Therefore, the real number system is ill-defined and FLT being formulated in it is also ill-defined. What it took to resolve this conjecture was to first free the real number system from contradiction by reconstructing it as the new real number system on three simple consistent axioms and reformulating FLT in it. With this rectification of the real number system, FLT is well-defined and resolved by counterexamples proving that it is false.
The second defect will follow.
E. E. Escultura : Aug 11, 2009 at 3:07 pm
The second defect of Wiles' proof.
2) The other fatal defect is that the complex number system that Wiles used in the proof being based on the vacuous concept i is also inconsistent. The element i is the vacuous concept: the root of the equation x^2 + 1 = 0 which does not exist and is denoted by the symbol i = sqrt(-1) from which follows that,
i = sqrt(1/-1) = sqrt 1/sqrt(-1) = 1/i = i/i^2 = -i or
1 = -1 (division of both sides by i),
2 = 0, 1 = 0, I = 0, and, for any real number x, x = 0,
and the entire real and complex number systems collapse. The remedy is in the appendix to [9]. In general, any vacuous concept yields a contradiction.
E. E. Escultura : Aug 11, 2009 at 3:14 pm
The references on Wiles' proof:
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A: Theory, Methods and Applications; online at Science Direct website
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
E. E. Escultura : Aug 22, 2009 at 11:35 pm
Minor correction to my post: The vacuous concept i leads to … 1 = 0, i = 0. Reference [9] was omitted from the list of references. It's the paper, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84. E. E. Escultura
E. E. Escultura : Aug 28, 2009 at 7:01 am
Summation of the Debate on the New Real Number System and the Resolution of Fermat’s last theorem – by E. E. Escultura
The debate started in 1997 with my post on the math forum SciMath that says 1 and 0.99… are distinct. This simple post unleashed an avalanche of opposition complete with expletives and name-calls that generated hundreds of threads of discussion and debate on the issue. The debate moved focus when I pointed out the two main defects of Andrew Wiles’ proof of FLT and, further on, the discussion shifted to the new real number system and the rationale for it. Naturally, the debate spilled over to many blogs and websites across the internet except narrow minded websites that accommodate only unanimous opinions, e.g., Widipedia and its family of websites, as well as websites that cannot stand contrary opinion like HaloScan and its sister website, Don’t Let Me Stop You.
E. E. Escultura : Aug 28, 2009 at 7:02 am
The most recent credible challenge to my positions on these issues was registered by Bart van Donselaar in the online article, Edgar E. Escultura and the Inequality of 1 and 0.99…, to which I responded with the article, Reply to Bart van Donselaar’s article, Edgar E. Escultura and the Donselaar’s paper has been set up:
http://www.reddit.com/r/math/comments/93n3i/edgar...
and the discussion is coming to a close as no new issues have been raised. Needless to say, none of my criticisms of the paper nor my positions on Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully.
E. E. Escultura : Aug 28, 2009 at 7:03 am
We highlight some of the most contentious issues of the debate.
1) Consider the equation 1 = 0.99… that almost everyone accepts. There are a number of defects here. Among the decimals only terminating decimals are well-defined. The rest are ill-defined or ambiguous. In this equation the left side is well-defined as the multiplicative identity element while the right side is ill-defined.
2) The second point is: David Hilbert already knew almost a century ago that the concepts of individual thought cannot be the subject matter of mathematics since they are unknown to others and, therefore, cannot be studied collectively, analyzed or axiomatized. Therefore, the subject matter of mathematics must be objects in the real world including symbols that everyone can look at, analyze and study collectively provided they are subject to consistent premises or axioms. Consistency of a mathematical system is important, otherwise, every conclusion drawn from it is contradicted by another. In order words, inconsistency collapses a mathematical system. Consider 1 and 0.99…; they are certainly distinct objects like apple and orange and to write apple = orange is simply nonsense.
E. E. Escultura : Aug 28, 2009 at 7:04 am
3) The field axioms of the real number system is inconsistent. Felix Brouwer and myself constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice, one of the field axioms. One version says that if a soft ball is sliced into suitably little piece and rearranged without distortion they can be reconstituted into a ball the size of Earth. This is a topological contradiction in R^3.
E. E. Escultura : Aug 28, 2009 at 7:05 am
4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root has been denoted by i = sqrt(-1). The notation itself is a problem since sqrt is a well-defined operation in the real number that applies only to perfect square. Certainly, -1 is not a perfect square. Mathematicians extended the operation to non-negative numbers. However, the counterexamples to the trichotomy axiom show at the same time that an irrational number cannot be represented by a sequence of rationals. In fact, a theorem in the paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149, says that the rationals and irrationals are separated.
E. E. Escultura : Aug 28, 2009 at 7:05 am
At any rate, if one is not convinced of the mischief that vacuous concept can play, consider this:
i .= sqrt(-1) = sqrt1/sqrt(-1) = 1/i = -i or i = 0. 1 = 0, and both the real and complex number systems collapse.
5) With respect to Andrew Wiles’ proof of FLT it has two main defects: a) Since FLT is formulated in the inconsistent real number system it is nonsense and, naturally, the proof is also nonsense. The remedy is to first remove the inconsistency of the real number system which I did.. b) The use of complex analysis deals another fatal blow to Wiles’ proof. The remedy for complex analysis is in the appendix to the paper, The generalized integral as dual to Schwarz Distribution, in press, Nonlinear Studies.
E. E. Escultura : Aug 28, 2009 at 7:06 am
6) By reconstructing the defective real number system into the contradiction-free new real number system and reformulating FLT in the latter, countably infinite counterexamples to it have been constructed showing the theorem false and Wiles wrong.
7) In the course of making a critique of the real number system some new results have been found: a) Gauss diagonal method of proving the existence of nondenumerable set only generates a countably infinite set; b) as of this time there does not exist a nondenumerable set; c) only discrete set has cardinality, a continuum has non..
The new real number system is a continuum, countably infinite, non-Hausdorff and Non-Archimedean and the subset of decimals is also countably infinite but discrete, Hausdorff and Archimedean. The g-norm simplifies computation considerably.
E. E. Escultura
E. E. Escultura : Sep 10, 2009 at 12:15 pm
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 – 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[6] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
E. E. Escultura : Sep 10, 2009 at 12:16 pm
[8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[9]Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[10]Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[11] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[12] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[13] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, India
http://users.tpg.com.au/pidro/
E. E. Escultura : Sep 17, 2009 at 5:56 am
CLARIFICATION ON THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura
Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:
1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.
E. E. Escultura
E. E. Escultura : Sep 17, 2009 at 5:57 am
2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].
3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)
for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.
E. E. Escultura
E. E. Escultura : Sep 17, 2009 at 5:58 am
4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.
The following references include references used in the consolidated paper [8] plus [2] which applies [8]
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 – 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
E. E. Escultura
E. E. Escultura : Sep 17, 2009 at 5:59 am
[3] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
[4] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
E. E. Escultura
E. E. Escultura : Sep 17, 2009 at 6:01 am
[7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[9] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A: Theory, Methods and Applications; online at Science Direct website
[10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
E. E. Escultura
E. E. Escultura : Sep 17, 2009 at 6:02 am
[11]Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[13] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[14] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, India
http://users.tpg.com.au/pidro/
Nick : Oct 6, 2009 at 9:46 am
The fact that this idiot "E E Escultura" is posting his mathematical theorems in a smashingtelly blog tells you all you need to know about his credibility.
What a dick.
E. E. Escultura : Oct 22, 2009 at 4:55 am
Name-calling reveals not only intellectual inadequacy and insecurity but also racism and emptiness at the top. BTW, there is no theorem here but historical account and clarification. E. E. Escultura
Phil J : Oct 29, 2009 at 9:11 pm
The culmination of work that led Andrew Wile's proof will shortly become known as the most expensive and long-winded wild-goose chase of all time. There is very simple proof currently circulating amongst a handful of mathematicians that shows that when the problem is deeply understood, a surprisingly simple solution is found that provides no support whatsoever for the traditional approach to this problem. In fact a fatal assumption has meant that researchers have gone off in the wrong direction. Furthermore the Beale Conjecture is related very closely to FLT but Wile's proof cannot show that.
So far no mathematician has been able to find fault with this truly simple proof but they don't want to entertain the nightmarish situation that Wiles and those before him have been chasing artifacts. Here is something to ponder – if you don't understand a problem, how can you solve it? Wile's solution makes a claim of having a deep relationship with ellipses but other than that can't offer anything that is simple and therefore close to the nature of the problem.
E. E. Escultura : Apr 30, 2010 at 6:35 am
CALL FOR A GRAND UNIFIED JOINT CELEBRATION
Materialist philosophers of all cultures must have pondered this question: what are the basic constituents of matter? The Greeks answered it with four constituents they found in nature: earth, water, fire and air. The Chinese added one more item – wood. Of course, they were not satisfactory and since then the search for the basic constituent of matter was in limbo for 5,000 years until in the 1950s inspired by the exciting developments in quantum physics particle physicists renewed the search with vigor by smashing the nucleus of the atom in pursuit of the basic irreducible elementary particles or building blocks of visible matter (since dark matter was unknown then). By the 1990s the search was a complete success with the discovery of the +quark (up quark) and ï€quark (down quark) and, earlier, the electron discovered by J. J. Thompson in 1897.
E. E. Escultura : Apr 30, 2010 at 6:39 am
They comprise every atom; a heavy isotope has at least one more additional stable elementary particle – the neutrino. Particle physicists have, indeed, found what they were looking for – the irreducible building blocks of visible matter – and whatever they have found beyond this discovery is a bonus for natural science and its applications, a bonus for mankind.
In the 1980s dark matter came to the fore with overwhelming evidence of its existence [6,7,8] and, using the new methodology of qualitative modeling that explains nature and its appearances in terms of natural laws [1,5], was established in 1997 [4] as one of the two fundamental states of matter the other ordinary or visible matter [2,5].
E. E. Escultura : Apr 30, 2010 at 6:42 am
That same year the building block of dark matter, the superstring, was discovered as the crucial factor in the solution of the gravitational n-body problem [4] and development of the grand unified theory (GUT). The latter has been established in a series of papers since 1997 and consolidated in [2]. There is only one basic constituent in view of the non-redundancy and non-extravagance natural principles [3] just as there is only one electron since all electrons have identical structure, properties, behavior and functions and differ only in locations. Moreover, it has been established that the superstring coverts to the basic elementary particles as agitated superstring [1,2,3]. In effect, this proves the superstring as the basic constituent of matter, dark and visible [1,2,3,4,5 ].
E. E. Escultura : Apr 30, 2010 at 6:44 am
This happy turn of events came without notice and fanfare but it is an important milestone for science that calls for a grand unified joint celebration by particle and theoretical physicists to mark these monumental achievements and the threshold of a new epoch for natural science and its applications. Whatever particle physicists have achieved beyond this discovery is a bonus for natural science and its applications, a bonus for mankind. Perhaps, a world congress of particle and theoretical physicists is appropriate on this momentous occasion.
References
[1] Escultura, E. E., The mathematics of the grand unified theory, Nonlinear Analysis, A-Series: Theory: Methods and Applications, 71 (2009) e420 – e431.
[2] Escultura, E. E., The grand unified theory, Nonlinear Analysis, A-Series: Theory: Methods and Applications, 69(3), 2008, 823 – 831.
E. E. Escultura : Apr 30, 2010 at 6:45 am
[3] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis, B-Series: Real World Applications, 11 (2009), 29 – 38.
[4] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear
Analysis, A-Series: Theory, Methods and Applications, 38(8), 521 – 532.
[5] Escultura, E. E., Superstring loop dynamics and applications to astronomy and
biology, Nonlinear Analysis, A-Series: Theory: Methods and Applications, 35(8), 1999, 259 – 285.
[6] Astronomy (a) August 1995, (b) January 2001, (c) June 2002.
[7] Science, Glow reveals early star nurseries, July 1998.
[8] Science, (a) Starbirth, gamma blast hint at active early universe, 282(5395), December, 1998, 1806; (b) Gamma burst promises celestial reprise, 283(5402),
January 1999; (c) Powerful cosmic rays tied to far off galaxies, 282(5391), Nov. 1998, 1969 – 1971.
Danny : May 21, 2012 at 4:30 am
Wow… Escultura and Phil J are nuts.
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