## Thursday, May 12, 2005

### P vs NP is for kids...

In our humble corner of the mathematical landscape we jump excitedly at each new proof that P = NP, P != NP, or even P = RP?. But for the brave souls who patrol the vast expanses of sci.math, much more enjoyment is to be had.

Dave Rusin maintains a page of some the "many delights available through the internet". Among them, the daring exploits of one Edgar Escultura, mathematics professor in the Phillipines, who slayed not one, but two mighty mathematical dragons.

In 1998 (search for Escultura), he disproved Fermat's Theorem, demonstrating that "trillions upon trillions" of numbers satisfying the equation exist. A lesser man would have balked at the existence of a contradictory proof; Prof. Escultura is not one of those men. Faced with a cruel and unrelenting system of numbers that mocked him, he invented a new one !

When Wiles announced his proposed solution of the problem in 1993, Escultura pointed out his main error as lack of knowledge of recent mathematical discoveries, particularly, the uncertainty in the behavior of large numbers and in dealing with infinite mathematical systems such as the natural numbers.

But Prof. Escultura is no one-trick pony.

Earlier, Escultura captured the last stronghold of physics with his
solution of the gravitational n-body problem posed by Marquiz de
Laplace at the turn of the 17th Century. [...] His paper, The Solution of the Gravitational n-Body Problem, is published by the journal, Nonlinear Analysis, Vol. 30, No. 8, pp. 5021 - 5032, Dec. 1997. The paper earned for him international recognition: membership in the American Association for the Advancement of Science; the 25-member Core of Experts which is the leading body of the International Federation of Nonlinear Analysts (IFNA); the Editorial Board of IFNA and the Global Organizing Committee of the Third World Congress of Nonlinear Analysts 2000 (WCNA 2000), 19 26 July 2000, Catania, Italy; and organizer and chair of the international mini- symposium on the topic, The El Niño
and its Impact: Drought and Turbulence, during the Congress.
Chair of a conference on El Niño ? member of AAAS ? Give the man a cookie !

p.s Why bring this up now ? For some reason, his "disproof" of Wiles' theorem has been doing the rounds on some blogs recently, prompted by an article in the Manila Times.

1. surely Prof. Escultura is a crank. even so..there certainly is no guarantee that Professor Wiles' proof is correct. indeed his proof had a gap in 1994? which was only corrected after some two years of hard work. still one cannot know for sure that there is an as yet undiscovered flaw in there somewhere....have you read Wiles proof? It is enormous!

Posted by Anonymous

2. The issue is context, as always. Surely I am not claiming that I *know* the Wiles proof to be correct. I rely on lots of work by many mathematicians that has gone into checking it. It is the particular nature of Prof. Escultura's "refutations" that sends off warning signals.

It is fairly easy to tell a crank proof from a real objection.

Posted by Suresh

3. Well, actually, to quote E.E.E., The new real line resolves all the problems, paradoxes, contradictions, counterexamples and unanswered questions of mathematics except the Bieberbach conjecture and Riemann hypothesis.'' :D
http://home.iprimus.com.au/pidro/
just check his site for no more than 10 seconds and you ll realise he is a total kook.

Posted by Anonymous

4. re:
Well, actually, to quote E.E.E., The new real line resolves all the problems, paradoxes, contradictions, counterexamples and unanswered questions of mathematics except the Bieberbach conjecture and Riemann hypothesis.'' :D
http://home.iprimus.com.au/pidro/
just check his site for no more than 10 seconds and you ll realise he is a total kook.
-----------------------------------------------------

Yeah and he's got a phd and who are you?

Posted by Anonymous

5. Much as I would like it to be otherwise, having a Ph.D is unfortunately not a cast iron guarantee that everything you say makes sense...

Posted by Suresh

6. I recommend that anyone who still considers Dr. Escultura statements valid in this area should check this out first:

Anatomy of a hoax  May 20, 2005
http://www.pcij.org/blog/?p=73

Posted by HiEv

7. Why don't you people just give Dr. Escultura's ideas a chance?

Dr. Escultura is my professor in my Natural Science classes and it has been an eye-opener for me.

Posted by rain

8. MY STRATEGY FOR CAPTURING FLT.

I wondered why FLT remained unresolved for centuries and concluded that its underlying fields –foundations, number theory and the real number system – are defective. In particular, number theory is flawed because its subject matter, the integers have no valid axiomatization. Therefore, I embarked on their critique-rectification that yielded the following:

1) There are sources of contradiction in mathematics including ambiguous and vacuous concepts, large and small numbers (depending on context), unbounded or infinite set and self-reference. Here is an example of vacuous concept: A triquadrilateral is a plane figure with three vertices and four edges. The Richard paradox is an example of self-reference: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is flawed, being self-referent.

2) Among the requirements for a contradiction-free mathematical space are the following:
a) It must be well-defined by consistent axioms and every concept must be well-defined by them. A concept is well-defined if its existence, properties and relationship with other concepts are specified by the axioms. A false proposition cannot be an axiom as it introduces inconsistency. So does a counterexample. For example, this proposition cannot be used as an axiom of any mathematical space: There exists a triangle with four edges.
b) The rules of inference (mathematical reasoning) must be specific to and well-defined by its axioms.
c) Any proposition involving the universal or existential quantifiers on infinite set is not verifiable and, therefore, cannot be used as an axiom for it would not endow certainty to the conclusion of a theorem.

3) The real number system does not satisfy the requirements for a contradiction-free mathematical space. In particular, the trichotomy axiom is false since it is equivalent to natural ordering which the real number system has none because most of its concepts are ill-defined. Moreover, Felix Brouwer constructed a counterexample to it (Benacerraf and Putnam, Philosophy of Mathematics, Cambridge U Press, p. 52. Therefore, the real number system is ill-defined or nonsense and FLT being fomulated in it is also nonsense. Consequently, to resolve FLT the real number system must be fixed first and FLT must be reformulated in it. Andrew Wiles failed to do this and his work collapses altogether.

4) It is alright to introduce ambiguity provided it is 'approximable" by certainty. For example, a nonterminating decimal is ambiguous since not all its digits are known but it can be approximated by a segment at the nth decimal digit at margin of error 10^-n.

5) The rectification is to build a new real number system R* with three simple axioms and two operations + and x: 1) R* contains the basic integers 0, 1, ..., 9, and the operations + and x are well-defined by 2) the addition and 3) multiplication tables of arithmetic that we learned in primary school. The rest of the elements of R* are the terminating decimals first which are later extended to the nonterminating decimals as standard Cauchy sequences.

A new real number is well-defined if every digit is known or computable, i.e., there is some rule or algorithm for determining it uniquely.

6) The new elements of the new real number system are the dark number d* = 1 – 0.99… = N – (N–1), N = 0, 1, … (the ordinary integers), and u* the equivalence class of divergent sequences. The integers are mapped into the decimal parts of the decimals; this is the needed rectification to number theory. Then the integers are isomorphically embedded in R* by their mapping into the integral parts of decimals. Moreover, the mapping 0 –> d*, N –> (N–1).99…, is an isomorphism of the integers into the new integers. They behave like the integers with respect to the additive and multiplicative operations. The well-defined real numbers are embedded in R* as terminating decimals (note that the well-defined real numbers are the terminating decimals. Nonterminating decimals cannot be added or multiplied because the operations + and x require the last element to carry out. In R* the nonterminating decimals are well-defined Cauchy sequences and the operations + and x are well-defined accordingly.

7) Then the counterexamples to FLT are as follows: Let x = (0.99...)10^T, y = d*, z = 10^T, where T is an ordinary integer, T = 1, 2, ... Then x, y, z satisfy Fermat's equation, for n > 2,

x^n + y^n = z^10.

Moreover, if k = 1, 2, ..., is ordinary integer, kx, ky, kz also satisfy Fermat's equation. They are the countably infinite counterexamples to FLT. They prove that FLT is false and Wiles is wrong.

The critique-rectification of the underlying fields of FLT, the construction of the counterexamples and applications of this new methodology, especially in physics, are developed in the following articles in renowned refereed international journals and conference proceedings:

[18] Escultura, E. E. (1996) Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications, Dynamic Publishers, Inc., Atlanta, 147 – 152.
[19] Escultura, E. E. (1997) The flux theory of gravitation (FTG) I. The solution of the gravitational n-body problem, Nonlinear Analysis, 30(Cool, 5021 – 5032.
[20] Escultura, E. E. (1998) FTG VII. Exact solutions of Fermat's equation (Definitive resolution of
Fermat's last theorem, J. Nonlinear Studies, 5(2), 227 – 254.
[21] Escultura, E. E. (1999) VIII. Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(Cool, 259 – 285.
[22] Escultura, E. E. (1999) FTG II. Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
[23] Escultura, E. E. (2000) FTG IX. Set-valued differential equations and applications to quantum gravity, Mathematical Research, Vol. 6, 2000, St. Petersburg, 58 – 69.
[24] Escultura, E. E. (2001) FTG X. From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
[25] Escultura, E. E. (2001) FTG XI. Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
[26] Escultura, E. E. (2001) FTG. XII. Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
[27] Escultura, E. E. (2001) FTG III: Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
[28] Escultura, E. E. (2001) FTG IV. Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 32(10), 1539 – 1551.
[29] Escultura, E. E. (2002) FTG V. The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[30] Escultura, E. E. (2003) FTG VI. The theory of intelligence and evolution, Indian J. Pure and Applied Mathematics, 33(1), 111 – 129.
[31] Escultura, E. E. (2003) FTG XVII: The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[32] Escultura, E. E. (2003) FTG XVIII. Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
[33] Escultura, E. E. (2001) FTG. XIV. The mathematics of chaos, turbulence, fractal and tornado breaker, deflector and aborter, Proc. Symposium on Development through Basic Research, National Research Council of the Philippines, University of the Philippines, 1 – 13.
[34] Escultura, E. E. FTG XIX. Recent results, new inventions and the new cosmology, accepted, J. Problems of Nonlinear Analysis in Engineering System.
[35] Escultura, E. E. FTG XV. The new nonstandard analysis and the intuitive calculus, submitted.
[36] Escultura, E. E. (2002) FTG VI. The philosophical and mathematical foundations of FLT’s resolution, rectification and extension of underlying fields and applications, accepted, J. Nonlinear Differential Equations.
[37] Escultura, E. E. (2002) FTG XXII. Extending the reach of computation, submitted.
[38] Escultura, E. E. (2003) The theory of learning and implications for Math-Science Education, submitted.
[39] Escultura, E. E. (2003) FTG XXIII. The complex plane revisited, accepted, Journal of Nonlinear Differential Equations.
[40] Escultura, E. E. (2002) FTG XXIV. Columbia: the crossroads for science, accepted, J. Nonlinear Differential Equations.
[41] Escultura, E. E. (2003) FTG XXV. Dynamic Modeling and Applications, Proc. 3rd International Conference on Tools for Mathematical Modeling, State Technical University of St. Petersburg, St. Petersburg.
[42] Escultura, E. E. (2004) FRG XXVII – XXVIII. Part I. The new frontiers of mathematics and physics. Part I. Theoretical Construction and Resolution of Issues, Problems and Unanswered Questions.
[43] Escultura, E. E. (2005) FRG XXVII – XXVIII. The new frontiers of mathematics and physics. Part II. The new real number system: Introduction to the new nonstandard analysis, Nonlinear Analysis and Phenomena, II(1), January, 15 – 30.
[44] Escultura, E. E. (2005) FTG XXVI. Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress of Nonlinear Analysts, Orlando, June 30 – July 7, 2004.
[45] Escultura, E. E. FTG. XXVII (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
[46] Escultura, E. E. Escultura (2006) FTG XXXIV. Foundations of Analysis and the New Arithmetic,
Nonlinear Analysis and Phenomena, January 2006.
[47} Escultura, E. E. FTG XXXV (2006) The Pillars of FTG and some updates, Nonlinear Analysis and Phenomena, III(2), 1 – 22.
[48] Escultura, E. E. FTG XXXVI (2006) The New Nonstandard Calculus, accepted, Nonlinear Analysis.

E. E. Escultura

Posted by E. E. Escultura

9. MY STRATEGY FOR CAPTURING FLT.

I wondered why FLT remained unresolved for centuries and concluded that its underlying fields –foundations, number theory and the real number system – are defective. In particular, number theory is flawed because its subject matter, the integers have no valid axiomatization. Therefore, I embarked on their critique-rectification that yielded the following:

1) There are sources of contradiction in mathematics including ambiguous and vacuous concepts, large and small numbers (depending on context), unbounded or infinite set and self-reference. Here is an example of vacuous concept: A triquadrilateral is a plane figure with three vertices and four edges. The Richard paradox (sometimes called Russell paradox) is an example of self-reference: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is flawed, being self-referent.

2) Among the requirements for a contradiction-free mathematical space are the following:
a) It must be well-defined by consistent axioms and every concept must be well-defined by them. A concept is well-defined if its existence, properties and relationship with other concepts are specified by the axioms. A false proposition cannot be an axiom as it introduces inconsistency. For example, this proposition cannot be used as an axiom of any mathematical space: There exists a triangle with four edges.
b) Since distinct mathematical spaces are well-defined only by their respective axioms, they are independent. Therefore, rules of inference (mathematical reasoning) must be specific to and well-defined by its axioms.
c) Any proposition involving the universal or existential quantifiers on infinite set is not verifiable and, therefore, cannot be used as an axiom for it would not endow certainty to the conclusion of a theorem.

3) The real number system does not satisfy the requirements of a contradiction-free mathematical space. In particular, the trichotomy axiom is false since it is equivalent to natural ordering which the real number system has none because most of its concepts are ill-defined. Moreover, Felix Brouwer constructed a counterexample to it (Benacerrap and Putnam, Philosophy of Math., Cambridge U Press, 1985, p. 52). A counterexample is a contradiction. Therefore, the real number system is ill-defined or nonsense and FLT being fomulated in it is also nonsense. Consequently, to resolve FLT the real number system must be fixed first and FLT must be reformulated in it. Andrew Wiles failed to do this and his work collapses altogether.

4) It is alright to introduce ambiguity provided it is 'approximable" by certainty. For example, a nonterminating decimal is ambiguous since not all its digits are known but it can be approximated by a segment at the nth decimal digit at margin of error 10^-n.

5) The rectification is to build a new real number system R* with three simple axioms and two operations + and x: 1) R* contains the basic integers 0, 1, ..., 9, and the operations + and x are well-defined by 2) the addition and 3) multiplication tables of arithmetic that we learned in primary school. The rest of the elements of R* are the terminating decimals first which are later extended to the nonterminating decimals as standard Cauchy sequences.

A new real number is well-defined if every digit is known or computable, i.e., there is some rule or algorithm for determining it uniquely.

6) The new elements of the new real number system are the dark number d* = 1 – 0.99… = N – (N–1), N = 0, 1, … (the ordinary integers), and u* the equivalence class of divergent sequences. The integers are isomorphically mapped into the decimal parts of the decimals and are subject to the axioms of R*; this is the needed rectification to number theory. Moreover, the integers are isomorphically embedded in R* by the mapping 0 –> d*, N –> (N–1).99…, into the new integers. Thus, the new integers behave like the integers and they are well-defined. The well-defined real numbers are embedded in R* as terminating decimals (note that the well-defined real numbers are the terminating decimals. Nonterminating decimals cannot be added or multiplied because the operations + and x require the last element to carry out. In R* the nonterminating decimals are well-defined Cauchy sequences and the operations + and x are well-defined accordingly.

7) Then the counterexamples to FLT are as follows: Let x = (0.99...)10^T, y = d*, z = 10^T, where T is an ordinary integer, T = 1, 2, ... Then x, y, z satisfy Fermat's equation, for n > 2,

x^n + y^n = z^10.

Moreover, if k = 1, 2, ..., is ordinary integer, kx, ky, kz also satisfy Fermat's equation. They are the countable counterexamples to FLT. They prove that FLT is false and Wiles is wrong.

The critique-rectification of the underlying fields of FLT, the construction of the counterexamples and applications of this new methodology, especially in physics, are developed in the following articles in renowned refereed international journals and conference proceedings:

[18] Escultura, E. E. (1996) Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications, Dynamic Publishers, Inc., Atlanta, 147 – 152.
[19] Escultura, E. E. (1997) The flux theory of gravitation (FTG) I. The solution of the gravitational n-body problem, Nonlinear Analysis, 30(Cool, 5021 – 5032.
[20] Escultura, E. E. (1998) FTG VII. Exact solutions of Fermat's equation (Definitive resolution of
Fermat's last theorem, J. Nonlinear Studies, 5(2), 227 – 254.
[21] Escultura, E. E. (1999) VIII. Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(Cool, 259 – 285.
[22] Escultura, E. E. (1999) FTG II. Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
[23] Escultura, E. E. (2000) FTG IX. Set-valued differential equations and applications to quantum gravity, Mathematical Research, Vol. 6, 2000, St. Petersburg, 58 – 69.
[24] Escultura, E. E. (2001) FTG X. From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
[25] Escultura, E. E. (2001) FTG XI. Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
[26] Escultura, E. E. (2001) FTG. XII. Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
[27] Escultura, E. E. (2001) FTG III: Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
[28] Escultura, E. E. (2001) FTG IV. Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 32(10), 1539 – 1551.
[29] Escultura, E. E. (2002) FTG V. The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[30] Escultura, E. E. (2003) FTG VI. The theory of intelligence and evolution, Indian J. Pure and Applied Mathematics, 33(1), 111 – 129.
[31] Escultura, E. E. (2003) FTG XVII: The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[32] Escultura, E. E. (2003) FTG XVIII. Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
[33] Escultura, E. E. (2001) FTG. XIV. The mathematics of chaos, turbulence, fractal and tornado breaker, deflector and aborter, Proc. Symposium on Development through Basic Research, National Research Council of the Philippines, University of the Philippines, 1 – 13.
[34] Escultura, E. E. FTG XIX. Recent results, new inventions and the new cosmology, accepted, J. Problems of Nonlinear Analysis in Engineering System.
[35] Escultura, E. E. FTG XV. The new nonstandard analysis and the intuitive calculus, submitted.
[36] Escultura, E. E. (2002) FTG VI. The philosophical and mathematical foundations of FLT’s resolution, rectification and extension of underlying fields and applications, accepted, J. Nonlinear Differential Equations.
[37] Escultura, E. E. (2002) FTG XXII. Extending the reach of computation, submitted.
[38] Escultura, E. E. (2003) The theory of learning and implications for Math-Science Education, submitted.
[39] Escultura, E. E. (2003) FTG XXIII. The complex plane revisited, accepted, Journal of Nonlinear Differential Equations.
[40] Escultura, E. E. (2002) FTG XXIV. Columbia: the crossroads for science, accepted, J. Nonlinear Differential Equations.
[41] Escultura, E. E. (2003) FTG XXV. Dynamic Modeling and Applications, Proc. 3rd International Conference on Tools for Mathematical Modeling, State Technical University of St. Petersburg, St. Petersburg.
[42] Escultura, E. E. (2004) FRG XXVII – XXVIII. Part I. The new frontiers of mathematics and physics. Part I. Theoretical Construction and Resolution of Issues, Problems and Unanswered Questions.
[43] Escultura, E. E. (2005) FRG XXVII – XXVIII. The new frontiers of mathematics and physics. Part II. The new real number system: Introduction to the new nonstandard analysis, Nonlinear Analysis and Phenomena, II(1), January, 15 – 30.
[44] Escultura, E. E. (2005) FTG XXVI. Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress of Nonlinear Analysts, Orlando, June 30 – July 7, 2004.
[45] Escultura, E. E. FTG. XXVII (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
[46] Escultura, E. E. Escultura (2006) FTG XXXIV. Foundations of Analysis and the New Arithmetic,
Nonlinear Analysis and Phenomena, January 2006.
[47} Escultura, E. E. FTG XXXV (2006) The Pillars of FTG and some updates, Nonlinear Analysis and Phenomena, III(2), 1 – 22.
[48] Escultura, E. E. FTG XXXVI (2006) The New Nonstandard Calculus, accepted, Nonlinear Analysis.