This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions to mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover these laws.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat's last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat's last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat's last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded: a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy. b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy. c) Identification of the defects of the real number system and the remedy. d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers. e) The remedy for the present flaw in extension of mathematical space. f) Characterization of undecidable propositions. g) The reconstruction of the real number system into the contradiction-free new real number system. g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website: http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
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.
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 the paper, The generalized integral as dual to Schwarz distribution.
Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < 1, N = 1, N > 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N. In general, any vacuous concept yields a contradiction
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. [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., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris. [11] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/ [12] 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 http://users.tpg.com.au/pidro/
Reply to Bart van Donselaar’s article, Edgar E. Escultura and the inequality of 1 and 0.999...
1) The reason Bart van Donselaar cannot see why 1 and 0.99… are distinct is he looks at them as concepts in one’s mind and missed what David Hilbert already knew almost a century ago that such concepts are ambiguous being unknown to others. 1 and 0.99… are distinct objects in the real world like orange and apple and to write the equation orange = apple is simply nonsense.
2) He could not understand why I “claim” that FLT is false and Wiles’ proof is incorrect since he says the proof is admired Worldwide (actually only four or five mathematicians do). I hope he has seen my article, Two fatal defects of Wiles’ proof of FLT, posted in several blogsites and websites.
3) He claims that constructivists have not found hard evidence of defects in standard mathematics. The evidence is just under his nose: Felix Brouwers’ counterexample to the trichotomy axiom, Putnam and Benacerraf, Philosophy of Mathematics, Cambridge University Press, 1985 and my own version in, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computation, 17(2009), 59 – 84.
4) He claims mathematicians (he probably means some mathematicians) are happy with traditional mathematics. I wish them continued bliss of innocence.
5) He doubts that I have solved the gravitational n-body problem. I did in the paper, The solution of the gravitational n-body problem, Nonlinear Analysis, 30(8), Dec. 1997, 521 – 532; the journal is a publication of Elsevier Science Ltd. based there in Amsterdam.
6) He claims he can compute with nonterminating decimals. His claim is based on unclear thinking. Can he add sqrt2 and sqrt3 and write the precise sum?
7) He also cannot understand why it is impossible to verify whether a nonterminating decimal is periodic or nonperiodic. Clue: the digits are infinite and we cannot look at all of them to check.
8) I notice lately, that Wiles’ supporters have done massive promotion of his proof including publication of some books about it. It will not prosper unless they address my specific criticisms of the proof point blank.
Conclusion.
The article is not well thought out and uses rumors and gossips. It quotes Alecks Pabico an amateur journalist who lost his job as a journalist for commenting on an issue he knows nothing about and writing about it that he posted in blogsites and websites across the internet.
Bart is unsure of his ideas, makes claims he cannot verify and resorts to name-dropping which makes me doubt if he, like Alecks, understands what he is writing about.
Being a newcomer in mathematics and science (my first publication is “The solution of the gravitational n- body problem”, Nonlinear Analysis, Series A, 30(8), Dec. 1997, 521 – 532), I was an underdog in contending ideas with the heavy weights of science and mathematics. Therefore, I had to disseminate my ideas broadly as quickly as possible. The first forum I posted my message in was SciMath, 1997, and the topic was the equality 1 = 0.99… I said that this was really nonsense and I’ll explain why later.
There was a howl of protest and hundreds of messages were posted in protest during the year. Some called me crackpot, lunatic, moron, etc. One even wrote my colleague (who sort of discovered me in mathematics), Prof. V. Lakshmikantham, a famous mathematician who founded the only rapidly expanding field of mathematics today, Nonlinear Analysis, founder and editor-in-chief of several scientific journals, and president and founder of the International Federation of Nonlinear Analysts, to tell him that he was a lunatic for associating with me. There were at least five such guys in SciMath. However, in due course they pulled enough rope to hang themselves with academically and are all quiet now.
Going back to 1 = 0.99…, it was David Hilbert who recognized almost a century ago that the concepts of individual thought, being inaccessible to others, are ambiguous and cannot be discussed, studied and analyzed collectively. Therefore, they cannot be the subject matter of mathematics. The proper subject matter of mathematics must be objects in the real world, e.g., symbols that we also call concepts that everyone can look at provided they are subject to consistent premises or axioms. Clearly, 1 and 0.99… are distinct objects like apple and orange and to say apple = orange is simply nonsense.
As it will turn out SciMath is the best forum in this category in terms of open participation and, naturally, diversity of ideas. The worst in this category, however, is Wikipedia along with its sister website Wikia. Wikipedia requires consensus on posted topic. In other words, it requires uniformity of thought. Wikia specifically bars original research. In effect, they block the progress of science and mathematics which do not thrive on consensus and their progress stands on original research. Between these two extremes the blogs and websites range from good to excellent in terms of diversity of ideas with the only exception of HaloScan and its sister website, DLMSY, which cannot stand contrary opinion. Consequently, they lose bloggers. I identify a few excellent ones in the category of in-betweens: False Proofs, MathForge, WorldPress and Faces of the Moon. I add Knowledgerush in terms of ease in posting – no no username and password which are easy to forget.
However, there is a forum that is a class by itself in terms of the quality and level of intellectual discussion: ISCID (International Society for Computing and Intelligent Design (?)). I recommend experts to visit this website.
E. E. Escultura Research Professor Lakshmikantham Institute for Advanced Studies and Departments of Mathematics and Physics GVP College of Engineering, JNT University, Visakhapatnam, AP, India E-mail: escultur36@gmail.com * URL: http://users.tpg.com.au/pidro/
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. (Main reference: Escultura, E. E., The new real new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84).
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 the paper, The generalized integral as dual to Schwarz distribution, in press, Nonlinear Studies.
Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < 1, N = 1, N > 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N. In general, any vacuous concept yields a contradiction.
12 Comments:
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
to mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover these laws.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat’s last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat's last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
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This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat's last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
This blog aims to be the clearing house of information and queries about the resolution of the centuries-old conjecture post by Fermat in 1637 known as Fermat's last theorem (FLT) and its off-shoot: the development of the new mathematics and physics. Although the threads, articles and posts in various blogs, websites and forums across the internet already provide an overview of the subject, I shall summarize them all right now and provide details later.
1) The failure to resolve FLT for 360 years is attributed to the defects of the underlying fields, namely, foundations, number theory and the real number system.
2) Critique-rectification of these fields was undertaken starting 1992 that yielded:
a) Identification of defects of foundations, particularly, present mathematical reasoning and their remedy.
b) The requirement for contradiction-free mathematical space, since any contradiction or inconsistency reduces a mathematical space to nonsense, and the remedy.
c) Identification of the defects of the real number system and the remedy.
d) The remedy for the major flaw of number theory: lack of valid axiomatization of the integers.
e) The remedy for the present flaw in extension of mathematical space.
f) Characterization of undecidable propositions.
g) The reconstruction of the real number system into the contradiction-free new real number system.
g) The countably infinite counterexamples to FLT.
3) The development of the new non-standard calculus whose base space is the new real number system.
4) Implications of the characterization of the characterization of undecidable propositions
for mathematical physics and the remedy for the flaw of present methodology called mathematical modelling.
5) That remedy is called dynamic modelling. While mathematical modelling EXPLAINS nature in terms of numbers, equations, functions, inequalities and statistical trends, dynamic modelling EXPLAINS nature in terms of its laws. Then the task of the physicists is to discover the laws of nature.
6) The discovery of the basic constituents of matter called superstring.
7) The solution of the 200-year old gravitational n-body problem in 1996.
8) The development of the flux theory of gravitation that now qualifies as grand unified theory or the theory of everything.
For background materials and references visit my website:
http://home.iprimus.com.au/pidro/
Comments, contributions, criticism, contrary opinions and debate are welcome.
E. E. Escultura
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.
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 the paper, The generalized integral as dual to Schwarz distribution.
Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < 1, N = 1, N > 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N. In general, any vacuous concept yields a contradiction
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.
[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., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier
Science, Ltd.), 2009, Paris.
[11] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[12] 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
http://users.tpg.com.au/pidro/
Reply to Bart van Donselaar’s article, Edgar E. Escultura and the inequality of 1 and 0.999...
1) The reason Bart van Donselaar cannot see why 1 and 0.99… are distinct is he looks at them as concepts in one’s mind and missed what David Hilbert already knew almost a century ago that such concepts are ambiguous being unknown to others. 1 and 0.99… are distinct objects in the real world like orange and apple and to write the equation orange = apple is simply nonsense.
2) He could not understand why I “claim” that FLT is false and Wiles’ proof is incorrect since he says the proof is admired Worldwide (actually only four or five mathematicians do). I hope he has seen my article, Two fatal defects of Wiles’ proof of FLT, posted in several blogsites and websites.
3) He claims that constructivists have not found hard evidence of defects in standard mathematics. The evidence is just under his nose: Felix Brouwers’ counterexample to the trichotomy axiom, Putnam and Benacerraf, Philosophy of Mathematics, Cambridge University Press, 1985 and my own version in, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computation, 17(2009), 59 – 84.
4) He claims mathematicians (he probably means some mathematicians) are happy with traditional mathematics. I wish them continued bliss of innocence.
5) He doubts that I have solved the gravitational n-body problem. I did in the paper, The solution
of the gravitational n-body problem, Nonlinear Analysis, 30(8), Dec. 1997, 521 – 532; the journal is a
publication of Elsevier Science Ltd. based there in Amsterdam.
6) He claims he can compute with nonterminating decimals. His claim is based on unclear thinking. Can he add sqrt2 and sqrt3 and write the precise sum?
7) He also cannot understand why it is impossible to verify whether a nonterminating decimal is periodic or nonperiodic. Clue: the digits are infinite and we cannot look at all of them to check.
8) I notice lately, that Wiles’ supporters have done massive promotion of his proof including publication of some books about it. It will not prosper unless they address my specific criticisms of the proof point blank.
Conclusion.
The article is not well thought out and uses rumors and gossips. It quotes Alecks Pabico an amateur journalist who lost his job as a journalist for commenting on an issue he knows nothing about and writing about it that he posted in blogsites and websites across the internet.
Bart is unsure of his ideas, makes claims he cannot verify and resorts to name-dropping which makes me doubt if he, like Alecks, understands what he is writing about.
E. E. Escultura
The best and the worst
Being a newcomer in mathematics and science (my first publication is “The solution of the gravitational n-
body problem”, Nonlinear Analysis, Series A, 30(8), Dec. 1997, 521 – 532), I was an underdog in contending ideas with the heavy weights of science and mathematics. Therefore, I had to disseminate my ideas broadly as quickly as possible. The first forum I posted my message in was SciMath, 1997, and the topic was the equality 1 = 0.99… I said that this was really nonsense and I’ll explain why later.
There was a howl of protest and hundreds of messages were posted in protest during the year. Some called me crackpot, lunatic, moron, etc. One even wrote my colleague (who sort of discovered me in mathematics), Prof. V. Lakshmikantham, a famous mathematician who founded the only rapidly expanding field of mathematics today, Nonlinear Analysis, founder and editor-in-chief of several scientific journals, and president and founder of the International Federation of Nonlinear Analysts, to tell him that he was a lunatic for associating with me. There were at least five such guys in SciMath. However, in due course they pulled enough rope to hang themselves with academically and are all quiet now.
Going back to 1 = 0.99…, it was David Hilbert who recognized almost a century ago that the concepts of individual thought, being inaccessible to others, are ambiguous and cannot be discussed, studied and analyzed collectively. Therefore, they cannot be the subject matter of mathematics. The proper subject matter of mathematics must be objects in the real world, e.g., symbols that we also call concepts that everyone can look at provided they are subject to consistent premises or axioms. Clearly, 1 and 0.99… are distinct objects like apple and orange and to say apple = orange is simply nonsense.
As it will turn out SciMath is the best forum in this category in terms of open participation and, naturally, diversity of ideas. The worst in this category, however, is Wikipedia along with its sister website Wikia. Wikipedia requires consensus on posted topic. In other words, it requires uniformity of thought. Wikia specifically bars original research. In effect, they block the progress of science and mathematics which do not thrive on consensus and their progress stands on original research. Between these two extremes the blogs and websites range from good to excellent in terms of diversity of ideas with the only exception of HaloScan and its sister website, DLMSY, which cannot stand contrary opinion. Consequently, they lose bloggers. I identify a few excellent ones in the category of in-betweens: False Proofs, MathForge, WorldPress and Faces of the Moon. I add Knowledgerush in terms of ease in posting – no no username and password which are easy to forget.
However, there is a forum that is a class by itself in terms of the quality and level of intellectual discussion: ISCID (International Society for Computing and Intelligent Design (?)). I recommend experts to visit this website.
E. E. Escultura
Research Professor
Lakshmikantham Institute for Advanced Studies and Departments of Mathematics and Physics
GVP College of Engineering, JNT University, Visakhapatnam, AP, India
E-mail: escultur36@gmail.com * URL: http://users.tpg.com.au/pidro/
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. (Main reference: Escultura, E. E., The new real new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84).
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 the paper, The generalized integral as dual to Schwarz distribution, in press, Nonlinear Studies.
Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < 1, N = 1, N > 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N. In general, any vacuous concept yields a contradiction.
E. E. Escultura
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