Tuesday, April 01, 2003

Counterexample to Fermat's Last Theorem Found!!!

There has been a really amazing development today on Fermat's Last Theorem. Noam Elkies has announced a counterexample, so that FLT is not true after all! His spoke about this at the Institute today. The solution to Fermat that he constructs involves an incredibly large prime exponent (larger that 10^20), but it is constructive. The main idea seems to be a kind of Heegner point construction, combined with an really ingenious descent for passing from the modular curves to the Fermat curve. The really difficult part of the argument seems to be to show that the field of definition of the solution (which, a priori, is some ring class field of an imaginary quadratic field) actually descends to Q. I wasn't able to get all the details, which were quite intricate...
So it seems that the Shimura Taniyama conjecture is not true after all. The experts think that it can still be salvaged, by extending the concept of automorphic representation, and introducing a notion of "anomalous curves" that would still give rise to a "quasi-automorphic representation".

This is an email I received indirectly from the late Gian-Carlo Rota dated April 2, 1994. The historical context: In June of 1993, Andrew Wiles announced that he had proven Fermat's Last Theorem but later that year a subtle bug was found which was not fixed until September of '94.
So in April of 1994 we didn't know whether Wiles' proof was valid and given the date was April 2 and how well the email was written, many mathematicians believed this message. But of course it was all an elaborate April Fools joke.

71 comments:

  1. 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.. In general, any vacuous concept yields a contradiction.

    E. E. Escultura
    Research Professor
    V. Lakshmikantham Institute for Advanced Studies
    GVP College of Engineering, JNT University

    ReplyDelete
  2. Response to the commentaries on FLT and my counterexamples to them.

    Since there is noticeable increase in commentaries about FLT, Wiles’ proof and my counterexamples, it is time to respond to some major points, present the foundational basis of my counterexamples and make a rejoinder on FLT.

    Constructivist mathematics in my sense has nothing to do with intuitionism. It simply avoids sources of ambiguity and contradiction in the construction of a mathematical system which are: the concepts of individual thought, ill-defined and vacuous concepts, large and small numbers, infinity and self reference. I have given examples in my posts in several websites of how these concepts yield contradictions. A contradiction or paradox in any mathematical collapses a mathematical system to sense since any conclusion from it is contradicted by another..

    Early in the 20th Century David Hilbert pointed out the ambiguity of individual thought being inaccessible to others and cannot be studied and analyzed collectively; nor can it be axiomatized as a mathematical system. Therefore, to make sense, a mathematical system must consist of objects in the real world that everyone can look at, study, etc., e.g., symbols, subject to consistent premises or axioms. A counterexample to an axiom or theorem of a mathematical system makes it inconsistent.

    This important clarification by Hilbert has not been grasped by MOST mathematicians, the reason for the popularity of the equation 1 = 0.99… How can 1 and 0.99… be equal when they are distinct objects? It’s like equating an apple to an orange. A lot of explaining is needed, if at all possible, to make sense out of this nonsense.

    It is true that the decimals are nothing new. In fact, they have their origin in Ancient India but until the construction of the contradiction-free new real number system nonterminating decimals were ambiguous, ill-defined. A decimal is defined by its digits and if we do not know those digits it is ambiguous; this is the case with any nonterminating decimal. So is an integer divided by a prime other than 2 or 5; the quotient is ill-defined. Thus, the concept of an irrational number is ambiguous but we did not realize it because all along we relied on traditions and did not realize that previous generations of mathematician could have made a mistake or that the world has changed and what was correct then is no longer so now.

    The dark number d* is the well-defined counterpart of the ill-defined infinitesimal of calculus. It is set-valued and a continuum that joins the adjacent predecessor-successor pairs of decimals under the lexicographic ordering into the continuum R*, the new real number system. The decimals, of course, are countable infinite and discrete.

    To dismiss difficult mathematics or physical theory is like sticking one’s head into the ground as the ostrich does. New ideas are often difficult initially, especially, when they grate one’s hard-earned achievements as they did in my case. If they are right they will pass the test of time. A number of my papers made it to the list of most downloaded papers at Elsevier Science, Ltd, Science Direct website since 2002. At any rate, I will be happy to clarify specific points in my work right here or my message board at http://users.tpg.com.au/pidro/

    With respect to FLT I have recently posted my rejoinder on several websites including Larry Freeman’s False Proof. I post it again here with slight editing to avoid redundancy:

    ReplyDelete
    Replies
    1. The following article debunks every false proof that 0.999... equals 1:

      https://www.filesanywhere.com/fs/v.aspx?v=8b696686586172b3b0a7

      John Gabriel
      http://johngabrie1.wix.com/newcalculus

      Delete
  3. 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. He missed what David Hilbert already knew almost a century ago that such concepts are ambiguous because they are not known to others. Therefore, they cannot be the subject matter of mathematics. 1 and 0.99.. are simply 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 “claimed” that FLT is false and Wiles’ proof of it 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 blogs and websites.

    3) He claims he can compute nonterminating decimals. Computation of decimal depends on its digits and most of the digits of a nonterminating decimal are unknown. This is based on imprecise thinking. At any rate, I would like to see how he did this impossible feat. We can only approximate a nonterminating decimal or approximate the result of computation with them.

    4) 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.

    5) I notice lately, that Wiles’ supporters have done massive promotion of his proof, some even writing books about it. Unless they address point blank my specific criticisms of the proof, they will not prosper.

    Conclusion.

    The article is not well thought out and uses rumors and gossips. For example, it quotes Alecks Pabico an amateur journalist who lost his job as a journalist for commenting on an issue he knows nothing about or writing an article about it which he posted in many blogs and websites across the internet.

    E. E. Escultura
    Research Professor
    V. Lakshmikantham Institute for Advanced Studies
    GVP College of Engineering, JNT University
    http://users.tpg.com.au/pidro/

    ReplyDelete
  4. 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 ones 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. SciMath stands out as the best forum for discussion of various mathematical issues from different perspectives. There was one regular at SciMath who did not debate me online but through e-mail. We debated for about a year and I learned much from him. The few who only had expletives and name-calls to throw at me are nowhere to be heard from.

    E. E. Escultura

    ReplyDelete
  5. There was one unsigned feeble attempt from the UP Mathematics Department to counter my arguments online. But it wilted without a response from the science community because it lacked grasp of what mathematics is all about.

    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 inequality of 1 and 0.99…; a website on the Donselaar’s paper has been set up:

    http://www.reddit.com/r/math/comments/93n3i/edgar_e_escultura_and_the_inequality_of_1_and/

    and the discussion is coming to a close as no new issues are being raised.

    E. E. Escultura

    ReplyDelete
  6. Needless to say, none of my criticisms of Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully on this website or across the internet. In peer reviewed publications there is not even a single attempt to refute my positions on these issues.
    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. The equation, therefore, is nonsense.

    E. E. Escultura

    ReplyDelete
  7. 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.
    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 pieces 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

    ReplyDelete
  8. 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 system 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, i.e., the union of disjoint open sets.
    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 and reformulate FLT in the consistent number system, the new real number system. 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

    ReplyDelete
  9. 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 none..

    8) 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

    ReplyDelete
  10. 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., 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/

    ReplyDelete
  11. Since I now know how to post a long comment, I now post my full reply to Bart van Donselaar's article by segments.E. E. Escultura

    ReplyDelete
  12. 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. He missed what David Hilbert already knew almost a century ago that such concepts are ambiguous being unknown to others. Therefore, they cannot be the subject matter of mathematics. The right subject matter for mathematics are objects in the real world everyone can see, e.g., symbols. 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). Well, an error is an error and I hope he has seen my article, Two fatal defects of Wiles’ proof of FLT, posted in several blogs and websites.

    E. E. Escultura

    ReplyDelete
  13. 3) He relies on dictionary definitions of concepts which is quite inappropriate for mathematics. Constructivism in my sense has nothing to do with intuitionism. It simply avoids sources of ambiguity and contradiction.

    4) He claims that constructivists have not found hard evidence of defects in standard mathematics. The evidences is just under his nose: Felix Brouwers’ counterexample to the trichotomy axiom, Putnam and Benacerraf, Philosophy of Mathematics, Cambridge University Press, 1985; I also have my own version in, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computation, 17(2009), 59 – 84.

    E. E. Escultura

    ReplyDelete
  14. 5) He thinks mathematicians (he probably means some mathematicians) are happy with traditional mathematics for there is nothing wrong with it. Well, I wish them continued bliss of innocence.

    6) He doubts that I solved the gravitational n-body problem. I did in the paper, The solution
    of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications,
    30(8), Dec. 1997, 521 – 532; the journal is a publication of Elsevier Science Ltd. based there in
    Amsterdam.

    7) He claims he can compute with nonterminating decimals. Try adding sqrt2 and sqrt3 and write the precise sum. I would like to see how he does this impossible feat. His claim is based on imprecise thinking.

    E. E. Escultura

    ReplyDelete
  15. 8) 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.

    9) He chastises me for writing difficult mathematics and physical theory. New ideas are initially difficult but if they are correct they will pass the test of time. Initial critics of my work had a hilarious time calling me a crackpot, lunatic, moron, etc., but where are they now? My posts had been picked up by many blogs and websites and my papers have been used by renowned publications such as the Encyclopedic Dictionary of Mathematics and Elsevier Science. A number of them made it to the top 25 most downloaded papers published by Elsevier Science, online at Science Direct archives. My book co-authored with Profs. V. Lakshmikantham and S. Leela released last March that applies the new real number system to mathematics, physics and other fields has now made it to #2 on World Scientific’s best sellers list in July from #48 last June. I really doubt if people buy books they don’t understand. Therefore, the problem of understanding the new real number system is elsewhere not in my work.

    E. E. Escultura

    ReplyDelete
  16. 10) 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. For example, 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 and posting it on blog 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

    ReplyDelete
  17. 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., 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
    http://users.tpg.com.au/pidro/

    ReplyDelete
  18. 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.

    2) From the definition of d* [], 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, … [].

    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.

    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.

    E. E. Escultura

    ReplyDelete
  19. 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.
    [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.
    [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.
    [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/

    ReplyDelete
  20. 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 indicating 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 0n = 0 and (0.99…)n = 0.99… since ^n = 1 for any integer n > 2.

    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…)10N, ((0.99,..)10)N + d* = 10N, N = 1, 2, … [8].

    3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10T,d*,10T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
    xn + yn = zn, (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.

    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.
    [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.
    [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.
    [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/

    ReplyDelete
  21. 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 ones 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. SciMath stands out as the best forum for discussion of various mathematical issues from different perspectives. There was one regular at SciMath who did not debate me online but through e-mail. We debated for about a year and I learned much from him. The few who only had expletives and name-calls to throw at me are nowhere to be heard from.

    E. E. Escultura

    ReplyDelete
  22. There was one unsigned feeble attempt from the UP Mathematics Department to counter my arguments online. But it wilted without a response from the science community because it lacked grasp of what mathematics is all about.

    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 inequality of 1 and 0.99…; a website on the Donselaar’s paper has been set up:

    http://www.reddit.com/r/math/comments/93n3i/edgar_e_escultura_and_the_inequality_of_1_and/

    and the discussion is coming to a close as no new issues are being raised.

    ReplyDelete
  23. Needless to say, none of my criticisms of Wiles’ proof of FLT or my critique of the real and complex number systems have been challenged successfully on this website or across the internet. In peer reviewed publications there is not even a single attempt to refute my positions on these issues.

    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. The equation, therefore, is nonsense.

    E. E. Escultura

    ReplyDelete
  24. 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.
    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 pieces 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

    ReplyDelete
  25. 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 system 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, i.e., the union of disjoint open sets.
    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.

    E. E. Escultura

    ReplyDelete
  26. 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 and reformulate FLT in the consistent number system, the new real number system. 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.

    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.

    E. E. Escultura

    ReplyDelete
  27. 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 none..

    8) 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

    ReplyDelete
  28. 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., 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/

    ReplyDelete
  29. To the admin,

    There has been inadvertent repetition of some of my posts. Please delete repeats. Thanks. E. E. Escultura

    ReplyDelete
  30. THE FINAL STRETCH IN THE CONSTRUCTION OF THE NEW REAL NUMBER SYSTEM R*: WELL DEFINING THE NONTERMINATING DECIMALS (for the first time)

    First we note that since a decimal is defined by its digits the only well defined decimals are the terminating ones. Nonterminating decimals are ill-defined or ambiguous because not all their digits are known. Therefore, the concept rational (and also irrational) is ambiguous because it is impossible to verify if its decimal representation is periodic since we cannot check all its digits being infinite. However, ambiguity can be contained by approximating it with certainty, e.g., by a terminating decimal (which has no ambiguity); such approximation is valid if the margin of error is known and can be made small as desired. Thus, while nonterminating decimals cannot be well defined we can contain its ambiguity to the point where we do algebraic operations with them and approximate the result with desired margin of error. Now we introduce the generating or g-sequence and its g-limit, a nonterminating decimal which has contained ambiguity (approximable by certainty).

    ReplyDelete
  31. As we raise n, the tail digits of the nth g-term of any decimal recedes to the right indefinitely, i.e., it becomes steadily smaller until it is unidentifiable. While it tends to 0 in the standard norm it never reaches 0 and is not a decimal since its digits are not fixed; ultimately, they are indistinguishable from the similarly receding tail digits of the other nonterminating decimals. In iterated computation when we are trying to get closer and closer approximation of a decimal the tail digits may vary but recede to the right indefinitely and become steadily smaller leaving fixed digits behind that define a decimal. We approximate the result by taking its initial segment, the nth g-term, to desired margin of error.
    .

    Consider the sequence of decimals,

    (delta^n(a_1a_2…a_k), n = 1, 2, …, (7)

    where delta is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (7) d-sequence and its nth term nth d-term. For fixed combination of delta and the a_j’s, j = 1, …, k, in (7) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (7) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.

    ReplyDelete
  32. Now, we allow delta to vary steadily in its domain and also the a_js along the basic integers (not simultaneously 0). Then their terms trace the tail digits of all the decimals and as n increases indefinitely they become smaller and smaller and indistinguishable from each other. We call their nonstandard limits dark numbers and denote by d* which is set valued, countably infinite and includes every g-limit of the nonstandard d-sequence (7). To the extent that they are indistinguishable d* is a continuum (in the algebraic sense since no notion of open set is involved). Thus, the tail digits of the nonterminating decimals merge and form the continuum d*.
    At the same time, since the tail digits of all the nonterminating decimals form a countable combination of the basic digits 0, 1, …, 9 they are countably infinite, i.e., in one-one correspondence with the integers. In fact, any set that can be labeled by integers or there is some scheme for labeling them by integers is in one-one correspondence with the integers, i.e., countably infinite. It follows that the countable union of countable set is countable. Therefore, the decimals and their tail digits are countably infinite. However, as the nth d-terms of (7) trace the tail digits of the nonterminating decimals they become unidentifiable and cannot be labeled by the integers anymore; therefore, they are no longer countable. In fact they merge as the continuum d*.

    ReplyDelete
  33. Like a nonterminating decimal, an element of d* is unaltered if finite g-terms are altered or deleted from its g-sequence. When delta = 1 and a_1a_2…a_k = 1 (7) is called the basic or principal d-sequence of d*, its g-limit the basic element of d*; basic because all its d-sequences can be derived from it. The principal d-sequence of d* is,

    (0.1)^n , n = 1, 2, … (8)

    obtained by the iterated difference,

    N – (N – 1).99… = 1 – 0.99... = 0 with excess remainder of 0.1;
    0.1 – 0.09 = 0 with excess remainder of 0.01;
    0.01 – 0.009 = 0 with excess remainder of 0.001;
    ………………………………………………… (9)

    Taking the nonstandard g-limits of the left side of (9) and recalling that the g-limit of a decimal is itself and denoting by d_n the d-limit of the principal d-sequence on the right side we have,

    N – (N – 1).99… = 1 – 0.99... = d_n. (10)

    Since all the elements of d* share its properties then whenever we have a statement “an element d of d* has property P” we may write “d* has property P”, meaning, this statement is true of every element of d*. This applies to any equation involving an element of d*. Therefore, we have,

    d* = N – (N – 1).99… = 1 – 0.99... (11)

    Like a decimal, we define the d-norm of d* as d* > 0.

    We state some theorems about R*.
    Theorem. The d-limits of the indefinitely receding (to the right) nth d-terms of d* is a continuum that coincides with the g-limits of the tail digits of the nonterminating decimals traced by those nth d-terms as the aks vary along the basic digits.
    Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs (each joined by d*); therefore, the g-closure R* of R is a continuum [9].
    Corollary. R* is non-Archimedean and non-Hausdorff in both the standard and the g-norm and the subspace of decimals are countably infinite, hence, discrete but Archimedean and Hausdorff.
    Theorem. The rationals and irrationals are separated, i.e., they are not dense in their union (this is the first indication of discreteness of the decimals) [7].
    Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99…, respectively [6].
    Theorem. An even number greater than 2 is the sum of two prime numbers.

    Remark. Gauss’ diagonal method proves neither the existence of nondenumerable set nor a continuum; it proves only the existence of countably infinite set, i.e., the off-diagonal elements consisting of countable union of countably infinite sets. The off-diagonal elements are not even well-defined because we know nothing about their digits (a decimal is determined by its digits). We state the following corollaries from our discussion: (1) Nondenumerable set does not exist; (2) Only discrete set has cardinality; a continuum has none.

    (This article is excerpted from Escultura, E. E., The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations 17 (2009), 59 – 84)

    E. E. Escultura
    Research Professor
    GVP - V. Lakshmikantham Institute for Advanced Studies
    and Departments of Mathematics and Physics
    GVP College of Engineering, JNT University
    Madurawada, Visakhapatnam. AP, India

    ReplyDelete
  34. SOME IMPORTANT INFORMATION ABOUT THE NEW REAL NUMBER SYSTEM

    We first note the sources of ambiguity in a mathematical space so that we can avoid or contain them; they are contained if their ambiguity is approximated by certainty, e.g., a nonterminating decimal which is ambiguous is approximated by its initial segment at the nth decimal place at margin of error 10^-n. We consider an ambiguous concept well-defined when its ambiguity is contained. The sources of ambiguity are: infinite set, large or small number (depending on context), self-reference, e.g., the barber paradox, vacuous concept, e.g., i = the root of the equation x^2 + 1 = 0, among the real numbers, ill-defined concept and statement involving ambiguous concept.

    1) The new real number system is built on the the elements 0 and 1 defined by the addition and multiplication tables (these are the three axioms).

    2) The basic digits 0, 1, …, 9 are built first, then the integers and the terminating decimals. They are the well defined decimals.

    3) Then the inverse operation to multiplication called division; the result of dividing a decimal by another if it exists is called quotient provided the divisor is not zero. Only when the integral part of the devisor is not prime other than 2 or 5 is the quotient well defined. For example, 2/7 is ill defined because the quotient is not a terminating decimal (we interpret a fraction as division).

    ReplyDelete
  35. 4) Since a decimal is determined or well-defined by its digits, nonterminating decimals are ambiguous or ill-defined. Consequently, the notion irrational is ill-defined since we cannot cheeckd all its digits and verify if the digits of a nonterminaing decimal are periodic or nonperiodic.

    5) Consider the sequence of decimals,

    (d)^na_1a_2…a_k, n = 1, 2, …, (1)

    where d is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (1) d-sequence and its nth term nth d-term. For fixed combination of d and the a_j’s, j = 1, …, k, in (1) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (1) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.

    ReplyDelete
  36. 6) As convention when d* appear in any equation or expression, it means that either is unaltered if d* is replaced by any dark number, i.e., element of d*. The dark number d* satisfies the following:

    N.99… - (N – 1)… = d*, N = 1, 2, …; if x is any nonterminating decimal different from zero, xd* = d*x = d*; (d*)^N = d* (1)

    7) Nonterminating decimals. Now we define a nonterminating decimal for the first time without contradiction and with contained ambiguity, i.e., approximable by certainty. We build them on what we know: the terminating decimals, our point of reference for all their extensions.
    A sequence of terminating decimals of the form,

    N.a_1, N.a_1a_2, …, N.a_1ª_2…a_n, … (2)

    where N is integer and the a_ns are basic integers, is called standard generating or g-sequence. Its nth g-term, N.a_1a_2…a_n, defines and approximates its g-limit, the nonterminating decimal,

    N.a_1a_2…a_n,…, (3)

    at margin of error 10^n. The g-limit of (2) is nonterminating decimal (3) provided the nth digits are not all 0 beyond a certain value of n; otherwise, it is terminating. As in standard analysis where a sequence converges, i.e., tends to a specific number, in the standard norm, a standard g-sequence, converges to its g-limit in the g-norm where the g-norm of a decimal is itself.

    8) Decimal integers. A nonterminating decimal of the form N.99… , N = 0, 1, …, is call decimal integer because the set of such decimals for all N is isomorphic to the integers, i.e., the integral parts of the decimals, under the mapping d* -> 0, N -> (N – 1).99…, N – 1, 2, … From the kernel of this isomorphism it follows that (0.99…)^N = 0.99… and ((0.99…)10)^N = (0.99…)10^N.

    ReplyDelete
  37. 9) We note these important results.

    a) Theorem. The d-limits of the indefinitely receding (to the right) nth d-terms of d* is a continuum that coincides with the g-limits of the tail digits of the nonterminating decimals traced by those nth d-terms as the a_ks vary along the basic digits.

    b) Theorem. The g-closure (closure in the g-norm) of the decimals is a continuum R*; this the new real number system; it is a continuum, countably infinite, non-archimedian and nonhausdorff but its subspace of decimals is also countably infinite, discrete, Archimedean and hausdorff/

    c) Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs each joined by the continuum d*.

    d) Theorem. The rationals and irrationals are separated, i.e., they are not dense in their union (this is the first indication of discreteness of the decimals).

    e) Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99… = d*, respectively.

    f) Theorem. An even number greater than 2 is the sum of two prime numbers (this used to be called Goldbach’s conjecture; now it has a proof in R*).

    g) The counterexamples to FLT. The exact solutions of Fermat’s equation, which are the counterexamples to FLT, are given by the triples (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, (4)

    for n = NT > 2. 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. One counterexample is, of course, sufficient to disprove a conjecture.

    Remark. Nondenmerable set does not exist. Cantor’s diagonal method generated only countable set. Any set which is a union of countable set or whose elements can be labeled by integers is countable. Only countable set has cardinality, a continuum has none.

    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, Nonlinear Studies 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, Neural,
    Parallel and Scientific Computations 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., 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
    and Departments of Mathematics and Physics
    GVP College of Engineering, JNT University

    ReplyDelete
  38. THE FASCINATING AND SUCCESSFUL SEARCH FOR THE BASIC CONSTITUENT OF MATTER

    For over half a century physicists have smashed the nucleus of the atom in search of the basic constituent of matter. What is the score so far? They have found only unstable elementary particles with half-life of split second. The basic constituent must not only be stable but nondestructible; otherwise, our universe would have collapsed a long time ago. On the contrary, it has not only existed for 8 billion years but also evolved to higher order. Just look at the biological laws that we now enjoy. They were non-existent a couple of billion years after the Big Bang for there were no biological species then to reveal them.

    Now, where is this atom-smashing leading to? The prospects are not bright. As the energy of the collider is raised more elementary particles are produced alright but they all disappear in split second. In fact, nothing appears nondestructible in the nucleus of the atom. This is the time to pause and change gears. Let us look at the present methodology of physics, quantitative modeling (formerly called mathematical modeling) that describes the appearances of nature mathematically its main tools being computation and measurement. This methodology has limitations for it cannot describe the very small or the unobservable such as latent energy. In fact, this is the reason it has left long standing problems unsolved, e.g., gravitational n-body and turbulence problems, and fundamental questions unanswered, e.g., what the basic constituent of matter and structure of the electron are.

    E. E. Escultura

    ReplyDelete
  39. Our remedy is the introduction of qualitative or non-quantitative modeling that explains nature and its appearances in terms of its laws. It goes much deeper than description of appearances by explaining the internal dynamics and interactions of a physical system. We include under physical systems all motions of matter including cosmic wave which is synchronized vibration of the medium. We shall find out what that medium is. Our strategy is to identify what we consider the most fundamental law of nature and proceed to find others consistent with it and if there is some phenomenon that appears to contradict it we find another natural law that reconciles them. This is always possible under our premise that there is order in our universe.

    Our choice for the fundamental law is the first law of thermodynamics which has a long history of applications and, therefore, verification. However, this is partial for it does not take latent energy into account. Therefore, we enrich and modify it as follows:

    Energy Conservation. In any physical system and its interactions, the sum of kinetic and latent energy is constant, gain of energy is maximal and loss of energy minimal.

    Then we notice from our experience that there are universal configurations and motions of matter. They are universal because they are optimizing with respect to the accumulation or disposition of energy. Therefore, they are expressions of energy conservation. We capture it in the next natural law.

    E E Escultura

    ReplyDelete
  40. Energy Conservation Equivalence. Energy conservation has other forms: order, symmetry, economy, least action, optimality, efficiency, stability, replicative-self-similarity (fractal), coherence, resonance, quantization, smoothness, uniformity, motion-symmetry balance, non-redundancy, non-extravagance, evolution to infinitesimal configuration, helical and related configuration, e.g., circular, helical, spiral and sinusoidal and, in biology, genetic encoding, reproduction and order in diversity and complexity of function, configuration and capability.

    Non-redundancy means that nature does not create another physical system with the same functions. The so-called third quark in the nucleus of an atom outside the proton discovered in 2004 joins two positive quarks, one from each of two protons; the negative quark joins the two positive quarks from the same proton. They do the same functions; therefore, they must be the same negative quark but in different places in the nucleus. This is an example of how qualitative mathematics solves physical problems without computation using natural laws alone. This is how we shall determine the structure of the superstring, primum (elementary particle), etc.

    E E Escultura

    ReplyDelete
  41. Observation through the Hubble says that matter steadily forms in the Cosmos first as cosmic dust that gets entangled in cosmological vortices then it collects around their eyes at the rate of one star per minute. In fact, there are “star nests” in the Cosmos that release stars rapidly [26,27]. Recently, Sky Cable BBC reported a baby galaxy (at its nascent phase). By energy conservation what appears as empty vacuum in the Cosmos is filled with matter that is not observable with present technology. Thus, we have this natural law.

    Existence of Two Fundamental States of Matter. There exist two fundamental states of matter: visible and dark; the former is directly observable, the latter is not.

    Our medium for observation is light and the reason for our failure to observe dark matter is it consists of pieces of matter of sizes finer than the finest wavelength of visible light which is of order of magnitude 10^(-14) meters. The appearance of matter in the Cosmos is due to conversion of dark to visible matter. With the existence of dark matter verified we can now ask this legitimate question: what does dark matter consist of? The answer: the superstring. For now it is just a name but we shall endow it with structure and properies as soon as we discover enough natural laws. First, we introduce physical concepts some of which tentative pending the discovery of the appropriate natural laws that well define them.

    E E Escultura

    ReplyDelete
  42. ENERGY is motion of matter; therefore, matter and energy are inseparable and neither pure energy nor pure matter exists. FLUX is motion of matter with identifiable direction at every point, e.g., water current. CHAOS is mixture of order none of which is identifiable, e.g., the onset of tropical cyclone on tropical depression. The immensity of colliding atoms makes it impossible to monitor or predict the trajectory of every atom but each atom is subject to the laws of natur (order). Every physical system vibrates due to the impact of cosmic waves coming from all directions in the Cosmos. Basic cosmic or electromagnetic waves are generated by the normal vibration of atomic nuclei and propagated in all directions across dark matter. We introduce the next natural law needed for the discovery of the superstring but also crucial for the study of earthly turbulence like typhoon and tornado.

    Flux-Low-Pressure Complementarity. Low pressure sucks matter around it and the initial chaotic rush of matter towards low pressure stabilizes into coherent (stable) flux; conversely, coherent flux induces low pressure around it.

    The only possible force in the Cosmos that can possibly destroy a superstring is the energetic cosmic wave because it resonates with it, i.e., its size has the same order of magnitude as the wave length. Now, what configuration does the superstring have to make it non-destructible? It must have a generalized nested fractal sequence configuration [25]. This means that the superstring has a closed circular helical configuration (like s lady's spring bracelet), by energy conservation and energy conservation equivalnce (explained below) containing a superstring traveling through its helical cycles (its motion as matter) called toroidal flux or flux torus at the speed of 7 x 10^22 cm/sec [1]. Then the latter superstring contains a superstring with the same motion and properties as the former, etc., ad infinitum. Since cosmic waves interacts only with the first term of the fractal sequence that alone can possibly be destroyed by it leaving the tail sequence still a generalized nested fractal sequence of superstrings, i.e., a superstring. Thus, the superstring survives each time the first term is destroyed by cosmic wave.

    E E Escultura

    ReplyDelete
  43. We define the superstring to be non-agitated if its cycle length (CL) is less than 10^(-16) meters, semi-agitated if 10^-16 < CL < 10^-14 meters and agitated or primum, unit of visible matter if 10^-14 meters < CL.

    We summarize the conversion of the superstring to its three phases: non-agitated, semi-agitated and agitated (visible) and determine its structure and behavior using the natural laws. All we know initially is that dark matter consists of superstrings. We knew nothing about it at the start except that it was a piece of matter.

    Cosmic waves traverse dark matter from all directions. When non-agitated superstring is hit by cosmic wave it gets thrown off and bounces against the other superstrings. When it loses imparted energy, it grinds to a halt as non-agitated superstring. But when it gets near its previous path it is sucked by it, by flux-low-pressure complementarity, and forms a loop. By energy conservation and energy conservation equivalence it evolves to helical, semi-agitated superstring, with the original superstring its toroidal flux traveling at staggering speed along its helical cycles, 7 × 1022 cm/sec [1]. As a superstring, this toroidal flux has toroidal flux, a superstring, etc. Thus, we have a generalized nested fractal sequence of superstrings.

    Another possibility is when a non-agitated superstring is hit by cosmic wave its first term as nested fractal sequence expands to a semi-agitated superstring. We summarize the structure of the superstring by the following natural law.

    E E Escultura

    ReplyDelete
  44. Seemi-superstring Formation. When suitable cosmic wave hits (agitates) a non-agitated superstring one of the following occurs: (a) its first term as nested fractal sequence expands and becomes a semi-agitated superstring with the rest of the sequence its toroidal flux; (b) it is projected into the first term of a new superstring with itself the toroidal flux or loses the energy imparted by the cosmic wave and remains dark.
    The next law governs conversion of dark matter to visible matter.

    Dark-to-Visible-Matter Conversion. When suitable shock wave hits a semi-agitated superstring one of these occurs: (a) the outer superstring breaks, its flux torus remaining non-agitated superstring; (b) a segment bulges into a primum, unit of visible matter.

    E E Escultura

    ReplyDelete
  45. While a semi or non-agitated superstring interacts neither with other superstrings nor with visible matter, the primum, being visible, does. Subjected to bombardment by cosmic waves from all directions its toroidal flux is thrust into erratic motion called spike within its neighborhood along the helical cycles [22] and pulls the superstrings around the primum into a vortex flux around its cylindrical eye along its axis at linear speed of 7 x 10^22 cm/sec, much greater than that of light. The vortex flux of a simple primum is measured as charge, its visible or kinetic energy. This makes the primum a magnet, and using the right-hand-rule with the index finger pointing in the direction of the toroidal flux, the thumb points to the north or N-pole of a positive primum like the positron, the anti-matter of the electron and it mirror image with respect to a plane between their equators normal to their common equatorial plane. Viewed from its N-pole, the toroidal and induced flux of the positron are counterclockwise, its charge positive. Naturally, its mirror image or anti-matter, the electron, has clockwise vortex rotation that provides it with negative charge, the unit of charge by convention, -1 or 1.6 × 10^-19 coulombs [22]. Another primum, the positive quark, has charge +2/3 and still another, the negative quark, charge -1/3 [24].

    Since the primum and its induced flux are visible matter (the latter detected by its charge) they are affected by centrifugal force. Thus, there is greater concentration of it along the equatorial plane. Then the primum’s profile is pointed sinusoidal arc of high even power of the sinusoidal curve, corresponding to the exponent m of our qualitative-computational model of a primum in flight below, so that the induced vortex flux is also discular like that of a cosmological vortex. This is only one form of duality between macro and quantum gravity.

    E E Escultura

    ReplyDelete
  46. It is clear that simple primum is charged and since the neutrino is neutral it is a coupled pair of prima of opposite but numerically equal charge, say -q and +q, so that -q +q = 0 and they neutralize each other’s flux and charge. The proton consists of two positive quarks joined by a negative quark at their equators on account of flux compatibility [22] (graphics in [5]). (Energy conservation requires that their axis be coplanar [22]). Thus, the charge of the proton is: 2/3 – 1/3 + 1/3 = +1. This means that there is a net coherent vortex flux around the cluster with the individual primal fluxes eddies in it. By flux compatibility the electron can attach itself to a positive quark of the proton at any point but energy conservation and energy conservation equivalence require it attaches itself between them beside the negative quark as the most stable position but pushes the negative quark a bit by flux compatibility so that their centers viewed from the N-pole form the vertices of a quadrilateral. In its interior are the coherent vortex fluxes of the positive quarks, negative quark and electron that make it a region of low pressure or depression. By flux low pressure complementarity this interior sucks neutral prima around it since charged prima are repelled by charged prima already in the cluster. Therefore, only suitably light neutral primum fits in and that is the neutrino. Thus, we have just composed the neutron consisting of a proton, electron and neutrino. Its charge is: +2/3 – 1/3 +2/3 – 1 + 0 = 0, i.e., neutral, and there is no net coherent vortex flux around it. The vortex flux of a coupled primum is also discular for the same reason as the simple primum’s is. The electron and positive and negative quarks are the basic prima because they are constituent of every atom. They are converted from dark matter to visible matter at enormous rate in the Cosmos and cellular membranes of every living thing, plant or animal [23,26,27].

    E E Escultura

    ReplyDelete
  47. We conclude this article by sharing the amusement of the superstring. Physicists are still looking for it in the atomic nucleus when it has been staring at us since 1811, the year Ernest Rutherford discovered the electron. As we have seen the electron is a primum, an agitated superstring.

    [1] Atsukovsky, V.A., General ether-dynamics. Simulation of the matter structures and fields on the basis of the ideas about the gas-like ether. Energoatomizdat, Moscow, 1990 (in Russian).
    [2] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications, 30(8), Dec. 1997, 521 – 532.
    [3] Escultura, E. E. (1999) Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(8), 259 – 285.
    [4] Escultura, E. E. (1999) Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
    [5] Escultura, E. E. (2001) From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
    [6] Escultura, E. E. (2001) Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
    [7] Escultura, E. E. (2001) Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
    [8] Escultura, E. E. (2001) Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
    [9] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
    [10] Escultura, E. E. (2003) Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
    [11] Escultura, E. E., (2005) Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress o f Nonlinear Analysts, Orlando, June 30 – July 7, 2004; Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, e519-e532.
    [12] Escultura, E. E., (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
    [13] Escultura, E. E., The Pillars of the new physics and some updates, Nonlinear Studies, 14(3), 2007, 241 – 260.

    E E Escultura

    ReplyDelete
  48. [14] Escultura, E. E., The physics of the mind, accepted, The Journal of the Science of Healing Outcome.
    [15] Escultura, E. E., The cosmology of our universe, submitted, Problems of Nonlinear Analysis in Engineering Systems.
    [16] Escultura, E. E., (2007) Dynamic Modeling and the new mathematics and physics, Neural, Parallel and Scientific Computations, 15(4), 2007, 527 – 538.
    [17] Escultura, E. E., The grand unified theory, contribution to the Felicitation Volume on the occasion of the 85th birth anniversary of Prof. V. Lakshmikantham: Nonlinear Analysis: TMA, 69(3), 2008, 823 – 831.
    [18] Escultura, E. E. The mathematics of the grand unified theory, in Nonlinear Analysis, C-Series: Theory, Mthods and Applications, 71 (2009) e420 – e431.
    [19] Escultura, E. E. Dynamic and mathematical models in physic, Proc. 5th International Conference on Dynamic Systems and Applications, June 30 – July 5, 2007, Atlanta, 164 – 169.
    [20] Escultura, E. E. (2004) Dynamic Modeling of Chaos and Turbulence, NA, TBA, 63(5-7), e519 – e532.
    [21] Escultura, E. E. The basic concepts and dynamics of quantum gravity with applications, in press, Nonlinear Studies
    [22] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis: B-Series: Real World Applications, 11 (2010), 29 – 38.
    [23] Escultura, E. E., Genetic alteration, modification and sterilization with applications to the treatment of genetic diseases, accepted, The Journal of the Science of Healing Outcomes.
    [24] Gerlovin, I. L. (1990) The Foundations of United Theory of Interactions in a Substance, Leningrad: Energoattomizdat
    [25] Lakshmikantham, V., Escultura, E. E. and Leela, S. The Hybrid Grand Unified Theory, Atlantis
    (Elsevier Science), 2009, Paris.
    [26] Science, Glow reveals early star nurseries, 281(5375), July 1998, 332 – 333.
    [27] Science, (a) Science, Starbirth, gamma blast hint at active early Universe, 282(5395), December, 1998, 1806; (b) Gamma burst promises celestial reprise, 283(5402), January 1999, 616; (c) Powerful cosmic rays tied to far off galaxies, 282(5391), Nov. 1998, 1969 – 1971 (100 million times reached in largest particle accelerators), 1023.

    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/

    ReplyDelete
  49. BACKGROUNDER ON THE GRAND UNIFIED THEORY (GUT) SOME BASIC INFORMATION by E. E. Escultura

    Why do problems in mathematics and physics defy solution or resolution for a long time? In mathematics the most famous unsolved problem was the 360-year-old Fermat’s conjecture known as Fermat’s last theorem (FLT) and in physics it was the 200-year-old Laplace or gravitational n-body problem. The author posed this question in 1988 after a 17-year absence from his mathematical career. Given that both problems appear to be very clearly stated he came to the conclusion that the difficulty lies in the inadequacy and other defects of their underlying fields. He then proceeded to first make a thorough critique of the underlying fields of FLT, namely, foundations, number theory and the real number system and he found, among others, that the real number system is inconsistent and is, therefore, ill-defined or ambiguous. Consequently, FLT being formulated in it is also ambiguous and cannot be resolved. He proceeded to construct the consistent new real number system on three simple axioms and reformulate FLT in it to make it clear and open to resolution. Indeed, FLT has countably infinite counterexamples in the new real number system.

    ReplyDelete
  50. What has FLT to do with GUT? The first major theorem in its resolution was the characterization of undecidable (unprovable) propositions that says, essentially, that a proposition is unprovable if it is ambiguous, i.e., involves ambiguous or ill-defined concepts. Being “ill-defined” is the negation of “well-defined” and a concept in a mathematical system is well-defined if its existence, properties or behavior and relationship with other concepts are specified by its axioms. To avoid ambiguity and contradiction (the latter often hides in the former) every concept in a mathematical space must be well-defined and in its construction the choice of the axioms is not complete until this requirement is achieved. When we have two distinct mathematical spaces every concept in one is ill-defined in the other since each mathematical space is well defined only by its axioms. A physical theory is a mathematical space whose axioms are laws of nature. In a mathematical space the axioms are man-made and have nothing to do with the laws of nature.

    In the present methodology of physics called quantitative modeling (formerly called mathematical modeling) natural phenomena are described mathematically and a physical problem is modeled by a mathematical problem so that the solution of the latter is attributed to the solution of the physical problem. Reasoning is purely by analogy since there is no causal relation between the physical and mathematical spaces concerned. This is the reason for the existence of long-standing unsolved problems and unanswered fundamental questions of physics like what the basic constituent of matter and the structure of the electron are.

    ReplyDelete
  51. The remedy for this inadequacy of methodology is qualitative or non-quantitative modeling (formerly called dynamic modeling) that explains nature or natural phenomena in terms of the laws of nature. While quantitative modeling describes the appearances of nature mathematically, qualitative modeling explains its internal dynamics and interactions including its appearances in terms of its laws. The former is based on computation, measurement and intuition, the latter on qualitative mathematics, rational thought and analysis. Qualitative mathematics includes the following routine activity of the mathematician or scientist:

    Making conclusions, visualizing, abstracting, thought experimenting, engaging in creative activity, intuition, imagination and trial and error to sift out what is more appropriate, negating what is known to gain some insights into the unknown, altering premises to draw out new conclusions, thinking backwards and all other techniques that yield results.

    Qualitative modeling alters the task of the scientist from computation and measurement to the search for the laws of nature. It was used for the first time to solve the gravitational n-body problem in 1997. The solution required the discovery of the basic constituent of matter, the superstring. It required 11 laws of nature to accomplish both. They where the initial laws of nature of GUT known as the flux theory of gravitation then.

    At present particle physicists are still smashing the nucleus of the atom in search of the basic constituent of matter, the superstring, which has been going on for over half a century. Actually, the superstring has been staring at us since 1811 when Ernest Rutherford discovered the electron. The electron is an agitated superstring. A non-agitated superstring is dark, i.e., its size is less than 10^(-14) meters. It is the basic constituent of dark matter, one of the two fundamental states of matter, the other being visible or ordinary matter. Dark matter is not observable with present technology and is known only by its impact on visible matter. When suitably agitated by cosmic waves the superstring expands to a primum, unit of visible matter such as the electron or positive or negative quark. These three prima are called basic prima because they are constituents of every atom. They are converted from dark matter at staggering rate in the Cosmos and in the cells of living things – plants or animals. In the Cosmos alone the prima form cosmic dust that get entangled into cosmological vortices and collect at their cores at the rate of one star per minute.

    ReplyDelete
  52. References

    [1] Escultura, E. E., The solution of the gravitational n-body problem, Nonlinear Analysis, Series A: Theory, Methods and Applications, 30(8), Dec. 1997, 521 – 532.
    [2] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
    [3] Escultura, E. E. (1999) Superstring loop dynamics and applications to astronomy and biology, J. Nonlinear Analysis, 35(8), 259 – 285.
    [4] Escultura, E. E. (1999) Recent verification and applications, Proc. 2rd International Conf.: Tools for Mathematical Modeling, St. Petersburg, vol. 4, 74 – 89.
    [5] Escultura, E. E. (2001) From macro to quantum gravity, J. Problems of Nonlinear Analysis in Engineering Systems, 7(1), 56 – 78.
    [6] Escultura, E. E. (2001) Quantum gravity, Proc. 3rd International Conference on Dynamic Systems and Applications, Atlanta, 201 – 208.
    [7] Escultura, E. E. (2001) Turbulence: theory, verification and applications, J. Nonlinear Analysis, 47(2001), 5955 – 5966.
    [8] Escultura, E. E. (2001) Vortex Interactions, J. Problems of Nonlinear Analysis in Engineering Systems, Vol. 7(2), 30 – 44.
    [9] Escultura, E. E. (2001) Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 32(10), 1539 – 1551.
    [10] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
    [11] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
    [12] Escultura, E. E. (2003) Macro and quantum gravity and the dynamics of cosmic waves, J. Applied Mathematics and Computation, 139(1), 23 – 36.
    [13] Escultura, E. E., (2003) Dynamic Modeling and Applications, Proc. 3rd International Conference on Tools for Mathematical Modeling, State Technical University of St. Petersburg, St. Petersburg.
    [14] Escultura, E. E., (2004) Problems and Unanswered Questions of physics and their resolution, Nonlinear Analysis and Phenomena, I(1), 1 – 26.
    [15] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
    [16] Escultura, E. E., (2005) Dynamic Modeling of Chaos and Turbulence, Proc. 4th World Congress of Nonlinear Analysts, Orlando, June 30 – July 7, 2004; Nonlinear Analysis, Volume 63, Issue 5-7, 1 November 2005, e519-e532.
    [17] Escultura, E. E., (2005). The theory of everything, Nonlinear Analysis and Phenomena, II(2), 1 – 45.
    [18] Escultura, E. E., (2006) Foundations of Analysis and the New Arithmetic, Nonlinear Analysis and Phenomena, January 2006.
    [19] Escultura, E. E., The Pillars of the new physics and some updates, Nonlinear Studies, 14(3), 2007, 241 – 260.
    [20] Escultura, E. E., The physics of the mind, accepted, The Journal of the Science of Healing Outcome.
    [21] Escultura, E. E., The cosmology of our universe, submitted, Problems of Nonlinear Analysis in Engineering Systems.
    [22] Escultura, E. E., (2007) Dynamic Modeling and the new mathematics and physics, Neural, Parallel and Scientific Computations, 15(4), 2007, 527 – 538.
    [23] Escultura, E. E., The grand unified theory, contribution to the Felicitation Volume on the occasion of the 85th birth anniversary of Prof. V. Lakshmikantham: Nonlinear Analysis: TMA, 69(3), 2008, 823 – 831.
    [24] Escultura, E. E. The mathematics of the grand unified theory, Nonlinear Analysis, A-Series: Theory, Methods and Applications, 71 (2009) e420 – e431.
    [25] Escultura, E. E. Dynamic and mathematical models in physic, Proc. 5th International Conference on Dynamic Systems and Applications, June 30 – July 5, 2007, Atlanta, 164 – 169.
    [26] Escultura, E. E. (2004) Dynamic Modeling of Chaos and Turbulence, NA, TBA, 63(5-7), e519 – e532.
    [27] Escultura, E. E. The basic concepts and dynamics of quantum gravity with applications, in press, Nonlinear Studies
    [28] Escultura, E. E., Qualitative model of the atom, its components and origin in the early universe, Nonlinear Analysis: C-Series: Real World Applications, 11 (2010), 29 – 38.

    ReplyDelete
  53. 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 elements 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 development of 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 matter. By the 1990s the search was a complete success with the discovery of the +quark (up quark) and quark (down quark) and the electron (discovered in 1897). They are basic as constituents of every atom; a heavy isotope has at least one more constituent – the neutrino. The particle physicists have, indeed, found what they were looking for – the irreducible building blocks of matter – and whatever they have found beyond these is a bonus for natural science.

    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]. That same year the building block of dark matter, the superstring, was discovered as the crucial factor for 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 was also established that the superstring coverts to the basic elementary particles as agitated superstring [1,2,3]. In effect, this established the superstring as the basic constituent of matter, dark and visible [1,2,3,4,5 ].

    This happy turn of events came without fanfare and was not even noticed 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. Perhaps, a world congress of particle and theoretical physicists is appropriate on this occasion.

    References

    [1] Escultura, E. E., The mathematics of the grand unified theory, Nonlinear Analysis,
    A-Series: Theory: Method and Applications, 71 (2009) e420 – e431.
    [2] Escultura, E. E., The grand unified theory, Nonlinear Analysis, A-Series: Theory:
    Method and Applications, 69(3), 2008, 823 – 831.
    [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: Method 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.

    ReplyDelete
  54. 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 quark (down quark) and, earlier, the electron discovered by J. J. Thompson in 1897. 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.

    ReplyDelete
  55. 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]. 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 ].

    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.
    [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.

    ReplyDelete
  56. BOOK REVIEW AND PROPOSED REMEDY FOR THE PROBLEMS OF PHYSICS

    By E. E. Escultura

    The book: THE TROUBLE WITH PHYSICS, Penguin Books, 2008

    By Lee Smolin

    The book is the most objective and comprehensive assessment of contemporary physics I have read. It assesses where physics is in the search for the grand unification of the forces and interactions of nature and identifies five great problems that must be resolved towards unification. They are stated below with this Reviewer's comments.

    Problem 1. Combine general relativity and quantum theory into a single theory that can claim to be the complete theory of nature.

    Reviewer's comment. By building a theory based on general relativity and quantum theory, it brings in their weakness and inadequacy and other limitations in the search for unification. There is a lot of ambiguity in their basic physical concepts such as matter, energy, charge and gravity. To know matter, for instance, we must know what matter consists of and this requires the discovery of the basic constituent of matter which contemporary physics has not done.

    Problem 2. Resolve the problems in the foundations of quantum mechanics either by making sense of the theory as it stands or by inventing a new theory that does make sense.

    Reviewer's comment. I agree with this formulation. However, I would broaden it to the problems of the foundations of natural science and mathematics, the latter being the language and tool of science.

    Problem 3. Determine whether or not the various particles and forces can be unified in a theory that explains them all as manifestations of a single fundamental entity.

    Reviewer's comment. I fully agree with this formulation.

    Problem 4. Explain how the values of the free constants in the standard model of particle physics are chosen in nature.

    Reviewer's comment. I agree with this formulation with some modification as follows: Explain the constants of nature, e.g., the Planck's constant h and the Hubble's constant, by providing details on how they were determined.

    Problem 5. Explain dark matter and energy. Or, if they don't exist, determine how and why gravity is modified on large scales. More generally, explain why the constants of the standard model, including dark energy, have the values they do.

    ReplyDelete
  57. Book review by E. E. Escultura (continued)

    Reviewer's comment. I would retain only the first sentence of the formulation since the rest of it is already moot and academic [42].

    The author proceeds to lay out in great detail the theories and models proposed so far with a graphic account of the twists and turns in the search for a complete theory. The proposed theories and models range from general relativity and quantum theory through the standard model of particle physics, string theories, a number of dimension theories, quantum gravity, gauge theory and superstring theory. In the end, he concluded that physics today is in total confusion, has accomplished nothing towards unification in almost three decades and a complete theory is nowhere in sight. He bewailed the fact that string theorists have almost exclusive access to research grants and top academic positions in physics for which he should be commended for being able to rise above the confines of this group to which he belongs.

    Now, he welcomes the seers to tell us what is wrong with physics and point the way towards unification. This will require, in my view, a critique of its foundations and the foundations of mathematics including its present methodology of quantitative modelling that describes the appearances of nature without providing insights into how nature works. This allows, at best, reasoning by analogy which has an obvious flaw: a bird that walks like a duck and quacks like a duck is not necessarily a duck.

    Reviewer's comment on rectification.

    1) We need to clarify physical concepts and distinguish them from mathematical concepts. The former have physical referents – physical objects – that exist in nature and are subject to its laws. Mathematical concepts are man-made that comprise its vocabulary as the language of science; they have no physical referents and are, therefore, not subject to natural laws, e.g., time, distance, dimension, function and equation. Among the physical concepts are matter, energy and physical systems such as the electron, light, electromagnetic wave, atom and galaxy; their existence are verifiable. For example, to clarify what matter is we need to know what it consists of which requires the discovery of its basic constituent.

    2) We need to improve the present sense of unification from being descriptive of the appearances of nature in terms of common physical concepts to explanation of how nature works in terms of its laws. This should include explanation of what the forces and interactions of nature and their nature and what physical systems or natural phenomena are.

    3) In view of the inadequacy of the present methodology of quantitative modelling of physics that describes the appearances of nature, we need the new methodology of qualitative modelling that explains nature and natural phenomena in terms of natural laws. Its main tool is qualitative mathematics, the complement of computation and measurement. It includes axiomatic systems and abstract mathematics and the search for the laws of nature.

    4) We need to clarify that mathematical physics, a collection of mathematical descriptions of nature and natural phenomena, is not theoretical physics. A physical theory is an axiomatic system where the axioms or basic premises are laws of nature and scientific reasoning is based on its axioms – laws of nature – that define it and conclusions drawn from them. This way of reasoning belongs to rational thought. Definition of physical concepts is based solely on its axioms. When the laws of nature that define a physical theory apply to the various disciplines of natural science it is called grand unified or complete theory. For sources and elaboration of these ideas the viewer is referred to the following references.

    ReplyDelete
  58. Book Review by E. E. Escultura (continued)

    References


    [1] Escultura, E. E., Diophantus: Introduction to Mathematical Philosophy (With probabilistic solution of Fermat’s last theorem), Kalikasan Press: Manila, 1993.
    [2] Escultura, E. E. Probabilistic mathematics and applications to dynamic systems including Fermat's last theorem, Proc. 2nd International Conference on Dynamic Systems and Applications: Atlanta, May 27 – 31, 1999, pp. 147 – 152.
    [3] Escultura, E. E. Bhaskar, T. G.; Leela, S., Laksmikantham, V., Revisiting the hybrid real number system, J. Nonlinear Analysis, C-Series: Hybrid Systems, May 2009, 3, 2, pp. 101 – 107.
    [4] Escultura, E. E. Extending the reach of computation, J. Applied Mathematics Letters, 2008, 21, 10, pp. 1074 – 1081.
    [5] Escultura, E. E. Exact solutions of Fermat’s equation (A definitive resolution of Fermat’s last theorem, J. Nonlinear Studies, 1998, 5, 2, pp. 227 - 254.
    [6] Escultura, E. E. Recent verification and applications, Proc. 2rd International Conference on Tools for Mathematical Modeling, St. Petersburg, 1999, 4, pp. 116 - 29.
    [7] Escultura, E. E. The generalized integral as dual of Schwarz distribution, invited paper, J. Nonlinear Studies.
    [8] Escultura, E. E. Set-valued differential equations and applications to quantum gravity, J. Mathematical Research, 2000, 6, St. Petersburg, pp. 221 - 224.
    [9] Escultura, E. E. The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 84.
    [10] Escultura, E. E. Introduction to Qualitative Control Theory, Kalikasan Press: Manila, 1991.
    [11] Escultura, E. E. The new mathematics and physics, J. Applied Mathematics and Computation, 2003, 138, 1, 145 – 169.
    [12] Escultura, E. E. Chaos, turbulence and fractal, Indian J. Pure and Applied Mathematics, 2001, 32,10, pp. 1539 – 1551.
    [13] Escultura, E. E. The mathematics of the grand unified theory, Proc. 5th World Congress of Nonlinear Analysts, J. Nonlinear Analysis, A-Series: Theory: Method and Applications, 2009, 71, pp. e420 – e431.
    [14] Escultura, E. E. The mathematics of the new physics, J. Applied Mathematics and Computations, 2002, 130, 1, pp. 149 - 169.
    [15] Escultura, E. E. Dynamic Modeling and the new mathematics and physics, J. Neural, Parallel and Scientific Computations (NPSC), 2007, 15, 4, PP. 527 – 538.
    [16] Escultura, E. E. The theory of intelligence and evolution, Indian J. Pure and Applied Math., 2003, 33, 1, PP. 111 – 129.
    [17] Escultura, E. E. The physics of the mind, accepted, J. Science of Healing Outcomes.
    [18] Escultura, E. E. The origin and evolution of biological species, J. Science of Healing Outcomes, 2010, 6-7, pp. 17 - 27.

    ReplyDelete
  59. I HAVE FOUND A SIMPLE CRITERION TO GET COUNTER EXAMPLES TO FLT AND ANY ONE CAN VERIFY MY PROOF

    1. x = k3^n + nk1h1, y = k2^n + nk1h1, z = k1^n - nk1h1 , if none of x,y,z is a multiple of n.
    (This can also be written in the simple form
    x= (k1^n - k2^n + k3^n)/2, y== (k1^n +k2^n - k3^n)/2, z = (k1^n +k2^n + k3^n)/2 )

    2. x = k3^n + nk2h2, y = k2^n + nk2h2, z = ((k1^n)/n) – nk2h2, if one and only one of x,y,z is a muliple of n

    where k1,k2,k3 are integers such that k3<k2<k1

    ReplyDelete
  60. I HAVE DEVELOPED A CRITERION TO FIND THE COUNTER EXAMPLE TO FLT during 2000 for which I got many awards and honours.

    The criterion are :

    1. x = k3^n + nk1h1, y = k2^n + nk1h1, z = k1^n - nk1h1 , if none of x,y,z is a multiple of n.
    (This can also be written in the simple form
    x= (k1^n - k2^n + k3^n)/2, y== (k1^n +k2^n - k3^n)/2, z = (k1^n +k2^n + k3^n)/2 )

    2. x = k3^n + nk2h2, y = k2^n + nk2h2, z = ((k1^n)/n) – nk2h2, if one and only one of x,y,z is a muliple of n


    where k1, k2 and k3 are positive integers such that k3<k2<k1

    ReplyDelete
  61. I have developed a criteria to get counter example to FLT and received various awards and honours during 2000 -2005.

    The criterion are :

    1. x = k3^n + nk1h1, y = k2^n + nk1h1, z = k1^n - nk1h1 , if none of x,y,z is a multiple of n.
    (This can also be written in the simple form
    x= (k1^n - k2^n + k3^n)/2, y== (k1^n +k2^n - k3^n)/2, z = (k1^n +k2^n + k3^n)/2 )

    2. x = k3^n + nk2h2, y = k2^n + nk2h2, z = ((k1^n)/n) – nk2h2, if one and only one of x,y,z is a muliple of n

    where k3<k2<k1 are natural numbers

    ReplyDelete
  62. Unni is also nuts. FLT has been proven to be true. The proof has been checked nd verfied by hundreds if not thousands of mathematicians. So any criterion for finding a counterexample is pointless, one cannot exist. It's like finding a criterion to find an integer k s.t. 0< k < 1, is can never be done.

    ReplyDelete
  63. Then you prove such ki doesn't exist. Then the proof will be much smaller and simpler than wiles.

    ReplyDelete
  64. So proving such criterion doesn't exist will prove the theorem with much smaller and simpler efforts than Wiles as to prove up to this I need only two A4 sheets using high school algebra. Can u prove such ki doesn't exist like there exist no integers between 0 and 1?

    ReplyDelete
  65. UnnI:

    you miss the point. Proving that there is no integer k satisfying 0<k<1 in no way proves or disproves FLT.

    THe point I make is that since FLT has be conclusively proven to be true, then any attempt to define criterion for a counterexample is pointless. You could create all sorts of criterion. Even if your criterion were valid, they cannot be satisfied and be a counter examply to FLT since no counter example to FLT can exist. Quite simply you are defining a set of rules that can only be satisifed by the null set. Maybe I didn't explain myself well. If you don't get what I am saying I can explain it further.

    Proving the criterion don't exist isn't the point. You could find all sorts of interesting facts about x,y,z satisfying FLT if you were to assume that such a solution exists. BUt since such a solution is impossible, the very things you set out to study do not in fact exists, rendering their study pointless.

    Dear lord, can someone else chime in here and help me?

    ReplyDelete
  66. Unni:

    I also ask, what awards and honours? I have serious doubt that any University or non-fringe mathematical institute or association or whatever would grant any awards to anyone based on counterexamples to FLT, since none can exist. Perhaps if in the course of looking for a counterexample to FLT someone were to discover something new, that might merit some attention. By something new, I mean some 'accidental' discovery that has value in and of itself.

    I have to ask, what is your mathematics background? Do you know what it means to prove something is impossible, or that something cannot exist?

    To draw an analogy, in earlier times alchemists searched for the philosphers stone. This stone would somehow turn lead into gold. Now they figured out all sorts of nice properties that this stone would have, but at the end of the day no such stone could exist, so these 'properties' were useless.

    My advice to you if you want to become a mathematician or want to be taken seriously by them, is to learn as much math as possible. Go to university, or educate yourself, but if you choose to educate yourself do not waste time with the nonsense of people like E Escultura, try cover the standard material from a typical undergrad program. By assuming that every problem can be sovled with just highschool algebra you are making a huge mistake. When all you have is a hammer, everything looks like a nail. If simple algebra were enough to tackle every problem then calculus, differential equations, topology, group theory, etc., would never have been needed or created. By learning as much as possible you allow yourself to bring as many tools as possible to bear on a problem, but you will also understand why this or that tool is inapropriate or not powerful enough to attack certain problems. My point is that simple algebra is to the study of mathematics as spelling is to literature; it is necessary to master, but not enough to produce anything meaningful.

    ReplyDelete
  67. A simple proof of Fermat’s last theorem:


    1)Fermat theorem has this equivalent theorem:

    X^n+Y^n ?= Z^n (1)

    (X, Y, Z :fractional-rational numbers, n: natural number >2)

    2) Let’s divide (1) by (Z-X)^n, then we can also prove that there exists this equivalent theorem:

    X’^n+Y’^n ?= Z’^n and Z’ =X’ +1

    (X’, Y’, Z’ :fractional-rational numbers, n: natural number >2)

    3) Please note that a theorem is a mathematical structure, in which the symbols are not important, then we can have this theorem:

    X^n+Y^n ?= Z^n and Z =X +1 (*)

    (X, Y, Z :fractional-rational numbers, n: natural number >2)

    This is, exactly, the theorem 1) with an additional condition Z=X+1.

    Then:

    a)Please check theorem 3) with any X that is an integer and with any Y that is a real fractional–rational number (such as 5/3, 15/7…: they are rational-fractional numbers that cannot be reduced to integers).

    b)Please check theorem 3) with any X that is a real fractional–rational number and with any Y that is an integer.

    This simple proof of Fermat theorem is clear enough.

    ReplyDelete
  68. See : Finding Numbers satisfying the condition of fermat
    http://www.iosrjournals.org/iosr-jm/pages/v7i4.html

    ReplyDelete