SUMMATION ON 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.

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. 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 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.
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 to nonsense. 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 are inconsistent. Felix Brouwer and I constructed counterexamples to the trichotomy axiom which means that it is false. Banach-Tarski constructed a contradiction to the axiom of choice of the field axioms, a variant of the completeness axiom. One version says: 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, a topological contradiction.
4) Vacuous concept generally yields a contradiction. For example, consider this vacuous concept: the root of the equation x^2 + 1 = 0. That root does not exist and yet it is 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 my paper, The new mathematics and physics, Applied Mathematics and Computation, 138(1), 127 – 149 [4], says that the rationals and irrationals are separated, i.e., the union of disjoint open sets (see also [6].
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. To hide this contradiction some mathematicians invented two supposed roots, the other i = -sqrt(-1) neither of which exists but choose the latter which does not yield a contradiction. There is no logical or mathematical reason for this choice other than to hide the contradiction. Both of them are nonsense anyway since they are both ill-defined.

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 [5].

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 no nondenumerable set exists; 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.

Finally, we note that all the issues about the new real number system, my critique of Wiles’ proof of FLT and my counterexamples to FLT to prove it false have been debated thoroughly in cyberspace during the last 12 years and ALL resolved COMPLETELY in my favor. Not a single hole has been punched on my entire work.

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