As strongly evidenced by how I finished it in a few sessions within a single 24-hour period, Simon Singh’s Fermat’s Last Theorem is an exciting book. When you are kept up for a good part of the night, reading a book about mathematics, you can generally tell that some very good writing has taken place. Alongside quick biographies of some of history’s greatest mathematicians – very odd characters, almost to a one – it includes a great deal of the kind of interesting historical and mathematical information that one might relate to an interested friend during a long walk.
xn + yn = zn
The idea that the above equation has no whole number solutions (ie. 1, 2, 3, 4, …) for x, y, and z when n is greater than two is the conjecture that Fermat’s Last Theorem supposedly proved. Of course, since Fermat didn’t actually include his reasoning in the brief marginal comment that made the ‘theorem’ famous, it could only be considered a conjecture until it was proven across the span of 100 pages by American mathematician Andrew Wiles in 1995.
While the above conjecture may not seem incredibly interesting or important on its own, it ties into whole branches of mathematics in ways that Singh describes in terms that even those lacking mathematical experience can appreciate. Even the more technical appendices should be accessible to anyone who has completed high school mathematics, not including calculus or any advanced statistics. A crucial point quite unknown to me before is that a proof of Fermat’s Last Theorem is also automatically a proof of the Taniyama-Shimura conjecture (now called a theorem, also). Since mathematicians had been assuming the latter to be true for decades, Wiles’ proof of both was a really important contribution to the further development of number theory and mathematics in general.
Despite Singh’s ability to convey the importance of math, one overriding lesson of the book is not to become a mathematician: if you manage to live beyond the age of thirty, which seems to be surprisingly rare among the great ones, you will probably do no important work beyond that point. Mathematics, it seems, is a discipline where experience counts for less than the kind of energy and insight that are the territory of the young.
A better idea, for the mathematically interested, might be to read this book.
On the subject of mathematical tragedies, it’s notable that on the Wikipedia page for “Taniyama–Shimura theorem” it simply says that “Taniyama died in 1958.”
In fact, he killed himself shortly before his wedding. His fiancee then did likewise, several days later.
The other very notable mathematical suicide is Alan Turing: mathematician and codebreaker who ended his life with a cyanide coated apple. Murdered mathematicians include: Archimedes, Hypatia of Alexandria, and Pythagoras. French mathematician Évariste Galois was slain in a duel.
Those interested in mathematical proofs should read this excellent article from The New Yorker on the Poincaré Conjecture.
On the subject of mathematical tragedies, Srinivasa Ramanujan died at 32.
1782^12 + 1841^12 = 1922^12
That joke/trick was on the Simpsons episode where Homer ends up in three dimensional space.
While true to ten significant figures, it isn’t actually an equality. As such, it does not disprove Fermat’s Last Theorem.
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.
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:
Rejoinder on FLT
1. Since every mathematical system is well-defined only by its axioms, universal rules of inference, e.g., formal logic, are irrelevant since they have nothing to do with the axioms.
2) The choice of axioms is arbitrary and depends on what one wants his mathematical system to do provided they are CONSISTENT since inconsistency collapses a mathematical system to nonsense. However, once the axioms are chosen the mathematical space becomes a deductive system where the truth or validity of the theorems rests solely on them.
3) The trichotomy axiom which is false in the real number system is true in the new real number system, a consequence of its lexicographic ordering.
4) To avoid ambiguity or error every concept must be well-defined, i.e., its existence, behavior or properties and relationship with other concepts MUST BE SPECIFIED BY THE AXIOMS. Thus, undefined concepts are allowed only INITIALLY but the choice of the axioms and the construction of the mathematical system are incomplete until every concept is WELL-DEFINED. Existence is important because vacuous concept often yields contradiction. We give another example of a vacuous concept: the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N.
5) There are other sources of ambiguity, e.g., large and small numbers due to limitation of computation and infinite set. The latter is ambiguous because we can neither identify most of its elements nor verify the properties attributed to them.
6) Another source of ambiguity is self-referent statement such as the barber paradox: the barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? A statement is self referent when the referent refers to the antecedent or the conclusion to the hypothesis. Unfortunately, the indirect proof is self-referent.
7) Of course, the real number system and, hence, FLT are ambiguous in view of the counterexamples to the trichotomy axiom by Felix Brouwer and this blogger and to the completeness axiom (a variant of the axiom of choice) by Banach-Tarski.
8) What do all these mean? FLT is nonsense being formulated in the inconsistent real number system. To resolve FLT the real number system must be freed from ambiguity and contradiction by constructing it on CONSISTENT axioms. Then FLT can be formulated in it and resolved.
8) To this end, I constructed the new real number system on the symbols 0, 1 and chose three consistent simple axioms that well-define them; then the integers and the terminating decimals are defined and using the latter the nonterminating decimals are well-defined for the first time.
9) To summarize: (a) the present formulation of FLT is nonsense; (b) to make sense of it the decimals are constructed into the contradiction-free new real number system; (c) then FLT is reformulated in it and (d) shown to be false by counterexamples. The counterexamples are given in a number of papers, especially, The real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 2009, pp. 59 – 84..
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 – 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Escultura, E. E. (1997) Exact solutions of Fermat’s equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[6] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[9] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[10]] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier
Science, Ltd.), 2009, Paris.
[11] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[12] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
The Status of the P Versus NP Problem
It’s one of the fundamental mathematical problems of our time, and its importance grows with the rise of powerful computers.
Lance Fortnow
When editor-in-chief Moshe Vardi asked me to write this piece for Communications, my first reaction was the article could be written in two words:
Still open.
When I started graduate school in the mid-1980s, many believed that the quickly developing area of circuit complexity would soon settle the P versus NP problem, whether every algorithmic problem with efficiently verifiable solutions have efficiently computable solutions. But circuit complexity and other approaches to the problem have stalled and we have little reason to believe we will see a proof separating P from NP in the near future.
Nevertheless, the computer science landscape has dramatically changed in the nearly four decades since Steve Cook presented his seminal NP-completeness paper “The Complexity of Theorem-Proving Procedures” in Shaker Heights, OH in early May, 1971. Computational power has dramatically increased, the cost of computing has dramatically decreased, not to mention the power of the Internet. Computation has become a standard tool in just about every academic field. Whole subfields of biology, chemistry, physics, economics and others are devoted to large-scale computational modeling, simulations, and problem solving.
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.
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 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.
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.
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.
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.
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/
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* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].
3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)
for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.
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/
This is beginning to look like weird spam…
“The Clay Mathematics Institute has announced its acceptance of Dr. Grigori Perelman’s proof of the Poincaré conjecture and awarded the first Millennium Prize. Poincaré questioned whether there exists a method for determining whether a three-dimensional manifold is a spherical: is there a 3-manifold not homologous to the 3-sphere in which any loop can be gradually shrunk to a single point? The Poincaré conjecture is that there is no such 3-manifold, i.e. any boundless 3-manifold in which the condition holds is homeomorphic to the 3-sphere. A sketch of the proof using language intended for the lay reader is available at Wikipedia.”