In 2005, there was a surprising article
in the Manilla Times. Apparently, a local mathematician had found a flaw in Wile's proof of Fermat's Last Theorem. Not only had he found a flaw in the proof, he had found a flaw in the real number system and Andrew Wile's himself had written an e-mail congratulating the professor, E. E. Escultura, on his discoveries.
Unfortunately, it turns out the Escultura article was based on a hoax
. The apparent e-mail from Wiles was a fake.
This past week, Escultura has been kind enough to post an explanation of his ideas
on my blog. In fairness to him, he was responding to an e-mail where I said that Escultura was "discredited." From my view, his response proves my point. In my first blog on false proofs, I will go through argument by argument and explain the flaws in Escultura's reasoning. I also provide a list of links for people who wish to see Escultura's postings on other blogs.
1) There are sources of contradiction in mathematics including ambiguous and vacuous concepts, large and small numbers (depending on context), unbounded or infinite set and self-reference. Here is an example of vacuous concept: A triquadrilateral is a plane figure with three vertices and four edges. The Richard paradox is an example of self-reference: The barber of Seville shaves those and only those who do not shave themselves; who shaves the barber? Incidentally, the indirect proof is flawed, being self-referent.:
(1) Paradoxes such as the Barber of Seville exist.
(2) It is also possible to define terms in a way that is contradictory or ambiguous such as triquadrilateral as a plane figure with three vertices and four edges which is impossible in Euclidean Geometry.Comments:
2) Among the requirements for a contradiction-free mathematical space are the following:
a) It must be well-defined by consistent axioms and every concept must be well-defined by them. A concept is well-defined if its existence, properties and relationship with other concepts are specified by the axioms. A false proposition cannot be an axiom as it introduces inconsistency. For example, this proposition cannot be used as an axiom of any mathematical space: There exists a triangle with four edges.
b) The rules of inference (mathematical reasoning) must be specific to and well-defined by its axioms.
c) Any proposition involving the universal or existential quantifiers on infinite set is not verifiable and, therefore, cannot be used as an axiom for it would not endow certainty to the conclusion of a theorem.
(1) To avoid contradictions, ambiguities, and paradoxes, we need to make sure:
(a) Each axiom is well-defined (consistent and composed of well-defined concepts)
(b) All conclusions are derived from the axioms.
(c) Any proposition involving universal (for all) or existential quantifiers (there exists) on an infinite set is not verifiable and therefore cannot be used as an axiom.Comments:
(a),(b) are pretty standard in mathematics so the only unusual point is (c).
From my view, (c) is a self-contradictory postulate because if we accept it, then we must reject it because it is making a statement about universals on an infinite set.
Further, (c) is not intuitive. people have no problem accepting statements such as:
All positive numbers are greater than 0.
All negative numbers are less than 0.
For any set of positive integers, there exists one integer which is the smallest.
Infinite parallel lines never intersect.
3) The real number system does not satisfy the requirements for a contradiction-free mathematical space. In particular, the trichotomy axiom is false since it is equivalent to natural ordering which the real number system has none because most of its concepts are ill-defined. Therefore, the real number system is ill-defined or nonsense and FLT being fomulated in it is also nonsense. Consequently, to resolve FLT the real number system must be fixed first and FLT must be reformulated in it. Andrew Wiles failed to do this and his work collapses altogether.
(1) Escultura claims the real number system is inconsistent because they depend on the axiom of trichotomy which is equivalent to "natural ordering" but the real numbers have no such ordering so Escultura claims that this is a contradiction.
(2) Because Fermat's Last Theorem depends on the real number system, this means that Fermat's Last Theorem is contradictory and does not make sense.
(3) To address Fermat's Last Theorem, it is first necessary to remove the contradiction in the real number system and then reformulate Fermat's Last Theorem.
(4) Wiles assumes that Fermat's Last Theorem is a coherent problem, therefore Wiles's proof is false.Comments:
Ouch. The reasoning here is pretty bad. Let's me start with the axiom of trichotomy.Real numbers and the Axiom of Trichotomy
The Axiom of Trichotomy (see here
) states that: for any two numbers x,y, there are three possible states:
(1) x is less than y
(2) x is equal to y
(3) x is greater than y
Escultura claims that this amounts to a "natural ordering." I assume that he means "well-ordering" (see here
) which states that for any nonempty subset, there exists a least element.
If he means "well ordering", then he is right that real numbers are not well-ordered. Consider the full set of real numbers. For any real number x, there is always a smaller number that also exists say x/2. To search for a smaller number leads to an infinite regression. In other words, the real numbers are not well-ordered.
Unfortunately, it is not clear how the Axiom of Trichotomy translates into well-ordering. Well-ordering is a statement about the existence of a number in a set. The axiom of trichotomy is a statement about a relationship between any two numbers that exist.
No surprise. Escultura fails to prove that the real numbers are inconsistent.Real Numbers and Fermat's Last Theorem
Fermat's Last Theorem is a statement about integers. It says nothing about real numbers so there is no reason that Fermat's Last Theorem needs to be reformulated.
Even so, it doesn't matter since he fails to show that the real numbers are inconsistent.
4) It is alright to introduce ambiguity provided it is 'approximable" by certainty. For example, a nonterminating decimal is ambiguous since not all its digits are known but it can be approximated by a segment at the nth decimal digit at margin of error 10^-n.
Escultura is now attempting to save the real number system. He says that while nonterminating decimals are ambiguous (for example, 1/3=0.33333....), we can approximate them at any point n that we would like.Comments:
This is completely false. Nonterminating decimals are not ambiguous. 1/3, 2/3 are clear and precise even if they lead to nonterminating decimals.
His claim of "ambiguity" refers to the fact that 0.9999... = 1 as shown by the following subtraction:
10 * 0.999999... = 9.99999....
1 * 0.999999... = 0.999999...
9*0.9999999... = 9
0.999999... = 9/9 = 1
But this is not ambiguous. This is a proof. Once again, asserting that something is ambigous (by his own method) is not sufficient. He must establish the inconsistency by showing a contradiction between axioms. It turns out that 2.999... = 3 etc. Just because a result is surprising does not mean that it is ambiguous or inconsistent.
5) The rectification is to build a new real number system R* with three simple axioms and two operations + and x: 1) R* contains the basic integers 0, 1, ..., 9, and the operations + and x are well-defined by 2) the addition and 3) multiplication tables of arithmetic that we learned in primary school. The rest of the elements of R* are the terminating decimals first which are later extended to the nonterminating decimals.
A new real number is well-defined if every digit is known or computable, i.e., there is some rule or algorithm for determining it uniquely. Note that the periodic decimals including the terminating decimals are well-defined new real numbers and the real numbers, the terminating decimals, are embedded in the new real number system. The integers are embedded isomorphically into the integral parts of the decimals and are, therefore, well-defined by the axioms of R*. This remedy’s the major flaw of number theory, namely, the fact that the integers have no adequate axiomatization.
The real number system can be rebuilt using the following axioms:
(1) a real number is composed of 0,1,2,3,4,5,6,7,8,9 plus a terminating decimal.
(2) a real number follows addition and subtraction as traditionally understood.Comments:
Ouch. This system is also characterized by the axiom of trichotomy so if we accept his claim about real numbers (which he does not prove), it is unclear how his new formulation avoids the claim.
He misses the whole point about axioms. The trick is coming up with axioms that lead to the properties of addition or multiplication. Just claiming that they are as we expect is failing to use the axioms as he claims are necessary. To see Tarksi's axiomitization of the real numbers, see here
But perhaps, the strongest argument against his approach is that it cannot handle irrational numbers. His formulation amounts the claim that all numbers are rational since any terminating decimal is statable as a fraction. This means that if we accept his assumptions, the square root of 2, pi, e, and any other irrational number does not exist because they are "ambiguous."
6) The new elements of the new real number system are the dark number d* = 1 – 0.99… - N – (N–1), N = 0, 1, … (the ordinary integers), and u* the equivalence class of divergent sequences. The mapping 0 – > d*, N – > (N–1).99…, where N = 1, 2, …, maps the integers isomorphically into the new integers which means that they have almost identical behavior, the only difference being that d* > 0.
Escultura believes that the real numbers can be saved by the following definitions:
(a) d* = 1 - 0.9999....
(b) N = (N-1)+0.99999....
(c) d* greater than 0Comments:
Unfortunately, Escultura's system is contradictory so must be rejected.
(1) Let d* = 1 - 0.9999....
(2) 10*d = 10 - 9.9999...
(3) 10d* - d* = (10-1) - (9.9999... - 0.9999...) = 9 - 9 = 0
(4) So, 9d* = 0
(5) But then d* = 0 which contradicts axiom (c).
Of course, his axioms are saved if we reject (c) and concede that d*=0 which, by the way, is consistent with standard mathematics.
Then the counterexamples to FLT are as follows: Let x = (0.99...)10^T, y = d*, z = 10^T, where T is an ordinary integer, T = 1, 2, ... Then x, y, z satisfy Fermat's equation, for n > 2,
x^n + y^n = z^n.
Moreover, if k = 1, 2, ..., is ordinary integer, kx, ky, kz also satisfy Fermat's equation. They are the counterexamples to FLT. They prove that FLT is false and Wiles is wrong.
Escultura has found what he believes is a counter example to Fermat's Last Theorem:
[(0.999...)10]^n + (d*)^n = 10^nComments:
Escultura misunderstand's the original problem of Fermat's Last Theorem (see here
). FLT holds that there are no solutions where xyz ≠ 0. Since I have shown that d* = 0, his counter example comes down to:
10^n + 0^n = 10^n.References to Escultura on the Web