E. E. Escultura
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) 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:
OK.
(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.
Summary:
(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.
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
So, that:
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.
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."
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 0
Comments:
Unfortunately, Escultura's system is contradictory so must be rejected.
Proof:
(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).
QED
Of course, his axioms are saved if we reject (c) and concede that d*=0 which, by the way, is consistent with standard mathematics.
Escultura has found what he believes is a counter example to Fermat's Last Theorem:
[(0.999...)10]^n + (d*)^n = 10^n
Comments:
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
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.
[Escultura]Summary
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:
OK.
[Escultura]Summary:
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.
[Escultura]
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.
Summary:
(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.
[Escultura]Summary:
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
So, that:
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.
[Escultura]Summary:
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."
[Escultura]Summary:
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 0
Comments:
Unfortunately, Escultura's system is contradictory so must be rejected.
Proof:
(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).
QED
Of course, his axioms are saved if we reject (c) and concede that d*=0 which, by the way, is consistent with standard mathematics.
[Escultura]Summary:
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^n
Comments:
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
- MathForge Discussion, June 13, 2006
- Anatomy of a Hoax, May 20, 2005
- Calmighty, Escultura Fermat Wiles Debacle Update
- Escultura's FLT Claim Clarification, June 2, 2005
posted by Larry Freeman at 11:35 AM
101 Comments:
Since a number is determined by its digits I require that to be well-defined every digit must be known or computable, i.e., it can be determined uniquely by some rule or algorithm. Thus, periodic decimals are well-defined; so are pi and the logarithmic base e since every digit of either can be computed from its series expansion.
To avoid ambiguity I specify that the real numbers are the decimals and developed them as the new real number system well-defined by the three axioms that I posted.
Note that nonterminating decimals cannot be added or multiplied because these operations need the last digits of the numbers concerned. We cannot even enter a nonterminating decimal into our calculator or pc because it can only register finite digits. We can only approximate their sum or product. That is the ambiguity or uncertainty I'm referring to. Try adding, for instance, sqrt2 and sqrt3 and write the sum. See if you get the sum which should be an irrational number.
There is no adequate axiomatization of the integers at this time. That is why even Andrew Wiles used real numbers tyring to prove FLT. So it's not true that Wiles' proof is confineed to the integers.
My remedy is to embed them isomorphically as the integral parts of the decimals so that they become well-defined by and subject to the three axioms of the latter.
E. E. Escultura
(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:
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.
Response:
Such prooposition is inadmissible as an axiom because it is not verifiable on infinite set and does not endow certainty to the the conclusion of a theroem. For exampe, if I want to verify that every element of an infite set has propert A, I start at some point and verify that it has that property; then I have to move to verify another one, etc. There is no way I can verify that EVERY element has that property because of the inexhaustibility of infinite set. Therefore, if this were an axiom, I cannot use it tp prove a theorm.
---------
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.
Response:
The statements are verifiable based on their definitions.
---------
For any set of positive numbers, there exists one number which is the smallest.
Response:
How about the open set (0,1)? Or, does this not follow from the definition positive numbers or numbers greater than O?
----
Infinite parallel lines never intersect.
Response:
Again, this follows from the definition of parallel lines.
E. E. Escultura
Nonterminating numbers are well defined.
Consider the common fraction (1/3), the constant pi, the constant e, or even the square root of 2. Each of these is a nonterminating decimal. Each of these is handled precisely by mathematics (though not handled precisely by most calculators).
Nonterminatings decimals do not cause problems just because they are nonterminating. The decimal system is just one of many ways to denote fractions and irrational numbers.
The integers are well defined by axioms. This is covered by any book on set theory, for example, Patrick Suppes' book (one of my favorites), Axiomatic Set Theory.
The real number system is well defined. It is silly to use a calculator as an example to argue that we need to throw out calculus, modern physics, and everything else which is based on the real number system.
Personally, I think that the early ambiguities in the foundations of calculus were well addressed by Cauchy (1789 - 1857). There's an excellent book on this (which you might be interested in) by Judith V. Grabiner called The Origins of Cauchy's Rigorous Calculus (Dover Edition).
-Larry
This is completely false. Nonterminating decimals are not ambiguous. 1/3, 2/3 are clear and precise even if they lead to nonterminating decimals.
Response:
Nonterminating decimals whose digits are known or computable are well-defined. They include periodid decimals, pi and the logarithmic base e.
---------
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
So, that:
0.999999... = 9/9 = 1
These are incorrect equations because one cannot add or multiply nonterminating decimals since these operations require the last digit on the right to carry out. Nonterminating decimals cannot even be entered into the calcular or pc, nor can they be added or multiplied by hand. They can only be approximated.
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.
Response:
The claim is not just surprising but wrong. The number 3 is only an approximation of 2.99... because one cannot actually add 2 and 0.99... since 0.99... is nonterminating.
E. E. Escultura
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.
Response:
As pointed out in previous post, d* > 0; xyz > 0 and the above claim is false.
------
References to Escultura on the Web
* MathForge Discussion, June 13, 2006
* Anatomy of a Hoax, May 20, 2005
* Calmighty, Escultura Fermat Wiles Debacle Update
* Escultura's FLT Claim Clarification, June 2, 2005
I have responded to the above blogs except the second one by Alex Pabico of PCIJ which contains nothing but nonsense because Pabico is is totally ignorant of the issue.
E. E. Escultura
Rebuttal of Opposition to my Resolution of FLT
Although I have responded to most of the mathematical points raised by Larry Freeman there are some areas that need clarification and stress.
1) One of the issues that have been discussed extensively across the internet since I raised it in 1997 is the question of whether 1 = 0.99… This was resolved by David Hilbert a century ago when he recognized that the concepts of individual thought cannot be the subject matter of mathematics since they are inaccessible to others and can neither be studied collectively nor axiomatized. It follows that intuition is not valid mathematical reasoning. Therefore, the objects of a mathematical space must be symbols in the real world that everyone can look at (we call them concepts also) well-defined by consistent axioms. A concept is well-defined if its existence, properties and relationships among themselves are specified by the axioms. Existence is stressed here because vacuous concepts (the defining expressions are vacuous) are ambiguous and a source of contradiction. An example of vacuous concept is the “triquadrilateral” that I posted earlier. An example of a vacuous proposition is this: “the greatest integer satisfies the trichotomy axiom”. It is easy to derive an absurdity from it such as: “the largest integer is 1”. BTW, the idea that an axiom is neither true nor false or that it is true by virtue of its being an axiom is nonsense. There are strict requirements for an axiom to avoid contradiction that reduces a mathematical space to nonsense.
2) Since 1 and 0.99… are distinct symbols they cannot be equal; therefore, d* = 1 – 0.99… cannot be zero contrary to Larry’s claim that d* = 0. Thus, the usual proof that 1 = 0.99… is flawed.
3) Since every mathematical space is well-defined only by its own axioms, distinct mathematical spaces are independent. Therefore, a concept in one space is ill-defined in another. It follows that any proposition or argument involving concepts from two distinct mathematical spaces is ill-defined. It follows further that Goedel’s incompleteness theorems are flawed because they involved concepts from two distinct mathematical spaces, the integers and the propositional calculus.
4) It follows from 3) that universal rules of inference including those provided by formal logic are not valid mathematical reasoning (since they apply to distinct mathematical spaces). Therefore, the rules of inference must be specific to and well-defined by the axioms of the given mathematical space. Moreover, every term or symbols must be well-defined because the introduction of undefined terms introduces ambiguity.
5) Infinite set has inherent ambiguity since not all its terms can be identified or enumerated or checked for its properties. Therefore, any proposition about the properties of EVERY element of an infinite set is ambiguous or unverifiable. For example, if I want to check that EVERY element of an infinite set has property A I must start with some element x and check that this is so. Then I check another element y, etc. It is clear that this is never completed. The same is true with the use of the existential quantifier, “there exists”. Therefore, the use of an axiom involving the universal or existential quantifier in an infinite mathematical space is inadmissible for it does not endow certainty to a theorem. The remedy: build a mathematical space on finite set; then well-define infinite set when needed.
6) In view of 5) the completeness axiom of the real number system is inadmissible for it introduces uncertainty on the infinite real numbers; this is one of its defects. Another ambiguity of the real number system is that its axioms do not specify the system of symbols they well-define: are they the triadic numbers, decimals, fractions, etc.? They are different systems of symbols having different properties. Therefore, they cannot be well-defined by the same set of axioms.
7) The most serious problem with the real number system is that the trichotomy axiom is false, a counterexample to it having been constructed by Felix Brouwer (Benacerraf and Putnam, Philosophy of Mathematics, Cambridge U Press, 1985). This makes it inconsistent and ill-defined.
8) Even ignoring 7) for the moment, addition and multiplication are well-defined only on terminating decimals because these operations require the last terms to carry out and nonterminating decimals do not have them. We can only approximate the sum and product of nonterminating decimals. Try adding sqrt2 and sqrt3 and you’ll run smack into this problem.
9) The integers do not form a well-defined mathematical space. The Peano postulates are essentially a definition and do not well-define a mathematical space. Other developments of the integers from set theory involve the axiom of choice which, in turn, involves the universal and existential quantifiers on infinite set. A remedy would be to embed them isomorphically into the decimals as their integral parts so they become well-defined as a subspace but this would require fixing the decimals first.
19) It follows from all of the above items that whether one considers FLT as a problem on the integers or the real numbers, it is ill-defined and does not make sense. The first crucial step in resolving FLT is to specify the real numbers as decimals and construct them as well-defined mathematical space on consistent axioms which I did here previously using three simple axioms. Then I well-defined the nonterminating decimals as standard Cauchy sequences. The resulting mathematical space is the new real number system (consisting of decimals) that contain the new integers d* and nonterminating decimals of the form N.99…, where N = 0, 1, …, are integers. Then I reformulate FLT as a problem on the real number system. Then the countable counterexamples to it exist and are already posted here proving that both FLT and Andrew Wiles” “proof” wrong;
The results here and their applications are published in over two-dozen refereed international journals; leads to references are in my website:
http://home.iprimus.com.au/pidro/
Finally, I commend Larry Freeman for keeping the discussion on a high plane.
Cheers.
E. E. Escultura.
I will respond point by point.
.999..., 1, and Axioms
Mr Escultura writes:
"Intuition is not valid reasoning." I think by this he means that mathematical arguments must be based on clear, precise axioms.
Mr Escultura, where are your axioms for integers? Where are your axioms for real numbers? Your whole argument seems to based on intuition.
I would propose that if your logic is sound that you should be able to find a contradiction in the standard axioms for integers (see here) and for real numbers (see here).
.999... and 1 are distinct symbols
This is not a very strong argument. 1/2 and 0.5 are distinct symbols but they are equal.
Likewise, 1/3 = 0.3333....
distinct mathematical spaces are independent
I am not sure where this conclusion comes from. I assume by "mathematical spaces", you mean a set conclusions based upon a set of axioms.
I believe that you are strongly mistaken in the assumption that everything is logically independent.
The most important breakthroughs in mathematics have come from finding relationships in what appeared to be logically distinct domains.
The real numbers for example provided a very rich reinterpretation of the proofs of the ancient Greeks. Descartes' clasic book The Geometry applies algebra back to geometry. This is in contrasts to Diophantus who used the ideas of geometry to examine problems in algebra. Likewise, calculus (see my analysis here) enabled us to find a definition for transcendental functions that made them independent of Euclid.
Problems with Peano's Postulates
The best way to show a problem with Peano's postulates is to provide a proof. Making a claim that does not rest on axioms violates your own methodology.
All axioms are good unless they involve a contradiction. This is the lesson from non-Euclidean geometry. You can talk about the axiom of choice, law of trichotomy, and other areas but unless you provide a rigorous mathematical proof of contradiction, there is no reason to agree with your conclusions.
Redefining numbers based on finite decimals
This seems to be a point that you keep returning too. You seem to take it on faith that what is clear to you is true and what is unclear to you must be wrong.
Conclusion
It seems to me Mr. Escultura, that you borrow what you like from the integers and real numbers and then discard what you don't like.
You talk about the need for axioms but you yourself don't provide a set of rigorous axioms and you don't provide a rigorous proof to attack the standard axioms.
I appreciate your energy and your desire to convince the world about your ideas but unless you are able to follow your own methodology of presenting a set of precise mathematical axioms and solid, mathematical proofs, no mathematician is going to take you seriously.
-Larry
Hi Jose,
You are right. I meant any "for any set of positive integers."
Thanks very much for pointing this out.
I have updated the page.
-Larry
Response to Larry Freeman.
Thank you for comments. Instead of responding to them directly, I have excerpted the relevant portions of a recent paper of mine that, I think, provide adequate response.
The concepts of individual thought cannot be the subject matter of mathematics since they are inaccessible to others and can neither be studied collectively nor axiomatized. Therefore, the subject matter of mathematics can only be symbols that can be studied collectively and well-defined by consistent axioms. This recognition of the subjectivity of thought and the remedy by way of axiomatization of the representation of thought by symbols are Hilbert’s main contributions to mathematics and, ultimately, science. Therefore, the decimals, triadics and the binary systems, being different symbols having different behaviour and cannot be well-defined by the same set of axioms. I built the decimals as a mathematical space on three axioms and call it the new real number system and have since them extended it to the new nonstandard analysis (the qualification “new” distinguishes it from Robinson’s Nonstandard analysis).
Title: Thought and Science: Introduction to Discrete Calculus
Abstract. The paper is a sequel to the article, The physics of the mind (Nonlinear Analysis and Phenomena, Vol. IV, No. 1). It summarizes the nature of our universe and devises appropriate mathematics for computation, measurement and development of physical theory. Appropriate mathematics includes the new real number system based on the rectification of foundations and its extension to what is called the new nonstandard or discrete calculus. The final section summarizes the novel features of the new mathematics called new nonstandard analysis and resolves and explains some of the paradoxes of mathematics.
1. Introduction
Thought resides in the mind and its representation as mathematical space is in the real world; it is the language of natural science. Each field of science is expressed as physical theory built on the laws of nature. Scientific knowledge as representation of thought is built and expressed as physical theory anchored on the laws of nature. The task of the theoretical physicist is to discover and that of the experimental physicist to verify them.
We stress that what happens in one’s mind and whether his representation of thought is accurate are solely his concern. The collective concern is that, for the purposes of science, the representation is consistent and ambiguity is approximable by certainty, both requirements being necessary to keep the mathematical integrity of the language of science which we take to be some mathematical spaces. It is necessary for the purposes of science that such mathematical space be deductive, i.e., while the choice of the axioms and symbols or concepts are inductive everything else is deductive, the rules of inference are specified by the axioms and the theorems and their conclusions are necessary consequences of the axioms. A single contradiction invalidates a mathematical space and ambiguity erodes the necessity of conclusions.
How does the mind comprehend nature and build physical theories that explain its behavior in the various disciplines of natural science? We use the wealth of information drawn from previous and contemporary study and collective experience, experimental results and empirical data. Sorting them out and taking a critical approach while at the same time observing regularity or patterns of behavior in physical processes, the mind learns or creates new concepts and principles (laws of nature) upon which it builds a physical theory to explain them. This is how a discipline of science is developed. The appropriate language for this purpose is mathematics, specifically, some mathematical space. Our focus here is the development of such mathematical space appropriate for physics and computing.
To determine the appropriate mathematics for physical theory we need some facts about nature, particularly, our universe. We need not start from ground level because we already have scientific knowledge to stand on, the flux theory of gravitation (FTG) developed in the Math-Physics series. A theory is necessary since experimental results and empirical data and their interpretation do not exist in a vacuum: the instruments and other scientific tools required to collect them are subject to and based on the laws of nature. FTG is at our disposal; at the same time the experimental results and empirical data are part of its verification.
The terminology is well thought out. For example, the term new nonstandard analysis stresses the fact that this is a critique of Robinson’s nonstandard analysis which is built on the real number system using formal logic. The new nonstandard analysis is built solely on the three axioms of the new real number system subject to the requirements of rectified foundations to insure consistency and minimize ambiguity in the sense that only ambiguity approximable by certainty is admissible.
Some major problems of mathematics are resolved. For instance, the present notion of limit in the sense of Bolzano-Weierstrass is quite ambiguous in view of the involvement of the universal and existential quantifiers. Any proposition about ambiguous set, e.g., infinite set, involving either quantifier is unverifiable and, hence, ambiguous. The notion of Cauchy limit which is computable is presented as the appropriate alternative and appropriate for the new nonstandard calculus. This is distinguished from the topological limit which remains valid for purposes of analysis.
2. The Universe and our universe
We summarize what we know about nature.
(1) There exist two fundamental states of matter – visible or ordinary and dark matter; visible matter is directly observable but dark matter is not, its existence and behavior indirectly verified by its impact on visible matter subject to the laws of nature. Conversion from one fundamental state to the other is also subject to the laws of nature.
(2) The basic constituent of dark matter is the superstring and the units of visible matter are the prima. Among the known simple prima are the electron, positron and positive and negative quarks; the proton, neutron and neutrino are coupled prima. Clearly, matter is discrete at both fundamental states of matter. Dark and visible matter forms the Universe. The so-called elementary particles of quantum physics are prima and there are over 170 of them, detected or postulated.
(3) Is motion discrete? Yes, in view of the quantization principle. Moreover, energy is motion of matter; therefore, matter and energy are inseparable and there is no such thing as pure matter or pure energy. Clearly, both the quantization law and Heisenberg’s uncertainty principle are consistent with the discreteness of matter.
(4) Energy is motion of matter; kinetic (visible) energy is motion of visible matter and latent (dark) energy that of dark and matter, respectively. Visible matter has dark component, a segment of superstring. This is the general form the first law of thermodynamics takes in FTG or the new physics that is incorporated in the energy conservation law.
(5) By flux-low-pressure complementarity dark matter is unbounded but Olber’s paradox rules out unbounded visible matter. Therefore, the Universe of dark matter is unbounded but visible matter in it is bounded.
(6) Our universe is a super…super galaxy 10^10 light years across [41]; its formation started with the explosion of a black hole 12 billion years ago called the big bang. That black hole was the destiny of the core of a previous universe. This explanation by FTG makes it the only physical theory that applies energy conservation to the big bang.
(7) Our universe is finite and expands at 10^20 km/sec which is accelerated at the rate of 10–10 km/sec [39]; its ultimate destiny is dark matter.
(8) There is evidence of the existence of universes other than ours (e.g., migrating galaxy clusters and collision of two galaxies one coming from a direction oblique to that of the other but our universe is quite special having been formed by a big bang.
(9) Most universes form as the result of the steady shrinking of the superstrings (by energy conservation), flux-low-pressure complementarity and the law of uneven development, among others, leading to the formation of fractal sets of low pressure regions that evolve into fractal sets of cosmological vortices of superstrings like galaxy, star, planet, moon and cosmic dust.
(10) Every cosmological vortex collects mass around the eye due to its suction. This is how the Sun and its planets formed as well as the metropolis of every galaxy. This process continues to this day. Cosmic dust forms due to generation and propagation of seismic waves at the core of cosmological vortices, by the micro component of turbulence that convert superstrings to prima, atoms and cosmic dust around it .
(11) Cosmic dust gets caught by cosmological vortices that collect it around their eyes and form stars at the rate of one star per minute; in 2005 scientists discovered a nascent galaxy in its initial phase of formation.
(12) The eye of every cosmological vortex, e.g., galaxy, is a region of calm and low pressure that de-agitates and shrinks the superstrings and prima at its boundary with the hot spinning collected mass around it. They form massive cluster of non-agitated superstrings called black hole. The existence of black holes at the eye of galaxies, including the Milky Way has been confirmed.
(13) The laws of nature are transitory, e.g., there were no biological laws immediately after the big bang, and they all vanish as our universe reaches its destiny – dark matter as black holes. (Natural laws are revealed and verified by motion of visible matter; when the latter vanishes so do the former).
3. Mathematical requirements of natural science
The above description of our universe sets the parameters for devising the appropriate mathematics for computation and measurement and development of scientific knowledge in the form of physical theory.
We build two mathematical spaces. The first is quantitative mathematical space for purposes of computation and measurement. When its concepts are physical concepts it is the mathematics of physics, its present language, especially, mathematical physics. Naturally, its principal component is computation. The other mathematical space is qualitative; the complement of computation. It includes abstract mathematical spaces such as axiomatic systems (e.g., topology and its three axioms and abstract algebra and its axioms). When its concepts are physical concepts and its axioms are the laws of nature it is called physical theory. This is a new field called theoretical physics. What used to be called theoretical physics is really mathematical physics. The principal task of the theoretical physicist is to find the laws of nature.
Given what we know about nature either mathematical space must meet the following requirements:
(1) The mathematical space and its concepts are well-defined; a concept is well-defined if its existence, behavior or properties and relationship with other concepts are specified by the axioms.
(2) It is discrete and the axioms are not only consistent but also free from ambiguity; otherwise, they would not endow certainty to theorems.
(3) Once the axioms have been chosen ambiguity may be introduced provided it is approximable by certainty.
(4) While the choice of the axioms and introduction of concepts are inductive proofs of theorems are deductive.
(5) It has the capability for approximating large and small numbers (depending on context).
(6) It has well-defined concepts of infinity and infinitesimal. We define a set to be countably infinite if it is both nonempty and unbounded, i.e., it can’t be contained in a finite set. This is the only known (well-defined) infinity.
(7) The rules of inference are specific to and well-defined by the axioms of a mathematical space.
There are other foundational facts, principles and requirements based on the critique-rectification of foundations undertaken in the Math Series; we summarize them as follows:
(1) Since every mathematical space is well-defined only by its own axioms, distinct mathematical spaces are independent.
(2) Therefore, the rules of inference must be specific to and well-defined by the axioms of the given mathematical space.
(3) It follows that formal logic, being universal, is not valid for the investigation of distinct mathematical spaces since it has nothing to do with their axioms, i.e., external to them.
(4) Therefore, any proposition involving concepts from two distinct mathematical spaces is ambiguous and undecidable (see characterization of undeciable propositions in the original resolution of FLT, Nonlinear Analysis, Vol. 5, No.. 1998). .
(5) In particular, proof involving mapping between two distinct mathematical spaces is flawed and this applies to Gödel’s incompleteness theorems. Such mapping is both ambiguous and flawed.
(6) Vacuous concept (vacuous defining expression) or proposition is contradictory, the essence of Perron paradox (a paradox is contradiction; e.g., Banach-Tarski paradox is contradiction in R^3, both geometrical and topological).
(7) Among the ambiguous concepts are large or small number (depending on context), infinity, nonterminating decimal and all ill-defined concepts. The ambiguity of large and small numbers stems from our inability to comprehend and compute them even with the most advanced technology. We can only approximate them by orders of magnitude.
(8) A concept in which the defining expression is self-referent as well as self-referent proposition (hypothesis and conclusion refer to each other) is ambiguous and contradictory. The Russell and Richard paradoxes and the indirect proof belong to this category of ambiguity and contradiction.
(9) An extension of a given mathematical space is external to it and is not subject to its axioms, this is the source of the contradiction in the complex number i; it is not well-defined. To avoid this problem the extension needs separate set of axioms to well-define it. Ref. [6,15,30] correct this problem of the complex numbers.
(9) Proposition involving the universal or existential quantifier on infinite set is ambiguous or unverifiable. For example, suppose we want to verify that every element of a certain infinite set has property A. We check an element if it has this property; if it has, we check another element, etc.; obviously, even if every element checked so far has this property the proposition is not established since the set is infinite. Moreover, anytime an element is found without this property the proposition collapses.
(10) Consider this proposition with the existential quantifier: The decimal expansion of has a row of 100 threes. True or not, it is not known but the probability that this is true is (1/9)100 which is ambiguous. In either case, the proposition is ambiguous. Both the axiom of choice and the completeness axiom belong to this category of ambiguous propositions.
(11) Finally, this question: is set theory necessary as the foundation of mathematics? If we want to build set theory we need some axioms to specify the behavior of its symbols. Then it becomes a mathematical space. For instance, if we introduce the concept of open set and specify the behavior of open sets by some axioms, e.g., the axioms of topology, then we have a topological space. We may introduce the concept of measure on subset and specify the properties of measure by some axioms then we have measure theory, another mathematical space. We may also introduce the axioms of Boolean algebra, etc. At any rate, like formal logic set theory, with its own axioms is external to other mathematical spaces and has no bearing on their behavior. Therefore, the answer is no. Moreover, a set is interesting only when it has structure, i.e., a subspace of some mathematical space.
4. The new real number system
Any search for the right mathematics for purposes of computation and measurement must start with the real number system for that is the only one in use in science today. Its 12 initial axioms are stated in [38]. The decimals form a concrete model of this abstract space of which the metric system is physical model. They have these desirable properties relative to the needs of science:
(1) Being digital, the decimals are discrete.
(2) Addition and multiplication are well defined on terminating decimals which are countably infinite.
(3) The scientific notation provides the capability for approximating large and small decimals without bounds; however, neither the mind nor the machine can distinguish large number from infinity (u*) since both of them are ambiguous; nor can the mind distinguish small number from the infinitesimal d* which are also ambiguous (it is shown in, The mathematics of the new physics, Applied Mathematics and Computation, Vol. 130, No. 1, 2003, that d* is set-valued; therefore, we take d* as the equivalence class of its Cauchy sequences).
(4) The decimal system is a universal language of science and yet it is specific in the sense that it is distinct from other numerals such as the triadics (base three) or binary system (base 2).
However, as real numbers there are problems with the decimals:
(1) The trichotomy axiom is false; Felix Brouwer constructed a counterexample to it, making the real number system inconsistent.
(2) The completeness axiom is ambiguous since it involves the existential and universal quantifiers in the supposedly infinite real numbers.
(3) Addition and multiplication are well-defined only on terminating decimals the reason we can only approximate them by terminating decimals in computation (only finite number of their digits can be entered in the calculator; moreover, addition and multiplication require the last digit to carry out which nonterminating decimals don’t have).
(4) In view of (1) and (2), the real number system is ill-defined and most of its concepts, e.g., the irrationals, are ill-defined.
Thus, the real number system including the decimals does not meet the requirements of a mathematical space and is, therefore, ill-defined. In fact, there is no known mathematical space that is free from paradoxes (contradictions). Is it then possible to have a contradiction-free mathematical space with, at worst, contained ambiguity, i.e., approximable by certainty? The answer is: yes and this is what the Math Series has accompliished; we summarize the results here. All that is needed is: avoid the sources of ambiguity and contradiction that we have already enumerated. For instance, we may first put in finite elements or symbols that we need so that we avoid ambiguity and don’t need to assume or prove their existence later. Then we extend them suitably to countably infinite elements. These are the parameters for the reconstruction of the decimals into the contradiction-free new real number system, R*, +, ×. Its axioms
are:
(1) R* contains the basic integers, 0, 1, …, 9 subject to
(2) The addition and
(3) Multiplication tables of arithmetic.
The axioms specify the behaviour of the basic integers. Then the operations + and x are extended to terminating decimals by first defining 10 = 1+ 9 and the rest of the terminating decimals are the linear sums of powers of 10 in descending order (extended Hindu-Arabic numerals). The nonterminating decimals are standard Cauchy sequences.
4. Summary of new features and explanation of some paradoxes
We summarize the new features of the new nonstandard calculus as introduction to the mathematical space of functions over the new real number system. We may also refer to it as discrete calculus.
(1) The dark number d* = 1 – 0.99…is the counterpart of the ill-defined infinitesimal of the calculus; its non-standard Cauchy sequence is, 0.1, 0.01, …; it mathematically models the fractal superstring, the basic constituent of dark matter one of the two fundamental sates of matter (the other is ordinary or visible matter).
(2) The unbounded number u* is the counterpart of infinity of calculus which mathematically models the unbounded Universe which contains local universes such as ours. Our universe is finite and consists of superstrings, agitated or non-agitated. An agitated superstring is called primum a unit of visible matter, e. g., electron, proton, etc.
(3) A terminating decimal is an initial segment of some standard Cauchy sequence.
(4) A nonterminating decimal is the Cauchy limit of its standard Cauchy sequence.
(5) The new real number system is countably infinite, discrete, consistent, has natural ordering, enriched beyond the real numbers by the dark number d* and unbounded number u* and non-Archimedean but the complement of the union of d* and u* is Archemedean.
(6) Although a function in the new nonstandard calculus is discrete-valued its graph in the new Cartesian plane (i.e., the Cartesian product R* x R* with the standard topology) is the same as its graph in the Cartesian plane since each missing value is a dark number which is not detectable.
(7) In calculus a function is smooth at a point if its derivative exists, i.e., its left and right derivatives are equal. A discrete function is, of course, non-smooth in calculus even if its left and right discrete derivatives are equal. We refer to it as regular. For example, at the vertical cusp of the catenoid the left and right discrete derivatives are both equal to u*.
(8) In calculus the existence of local maximum or minimum requires the existence of the derivative at either point. This is not the case with discrete functions. For example, the vertical cusp of the schizoid is its maximum even if its left and right discrete derivatives there are not equal.
(9) Since the left and right discrete derivatives are independent, we distinguish the left from the right maximum or minimum of a function. If the left derivative at a point is finite and positive then it has a left maximum there; if it is negative, it has a left minimum there. Analogous statement holds for the right maximum and minimum.
(10) A discrete function has both maximum and minimum regardless of the existence of discrete derivative. Take a segment, and suppose that its left maximum M_L has been computed as Cauchy sequence in accordance with the scheme we have devised. Then the function M_L – f(x) is maximum when f(x) is minimum. Simply compute the value of the function along sequence of midpoints that gets closer and closer and using the sequence of truncations of both ML and sequence of values of f(x) corresponding to the midpoints of those segments to find the Cauchy sequence for max(M_L – f(x)). Since M_L is constant, we can find the Cauchy sequence for the minimum mL. Thus there will be finite local left minima and maxima for the subintervals. Therefore, we can find the absolute maximum and minimum for f(x) on [a,b].
(11) Some functions have countably infinite maxima and minima, e.g., infinitesimal zigzag and rapid oscillation near the origin.
(12) The article, Death of proof, Scientific American, Oct. 1983, presents several counterexamples to the generalized Jourdan curve theorem for n-sphere where in each case a continuous curve has points in both the interior and exterior of the n-sphere, n = 2, 3, … . The explanation is: what was thought to be a continuum (continuous image of an arc) is actually discrete and misses the n-sphere.
(13) The book, Proofs and Refutations, by I. Lakatos, presents counterexamples at every step in Cauchy’s proof of Euler’s conjecture on the vertices, edges and faces of a polyhedron. This paradox is due to the ambiguity of infinite set, the set of polyhedra being infinite so that there may be an exception to any proposition about it. Another flaw in Cauchy’s proof is the use of mapping of R^3 to R^2 which are distinct mathematical spaces. It is also this ambiguity of infinite set that yields the Banach-Tarski paradox.
(Those interested in the further development of the new mathematics and physics may subscribe to the journal, Nonlinear Analysis and Phenomena or purchase specific issues)
E. E. Escultura
Correction to the latest Escultura post:
2(7) The acceleration of our universe should read, "10^-10".
3(10) "?" should read "pi" and
"(1/9)100" should read "(1/9)^100.
All citations should be deleted.
Insertion to Section 4:
The new integers are d* and N.99..., where N = 0, 1, 2, ..., are integers. They are isomorphic to both the natural numbers and the integers, i.e., the integral parts of the decimals.
E. E. Escultura
Although most of Larry’s comments are covered by my previous post I would like to make some insertion and clarification.
1) On the trichotomy axiom: Felix Brouwer constructed counterexample to it in, Putnam and Benecerraf, Philosopy of mathematics, Cambridge U Press, 1985, p. 52.
2) Any counterexample to an axiom or principle of a mathematical space is a contradiction, an inconsistency. Brouwer’s counterexample shows the real number system inconsistent, ill-defined, ambiguous, nonsense.
3) To the extent that FLT is formulated in the real number system, it is also ambiguous and the only way to make sense of it is to fix the underlying fields of foundations, and the real number system which I did.
4) The trichotomy axiom has nothing to do with well-ordering which is a different matter. If you look closely, it is equivalent to does not exist in the real number system.
5) I never attempted to save the real number system; I dropped and replaced tem by the new real number system completely under different set of consistent axioms. The axioms have no ambiguity; then I introduced ambiguity approximable by certainty to avoid contradiction.
6) The term “real number” is ambiguous; I specified it as “decimal” which is distinct from other mathematical object such as triadic or binary number or fraction. The metric system is a physical model of the new real number system. A decimal is well-defined by its digits; if you don’t know some digits you don’t know and can only approximate it. That is the ambiguity of a nonterminating decimal. Since you neither know nor can you compute ALL digits of an irrational number, it is ambiguous; that’s why you can’t add, subtract or multiply them (since this operation requires the last digit), you can only approximate their sum, difference or product. Can’t even enter All their digits on the calculator or computer (meaning, if you want to use a nonterminating decimal for computation, you must throw away a little bit of it).
7) If you disagree with 6) try adding sqrt2 and sqrt3 and write the sum (don’t approximate); if you can’t that is due to the ambiguity of these irrationals.
8) In the new real number system the well-defined nonterminating decimals are standard Cauchy sequences. Then the dark number (not a decimal) d* = 1 – 0.99… = N+1 – N.99… has the non-standard Cauchy sequence, 0.1, 0.01, 0.001, …; it is the counterpart of the dill-defined infinitesimal of the calculus.
9) If you look closely, there is an error in the following:
a) “9*0.9999999... = 9, 0.999999... = 9/9 = 1,
9*0.9999999... = 9, 0.999999... = 9/9 = 1.
b) “Let d* = 1 - 0.9999...., 10*d = 10 - 9.9999...,
10d* - d* = (10-1) - (9.9999... - 0.9999...) = 9 - 9 = 0,
9d* = 0, then d* = 0.”
Calculate step by step and you’ll see the error.
10) Mathematically, the equation 1+ 0.99… = 1.99… is wrong since we cannot add nonterminating decimal to 1; however, from 8), 1+ 0.99… = 1 + (1 – d*) = 2 – d* = 1.99… and we can use the first equation as definition.
11) Strictly, the equation 1/3 = 0.33… is wrong since the left side is not a decimal. However, we can define 1/3 as the quotient in the division of 1 into 3; then the equation is correct. (The integers N, N = 0, 1, …, are decimals in view of the isomorphic embedding N -> N.00…; this is a different case from the equation 1 = 0.99… which is wrong since they are both decimals but distinct from each other. We generally ignore this error because we are not used to this new level of precision require of mathematics.
13) The best way to show that there is a problem with the Peano postulates is to observe that no number theorist relies on them; they use other tools, e.g., the real number system. The reason: they are ambiguous. The natural numbers are not well-defined by the Peano postulates. In other words, the Peano postulates do not satisfy the requirements of a mathematical space specified in the previous post.
14) Re axioms: It’s not a matter of accepting or rejecting them because everyone has the right to choose his axioms. The concern of a mathematician is that they are consistent; otherwise, they are nonsense.
15) As far as I know all existing nonfinite set theory is ambiguous.
12. General remark:
1) Try to list the items you disagree with so that we can debate them.
2) My entire works in math and physics are well-thought out and I bet my life on their consistency (except possibly for typos and inadvertence)
3) My work in mathematics (the new non-standard analysis) has just scratched the tip of the iceberg, so to speak, and everyone is welcome to join in; there is nothing else beyond my theoretical work in physics except verification and applications, especially, the generation of new technology. New technology involves conversion of dark (latent) energy (matter) to visible (kinetic) energry (matter).
4) I’m not just a local mathematician; I have over two-dozen publications in renowned refereed international journals and conference proceedings.
3) I have a Ph. D. from the University of Wisconsin and trained under the famous British mathematician, L. C. Young (my advisor), whose parents, W. H. and G. C. Young, were themselves famous mathematicians.
4)To dispel the rumor that my 2005 nomination for the Nobel Prize for physics was a hoax, see the article in the Science and Technology website:
http://www.deccanherald.com/deccanherald
/Dec132005/snt1734020051212.asp
E. E. Escultura
Minor corrections:
"dill-defined" should read "ill-defined".
Insert in 4) between "... equivalent to" and "does not ...":
"linear or natural ordering which"
FUNDAMENTALS OF THE GRAND UNIFIED THEORY
1. The first milestone of the theory was the introduction of qualitative or noncomputational modeling that explains nature and natural phenomena in terms of the laws of nature used for the first time to solve the gravitational n-body problem (Nonlinear Analysis, 1997); it was instrumental in the discovery of the superstring, basic constituent of matter, required for the solution.
2. Dark matter is one of the two fundamental states of matter, the other visible or ordinary matter. It is dark because it is not diretly observable with present technology using light as medium and is known only indirectly by its impact on visible matter. It comprises over 95% of our universe.
3. The superstring is generalized nested fractal sequence of superstrings, where the first term is a helical loop like a lady’s spring bracelet. The first term contains a superstring, its toroidal flux, that travels through the helical cycle at the speed of 7 x 10^22 cm/sec. When the cycle length (CL) of its segment is < 10^-16 meters it is called non-agitated; when 10^-16 < CL < 10^-14, semi-agitated, when CL > 10^-14, agitated. It is this nested fractal structure that makes the superstring indestructible and qualify as basic constituent of matter. For throrough discussion of its structure see the book, Grand Hybrid Unified Theory; for detailed account of its discovery, see the paper, Qualitative model of the atom, its components and origin in the early universe, in press, Nonlinear Analysis, online at Science Direct website.
4. Cosmic waves. Everything vibrates due to the impact of cosmic waves coming from all directions in the Cosmos. The normal vibration of the atom vibrates its nucleus that generates Type I or basic cosmic or electromagnetic waves. Basic cosmic wave serves as carrier of photon, radio or television signals and thought or signal from the sense organ or the gene, i.e., information encoded by neural vibration or vibration of the receptor of a sense organ or normal vibration of the gene. In general we call brain wave basic cosmic wave superposed by or encoded with the vibration characteristics of a living cell. In the cortex they light up the tips of neural dendrite that send them outward across dark matter in all directions like radio transmitters antenna do with radio signals. A wave of any kind is the synchronized vibration of the the constituents of the medium. The other two types of cosmic waves are type II seismic waves composed from basic cosmic waves at the interface of turbulence and type III seismic waves also composed from basic cosmic waves and generated by powerful explosion such as nuclear explosion and lightning.
5. Nature of superstring. Dark matter consists of non-agitated and semi-agitated superstrings. When a non-agitated superstring is hit by cosmic wave, the first term of its fractal sequence may expand to semi-agitated superstring or be projected to bounce among the other suprestrings in its neighborhood and grinds to a halt when it loses impacted energy or its path forms a loop, by flux-low-pressure complementarity, and stabilizes into a semi-agitated superstring. When a semi-agitated superstring is hit by cosmic wave, its first term may break, its toroidal flux remaining in dark matter as non-agitated superstring or a segment may bulge and become agitated to become a primum, unit or visible matter like the electron or the positive and negative quarks. Left alone (no agitation) the superstring shrinks steadily, by energy conservation; this is a very very long process like what happens in the so-called neutron star.
6. Agitated or non-agitated, the speed of the toroidal flux of the superstring, 7 x 10^22 cm/sec, is constant, a constant of nature, by energy conservation, including electric current as flux of superstrings. Note that this is motion of matter 1012 times the speed of light.
7. The primum. Since the toroidal flux of a primum is being hit by cosmic waves from all directions it is thrust into erratic motion called “spike” in the neighborhood of the helical cycles as it is being pulled by the toroidal flux at the speed of 7 1022 cm/sec. This motion pulls the superstrings around the primum into a vortex flux with eye along the primal axis. We call it the induced vortex flux of the primum. Since the primum cannot be separated from its flux, we shall refer to the superstring and its induced vortex flux as the primum and its vortex flux is part of its dynamics. Being motion of matter it is energy measured as static electricity or charge. By convention, the electron has charge 1, the unit charge. Since its vortex flux is matter in motion it is energy which equals 1.6 × 10−19 coulombs. The other basic prima are the positive and negative quarks; their charges are +2/3 and 1/3, respectively. They are the basic prima being the constituent of every atom. They are produced at rapid rate in the Cosmos and in every cosmological body and are produced by the gene in the cellular membrane of every plant or animal.
8. The eye of the primum. By a natural law called flux-low-pressure complementarity, the eye of a primum sucks non-agitated superstrings, by flux-low-pressure complementarity [11] that steadily accumulate there. They are, in fact, the major source of energy in the nuclues (coupled prima) released in nuclear explosion when they are converted to photons and prima besides the shock waves generated by the explosion. In quantum physics the energy released in nuclear explosion, e.g., atom bomb explosion, is attributed to the difference in mass between the constituent nucleons when the nucleus splits. This difference is due to the loss of negative quarks that binds the protons in the nucleus which is quite minute and does not match the explosive power of the atom bomb. That this binding force is quite small is revealed by the fact that a neutron of very low energy. 0.25 calories, is sufficient to split the nucleus of the uranium 235 atom.
9. The primum as magnet. A simple primum is a magnet, its magnetic field determined by the vortex flux of superstrings around it. We use the right hand rule of electricity and magnetism to determine its polarity. For a positive primum, when the index finger points in the direction of flux the thumb points to the north pole or N-pole. Thus, viewed from the N-pole its induced flux is counterclockwise. We shall refer to this magnet as having positive polarity. Otherwise, the flux is clockwise, the primum negative and the polarity reversed – negative.
E. E. Escultura
The *real number system* has absolutely nothing to do with the *decimal numeral system*.
But both are well defined. The only source of problems is E.E.E's hypothesis that a number must have a unique representation. Maybe this could be enforced by additional conventions, but its not needed. (In some sense it amounts to forbid considering infinite sums.)
Anyway, the issue of writing a number in the base-10 positional system is completely irrelevant for Wiles' proof.
(Even if he may have used the standard abbreviations for the natural numbers, 0={}, 1={0}, 2={0,1},... and the usual identification of a natural number with its image by the canonical injection into the integers, and so on for fractions, reals, and complex numbers.)
EEE’s Response to MFH:
If by the “real number system” you mean the mathematical space defined by the field axioms then it is not well-defined because the field axioms are inconsistent since one of them, the trichotomy axiom, is false by virtue of the counterexamples constructed by Brouwer and myself (The new real number system and discrete computation and calculus, accepted, Neural, Parallel and Scientific Computations).
The decimal system is not well-defined as a mathematical space unless its axioms are identified. To be well-defined and avoid ambiguity a real number must be identified or represented by a unique symbol. This is not a hypothesis. I don’t forbid infinite sum; I just require that it be approximable by something well-defined.
I do not require numbers to be represented by base 10 place value system; I admit binaries, triadics and set of fractions but they are distinct number systems. Each of them can be a mathematical space provided its axioms are identified and consistent. I agree that this issue has nothing to do with Wiles’ proof.
BTW, the complex number system used in Wiles’ proof collapses in view of this contradiction:
i = sqrt(-1) = sqrt1/-1) = 1/i = -i or 2i = 0 or i = 0, from which follows, 1 = -1 or 1 = 0, 2 = 0, …, and, for any real number x = 0. The remedy is in the paper, The Generalized Integral as Dual of Schwartz Distribution, in press, Nonlinear Studies.
E. E. Escultura
You use the "rule" sqrt(a/b)=sqrt(a)/sqrt(b) without saying what "sqrt" means.
A priori "sqrt" is a well-defined function from R+ to R+ (where R+ are the nonnegative reals). Only in this case the above rule is valid.
Every teacher and every textbook on the subject insists on that, and every serious student knows that. If you use "rules" in situations where they are not valid, then of course you can produce any contradiction.
Response to MFH:
x = sqrt of a if a is the solution of the equation x^2 = a or x^2 - a = 0. This is nonsense when a < 0. It is the same nonsense present in the definition of i as the solution of the equation x^2 + 1 = 0 or i = sqrt(-1) for it violates your definition of sqrt as a function from the non-negative reals to the non-negative reals.
On another matter, I appreciate Larry’s thorough response and I hope I have responded as thoroughly. Just to clarify, my objection to Peano’s postulates is their inadequacy. Number theorists cannot do much with them and must use the real numbers in their pursuit. They consist mainly of the undefined terms 0, 1, the addition and multiplication binary operations and the definition of successor.
It suffices to present a counterexample to prove a set of axioms inconsistent.
E. E. Escultura .
E> x = sqrt of a if a is the
E> solution of the equation
E> x^2 = a or x^2 - a = 0.
There is no such thing as "THE solution to x^2=a".
There are several solutions, so the "function" that yields these solutions does not have a unique value. You tried to construct a contradiction by chosing one of the solutions, and later on another solution, and pretending they must be equal. (Actually what you wrote is equivalent to:
-1 = sqrt(1) (since (-1)^2=1) and
+1 = sqrt(1) (since (+1)^2=1),
therefore -1 = +1.
> This is nonsense when a < 0.
As you see from the above, the same nonsense can be produced for a = 1 > 0.
E> It is the same nonsense present
E> in the definition of i as the
E> solution of the equation
E> x^2 + 1 = 0
This equation can well have one or more solutions, if you specify what is meant by ^2 (i.e. multiplication) and by +1.
E.g. in Z/2Z it has a single solution, x=1. In C it has 2 distinct solutions, i and -i, which are well defined elements of C=R² or C=R[X]/(1+X²).
E> or i = sqrt(-1)
This is indeed nonsense, or at least an abuse of notation which must not be used unless one clarifies what it means to write this.
E> for it violates your definition
E> of sqrt as a function from the
E> non-negative reals to the
E> non-negative reals.
Indeed. (But it is not "my" definition, but that of all textbooks on the subject.)
Response to MFH.
You admit that the equation i = sqrt(-1) is nonsense but excuse yourself by saying, "But it is not "my" definition, but that of all textbooks on the subject."
Is it not our responsibility as mathematicians or teachers to point out and correct errors in mathematics wherever and whenever they appear?
BTW, the remedy to this particular error is in the appendix to the paper, "The generalized integral as dual of Schwartz distribution, in press, Nonlinear Studies.
E. E. Escultura
I would like to introduce another false proof: Gauss' diagonal method of proving the uncountability of the decimals. The off-diagonal elements generated are still countable. Moreover, by the lexicographic ordering of the decimals they are shown to be countably infinite.
Moreover,since in this ordering every predecessor-successor pair of decimals are adjacent, i.e., they differ by d* which is a continuum, the new real number system (the pairwise union of such decimals) which consists of the decimals joined at their tails by d* form a continuum (Refereence: The Hybrid Grand Unified Theory, co-authored with V. Lakshmikantham and S. Leela and published by Atlantis, a division of Elsevier Science, Ltd.; released Feb. 2009)
Another false proof is Goedel's proof of any of his incompleteness theorems since it involves mapping between two independent spaces, the integers and the propositional calculus; they are independent because each is defined by its own axioms. Therefore, a concept in one is ill-defined in the other. Since the proof involves concepts from these space, it is ambiguous and does not make sense.
E. E. Escultura
With the noticeable increase in commentaries about FLT, it is time to provide the oundational basis of my counterexamples.
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.
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. Inconsistency collapses a mathematical system since any conclusion drawn from it is contradicted by another. A counterexample to an axiom or theorem of a mathematical system makes it inconsistent.
This 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.
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.
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.
References
[1] 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.
[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. (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., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier
Science, Ltd.), 2009, Paris.
[9] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Visakhapatnam, AP, India
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 [9] In general, any vacuous concept yields a contradiction.
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[4] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[5] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[6]. Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[7] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[8] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[9] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[10]] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier
Science, Ltd.), 2009, Paris.
[11] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[12] Kline, M., Mathematics: The Loss of Certainty, Cambride University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
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
Needless to say, none of my criticisms of my positions on 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.
E. E. Escultura
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 piece 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.
E. E. Escultura
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
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
s
I am allowed limited space to respond so I must split my response into multiple posts.
Sir, your arguments are deeply flawed.
1) i = sqrt(-1) = -i
By your reasoning I could write 3 = sqrt(9) = sqrt(-3 * -3) = -3. Even a highschool student knows this is an invalid deduction. You seem to forget that sqrt is a multivalued function which has two values for real number x > 0.
Further your claim that the reals collapse since i = sqrt(-1) is also unfounded. Sure there is no real number solution to x^2 + 1 = 0, but similarily there is no rational solution to x^2 - 2 = 0. The rationals were extended to the reals by the introduction of the irrationals to deal with this, in the same spirit the reals were extended to the complex numbers to deal with x^2 + 1 = 0. A statement outside the scope of real numbers can not demonstrate any inconsistency within the reals. That is, any statement that can not be formulated using the language of the reals, has no bearing on the real number system. THat is akin to me saying apple = 7, thus the reals are flawed. You confuse the real number system and the complex number system. The axionatization of the real numbers in no was involves the notion of he complex number i. Asking what sqrt(-1) is equal to within the reals is as meaningless as asking what the answer to 3+cookie is in the real numbers. It is simply undefined there.
So seem to confuse two distinct functions sqrt:[0,oo) -> R (note oo = infinity) and SQRT:C -> C. It is true that SQRT and sqrt agree on [0, oo), however sqrt(a + bi) is meaningless the function is not defined for a complex number, only numbers in [0,oo)
Further complex numbers can be defined in a completely unambiguous way via ordered pairs. Where 1 is shorthand for (1,0) and i is shorthand for (0,1) and a+bi is shorthand or (a,b), and finally X^2 + 1 = 0 is shorthand for (x,y)*(x,y) + (1,0) = (0,0). In this system addition is defined as follows: (a,b) + (c,d) = (a+b, c+d) and multiplication as: (a,b)*(c,d) = (ac - bd, ad + bc). Thus solving X^2 + 1 = 0 equates to solving (x,y)*(x,y) + (1,0) = (0,0). This in turn is equivalent to solving (x^2 - y^2 + 1, 2xy) = (0,0), which is equivalent to solving the two simultaneous equations x^2 - y^2 + 1 = 0 AND 2xy = 0. You must be able to see these have solutions x=0 and y=1, thus (0,1) is the desired solution to the above. Check it: (0,1)*(0,1) + (1,0) = (0*0 - 1*1, 0*1 + 0*1) + (1,0) = (1 - 1, 0) = (0,0).
In this manner complex numbers are defined rigorously without any appeal to the statement i = sqrt(-1). I challenge you to find any inconsistency with this definition.
2) Godel's incompleteness theorem
Godel did not confuses two different mathematical spaces. Within the system of formal logic he created a language, then he axiomatized the natural numbers using this language, then specified certain rules of deduction. Then within the framework of the given language, axioms, and rules of deduction he created a statement that is undecidable.
I will also make another damning point. Let us assume for a moment that your new axiomatization of the reals is consistent. Then it serves to axiomatize the integers as well, since they are a subset of the reals. Now in the construction of the undecidable statement from Godels incompleteness theorem, he used only the properties of integers. SO assuming your axiomatization works and adds nothings unneccesary, ie: you have an axiom equivalent to each of the standard axioms, then THE EXACT SAME STATEMENT WILL BE UNDECIDABLE WITHIN YOUR SYSTEM.
I ask you sir, without the tools of logic how is one to DO mathematics? ALthough we do not adhere to the strict formalism, I assure you what we do is valid in a formal system, and all accepted proofs could be translated into a formal language. How are we to make deductions without logic? Proofs could never be completed as there would be no way to make a deduction. Do you not understand that logic is the tool we use to do mathematics? Much like a painter uses a brush to produce art, we use logic to produce math. However the brush is not the painting, and similarily formal logic is not the math, do not confuse the two.
Response to Keven regarding i:
1) The problem here is that sqrt(-1) is ill-defined, nonsense, and you can make any conclusion from it, true or false or nonsense just as in formal logic where a false statement implies a false statement or a true statement. Thus, even from formal logic both conclusions sqrt(-1) = i and sqrt(-1) = -1 are valid.
2) Another point is that, if you recall, the symbol sqrt(-1) is defined as the root of the equation x^2 + 1 = 0 which does not exist. Therefore, the concept sqrt(-1) is vacuous and vacuous and ill-defined concepts belong to the same category of nonsense.
Actually, the operation sqrt is well defined in the real number system ONLY when applied to a perfect squre. Even applying sqrt to a prime is pushing beyond the its domain of validity.
What these all mean is that we need a whole lot of rectification of foundations and, naturally, the real and complex number system.
E. E. Escultura
More on i.
From the equation i = sqrt(-1) = -i, we have, 1 = -1, 1 = 0 and i = 0 from which follows that both the real and complex number systems reduce to the singleton <0>.
(BTW, the statement, 3 = sqrt(9) = sqrt(-3 * -3) = -3, is also valid following your formal logic.
On mathematical systems, extension and FLT.
1) David Hilbert made the valid observation almost a century ago that the subject matter of mathematics cannot be the concepts of individual thought since they are inaccessible to others and, therefore, cannot be studied, examined and analyzed collectively and objectively.
2) Therefore, to make sense, the subject matter of mathematics must be objects in the real world that everyone can look at including symbols (which may represent thought), which we shall also call concepts, subject to consistent premises or axioms. As part of the rectification of foundations we require the following:
a) Every concept must be well-defined by the axioms (explained elsewhere in my posts here) and the choice of the axioms is not complete unless this requirement is satisfied.
b) Sources of ambiguity must be avoided or contained (identified and explained elsewhere on this web)
3) Two distinct mathematical systems are independent since they are well-defined only by their respective axioms. Therefore, a concept in one is ill-defined, ambiguous in the other.
4) It follows from 3) that validity of conclusion in a mathematical system rests solely on its axioms.
5) Ambiguous concepts include ill-defined and vacuous concepts and a proposition involving ambiguous concept is ambiguous.
At this point it should be clear why Goedel's incompleteness theorems are flawed.
6) The undecidable propositions are characterized as ambiguous propositions in my original work on FLT (Nonlinear Studies, 5(2), pp. 227 - 254).
7) Extension of a mathematical space belongs to its complement and is distinct from it; therefore, it is not covered by its axioms and requires a separate set of axioms to well-define it.
8) The real number system is not well-defined (explained elsewhere on this web); therefore, FLT being formulated in it is also ambiguous, nonsense.
9) The resolution of FLT required its reformulation in the consistent new real number system where counterexamples to it are constructed (Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
E. E. Escultura
Dear EEE,
everybody agrees that writing sqrt(-1)=-i is WRONG and of course can be used to deduce any other wrong statement. NOBODY claims that the "equation" sqrt(-1)=-i holds, you are the only one to write this, and you accuse others of being wrong for this reason!
Once again, sqrt is a (well defined, bijective) function from [0,+oo) to [0,+oo). PERIOD.
The symbol "i" represents the element (0,1) of C = R^2 equipped with the well known rules for addition and multiplication, which makes it a field containing R as a subfield via the embedding (injective ring morphism) x -> (x,0), and in that sense the real number -1 is an element of C and you have i^2 = -1.
There is NO inconsistency! If someone like you uses "rules" which do not exist, this is not a problem of the theory, but a problem of that person.
Reply to MFH:
If you read my post correctly you will understand that I simply pointed this out: introducing the vacuous concept i into the complex number system makes it inconsistent with the conclusions 1 = 0 and i = 0 that follow from the definition of this vacuous concept i.
The Hamiltonian development of the complex number system that you mentioned is a good attempt at rectification but it still suffers from the defects of the real number system. The full rectification is the paper, The complex vector plane, developed in the appendix to my paper, The generalized integral as dual of Schwarz distributions, in press, Nonlinear Studies; it is also summarized in my paper, The mathematics of the grand unified theory, Nonlinear Analysis, A -Series, 71(2009), e420 – e431.
Omission: The paper, Escultura, E. E., The new real number system and discrete computation and calculus,
17(2009), pp. 59 – 84, appeared in the journal, Neural, Parallel and Scientific Computations, published by Dynamic Publishers.
MFH:
While I disagree with EEE, I must also disagree with you. sqrt(-1) = -i is a valid statement. CHeck it: (-i)*(-i) = (-1*i)*(-1*i) = (-1)*(-1)*i*i = 1*i*i = i*i = -1. The point is is that Positive square root is a bijection, however sqrt(z) is a multivalued function on the complex plane. The fundamental theorem of algebra demonstrates this is possible as any nth degree polynomial with complex coeefecients has exactly n roots in the complex domain, thus z^2 - k = 0 has to complex roots, which are complex conjugates.
There is a well known theorem that states that if z is a root of a complex polynomial then z_bar, the complex conjugate of z is also a root.
About the notion of a complex number as an ordered pair, I completely agree, Escultura completely misses the point. He doesn't realize that while sqrt(-1) has no real number value, but misses the point that i was enver claimed to be real, it is in the complex domain.
EEE;
Sir you are clearly mistaken. You gloss over the fact that sqrt is a multivalued function of a complex variable. The conclusion i = -i would only be valid if sqrt were not multivalued, but it is. And no, with formal logic 3 = sqrt(9) = sqrt(-3*-3) = -3 is NOT VALID. WHy do you fail to accept that sqrt is a multivalued function with 2 distinct roots?
you said: "Thus, even from formal logic both conclusions sqrt(-1) = i and sqrt(-1) = -1 are valid."
That is exactly my point, they are both valid statements, but they are not equal. sqrt(9) = 3 and sqrt(9) = -3, but 3 IS NOT EQUAL TO 3. Take a minute to think about this.
EEE:
Your counter examples to FLT are also invalid. The complete statement of FLT is for all integer values of x,y,z s.t. no x,y or z is equal to zero, and for all integer n>2, then x^n + y^n != z^n (!= mean not equal)
So the point is not to find trivial solutions of the form x^n + 0^n = z^n. Now let us consider your number system, where you claim that your integers and d* are isomorphic to the standard integers. You also state that f:R* -> R is the isomorphism, them f(d*) = 0. Now let us examine your solution x^n + ((d*) * y)^n = x^n, this is isomorphic to a trivial solution.
f(x^n) + f([(d*) * y]^n) = f(x^n)
f(x)^f(n) + f(d*)^n * f(y)^f(n) = f(x)^f(n)
f(x)^f(n) + 0^n * f(y)^f(n) = f(x)^f(n)
f(x)^f(n) = f(x)^f(n)
This clearly shows that your counterexample is acutally equivalent to a trivial case. If FLT were to be translated into your number system the added statement x,y, & z all must satisfy x != d* * k, here k is any other integer, and similarily for y & z.
Furhter to this, since the standard reals and yours are both vector spaces, thus they both have a zero element. If f is any isomorphism from R to R*, then f(0) must be the zero element in R*, but you claim f(0) = d* and still you claim that d* is not the zero element. Right away your notion that d* is not the zero element, but is sent to zero by your isomorphism is self contradictory.
If your R* and R were isomorphic, then any solution in your system would have a corresponding solution in the standard reals, and I have shown that the corresponding solution is a trivial solution.
Further since you assume that the integers as a subset of the reals are flawed, but assert that your integers are isomorphic to them, then your system is automatically flawed as well.
And finally, forget the reals, what about the integers? You have never demonstrated any flaw with the integers, and FLT is posed as a statement about integers. Even if we were to accept, which we don't, your notion of irrationals being ambiguous, the integers do not fall into this category.
How do you address this?
Keven,
it is better to define sqrt as a /function/ from R+ to R+.
The inverse relation of z -> z^2 defined on C is /not/ a function.
(I know about functions on the complex plane, and in particular the complex valued logarithm, and introducing a cut, etc, but the best is to say that there is no uniquely defined function "sqrt" on C.)
Of course you can introduce set-valued functions, which assiciate the set of solutions to an equation depending on a parameter.
But then you cannot write sqrt(-1) = -i.
you must write SQRT(-1) = { -i, i }.
As soon as you allow to write sqrt(-1) = -i without explanations on branch cuts or so, but only with the "proof" that (-i)^2 = -1, then you get a contradiction because the same "proof" proves that sqrt(-1) = +i, and equality is no more transitive.
note also that there is something wrong with your statement
"If z is a root of a complex polynomial then z_bar, the complex conjugate of z is also a root. "
(e.g. the polynomial P(z)=z^2-2i has as roots z=1+i and its opposite, but not the conjugate 1-i.)
It is left as an exercise to the reader to find the correct formulation :-) !
MHF:
Fair enough, I should have specified that if P(z) is a polynomial in one complex variable with real valued coeffecients, then if z is a root of P(z) = 0, then z_bar is also a root. Point taken.
I'm familiar with complex analysis, I wasn' attempting to get to into technical discussion on multivalued funcions and principle banches and so on. My idea was to explain to EEE why his method of proving i = -i is invalid, and his criticism of complex variables is vacuous. Thats what I was trying to do with my 3 = -3 rant. I argued above about confusing sqrt:[0,oo)->[0,oo), vs sqrt':[0,oo)->R, vs SQRT:C->C and that they are different things, although they agree on [0,oo) or {a + bi | a >=0 b = 0}. I don't think he makes the distinction between surjection, injection, bijection.
I think we are pretty much on the same side of things here. Perhaps I shouldn't have used the notation sqrt, I should have said solutions to the eqn x^2 - k = 0.
Other than that, any thoughts on my statement that any of his conterexamples to FLT are actally equivalent to a trivial solution?
My 0 button is giving me problems, so if you see P(Z) = o, understand it as P(Z) = 0.
E. Escultura:
I forgot to ask you for one thing. I've seen you all over the internet claiming to have constructed a counter example to the trichotomy law, but I've never seen you explicitly state it. Please do tell, enlighten us all, show me one concrete example where the law fails to hold.
I've also searched for Felix Brouwer, but all I could find were posts by you talking about him, nothing else. I also saw you state somewhere that Felix Brouwer is the mathematician of Brouwers Fixed Point theorem fame. I'm sorry to tell you this, you are mistaken. Luitzen Egbertus Jan Brouwer, more commonly LEJ Brouwer or simply Jan Brouwer proved the fixed point theorem, NOT Felix Brouwer.
I'm also awaiting a response to my assertion that your counter example to FLT is actually equivalent to a trivial solution. One requiremet for FLT is that x*y*z != 0 (!= means not equal). So consider your solution where y = k*(d*) and your own statement that your integers are isomorphic to the standard intgers with f:d* -> 0 where f is your isomoprhism. Then f(k * (d*)) = f(k)*f(d*) = f(k)*0 = 0, thus making it a trivial solution.
Reply to Keven:
My version of the counterexample to the trichotomy axiom appears in,
Escultura, E. E. The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations (published by Dynamic Publishers), 2009, 17, pp. 59 – 84,
and my counterexample to FLT is also in this paper.
Felix Brouwer's version of the counterexample to the the trichotomy axiom is in, Benacerraf, P. and Putnam, H. Philosophy of Mathematics, Cambridge University Press: Cambridge, 1985.
Escultura:
I have no way to get my hands on whatever obscure journal your supposed counter example is published in. Why not juts show us this counter example right here?
I also would like to ask why you chose to completely ignore my demonstration that your counter example to FLT is actually trivial?
A real mathematician would either show me how I am wrong, or admit they were wrong. By showing me I am wrong I don't mean cutting and pasting your earlier writing and telling me to read it again. I mean argue against my point. It seems to me that I have shown your counter example to be equivalent to a trivial example, and thus does not stant as a disproof of FLT.
Just incase you don't know what I am refering to, I'll quickly outline it again. You claim your integers are isomorphic to the standard integers. If f:R*->R is this isomorphism, you also claim f(d*) = 0.
Here are the problems with that. You claim d* is not zero, but if f(d*) = 0, that implies d* IS the zero element in your system.
As for your counter example consider the following.
x^n + (d* * y)^n = z^n
Since you claim they are isomorphic then I can write:
f(x^n + (d* * y)^n) = f(x^n) ->
f(x)^f(n) + f(d* * y) ^f(n) = f(x) ^f(n) ->
f(x)^f(n) + [f(d*)f(y)]^f(n) = f(x)^f(n) ->
f(x)^f(n) + [0 * f(y)]^f(n) = f(x)^f(n) ->
f(x)^f(n) + 0^f(n) = f(x) ^ f(n) ->
f(x)^f(n) = f(x)^f(n)
This is a trivial example and thus not a disproof of FLT.
Either admit that you are wrong, or prove that I am wrong.
Reply to Keven:
Elsevier Science and Dynamic Publishers which publish most of my works are obscure only to those who do not know how to access scientific publications (both of them have websites of their journals). Books published by universities, e.g. Benacerraf and Putnam's Philosophy of Mathematics (which has Brouwer's Counterexamples), are available at the US Library of Congress. If you cannot access mathematical works then you are not a mathematician and will never understand these things anyway.
At any rate, you are simply telling us here what you know which has nothing to do with the counterexamples. Moreover, effective refutation of my work requires publication in scientific journals and cannot be taken from the flat of one's foot.
More on the counterexamples to FLT
Construction of the counterexamples to FLT required:
1) Rectification of the defects of the real number system, particularly, the inconsistency of its field axioms.
2) The rectification is the reconstruction of the new real number system on three simple but consistent axioms.
3) Reformulation of FLT in the new real number system to make sense.
4) Construction of the counterexamples to FLT in the new real number system.
All of these amount to 25 pages of the specified paper which cannot be reproduced here.
BTW, the kernel of a morphism is not necessarily 0.
Rreply to Keven's and MFP"s posts that I missed re complex numbers:
The concept: the root of the equation x^2 = -1 is clear only in the domain of real numbers because at this point the complex number system is not well defined yet, i.e., the concept i is not defined yeat. The problem here is that an extension of a mathematical space is NOT well-defined by the axioms of the latter. It requires a separate set of axioms consistent with the original space that was extended (see the section on Extension of the paper, The new real number system and discrete computation and calculus, cited above).
Further comments:
While the Hamiltonian development of the complex number system is valid (except for the fact that it suffers from the defects of the real number system) it has never been operative. I have never seen a paper in complex analysis that uses the Hamiltonian system. All authors revert back to the usual notation z = a + bi. If anyone knows otherwise, I would be interested.
The journals you publish in, like nonlinear studies, have so little respect that they are not inluded in jstor. I have access to jstor as I am at a rather large University. I searched the database for your name, only one paper was found: "The Roots of Backwardness: An Analysis of the Philippine Condition" Society & Science 1974. I checked around as well, most universities do not subscribe to the joke journals you publish in, because they are joke journals. So you're whole argument about me not having access to scientific journals is a non starter.
Forgot to mention, the website for Dynamic Publishers doesn't even work, the search link goes to a 404 not found error. The journal is a joke for hacks that couldn't publish in a real journal.
ABout the kernel of a morphism not being zero, you are so wrong as far as it applies to your arguemnt. The reals are a vector space, and since you claim your reals are isomorphic to them, your reals must also a vector space. If two vector spaces a isomorphic then the image of the zero element in the first space is the zero element of the second space. You are just wrong. You yourself stated that f(d*) = 0, so f^-1(0) = d*. Therefor d* must be the zero element in your space. Do I really need to prove this to you? Do tell, since an isomorhism is a bijection and f(d*) = 0, then where does zero map to, what is f(0)??? I think you need to go back and take a first year linear algebra course to refresh yourself.
Further you completely ignore my argument that shows your "counter examples" are actually equivalent to trivial examples in the standard reals. You dance around this and choose to ignore it since you know it clearly proves you wrong. Why don't you answer this critizism?
About your argument that any refutation of your work must appear in a journal, you are wrong. A proof is a proof, it need not be published to be true. Perhaps you have heard of Grigori Perelman, he proved the Poincaure conjecture. He did not submit it to a printed scientific journal, I suppose according to you that his proof was invalid then.
About the complex numbers being ill defined, once again you are wrong. They are well defined, the error is only in your mind I'm afraid.
About the book "Philosophy of Mathematics", perhaps if you took the time to read the books you try to cite you might find out that FELIX Brouwer never contributed to that book, L.E.J Brouwer did. Why do you continually confuse these two men? I read the contributions of LEJ BRouwer to the book, nowhere does he construct any counterexample to the trichotomy law for real numbers. Are you just a liar hoping nobody will verify your claims?
The Hamiltonian notion of complex numbers as an ordered pair is equivalent to the standard notation z= a + bi. Further it is used quite often, perhaps you might want to look into the Cayley-Dickson construction.Try doing even a little research before opening your mouth next time. You make yourself look like a fool.
The flat of my foot? What does that even mean? Do you know the real reason nobody has refuted you in a journal? It is because your work is so trivially unimportant that nobody would waste their time. Even if somone were to refute your work, no journal would publish a counter proof to garbage to begin with. You've already been widely discredited all over the internet.
If my arguments come from the flat of my foot, then yours plainly come right out of your ass. You've clearly demonstrated to everyone who reads this that you lie, misdirect, hide from arguemnts you don't like, and don't even understand basic linear algebra. You know, I never started with the personal attacks, but since you started allow me to respond in kind. Quite franky you are a hack. The rest of the mathematical community deems you a crank. You have zero credibility given your lies about the Nobel prize nominations. You lied about Dr. Wiles conceding you were right, then when Dr. Wiles stated he had no idea who you are you lied some more and claimed is email account was hacked. You work at some backwater university that nobody has ever heard of (is it even a university???) and certainly is not center for mathematical thought. To be completely honest you are a sad old man who never had what it takes to contribute meaningful work the the field of mathematics, so now you resort to sensational claims in some attempt to secure your fifteen minutes of fame. You will be remembered, as a crank and a disgrace to your people. It's truly pathetic.
I forgot to mention, I looked at this supposed counter example to the law of trichotomy. The idea is to construct a number P that agrees with Pi at every decimal place, until you reach a string of 100 zeros in the decimal expansion of Pi. When you reach a string of 100 zeros then you change the value of P by this or that rule at the decimal place where the string of 100 zeros in Pi begins, and there is a different rule depending on whether this string begins at an even or odd numbered decimal place. Now since as of yet we do not know if there is ever a string of 100 zeros in Pi, we cannot know yet all the digits of P, thus we do not know if N = Pi - P is positive, negative, or zero. If there is no string of 100 zeros ever, then P = Pi, if there is a string of 100 zeros then depending where the string begins either N < 0 or N > 0. This was thought to be a contradiction to the trichotomy law, but on further inspection it is not. This is a classic example of arguments for intuitionism, and is mostly rejected by mathematicians these days. I will also add that the number constructed here N is computable, so there is no giant mystery at all, we can calculate it to any degree of accuracy required.
Your entire argument boils down to the fact that irrationals have a non-terminating non-repeating decimal expansion, and this is somehow a problem for you since this decimal expansion can never be written down. I assure you this is not a problem for mathematicians these days, irrationals are not a source of contradiction or paradox. The construction of the reals via Dedekind cuts easily dispenses with your sort of problem. You could also define reals using Cauchy sequences, either way your construction is nowhere near rigorous or consistent and would fall victem to the same "problems" you find with the current systems.This sort of arguemnt is old and tired, is has been dispensed with long ago. Constructivism is generally rejected now, maybe it's time you cuaght up on current thought.
Reply to Keven
Now you cry mommy!!! Didn't you call me a crank, an insecure little man? Unless, of course, you have short memory.
Anyway, let me respond to your posts.
1) You just name drop and quote others because you have nothing of your own. That is what I mean by taking staff from the flat.
2) Your pronouncements are not work a grain of salt unless they appear in the only arbiter of scientific truth and merit, namely, the network of peer reviewed scientific journals. Naturally, it follows that refutation of scientific papers is nothing unless it is published in such network.
3) Felix Brouwer and LEJ Brouwer are the same person. His entry to fame in mathematics is the fixed point theorem.
4) The notation z = a + bi is nonsense because of the involvement of the vacuous concept i. If you claim that this representation of complex numbers is equivalent to Hamilton's representation by ordered pairs then you are, in effect, claiming that the latter is also nonsense to which I disagree.
5) Where is your proof that my nomination was a hoax?
5) I have over 50 publications in renowned peer reviewed journals; where are yours if any? I still notice your difficulty accessing scientific journals.
6) Only a racist would call the renowned Jawarhalal Nehru Technical University a backwater university.
Correction to reply to Keven:
"staff" should be "stuff"
"work" shuld be "worth"
Unpublshed mathematical rumbling academic pollution.
More correction:
insert "is" after "rumbling"
Remark for Kevin,
I just noticed that the comments you sent me by email are not posted here but I have posted my reply anyway.
I notice that you misunderstood many of my posts. For example, your post on the counterexample to the trichotomy axiom has nothing to do with Brouwer's or mine.
You need to familiarize yourself with basic mathematical stuff to pariticipate effectively in mathematical discussion.
Question for Keven:
How do you verify that a nonterminating decimal has repeating or nonrepeating decimal expansion?
Arguement B)
You claim to have found a counter example to FLT of the form X^n + (d*)^n = z^n. Now since you claim f is an isomorphism from R* to R I can write:
f(x^n + (d*)^n) = f(z^n) ->
f(x^n) + f(d*^n) = f(z^n) ->
you claim (d*) * (d*) = (d*) in one of your posts thus I can continue
f(x^n) + f(d*) = f(z^n) since you claim f(d*) = 0 I can continue
f(x^n) + 0 = f(z^n) ->
f(x^n) = f(z^n) -> x^n = z^n
Thus I have shown your "counter example" is equivalent to a trivial solution in the standard reals, it does not meet the criteria of FLT and is thus not a counter example.
10) So there you go. THose are two very precise mathematical arguments against your claims. No handwaving, no personal attack, just precisely stated mathematical arguments. So are you going to answer them, or just evade the point and rant on as usual?
11)As to your question "How do you verify that a nonterminating decimal has repeating or nonrepeating decimal expansion?". First I do not see how this is relevant, and further it isn't a well posed question. Do you mean "Given a nonterminating number x, how do you dertermine if it has repeating or nonrepeating decimal expansion?" Or do you mean "How do you prove there are a)numbers with nonrepeating nontermination decimal expansion, and b)numbers with repeating but nonterminating decimal expansion?"
12) About he post I made regarding Brouwers attempt to construct a number that serves as a counter example to the trichotomy law, that is exactly the method LEJ Brouwer tried. That sort of arguement fails on a few levels, one is that it is an intuitionist argument and that school of thought is largely dismissed. You claim that Felix Brouwer is LEJ Brouwer (which he is not, you simply got the names mixed up), and you always refer to this counter example but never state it, so I simply posted it so anyone reading will know what you are talking about. Once again, this is exactly the method LEJ Brouwer used to try to find a counterexample to the trichotomy law, and it fails.
More comments on Keven's posts:
I studied your post and here is what I found:
You do not refute my arguments. Instead, you state your opinion that counters I have established, e.g., my counterexample to FLT or the trichotomy axiom. Worse, you have not even read my original counterexamples.
Consequently, I just have to ask you direct questions. I raised one question already.
Now, I have another request: Point to a specific mathematical contradiction in my posts on this page. State what you think the contradiction is and prove that it is, indeed, a contradiction.
Reply to your argument B:
I never claimed that R* (the new real number system) is isomorphic to R (the real number system). That is ridiculous. I can stand your shortness of memory but this is just too much. The rest of your rumblings falls on the wrong premise.
Since you did not understand my question let me be even more specific: Prove that the decimal expansion of pi is nonrepeating.
On item 12) of your post:
It is clear that you never saw the original statement of Brouwer's counterexample to the trichotomy axiom the reason you rumbled about other things. You are just gussing from what I have post. This is no way to engage in serious mathematical debate. It borders on futility.
More...
Anytime you you write a statement with vacuous and other symbols is manipulation of symbols because you cannot argue with vacuous symbol which does not exist.
I noticed that several of your posts and mine are not displayed. I'll just wait until they are displayed before I respond because it would be confusing to the viewers.
Something strange is going on, some of my posts are getting posted, but disappearing.
If you did not claim that R and R* are isomorphic then I am mistaken, perhaps I read someone else saying that in a discussion of your work.
I did check however, and you did say your integers are isomorphic to the standard integers. Thus let f:Z* -> Z be the isomorphism. Then the following does still hold. Convenient how you ignored this part.
Arguement B)
You claim to have found a counter example to FLT of the form x^n + (d*)^n = z^n. Now since you claim f is an isomorphism from Z* to Z I can write:
f(x^n + (d*)^n) = f(z^n) ->
f(x^n) + f(d*^n) = f(z^n) ->
you claim (d*) * (d*) = (d*) in one of your posts thus I can continue
f(x^n) + f(d*) = f(z^n) since you claim f(d*) = 0 I can continue
f(x^n) + 0 = f(z^n) ->
f(x^n) = f(z^n) -> x^n = z^n
Thus I have shown your "counter example" is equivalent to a trivial solution in the standard reals, it does not meet the criteria of FLT and is thus not a counter example.
Why do you want me to prove Pi is irrational? There are already quite a few well known proofs. It was originally proved by Lambert, since then Niven has offered an elentary proof which appears in the classic text: Calculus 3rd ed by Michael Spivak. Are you claiming that Pi is not irrational and the decimal expansion of Pi repeats?
12) I asked you on more than one occasion to offer this alleged proof by Felix Brouwer, yet you refused. I tried searching for Felix Brouwer, but could find nothing other than posts you had made where you mention him. I looked for LEJ Brouwer and found the stuff I posted. What do you want, I've asked you to post it yet you refuse, what am I supposed to be a mind reader?
I've read of LEJ Brouwers attempts at a counter example, they seems to involve non computable numbers. This train of thought does not work though, just because a number is not computable does not mean it is a counter example to the trichotomy law.
I never used any "vacuous symbols" so what are you talking about. Now if you never claimed R and R* are isomorphic I retract my comments on the zero element. However my other arguement still stands.
Correction:
"Internal" should be "International" and "Benntham" should be "Bentham"
I'll particularly wait for your proof that the decimal expansion of pi is nonrepeating. This is important because you have been using "nonterminating decimal" in your claims and if this concept is not well-defined then your claims means nothing.
I am not going to repeat a proof that Pi is irrational here. There are many well known proofs. The notion of non terminating non repeating decimal is very well defined.
Forget the decimal crap, I don't care about this trichotomy stuff anymore.
Either argue against the following or drop it. I've posted it way to many times and you never once say anything about it. I'm tired of this.
You say your integers are isomorphic to the standard integers. Thus let f:Z* -> Z be the isomorphism. Then the following does still hold.
Arguement B)
You claim to have found a counter example to FLT of the form x^n + (d*)^n = z^n. Now since you claim f is an isomorphism from Z* to Z I can write:
f(x^n + (d*)^n) = f(z^n) ->
f(x^n) + f(d*^n) = f(z^n) ->
you claim (d*) * (d*) = (d*) in one of your posts thus I can continue
f(x^n) + f(d*) = f(z^n) since you claim f(d*) = 0 I can continue
f(x^n) + 0 = f(z^n) ->
f(x^n) = f(z^n) -> x^n = z^n
Thus I have shown your "counter example" is equivalent to a trivial solution in the standard reals, it does not meet the criteria of FLT and is thus not a counter example.
Here is another simple proof that d* = 0, and this doesn't involve multiplying non termination decimals. It's a simple identity and a little formula manipulation. Since you claim (d*)*(d*) = d* I can write
(1 - d*) = (1 - d*) ->
(1- d*) - d* + d* = (1 - d*) ->
1 - 2d* + d* = (1 - d*) ->
1 - 2d* + (d*)^2 = (1 - d*) ->
(1 - d*)(1 - d*) = (1 - d*) ->
now we cancel one factor of (1 - d*) from each side giving
(1 - d*) = 1 ->
-d* = 0 ->
d* = 0
That was done entirely in your number system R*. I did not multiply any non terminating deciamls or anything else you don't seem to like.
Reply to Keven:
You have the wrong arithmetic for d*. You need to really look at the original paper to even debate this issue sensibly. 1 - d* = 0.99...; d* is not a real number, does not exist there and has a different arithmetic. Moreover, the new integers are the decimal integers, i.e., decimals of the form N.99..., where N = 0, 1, ....
The claim you are referring to is this: that the integers are isomorphic to my decimal integers (also called new integers). Your argument here does not hold water because you did not see the original theorem in my paper.
Now, you gave up on the counterexample to trichotomy axiom because you could not find a flaw in it.
Then you must give up your challenge to my counterexamples to FLT because the counterexample to the trichotomy axiom is central to them for the following reasons:
1) The counterexample to the trichotomy axiom proves that the field axioms of the real number system are inconsistent.
2) Therefore, the real number system is nonsense. Since FLT is formulated in the real number system it is nonsense, too. This should be sufficient to dispose this problem and Wiles' "proof", too.
3) But a mathematician is not satisfied with a dud. I went further: I fixed the real numbers by embedding them in the consistent new real number system, formulated FLT in it and resolved it with countably infinite counterexamples. (I really needed only one for purposes of resolving FLT but a true mathematician is never a minimalist)
There you go. Like the others you failed to punch a hole to my counterexamples to FLT. So, back to square one.
I assume that by "irrational" you mean a real number that has nonrepeating decimal expansion. I also assume that you consider pi irrational. If my assumptions are correct, I challenge your claim that the notion of irrational is well defined for just one case by requesting you to prove that the decimal expansion of pi is nonrepeating.
I made a trivial mistake above, I wrote 0 when I should have wrote Z, since Z is the zero element in your system. It should read:
MM - Md* = (M - Md*) ->
MM - MMd* - MMd* + MMd* = (M - Md*) ->
MM - MMd* - MMd* + MM(d*)(d*) = (M - Md*) ->
(M - Md*)(M - Md*) = (M - Md*) ->
(M - Md*)(M - Md*) = M(M - d*) ->
(M - Md*) = M
-Md* = Z ->
d* = Z
Thus I have proved that in your system d* is the zero element.
Reply to Keven:
No, the zero element in my system is 0 and d* is a new element in my system that does not exist in the real number system.
As I told you earlier, you cannot refute my work by simple manipulation of symbols. You need to grasp the concepts fully and you cannot rely on hearsay (that someone has proved this or that theorem). You need critical thinking, the core value of mathematics and science along with its complement, creativity.
Are you serious? I just PROVED THAT d* IS THE ZERO ELEMENT IN YOUR SYSTEM.
Take a look at it again. It is a perfectly valid proof within your system. The only assumptions I made are that your system has multiplicative identity and a zero element. Are you saying that your system does not have a multiplicative identity or zero element?
I have not relied on any hearsay. WHere do you come up with these accusations???
I constructed a perfectly valid proof within your system to prove that d* MUST be the zero element. It does not matter if you like it or not, the proof is right there.
I noticed you still have not responed to my proof that your "counter example" to FLT is equivalent to a trivial example in the standard integers.
If you are so sure I am wrong, then show me exactly where I am wrong in the following:
I will work in your system. Let M stand for the multiplicative identity in your system and let Z stand for the zero element in your system. This means for any integer N in your system M*N = N and N + Z = Z. In your paper you state (d*)x(d*) = d*
M - Md* = (M - Md*) -> (identity)
M(M - Md*) = M(M - Md*) -> (multiply both sides by multiplicative identity)
MM + (MMd* - MMd*) + MMd* = M(M - Md*) -> (adding zero to left side)
MM - MMd* - MMd* + MM(d*) = M(M - Md*) -> (remove brackets)
MM - MMd* - MMd* + MM(d*)(d*) = M(M - Md*) -> (subsititute (d*)(d*) = d* on Left side)
(M - Md*)(M - Md*) = M(M - Md*) -> (factor)
(M - Md*) = M ->(cancel common factors (M - d*) )
M - Md* = M ->(remove brackets from left side)
-Md* = Z ->(subtract M from boths sided)
-d* = z ->(Md* = d* since M is multiplicative identity)
Thus -d* = Z, d* is the zero element. I have done this entirely inside your system, I only used numbers from your system. I have PROVED that d* is the zero element of your system. If you disagree show me EXACTLY where I made a mistake in the above. Contrary to what you claim, I just did refute your work by simple manipulation of symbols. I challenge you to show me EXACTLY what step in the above is wrong.
Reply to Keven,
Your latest comment is not posted here but I'll respond to your email.
1) The additive and multiplicative elements of my system are still 0 and 1 since the real number system is a subset of my number system. The dark number d* is not a zero element in my system; in fact, d* > 0.
2) You claimed that pi has been proved to be irrational by others. So your claim is bssdrf on hearsay (this is only one of the instances where you relied on the work of others). That is why I wanted you to prove that pi is an irrational but you need to first define what an irrational is.
3) Your proof is not valid because you are using standard arithmetic (computations with the operations + and x of the real numbers) but my syestme has a different arithmetic, i.e., different additive and multiplicative operations. Moreover, d* is not even a real number and you cannot apply standard arithmetic to d*. It is true that (d*)(d*) = d* just as (0)(0)= 0 in view of the isomorphism between the decimal integers and the integers.
4) Your "proof" that my counterexample is trivial or equivalent to a trivial solution is based on your incorrecct premise that d* = 0 which is false.
5) The first statement in your "proof" is already trivial since M = 1 and Z= 0. Naturally, the rest of it cannot be profound.
I tell you, you can't discuss my system sensibly unless you READ and GRASP the original paper.
1) Perhaps you don't understand what I did. I PROVED in your system that d* MUST be the zero element. The only possibly way that my proof could fail is if your system does not have a multiplicative identity or additive identity.
2) I have not reproduced the proof that Pi is irrational because the is a text based blog and does not support mathematical symbols. I would need various mathematical symbols such as the integration symbol which I can not reproduce here. You claim I rely on hearsay, I relied on a proof. If you think a proof is hearsay, then you fundamentally misunderstand mathematics.
3) I framed my proof entirely in your system. I used your arithemetic, the ONLY assumption I used was the existence of a multiplicative and additive identity and existence of an additive inverse.
4) My proof that your counter eample is equivalent to a trivial solution is NOT based on d* = 0. My proof that your counter eample is equivalent to a trivial solution is based on the fact that YOU said f(d*) = 0 where f is YOUR isomorphism. Maybe you should reread what I wrote.
5) The fact that M = 1 and Z = 0 do not change my proof at all. I used the symbols M and Z so that you would not think I am mixing the standard reals and your reals. If you take my proof and swap 1 for M and 0 for Z it still holds. I never claimed my proof was profound, it's actually rather simple, but it is valid.
I have read your paper. I do understand your paper, what I am trying to show you is that there are inconsistencies within the system you define in your paper.
Let us discuss one thing at a time, so for the moment lets concentrate on the following proof, we will get back to your FLT counter example later. Lets try this again.
Let + mean the addition operation IN YOUR SYSTEM
Let X mean the multiplication operation IN YOUR SYSTEM, so if I write MMd* understand it to mean M X M X d*
Let Z be the zero element IN YOUR SYSTEM (if Z = zero this does not change the validity of the proof)
Let M be the multiplicative identity IN YOUR SYSTEM (if M = 1 this does not change the validity of the proof)
I will use ONLY operations in your system, this entire proof is written only in your system. I will not use any standard reals or the standard additiion or multiplication operations, I will only use the operations from your system.
[1] M - Md* = (M - Md*) -> (identity)
[2] M(M - Md*) = M(M - Md*) -> (multiply both sides by multiplicative identity)
[3] MM + (MMd* - MMd*) + MMd* = M(M - Md*) -> (adding MMd* - MMd* is adding zero to left side)
[4] MM - MMd* - MMd* + MM(d*) = M(M - Md*) -> (remove brackets)
[5] MM - MMd* - MMd* + MM(d*)(d*) = M(M - Md*) -> (subsititute (d*)(d*) = d* on Left side)
[6] (M - Md*)(M - Md*) = M(M - Md*) -> (factor)
[7] (M - Md*) = M ->(cancel common factors (M - d*) )
[8] M - Md* = M ->(remove brackets from left side)
[9] -Md* = Z ->(subtract M from boths sided)
[10] -d* = z ->(Md* = d* since M is multiplicative identity)
Take the time to actually read and think about the above proof. I have numbered the steps, so if you think there is a problem tell me at which step. Tell me exactly why the operation from line [i] to line [i+1] is not valid in your system.
Reply to Keven:
1) I understand precisely what you are saying. You did not prove anything about d* because you are applying standard computation of the real number system on d* which is not even a real number. That is basic mathematical error.
2) I want you to reproduce the proof that pi is nonrepeating to establish your claim. You can use USENET notation or the English language(symbols are only short cuts). Otherwise, otherwise, you would be using the hearsay that pi had been proved to be irrational according to what you heard or read about it.
3) Not true (see my comment 2) above)!
4) This is another obvious error. if f is an isomorphish and f(d*) = 0, it does not mean that d* = 0. It simply means that d* is in the kernel of f which need not be 0.
5) First, your claim that my counterexample is equivalent to a trivial solution in the standard reals does not hold water because (a) it brings in d*, which is not even a real number, into your standard calculation where (b) you assume d* = 0, which is not based, on an incorrect "proof". This whole item of your comment collapses altogether. I have actually brought this out already in my previous remark.
Glad to know that you read the paper. Now, I want you to pinpoint at least one inconsistency in it. That is all you need to do to demolish my paper. You don't have to prove anything and ask me what is wrong with your proof because even if there is nothing wrong with your proof it will not accomplish your mission: to find an inconsistency in my counterexample to FLT. It is not the way we do things in mathematics.
Correction:
The "," in item 5) shouldd be placed after "not" instead of after "based".
Let + mean the addition operation IN YOUR SYSTEM
Let X mean the multiplication operation IN YOUR SYSTEM, so if I write MMd* understand it to mean M X M X d*
Let Z be the zero element IN YOUR SYSTEM (if Z = zero this does not change the validity of the proof)
Let M be the multiplicative identity IN YOUR SYSTEM (if M = 1 this does not change the validity of the proof)
I will use ONLY operations in your system, this entire proof is written only in your system. I will not use any standard reals or the standard additiion or multiplication operations, I will only use the operations from your system.
[1] M - Md* = (M - Md*) -> (identity)
[2] M(M - Md*) = M(M - Md*) -> (multiply both sides by multiplicative identity)
[3] MM + (MMd* - MMd*) + MMd* = M(M - Md*) -> (adding MMd* - MMd* is adding zero to left side)
[4] MM - MMd* - MMd* + MM(d*) = M(M - Md*) -> (remove brackets)
[5] MM - MMd* - MMd* + MM(d*)(d*) = M(M - Md*) -> (subsititute (d*)(d*) = d* on Left side)
[6] (M - Md*)(M - Md*) = M(M - Md*) -> (factor)
[7] (M - Md*) = M ->(cancel common factors (M - d*) )
[8] M - Md* = M ->(remove brackets from left side)
[9] -Md* = Z ->(subtract M from boths sided)
[10] -d* = z ->(Md* = d* since M is multiplicative identity)
I have pinpointed an inconsistency in your paper with the above. What part of it do you not understand? I used YOUR addidtion operation, I used YOUR multiplication. It did accomplish my goal,it demonstrates that your counter example to FLT is constructed in an inconsistent framework.
Let me ask you these few questions.
A) Does your system have a multiplication operation?
B) If it does does it have a multiplicative identity? Does it have an addition operation?
C) If it does, does is have a zero element?
D) Does every number Y have an additive inverse (-Y) such that Y + (-Y) = ZERO?
If your answer to all the above is YES, then my proof clearly demonstrates an inconsistency in your system. If the anwser to any of those is NO then your system cannot describe the integers. WHich one is it? This is getting tiring, I have clearly proved that your system is inconsistent, but you refuse to acknowledge it.
If you want to convince anyone at all that you are right, then answer A-D. Then demonstrate EXACTLY which line number I made a mistake on. Make sure you read the lines above the actual proof.
Once we are done with this part we will return to my other arguments against your counter example. FOr the moment let us concentrate on the above proof.
Oh yeah I checked with two people at Jawarhalal Nehru Technical University, as far as they can tell you are NOT a member of the faculty. It seems that at one point in time you were a librarian. You were not and are not in the Math, Physics, or Engineering departments. Unless perhaps both people I contacted there lied to me?
Reply to Keven:
All your questions from A to D are answered affirmatively. The additive and multiplicative operations are well defined by the second and third axioms of the new real number system. The inverses are also well defined. Since the real number system is isomorphically embedded in the new real number system the additive and multiplicative identities of the latter coincides with those of the former.
ALL your "proofs" are NOT VALID because they apply the additive and multiplicative operations of the real number system to elements of the new real number system which are not real numbers. That is sloppy mathematics to say the least.
JNT University is the only State University of the State of Andhra Pradesh, India consisting of 472 Colleges as of last count and two autonomous campuses, one in Hyderabad (flagship campus) and the other in Kakinada. The V. Lakshmikantham Institute for Advanced Studies and the Departments of Mathematics and Physics where I am a research professor and inventor are in the GVP College of Engineering in Madurawada, Visakhapatnam, AP, India affiliated with the Kakinada Campus. Poor fellows, you accused them as liars.
If you were a mathematician (i.e., published mathematician)you would not have missed me as an accomplished mathematician and physicist of international stature through the network of peer reviewed scientific publications and my keynote addresses at international conferences.
I think this dialogue has reached the point of desperation and futility. I could have dismissed you as a nuissance since you have not scored a single point in this debate but I am generally magnanimous to the less endowed.
Correction:
JNT University is the only statewide private university in the state of Andhra Pradesh, India.
About "Poor fellows, you accused them as liars.". I didn't accuse anyone of anything, That is the response I got from emails I sent to the school. I also found a faculty list categorized by department, your
name was NOT listed in the math, physics, or engineering departments, it was listed under "LIbrary". Your position in the school is irrelevant to the discussion, the only reason I ever brought it up was
because you tried to attack my credentials, which I have never precisely stated, alluded to, or used to support anything I have said, so I thought it fair to question yours. From now on I will agree to not
make credentials part of this discussion if you will agree to the same, we will discuss the mathematics and nothing else, agreed?
Now on to the mathematics. You answered yes to all of the following questions.
A) Does your system have a multiplication operation?
B) If it does does it have a multiplicative identity? Does it have an addition operation?
C) If it does, does is have a zero element?
D) Does every number Y have an additive inverse (-Y) such that Y + (-Y) = ZERO?
So the following are valid in yur system.
A') Let X represent the multiplication operation IN YOUR SYSTEM
If I write MMd* understand it to mean M X M X d*
B') Let M represent the multiplicative identity IN YOUR SYSTEM
Let + represent the addition operation IN YOUR SYSTEM
C') Let Z represent the zero element IN YOUR SYSTEM
D') Since every number Y has as additive additive inverse (-Y) such that Y + (-Y) = zero, adding [Y + (-Y)] to either side of an equation is a valid step that changes nothing. So if you have an equation:
LHS = RHS then LHS + [Y + (-Y)] = RHS is still valid. (LHS = left hand side, RHS = right hand side)
Thus the following proof uses ONLY numbers IN YOUR SYSTEM. It also uses ONLY operations IN YOUR SYSTEM. If
[1] M - Md* = (M - Md*) -> (identity)
[2] M(M - Md*) = M(M - Md*) -> (multiply both sides by multiplicative identity)
[3] MM + (MMd* - MMd*) + MMd* = M(M - Md*) -> (adding MMd* - MMd* is adding zero to left side)
[4] MM - MMd* - MMd* + MM(d*) = M(M - Md*) -> (remove brackets)
[5] MM - MMd* - MMd* + MM(d*)(d*) = M(M - Md*) -> (subsititute (d*)(d*) = d* on Left side)
[6] (M - Md*)(M - Md*) = M(M - Md*) -> (factor)
[7] (M - Md*) = M ->(cancel common factors (M - d*) )
[8] M - Md* = M ->(remove brackets from left side)
[9] -Md* = Z ->(subtract M from both sides)
[10] -d* = z ->(Md* = d* since M is multiplicative identity)
QED
You claim this proof is not valid since I apply the addition and multiplication operations of real numbers to elements of your system. If you read A') - D') above you will see that I clearly state I am using
the addition and multiplication operations of YOUR SYSTEM. Why do you insist I am using operations from the standard real number system to elements of your system when I clearly state that I am using the
operations FROM YOUR SYSTEM? Just read A') - [10] carfully, make sure you are clear that I am using only operations IN YOUR SYSTEM.
So I ask you, how can you possible justify saying "ALL your "proofs" are NOT VALID because they apply the additive and multiplicative operations of the real number system to elements of the new real number system which are not real numbers." when I have so clearly stated I am working entirely within the framework you have constructed?
Reply to Keven
I NEVER worked as a librarian at JNTU, not a chance. You tried to locate me in JNTU and when the two fellows could not confirm my presence there you were ready to call them liars if it turns out that I am, indeed, a member of the faculty which is true. They deserve an apology. My institutional affiliation is indicated in my earlier posts above and in many of my scientific papers (of course, you were not aware of them). It has a website that you can check.
Your credential is essential to this debate because if you are not a mathematician you will never understand the subject just as you already admitted. Thus, this debate has become an exercise in futility.
In fact, many instances in your arguments reveal inability to understand basic mathematics and what I am saying. For example, your understanding of the Brouwer counterexample to the trichotomy axiom and my counterexample to FLT are completely off tangent. So is your discussion of the concept of irrational. Most revealing is the fact that although I have already explained that since the real number number system is embedded isomorphically in the new real number system, their additive and multiplicative identiities and operations coincide or are the same in their intersection. You still keep asking the same questions that I have already answered. Moreover, labeling the additive and multiplicative operations of the real number system as the additive and multiplicative operations in my system does not make them so. And yet you continue with your fallacious arguments. These are fundamental issues any true mathematician would know. It is this lack of proper credentials that deprived you of the relevant information that would have told you that I AM NOT AN IMPOSTOR and you need not have gone to the trouble of checking my institutional affiliation and acade,oc credentials.
I think you will be better off to apply your present skill to more appropriate endeavor and forget that you ever got involved in this kind of discussion because the longer you pursue it the more resounding the fall. I will now dismiss you now as a nuissance unless you reveal your true identity.
At any rate, we should both thank Larry for keeping this website on a high plane by deleting the abusive language in your earlier posts.
Correction: "academic credentials"
I will make this very easy for you. Here are the exact assumption that I made:
A) The is a multiplication operation X in your system
B) There is a mulitplicative identity M s.t. for all N in your system MXN = N
C) There is an addition operation + in your system
D) There is a zero element in your system Z, such that for all N in your system Z + N = N
E) For every element N of your system there is an additive inverse (-N) s.t. N + (-N) = Z
F) Multiplication distributes over addition, so for all A, B, N in your system NX(A + B) = NxA + NXB
E) d*xd* = d*
Thats it, that is the complete list of properties of addition and multiplication that are required to prove that there is an internal contradiction in your system, namely that you cliam d* > 0, but I can clearly prove d* = 0.THose are the ONLY properties needed. I stress those are the ONLY properties needed for my proof.
If you can clearly state which of those properties fails to hold in your system then I will admit I am wrong. If you cannot make this statement will you admit you are wrong? There are only seven choices, if even one of them fails then my argument fails. All you have to do is demonstrate that one of (A)-(E) fails. This should be childs play to a mind as supposedly as brilliant as yours. So go ahead, try to prove me wrong.
You can go on trying to make personal attacks, but everyone who reads this will know that if you cannot answer this challenge then your system must be inherently flawed.
Reply to Keven:
There is really no need to go through the "sopdhistication" of your arguments. As I have explained many times, the additive and multiplicative identities of the new real number system are the same as those of your real number system, namely, 0 and 1, and its additive and multiplicative operations coincide with those of the real number system IN THEIR INTERSECTION which is the real number system since the latter is isomorphically embedded in the former. Therefore, A) - D) are trivial restatements of what I have already explained and trivial applications of the relevant field axioms of the real number system that do not merit any score.
Statement E) is nonsense because it applies the multiplicative operation of the real number number system to d* which is not even a real number, another blunder on your part.
Naturally, the rest of your post is nonsense because it rests on your blunder. You are beyond personal attack because you are hiding comfortably behind your username.
So, what's new?
More for Keven:
BTW, in case you have not realized it yet, you are living in your daily world of contradictions by trying hard to prove something that you don't understand.
Let me make sure that I understand your problem with what I did. You claim d*Xd*=d* makes no sense. So you are saying you can not apply a multiplication operation to d* in your system?
What operation are you using in your alleged counter example to FLT when you use (d*)^10. Am I to understand that (d*)^10 does not mean d*xd*xd*x...xd*?
The additive and multiplicative operations of the real number system are defined only in this system, never in its complement. Every mathematical system including its concepts and operations are well defined ONLY by its axioms. Therefore, operations and elements of its complement including its extension are not well defined by them. In the case of the new real number system it is even worse for you since, as you admitted, you did not understand it. Regarding your last question, d*xd*x...xd* is nonsense because x is an operation in the real number system but d* is not a real number.
This line of discussion has reached a dead end. I don't want a monologue. How about proving that pi is irrational? No bluffing please!
I am willing to share what I know but you need to be humble a bit. You don't just come here and call me a crank insecure little man.
Re your last question, the operations I used in constructing my counterexamples to FLT are, naturaally, operations in the new real number system.
Re: E.E.E.
"Your pronouncements are not work a grain of salt unless they appear in the only arbiter of scientific truth and merit, namely, the network of peer reviewed scientific journals."
If he can prove his arguments from a set of consistent axioms, they are worth the same as any published proof produced from similarly consistent axioms. I hardly see how publication makes a mathematical proof more or less "valid". Surely only the logic used from step to step decides that?
Also, I may be incorrect, but FLT stated
x^n + y^n != z^n
requires x, y, and z all to be positive integers, doesn't it?
Show that (1 - 0.999...) is an integer, and maybe I'll bother to look up your paper. Otherwise what you have is a proof of another (trivial) theorem. Well done.
Reply to Euan117:
You missed the whole point here. The integers as real numbers are nonsense because the axioms of the real number system are inconsistent (from which follows that the original formulation of FLT is also nonsense). That is why I had to fix it first by reconstructing it into the new real number system well defined by only three axioms. Such construction along with the critique of the real number system and the counterexamples to FLT appears in the network of peer reviewed scientific journals. They cannot be replicated by blogposts. Therefore, there is no way to avoid those journals if one wants to stay in the game.E. E. Escultura
Re: E.E.E's statements.
I may not be in your field of mathematics but i sure know your statement:
"Since 1 and 0.99… are distinct symbols they cannot be equal; therefore, d* = 1 – 0.99… cannot be zero contrary to Larry’s claim that d* = 0. Thus, the usual proof that 1 = 0.99… is flawed."
is incorrect. 0.99... and 1 are just different representations of each other. The recurring decimal goes on infintely, likewise 3=3.0000..., 1-0.99... is equal to 0. i can't understand how you disagree. If you would like to explain your reasoning behind this i would much appreciate this. Thanks.
Reply to Luke.
In the last couple of years I have published several books and papers that fully elaborate and explain my critique of current mathematics among which are:
1) Escultura, E. E. The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 84.
2) Escultura, E. E.,The Mathematics of the Grand Unified Theory, Chapter II of Scientific Natural Philosophy, Bentham Science Publishers, 2011, pp. 10 - 59: http://www.benthamscience.com/ebooks/9781608051786/index.htm
At any rate, let me offer a brief explanation, again, on this website.
Davide Hilbert recognized a century ago that the concepts of individual thought cannot be the subject matter of mathematics because they are not accessible to others and, therefore, can neither be studied collectively nor axiomatized. Therefore, its subject matter can only be objects in the real world that everyone can look at and examine such as symbols provided that their behavior, propeties, etc., are subject to consistent premises or axioms. The objects 1 and 0.99...are certainly distinct and to say that 1 = 0.99... is like saying "apple = orange" which is absurd. This is only one of the many flaws in the real number system. As remedy, I constructed it into the New Real Number System on only three consistent axioms.
The only well-defined numbers are the rational numbers. Repeated decimals are ill-defined representations of numbers.
The following articles:
Proof that 0.999... is not equal to 1
AND
Magnitude and number
can be found at:
https://www.filesanywhere.com/fs/v.aspx?v=8b69658b5d62707cb3a5
http://johngabrie1.wix.com/newcalculus
John Gabriel
Hi John,
Thank you for the information and welcome to the club.
In the scale of 1 to 10 (ten highest) I rate you 10 on creativity. Congratulations! Are you a Filpino?
Hi,
No, I am Greek. But it is always refreshing to read about others who think for themselves. And while I don't necessarily agree with everything that others say, I always believe it is worth a first, second and third look. I never close the door to any knowledge and by so doing, consider it past further investigation. You have some very interesting ideas. Perhaps all I can suggest is that you try to well-define all of these so that they can withstand the arrows of academic criticism and ignorance. :-)
Nice to make your acquaintance Dr. Escultura!
Hi John,
I feel that I have well defined all the concepts in my work. In fact, one of my requirements of a mathematical space is: every concept is well defined by the axioms. In other words, the traditional practice of admitting undefined symbols (aside from linguistic artifacts) is inadmissible for it introduces ambiguity in a mathematical space. While undefined symbols (concepts)may be introduced initially, the the choice of the axioms in the construction of a mathematical space is not complete until all the concepts are well defined. Therefore, I would appreciate it if someone points to a concept in my mathematics that is ill-defined so that I can quickly define it.
I suppose that by rational you mean the quotient of two integers or a fraction where both the numberator and denominator are integers. If that is so, I would point out that when the divisor or denominator has a prime factor other than 2 or 5, the fraction is ill-defined for the division when carried out yields a nonterminating decimal in which case not all its digits are known; this is an ambiguity. Only a rational that equals a terminating decimal is well defined.
Details about my work (e.g., the new real numbers system in pdf format) are found on my website,
http://users.tpg.com.au/pidro/
I commend your originaliy and creativity and I suggest you publish your work in new and upgraded peer reviewed journals, say, not more than 40 years old, so that it will rise above the bloggers and will not remain burried in the archives.
I really welcome this exchange of ideas.
Cheers,
EEE
Larry, it seems to me that the proof given that 1 = .999..... is flawed and cannot therefore, as I see it, be used by anyone to draw deep philosophical deductions about number groups that rely on it. It has been assumed that the number of 9's after the decimal point is the same after multiplication as it was before and therefore cancel out after subtraction. So it has been assumed that infinity -1 is equal to infinity but it can be demonstrated that this is not so by [1] a practical demonstration & [2] numerically on an electronic calculator in similar manner to the given proof.
@Alastair Bateman: It is very refreshing to read a comment from another like-minded academic.
Truth is, I don't know of any philosophical or mathematical deductions about number groups that have relied on this fallacy.
http://thenewcalculus.weebly.com
Larry, for my own peace of mind I would like to clarify my comment that I considered 1 = .999... as a proof was flawed. I am well aware that 0.999... is an infinite decimal expansion for 1 using precisely the same algorithm used for infinite decimal fractions. As such I considered it an approximation rather than an equality and as such feel that it should be accompanied by a statement such as ' When the number of decimal places tends to infinity then 0.999... approaches closer and closer to 1. Perhaps I was being just a bit to pedantic!!
To John Gabriel, thanks for the accolade but no I'm not an academic, just a member of 'Joe Public' who in my opinion have had the wool pulled over their eyes regarding a simple understanding of FLT, which is that z^2 = y^2 + x^n which of course is just Pythagoras's equation z^2 = y^2 + [x^(n/2)]^2. This might be considered an explanation rather than a proof but it was as far as I'm concerned what prompted Pierre de Fermat to make his marginal note and not a 'Eureka' moment as is often stated.
This comment has been removed by the author.
"Note that nonterminating decimals cannot be added or multiplied because these operations need the last digits of the numbers concerned."
What are you talking about? 1/3 and 2/3 result in "nonterminating decimals" yet are easily added and multiplied. 1/3 + 2/3 = 1. There, and I did it.
Or are you claiming that 1/3 is not equal to 0.33333...?
Your first paragraph says that periodic decimals are well-defined yet you still maintain that they cannot be added or multiplied because they are unterminating,
"We cannot even enter a nonterminating decimal into our calculator or pc because it can only register finite digits"
So what? How does the inadequacy of a user interface mean that a number is invalid or ambiguous. First, try to compute 100000! on a calculator. I don't know of any calculator that can calculate or Display such a large result. Does this limitation of the calculator mean that 100000! does not result in a well-defined unambiguous result? Of course not. Using your logic, an abacus user can similarly use that instrument's limitations to invalidate even more numbers.
Secondly, many calculators and software allow users to enter fractions directly. For example, the Texas Instruments TI-30X IIS, which only costs $10US.
The bottom line is that the inadequacy of technology -- or our language -- has no bearing on whether a quantity is well-defined or manipulatable mathematically.
@Bryguy:
"What are you talking about? 1/3 and 2/3 result in "nonterminating decimals" yet are easily added and multiplied. 1/3 + 2/3 = 1. There, and I did it."
You did it with 1/3 and 2/3, NOT 0.333... and 0.666... BIG DIFFERENCE!
All the operations of arithmetic are finite. That these fractions have repeating patterns is a property of the fact that they are NOT measurable in the given base. In fact, only an idiot would think that 0.333... is the measure of 1/3 in base 10. A well known number theorem prevents this folly:
Given any p/q, and some base b, then p/q is measurable in base b, IF and ONLY IF, base b contains ALL the prime factors of q.
Gee, that theorem means you can't represent 1/3 in base 10.
But, you may say, 0.333... is the infinite and unique radix representation of 1/3 in base 10. To which I respond:
You're an idiot because infinity is a junk concept. I can show you innumerably many rational numbers with exactly the same digits in the same first n places as 0.333..., no matter how large your n gets. So I'm trying to stimulate your peanut brain to think and arrive at the conclusion that even 0.333... if it were possible, would never be a unique representation. Contrary to the bullshit you have been brainwashed to believe in real analysis by your orangutan teachers, there is NO such thing as an irrational number and hence no real number.
Read my article to learn how Euler blundered:
https://www.linkedin.com/pulse/eulers-worst-definition-lim-john-gabriel
Before you post again, try THINKING!
I am the great John Gabriel, discoverer of the first and only rigorous formulation of calculus in human history:
http://thenewcalculus.weebly.com
https://www.youtube.com/channel/UClBbBVLs3M-d3dNgU4Vop_A
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