Thursday, May 17, 2007
JSH: What a paper
Ok, so for those who don't know, all the arguing that I do actually has a purpose as when I talk things out I learn better ways to explain and sometimes, I gain a better understanding myself of my own research.
That is what has happened in the last few days as yes, I do write papers, and as I've talked about how I use identities, which I call tautological spaces, and subtract from them, I've had various ideas which have gained fruition in my latest paper.
What I finally realized is that the question of what ring you are in had to ultimately rest on what was being subtracted from the identities as the identities cannot change the ring!
Identities are just like x=3Dx, or x+y =3D x+y and they are ALWAYS TRUE so they CANNOT CHANGE THE RING, so if your ring is changing then the identity cannot change it.
With that focused in my mind I could go back to an early draft of the current paper, and reverse the techniques I use to find what was being subtracted from the identities, and amazingly enough it was very particular about when it could be in the ring of algebraic integers.
Here it is:
r^2 + rs =E2=88=92 (2 + 2xt + tQ(x))st + s^2 =3D (2x + Q(x))t^2
That is the conditional that I have been using without knowing it before as I hadn't reversed to get it, which shows when you can have r, s and t all in the ring of algebraic integers with algebraic integer x.
To get the full paper go to my Extreme Mathematics group:
http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper
Also there is a link to my original paper that got published and retracted, where I was focused on cubics, so it's more complicated, and I didn't give how I derived key expressions, though now I can tell you I derived them from what I call tautological spaces—identities—which are true in the ring of algebraic integers.
So that's all. That closes the door. No longer can anyone argue that there is any other possibility than that the ring of algebraic integers is flawed and my research is clearly correct.
The consequences are HUGE as over a hundred years of number theory is affected.
Identities people. If you have any real mathematical bones within you, then you cannot believe that identities are controlling the ring, so what I'm subtracting from them MUST be controlling the ring, and given that you can start in the ring of algebraic integers and provably be pushed out of it, it MUST be true that the ring of algebraic integers has special problems.
No room for doubt—if you really care about what is mathematically true.
Now I've been arguing over this for years, and I'm writing papers, even got published and I'm explaining, and trying, so it's not about me. If this paper cannot convince you, then it's about you.
[A reply to someone who wrote that if the paper written by Wiles about FLT cannot convince James, then it's not about Wiles, it's about James.]
Except I've specifically addressed problems with Wiles's paper, having actually looked it over.
Posters in the past have claimed problems with my research and for years I have worked at answering them, which is what this latest paper I'm announcing does.
I've removed several areas from previous arguments where posters went after my research, eliminating talk of constant terms, values at 0, and I've derived my important polynomial expressions completely to emphasize that I subtract from identities and to show what is being subtracted from identities.
Of course identities are expressions like
x=x
or
x+y = x+y
which cannot of themselves change the ring, as they are always true.
The nature of the current paper is to directly answer any and all objections raised against this approach by giving a lot of detail.
Wiles in contrast has a paper I can tear apart in multiple ways.
His research is just wrong, as I've repeatedly explained before in talking about the errors in it.
[A reply to someone who wrote that obviously James doesn't have the necessary background knowledge to evaluate Wiles' argument.]
Well there are multiple ways including this latest paper which tears away the tools that people like Wiles use, as they rely on flawed beliefs about the ring of algebraic integers.
Now I think support of Wiles despite my latest paper is just simple hero worship.
I wrote that paper quite deliberately not only to better understand my own work, but also to shut down objectors like yourself.
Wiles does not have a proof. My own research pulls the rug out from under the techniques he relied upon and I use simple enough mathematics that the cry against my training is answered by my saying, check the math.
It is just such a good feeling to have that paper done! And I was myself surprised by a couple of things, as I deduced you should be able to reverse technique to get to the base conditional to handle the generalized argument I used, but I was surprised by what it looked like!
r^2 + rs − (2 + 2xt + tQ(x))st + s^2 = (2x + Q(x))t^2
That's not what I would have guessed was the underlying expression, which I call the conditional.
But what it does is control the factorization of
175x^2 − 15x + 2 = 2(f(x) + 1)* (g(x) + 1)
by picking the functions f(x) and g(x) by what Q(x) is chosen so that every possible factorization is covered.
The addition to the state of the art being the consideration of non-polynomial factors, as before my research, what mathematicians would think to factor that quadratic into anything other than polynomial factors?
And it is just remarkable that the full set of possible functions f(x) and g(x) are controlled by the expression
r^2 + rs − (2 + 2xt + tQ(x))st + s^2 = (2x + Q(x))t^2
as who would have guessed that?
The paper is a fascinating one, and I'm finding myself going over it again and again, which is also a good thing as I'm still doing changes to it, as I clarify and correct any errors I might find.
It is just so incredible that such complexity was wrapped up in factoring even simple quadratics, if you cover ALL possible factorizations versus being focused on polynomial ones, yet still all of the complexity—an infinite amount as there are an infinite number of factorizations—can be handled by a single expression.
The advance to mathematics is akin to when people learned to go from factoring integers into prime factors, to learning that there were non-rational factors as well.
The leap is bringing to algebra and polynomials what was brought to numbers and integers before.
Ok, yeah, I'm going on and on, but it is just so cool.
[A reply to someone who asked James to name an accepted theorem about the algebraic integers which Wiles uses and which is false.]
The flaw in the ring of algebraic integers allows a person to "prove" things that are not true based on factors results in that ring that give a false implication.
Once you realize that ring is flawed then it IS possible for the root of monic quadratic with integer coefficients to be both non-rational, and have some prime number like 7 as a factor, while the other root does not.
Once you realize that then you realize that Galois Theory tells you nothing more about non-rationals than it does about rationals, so you lose Galois Theory for any real number theory research.
So, the short answer is, my research takes away ANY value in Galois Theory for number theory research.
It just cannot tell you anything more about non-rationals than it can about rationals.
And take away Galois Theory, and you take away Wiles's argument.
Denial is not about mathematics. It's not even hard at this point to understand what's wrong with the ring of algebraic integers as the evens with 2 and 6, where 3 is excluded if you only consider evens so then 2 is coprime to 6, gives the full gist of the problem. So there is an easy example to help you grasp how the ring is flawed—it arbitrarily excludes numbers like if you say take only evens so that you find that 2 is coprime to 6.
The specifics are that the definition of algebraic integers as roots of monic polynomials with integer coefficients arbitrarily excludes numbers in a way that provably can lead to "proof" that is not actual proof.
So it's a way to believe you have proved just about anything, and Wiles is just one more person in a long line of people who have used wrong mathematical ideas and believed he had a proof when he did not, and it does not matter how many other people believed him or believe in him as the mathematics does not care.
If you think for one moment that mathematical truth cares about the misery you might feel now after reading over my paper and realizing there is no way to object then you are so far from being a real mathematician that you are getting a major favor by learning how wrong you have been before.
Mathematical discovery is about the search for truth, so sometimes people wander down the wrong path, and can do so for a while, but if you believe that truth is actually important, then you can be thankful when you are turned back.
But that is not necessarily easy.
And if you want easy, go find something else to do.
The math tells the tale. My paper closes the door on mathematical objections.
If you wish to turn to human interest ones that is a choice you can make. Worry about how you feel or how everyone should feel and curse mathematical discovery.
And then think about where our species would be if people like you ever won.
You tried to win here and hold back number theory. If people like you had won thousands of years ago we would not have algebra today.
And without algebra we would not have human civilization.
You have to lose no matter what you feel so that there IS a future.
These kind of battles are always about the fate of the species.
It's for all the marbles. When ignorance wins, we all lose.
That is what has happened in the last few days as yes, I do write papers, and as I've talked about how I use identities, which I call tautological spaces, and subtract from them, I've had various ideas which have gained fruition in my latest paper.
What I finally realized is that the question of what ring you are in had to ultimately rest on what was being subtracted from the identities as the identities cannot change the ring!
Identities are just like x=3Dx, or x+y =3D x+y and they are ALWAYS TRUE so they CANNOT CHANGE THE RING, so if your ring is changing then the identity cannot change it.
With that focused in my mind I could go back to an early draft of the current paper, and reverse the techniques I use to find what was being subtracted from the identities, and amazingly enough it was very particular about when it could be in the ring of algebraic integers.
Here it is:
r^2 + rs =E2=88=92 (2 + 2xt + tQ(x))st + s^2 =3D (2x + Q(x))t^2
That is the conditional that I have been using without knowing it before as I hadn't reversed to get it, which shows when you can have r, s and t all in the ring of algebraic integers with algebraic integer x.
To get the full paper go to my Extreme Mathematics group:
http://groups.google.com/group/extrememathematics/web/non-polynomial-factorization-paper
Also there is a link to my original paper that got published and retracted, where I was focused on cubics, so it's more complicated, and I didn't give how I derived key expressions, though now I can tell you I derived them from what I call tautological spaces—identities—which are true in the ring of algebraic integers.
So that's all. That closes the door. No longer can anyone argue that there is any other possibility than that the ring of algebraic integers is flawed and my research is clearly correct.
The consequences are HUGE as over a hundred years of number theory is affected.
Identities people. If you have any real mathematical bones within you, then you cannot believe that identities are controlling the ring, so what I'm subtracting from them MUST be controlling the ring, and given that you can start in the ring of algebraic integers and provably be pushed out of it, it MUST be true that the ring of algebraic integers has special problems.
No room for doubt—if you really care about what is mathematically true.
Now I've been arguing over this for years, and I'm writing papers, even got published and I'm explaining, and trying, so it's not about me. If this paper cannot convince you, then it's about you.
[A reply to someone who wrote that if the paper written by Wiles about FLT cannot convince James, then it's not about Wiles, it's about James.]
Except I've specifically addressed problems with Wiles's paper, having actually looked it over.
Posters in the past have claimed problems with my research and for years I have worked at answering them, which is what this latest paper I'm announcing does.
I've removed several areas from previous arguments where posters went after my research, eliminating talk of constant terms, values at 0, and I've derived my important polynomial expressions completely to emphasize that I subtract from identities and to show what is being subtracted from identities.
Of course identities are expressions like
x=x
or
x+y = x+y
which cannot of themselves change the ring, as they are always true.
The nature of the current paper is to directly answer any and all objections raised against this approach by giving a lot of detail.
Wiles in contrast has a paper I can tear apart in multiple ways.
His research is just wrong, as I've repeatedly explained before in talking about the errors in it.
[A reply to someone who wrote that obviously James doesn't have the necessary background knowledge to evaluate Wiles' argument.]
Well there are multiple ways including this latest paper which tears away the tools that people like Wiles use, as they rely on flawed beliefs about the ring of algebraic integers.
Now I think support of Wiles despite my latest paper is just simple hero worship.
I wrote that paper quite deliberately not only to better understand my own work, but also to shut down objectors like yourself.
Wiles does not have a proof. My own research pulls the rug out from under the techniques he relied upon and I use simple enough mathematics that the cry against my training is answered by my saying, check the math.
It is just such a good feeling to have that paper done! And I was myself surprised by a couple of things, as I deduced you should be able to reverse technique to get to the base conditional to handle the generalized argument I used, but I was surprised by what it looked like!
r^2 + rs − (2 + 2xt + tQ(x))st + s^2 = (2x + Q(x))t^2
That's not what I would have guessed was the underlying expression, which I call the conditional.
But what it does is control the factorization of
175x^2 − 15x + 2 = 2(f(x) + 1)* (g(x) + 1)
by picking the functions f(x) and g(x) by what Q(x) is chosen so that every possible factorization is covered.
The addition to the state of the art being the consideration of non-polynomial factors, as before my research, what mathematicians would think to factor that quadratic into anything other than polynomial factors?
And it is just remarkable that the full set of possible functions f(x) and g(x) are controlled by the expression
r^2 + rs − (2 + 2xt + tQ(x))st + s^2 = (2x + Q(x))t^2
as who would have guessed that?
The paper is a fascinating one, and I'm finding myself going over it again and again, which is also a good thing as I'm still doing changes to it, as I clarify and correct any errors I might find.
It is just so incredible that such complexity was wrapped up in factoring even simple quadratics, if you cover ALL possible factorizations versus being focused on polynomial ones, yet still all of the complexity—an infinite amount as there are an infinite number of factorizations—can be handled by a single expression.
The advance to mathematics is akin to when people learned to go from factoring integers into prime factors, to learning that there were non-rational factors as well.
The leap is bringing to algebra and polynomials what was brought to numbers and integers before.
Ok, yeah, I'm going on and on, but it is just so cool.
[A reply to someone who asked James to name an accepted theorem about the algebraic integers which Wiles uses and which is false.]
The flaw in the ring of algebraic integers allows a person to "prove" things that are not true based on factors results in that ring that give a false implication.
Once you realize that ring is flawed then it IS possible for the root of monic quadratic with integer coefficients to be both non-rational, and have some prime number like 7 as a factor, while the other root does not.
Once you realize that then you realize that Galois Theory tells you nothing more about non-rationals than it does about rationals, so you lose Galois Theory for any real number theory research.
So, the short answer is, my research takes away ANY value in Galois Theory for number theory research.
It just cannot tell you anything more about non-rationals than it can about rationals.
And take away Galois Theory, and you take away Wiles's argument.
Denial is not about mathematics. It's not even hard at this point to understand what's wrong with the ring of algebraic integers as the evens with 2 and 6, where 3 is excluded if you only consider evens so then 2 is coprime to 6, gives the full gist of the problem. So there is an easy example to help you grasp how the ring is flawed—it arbitrarily excludes numbers like if you say take only evens so that you find that 2 is coprime to 6.
The specifics are that the definition of algebraic integers as roots of monic polynomials with integer coefficients arbitrarily excludes numbers in a way that provably can lead to "proof" that is not actual proof.
So it's a way to believe you have proved just about anything, and Wiles is just one more person in a long line of people who have used wrong mathematical ideas and believed he had a proof when he did not, and it does not matter how many other people believed him or believe in him as the mathematics does not care.
If you think for one moment that mathematical truth cares about the misery you might feel now after reading over my paper and realizing there is no way to object then you are so far from being a real mathematician that you are getting a major favor by learning how wrong you have been before.
Mathematical discovery is about the search for truth, so sometimes people wander down the wrong path, and can do so for a while, but if you believe that truth is actually important, then you can be thankful when you are turned back.
But that is not necessarily easy.
And if you want easy, go find something else to do.
The math tells the tale. My paper closes the door on mathematical objections.
If you wish to turn to human interest ones that is a choice you can make. Worry about how you feel or how everyone should feel and curse mathematical discovery.
And then think about where our species would be if people like you ever won.
You tried to win here and hold back number theory. If people like you had won thousands of years ago we would not have algebra today.
And without algebra we would not have human civilization.
You have to lose no matter what you feel so that there IS a future.
These kind of battles are always about the fate of the species.
It's for all the marbles. When ignorance wins, we all lose.
# posted by James Harris @ 02:46 
