Friday, September 10, 2010
JSH: Using rationals to understand the error
Mathematics is easy, explaining to people who hate a result, can be damned hard. One bizarre thing that has emerged in the last couple of days is that using the ring of integers and also the field of rationals with 9 instead of 7, can block all the various tricks that posters have used to try and obfuscate a stunning error at the heart of modern number theory.
Damn strange though. Problem seems to be that a lot of people focus on 7 being prime, when that is irrelevant. So I use 9, and it flummoxes posters. Oddity of the human brain. So now I'll explain one of the most far-reaching and tragic errors in human thought.
Consider in integers:
9(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)
where g_1(0) = g_2(0) = 0, and one of the f's equals 0, at x=0, and P(x) is a quadratic with integer coefficients.
The 9 is forced into one of the f's, though symmetry actually means you do not know which one.
By the distributive property, if you have 9(g_1(x) + 1) = (9*g_1(x) + 9) = (f_1(x) + 9) choosing f_1 to match indices, you know that x=0 is not a special case, and that the result holds for all x.
The issue may seem trivial, but those people who argue with me incessantly want the 9 to be able to split up. So that maybe one of the f's has 3 as a factor, for certain values of x, but that is nonsensical by the distributive property.
One way to FORCE a split is to use something like:
9(g_1(x) + 1)(g_2(x) + 2) = (3f_1(x) + 9)(3f_2(x) + 9) = 9*P(x)
which pushes you out of integers and into rationals.
Notice that FORCING 3 to be a factor of both, splitting up the 9 by human conscious effort, makes the mathematics shift out of the ring of integers, as it has no choice. You've then given it no choice.
So what does any of the above have to do with error?
Well if you understand why 9 is a factor of just one of the f's and realize it's not about factorization, but about the distributive property, well the distributive property is valid all the way up to the field of complex numbers, so the result should hold all the way up to the field of complex numbers, and it does hold in the field of complex numbers.
But it does NOT always hold in the ring of algebraic integers.
Proving that took some clever algebraic manipulations, where I used non-polynomial factors.
And that is the easy explanation.
So the result holds by the distributive property through the complex plane with one exception being the ring of algebraic integers. Explaining exactly what happens there is where things get actually interesting as that ring has to do some clever things to stick with mathematical rules!
And those things are not hard to understand. The 9 can't split up, so the ring of algebraic integers masks it with factors that behave as units, though because of an arbitrary rule they are NOT units in the ring of algebraic integers.
What arbitrary rule?
Well the rule that algebraic integers are roots of monic polynomials with integer coefficients induces some subtle requirements, for instance it prevents any non-rational unit in the ring of algebraic integers from being the root of anything but a monic polynomial with integer coefficients where the last is 1 or -1.
Notice integers do not have that problem as consider:
x^2 + 3x + 2 = (x+2)(x+1)
where a unit is paired with a non-unit in the ring of integers.
The corresponding case with non-rational algebraic integer solutions is NOT allowed by the ring of algebraic integers.
And that is weird. It turns out that having only roots of monic polynomials with integer coefficients leaves out numbers, and is a display property, which leads us to Galois Theory.
Galois Theory is correct in showing you what solutions can be displayed by integers, +, -, and radicals. And is correct in telling you when numbers cannot be so displayed. But that is just a display issue.
To the mathematics it's a non-issue as the math doesn't see sqrt(7). It sees two numbers that give 7 when squared. Human beings cannot get beyond the square root there, but may say +/- sqrt(7) or put up rational approximations like:
sqrt(7) approximately equals: 2.64575131
So to us it's a big deal if we can use radical or not, but to the mathematics it means nothing, so the mathematics doesn't see Galois Theory as doing anything at all.
From today's perspective though it kind of makes sense that over a hundred years ago to the mathematicians then, it may have seemed a big deal whether or not you could actually display a solution, which is why the quintic proof was a big deal, but to us it really doesn't mean much. Like, even with the general solution for cubics or quartics the answers you get are fairly useless and can actually be integers, obscured by radicals.
So why would people argue with me incessantly for years—I first discovered this problem in 2002—and modern mathematicians continue as if the issue does not exist?
Good question! Supposedly they wouldn't. But they have. More remarkably as my body of research has grown the protests by Usenet posters against it have remained steady, but spread to fairly odd things like claiming that Google search results are meaningless.
And those things are scary in a way. We have a world of 6.8 billion people. A lot of them are now on-line. In that world, my research is often coming out as #1. I look at search results where my ideas are at the top out of hundreds of millions of searches, and it's a weird feeling.
So what gives? How come I'm not famous? My best guess is that, um, I actually am? But that social order is more important than individual human beings. For instance, how many of you are worth, say, your country? None of you are, which is why we have soldiers. People go to fight to die for the good of the whole.
The disruptive social impact of this result—like on people's opinions of universities and university professors—seems for the moment to be balanced against the value society sees in keeping the current ideas about same in place!
So it's more important to world society I'm guessing to keep universities valued as they are, rather than have correct mathematics in this area, and the students sacrificed in mathematics aren't nearly important enough in that equation.
And I'm not losing anything myself as I have the freedom to keep researching and live a normal life, when by all rights I should be one of the most famous people on the planet now, and living a really abnormal one.
So the sacrifices here are the college students. The world considers them less valuable than the social systems that are in place, but over time the information is going to be more valuable. But it's up to the world to balance that equation.
Worse though for me is that I get bigger as a historical figure the longer it takes! So I'm less inclined to bother with it all anyway, but am probably stuck at this point.
So yeah, your world is smarter than any and all of you. And it has a world to run. None of you are worth the entire world in the balance so any one or all of you can be potentially sacrificed by your world for its own benefit.
After all, it can always make more people!
The cold calculation is not refutable.
Damn strange though. Problem seems to be that a lot of people focus on 7 being prime, when that is irrelevant. So I use 9, and it flummoxes posters. Oddity of the human brain. So now I'll explain one of the most far-reaching and tragic errors in human thought.
Consider in integers:
9(g_1(x) + 1)(g_2(x) + 2) = (f_1(x) + 9)(f_2(x) + 9) = 9*P(x)
where g_1(0) = g_2(0) = 0, and one of the f's equals 0, at x=0, and P(x) is a quadratic with integer coefficients.
The 9 is forced into one of the f's, though symmetry actually means you do not know which one.
By the distributive property, if you have 9(g_1(x) + 1) = (9*g_1(x) + 9) = (f_1(x) + 9) choosing f_1 to match indices, you know that x=0 is not a special case, and that the result holds for all x.
The issue may seem trivial, but those people who argue with me incessantly want the 9 to be able to split up. So that maybe one of the f's has 3 as a factor, for certain values of x, but that is nonsensical by the distributive property.
One way to FORCE a split is to use something like:
9(g_1(x) + 1)(g_2(x) + 2) = (3f_1(x) + 9)(3f_2(x) + 9) = 9*P(x)
which pushes you out of integers and into rationals.
Notice that FORCING 3 to be a factor of both, splitting up the 9 by human conscious effort, makes the mathematics shift out of the ring of integers, as it has no choice. You've then given it no choice.
So what does any of the above have to do with error?
Well if you understand why 9 is a factor of just one of the f's and realize it's not about factorization, but about the distributive property, well the distributive property is valid all the way up to the field of complex numbers, so the result should hold all the way up to the field of complex numbers, and it does hold in the field of complex numbers.
But it does NOT always hold in the ring of algebraic integers.
Proving that took some clever algebraic manipulations, where I used non-polynomial factors.
And that is the easy explanation.
So the result holds by the distributive property through the complex plane with one exception being the ring of algebraic integers. Explaining exactly what happens there is where things get actually interesting as that ring has to do some clever things to stick with mathematical rules!
And those things are not hard to understand. The 9 can't split up, so the ring of algebraic integers masks it with factors that behave as units, though because of an arbitrary rule they are NOT units in the ring of algebraic integers.
What arbitrary rule?
Well the rule that algebraic integers are roots of monic polynomials with integer coefficients induces some subtle requirements, for instance it prevents any non-rational unit in the ring of algebraic integers from being the root of anything but a monic polynomial with integer coefficients where the last is 1 or -1.
Notice integers do not have that problem as consider:
x^2 + 3x + 2 = (x+2)(x+1)
where a unit is paired with a non-unit in the ring of integers.
The corresponding case with non-rational algebraic integer solutions is NOT allowed by the ring of algebraic integers.
And that is weird. It turns out that having only roots of monic polynomials with integer coefficients leaves out numbers, and is a display property, which leads us to Galois Theory.
Galois Theory is correct in showing you what solutions can be displayed by integers, +, -, and radicals. And is correct in telling you when numbers cannot be so displayed. But that is just a display issue.
To the mathematics it's a non-issue as the math doesn't see sqrt(7). It sees two numbers that give 7 when squared. Human beings cannot get beyond the square root there, but may say +/- sqrt(7) or put up rational approximations like:
sqrt(7) approximately equals: 2.64575131
So to us it's a big deal if we can use radical or not, but to the mathematics it means nothing, so the mathematics doesn't see Galois Theory as doing anything at all.
From today's perspective though it kind of makes sense that over a hundred years ago to the mathematicians then, it may have seemed a big deal whether or not you could actually display a solution, which is why the quintic proof was a big deal, but to us it really doesn't mean much. Like, even with the general solution for cubics or quartics the answers you get are fairly useless and can actually be integers, obscured by radicals.
So why would people argue with me incessantly for years—I first discovered this problem in 2002—and modern mathematicians continue as if the issue does not exist?
Good question! Supposedly they wouldn't. But they have. More remarkably as my body of research has grown the protests by Usenet posters against it have remained steady, but spread to fairly odd things like claiming that Google search results are meaningless.
And those things are scary in a way. We have a world of 6.8 billion people. A lot of them are now on-line. In that world, my research is often coming out as #1. I look at search results where my ideas are at the top out of hundreds of millions of searches, and it's a weird feeling.
So what gives? How come I'm not famous? My best guess is that, um, I actually am? But that social order is more important than individual human beings. For instance, how many of you are worth, say, your country? None of you are, which is why we have soldiers. People go to fight to die for the good of the whole.
The disruptive social impact of this result—like on people's opinions of universities and university professors—seems for the moment to be balanced against the value society sees in keeping the current ideas about same in place!
So it's more important to world society I'm guessing to keep universities valued as they are, rather than have correct mathematics in this area, and the students sacrificed in mathematics aren't nearly important enough in that equation.
And I'm not losing anything myself as I have the freedom to keep researching and live a normal life, when by all rights I should be one of the most famous people on the planet now, and living a really abnormal one.
So the sacrifices here are the college students. The world considers them less valuable than the social systems that are in place, but over time the information is going to be more valuable. But it's up to the world to balance that equation.
Worse though for me is that I get bigger as a historical figure the longer it takes! So I'm less inclined to bother with it all anyway, but am probably stuck at this point.
So yeah, your world is smarter than any and all of you. And it has a world to run. None of you are worth the entire world in the balance so any one or all of you can be potentially sacrificed by your world for its own benefit.
After all, it can always make more people!
The cold calculation is not refutable.
# posted by James Harris @ 22:28 
