Sunday, October 26, 2003
Explaining math definition problem
I'm an independent researcher, which means that I use my own funding, and my own direction to go out and see what knowledge I can obtain. Some of my research has been in the area of mathematics.
Getting important research findings is one thing, and getting them noticed, is another.
At least here on Usenet I can talk freely to people around the world.
What I'd like to explain is my disturbing and to me fascinating finding of a problem with a math definition that's over a hundred years old. In looking over various replies to my previous posts on this subject, I've seen assertions that definitions can't cause problems, which is something that I can address quickly at the start.
Over a hundred years ago, the great German mathematicians Karl Gauss played with numbers of the form a+bi, where 'a' and 'b' are integers. In his honor they were later called gaussian integers, though a number like 1+2i is not an integer. The "gaussian" up-front is important. Later mathematicians came up with other numbers they called algebraic integers, which include gaussian integers.
They thought they'd found THE set, or superset you might call it, which includes all numbers with certain special properties of integrality.
The most important property to point out is the ability to have primeness between numbers.
For instance, with integers, 2 and 3 are coprime, that is, they don't share non-unit factors, that is, factors other than 1 or -1, with each other.
Just be clear here, factors of 1, are called units or unit factors.
But notice that with rationals, you have 2(3/2) = 3, so 2 and 3 do share a factor and are not coprime in that ring, which is typically called a field because every element except 0, has a multiplicative inverse.
What Gauss had started considering, which other mathematicians extended, was the idea of sets of numbers where you kept interesting properties of the set of integers, like being able to say two numbers were coprime.
What I've found is a problem with their set of algebraic integers, as unfortunately, despite what many mathematicians think, it's too small.
That's it. The definition they use is too small to do what they think it does, which is include all these interesting numbers with special properties.
But because they think it's big enough, mathematicians have an error in their discipline based on their false assumption, as they've come up with more arguments based on that assumption, which then aren't actually proven.
It's like when the Greeks with their word "atom" thought they had the smallest thing, and later our civilization used it, and broke atoms apart, though part of the definition is that they are indivisible, as people can define things, and later refine their definitions.
Now my research finding isn't hard to show quickly in broad strokes.
On of my important analysis tools is a simple technique to factor polynomials into non-polynomial factors.
For instance, with the polynomial
P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078
that technique gives you
P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3
so I can factor to get
P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7).
where the a's are the roots of the cubic
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x).
Now despite the complexity, you can rely on simple ideas still, by noticing that setting x=0, pulls out constant terms, as
P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22)
as the cubic defining the a's with x=0 is
a^3 - 3a^2, which has roots, 0, 0 and 3.
You may not realize it, but what you just saw is revolutionary, both in the special techniques, and most importantly with the consequences that quickly follow.
That's because P(x) has another special feature as
P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22)
where that 49 is just begging to be divided off, which gives, of course,
P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22.
But remember, my three factors with the a's from before had constant terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, 1, and 22, which is the result that is so earth shattering.
Here the principle is like if you have
S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1)
in that setting x=0 gives you constant terms within the expression, which you can conveniently, also look at to see how it works.
S(0) = (7(0) + 7)(0 + 1) = 7(1).
The point is that the 7 is constant, so x's value means nothing to it.
So from before with
P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22)
I know that dividing through by 49, it must go like
P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22)
and as the constant terms are independent of the value of x, it MUST be that in general
P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7).
The problem now though is that conclusion can be used to show that unequivocally beyond any reasonable doubt the definition of algebraic numbers is TOO SMALL, as at times 5a_1/7 and 5a_2/7 are not included.
You see, they get left out, which is a problem because from the assumption of mathematicians, they should be included, if the ring of algebraic integers is the ring that mathematicians thought it was.
Some of you may find yourselves fearful of using your own mathematical understanding, if you realize I'm right, and then realize that mathematicians are disputing the result, especially if you see posters tossing out far more complicated math in reply to my post, but remember, math isn't magic.
Logic rules mathematics, so look for what makes sense. And remember that you can't assume that posters are on your side. I don't want you to assume that I'm on your side either.
You see, I don't need you to assume anything, as what I need you to do is check.
While some mathematicians may erroneously believe now that it's in their interest to hide the problem I've revealed, that mistake in thinking does not help the rest of the world. After all, what good does it do everyone else for mathematicians to hide their definition problem?
What's in it for you?
Getting important research findings is one thing, and getting them noticed, is another.
At least here on Usenet I can talk freely to people around the world.
What I'd like to explain is my disturbing and to me fascinating finding of a problem with a math definition that's over a hundred years old. In looking over various replies to my previous posts on this subject, I've seen assertions that definitions can't cause problems, which is something that I can address quickly at the start.
Over a hundred years ago, the great German mathematicians Karl Gauss played with numbers of the form a+bi, where 'a' and 'b' are integers. In his honor they were later called gaussian integers, though a number like 1+2i is not an integer. The "gaussian" up-front is important. Later mathematicians came up with other numbers they called algebraic integers, which include gaussian integers.
They thought they'd found THE set, or superset you might call it, which includes all numbers with certain special properties of integrality.
The most important property to point out is the ability to have primeness between numbers.
For instance, with integers, 2 and 3 are coprime, that is, they don't share non-unit factors, that is, factors other than 1 or -1, with each other.
Just be clear here, factors of 1, are called units or unit factors.
But notice that with rationals, you have 2(3/2) = 3, so 2 and 3 do share a factor and are not coprime in that ring, which is typically called a field because every element except 0, has a multiplicative inverse.
What Gauss had started considering, which other mathematicians extended, was the idea of sets of numbers where you kept interesting properties of the set of integers, like being able to say two numbers were coprime.
What I've found is a problem with their set of algebraic integers, as unfortunately, despite what many mathematicians think, it's too small.
That's it. The definition they use is too small to do what they think it does, which is include all these interesting numbers with special properties.
But because they think it's big enough, mathematicians have an error in their discipline based on their false assumption, as they've come up with more arguments based on that assumption, which then aren't actually proven.
It's like when the Greeks with their word "atom" thought they had the smallest thing, and later our civilization used it, and broke atoms apart, though part of the definition is that they are indivisible, as people can define things, and later refine their definitions.
Now my research finding isn't hard to show quickly in broad strokes.
On of my important analysis tools is a simple technique to factor polynomials into non-polynomial factors.
For instance, with the polynomial
P(x) = 14706125 x^3 - 900375 x^2 + 17640 x + 1078
that technique gives you
P(x)= 7^2(2401 x^3 - 147 x^2 + 3x) (5^3) - 3(-1 + 49 x )(5)(7^2) + 7^3
so I can factor to get
P(x) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7).
where the a's are the roots of the cubic
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x).
Now despite the complexity, you can rely on simple ideas still, by noticing that setting x=0, pulls out constant terms, as
P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22)
as the cubic defining the a's with x=0 is
a^3 - 3a^2, which has roots, 0, 0 and 3.
You may not realize it, but what you just saw is revolutionary, both in the special techniques, and most importantly with the consequences that quickly follow.
That's because P(x) has another special feature as
P(x) = 49(300125 x^3 - 18375 x^2 + 360 x + 22)
where that 49 is just begging to be divided off, which gives, of course,
P(x)/49 = 300125 x^3 - 18375 x^2 + 360 x + 22.
But remember, my three factors with the a's from before had constant terms of 7, 7 and 22, so dividing by 49 must give constant terms of 1, 1, and 22, which is the result that is so earth shattering.
Here the principle is like if you have
S(x) = 7x^2 + 14x + 7 = (7x+7)(x+1)
in that setting x=0 gives you constant terms within the expression, which you can conveniently, also look at to see how it works.
S(0) = (7(0) + 7)(0 + 1) = 7(1).
The point is that the 7 is constant, so x's value means nothing to it.
So from before with
P(0) = (5(0) + 7)(5(0) + 7)(5(3) + 7) = 7(7)(22)
I know that dividing through by 49, it must go like
P(0)/49 = (5(0)/7 + 1)(5(0)/7 + 1)(5(3) + 7) = 1(1)(22)
and as the constant terms are independent of the value of x, it MUST be that in general
P(x)/49 = (5a_1/7 + 1)(5a_2/7 + 1)(5a_3 + 7).
The problem now though is that conclusion can be used to show that unequivocally beyond any reasonable doubt the definition of algebraic numbers is TOO SMALL, as at times 5a_1/7 and 5a_2/7 are not included.
You see, they get left out, which is a problem because from the assumption of mathematicians, they should be included, if the ring of algebraic integers is the ring that mathematicians thought it was.
Some of you may find yourselves fearful of using your own mathematical understanding, if you realize I'm right, and then realize that mathematicians are disputing the result, especially if you see posters tossing out far more complicated math in reply to my post, but remember, math isn't magic.
Logic rules mathematics, so look for what makes sense. And remember that you can't assume that posters are on your side. I don't want you to assume that I'm on your side either.
You see, I don't need you to assume anything, as what I need you to do is check.
While some mathematicians may erroneously believe now that it's in their interest to hide the problem I've revealed, that mistake in thinking does not help the rest of the world. After all, what good does it do everyone else for mathematicians to hide their definition problem?
What's in it for you?
# posted by James Harris @ 11:07 
