Thursday, January 01, 2004
JSH: Understanding the math
My research can be difficult to understand, so I thought I'd try out yet another way of explaining it. Some of you may have figured out that I test out explanations on Usenet for use elsewhere, to refine my own understanding, or just in case someone out there might finally get it.
Now then, again here's my discovery:
(5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)
where b_3(x) = a_3(x) - 3 and the a's are roots of
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)
and when x=0, a_1(0) = a_2(0) = b_3(0) = 0.
In that form it's hard to understand what follows next unless you pay attention to what you have, specifically that cubic defining the a's.
I can get it because of the symmetry of
(5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)
where I've gone ahead and substituted a_3(x) back in to replace b_3(x), and it's important that you focus on that symmetry.
It's that symmetry which allows the cubic
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)
to define ALL the a's, but something happens when I divide by 49.
Then the symmetry is broken. Without that symmetry it's impossible to find a SINGLE cubic to handle what results when you divide both sides by 49.
That's important because it's why the functions are NOT algebraic integer functions!!!
Now then, I'll recap. Symmetry allows the a's to be defined by a cubic, which shows them to be algebraic integer functions, but dividing by 49 *breaks* that symmetry, taking away the ability to find some cubic to define the results, which proves that the resulting functions are not algebraic integer functions.
(5 b_1(x) + 1)(5 b_2(x) + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22
where the b's are roots of
b^3 + ? b^2 + ? b - (2401 x^3 - 147 x^2 + 3x)
and when x=0, b_1(0) = b_2(0) = b_3(0) = 0.
My point is that the second and third coefficients are impossible to define in general.
You may find them for some particular x, but in general, they are forever hidden from you.
Notice that doing that substitution with a_3(x) for b_3(x) gives me
(5 b_1(x) + 1)(5 b_2(x) + 1)(5 a_3(x) + 7) = 300125 x^3 - 18375 x^2 - 360 x + 22
but you have broken symmetry since the other constant terms are 1 and 1, so you're still stuck.
Now by emphasizing what happens after 49 is divided from both sides I'm trying to get at least some of you to face the mathematical realities here, and I've made other posts pointing it out as well.
Courage in mathematics might sound new to you as an idea, but those who came before you had courage. That's how they learned of sqrt(2), and sqrt(-1), and much other mathematics that some refused, fought, and bitterly attacked as they lacked that courage.
Of course, the mathematics should win, assuming that humanity survives long enough, but regardless each of you now faces an individual test of your own courage. It's not yet in the history books this time, where you can read and just imagine that you'd never have fought over sqrt(2) or refused to accept sqrt(-1) as an "imaginary" number.
Here it's the present, and your test of courage is now.
Oh, and be sure to check my blog archives:
http://mathforprofit.blogspot.com/2003_10_01_mathforprofit_archive.html
And hey, if you're moping and miserable because mathematics tests you, then maybe, if you think you're a mathematician, you might want to try a different field.
[A reply to David C. Ullrich.]
Well sure, delete out the argument which settles the issue, and then come back and claim that it's not settled.
Now math history is full of people like David Ullrich, a guy with a title, fighting against some new idea. In the past it was sqrt(-1), as mathematicians fought against an idea they thought of as silly.
The hallmark of such people is a refusal to follow mathematical logic. You give a detailed mathematical position, and they just delete it all out, and claim it's not what it is.
Yet the facts speak for themselves:
My work depends on advanced mathematical tools for factoring polynomials, where I use one polynomial to factor another polynomial into non-polynomial factors.
I'd be curious if anyone can find anything like it in the recorded history of mathematics.
Along with that work, I have my prime number research, where I have a partial difference equation used to count prime numbers—a first in math history.
People like David Ullrich are the anti-knowledge crowd. Those people who know just enough to feel comfortable in a complex world. The kind of people who say, "no more knowledge".
Ullrich probably feels very comfortable with whatever mathematical knowledge he has. He can pay his bills using the money given him by taxpayers, as he's a professor at Oklahoma State University. He probably has lots of feedback from various people in society to make him feel good about himself.
What I represent is the unknowable future. The power that comes no matter how comfortable people are, no matter how satisfied they are with themselves or their positions—the power of change.
In that sense I'm a force of Nature, a force of the Universe, a living emodiment of change itself, so I understand the fear I can induce in people like David Ullrich.
But the spirit of mathematics is ultimately about what's not known, what's beyond what's known today, as that defines the work of the future.
I stand here part of a great tradition of discoverers, people who got past the Ullrich's to push forward knowledge so that today you have sqrt(2), i, e, e^x, planes, trains, automobiles, spacecraft, and computers.
It's too easy to look at the past as if it were easy. As if new ideas were just accepted because they were right, and could be proven logically, mathematically, or practically, so that you stand by today, forcing yet another discovery to break through.
But that is your loss, as my strength comes from the knowledge I have, the knowledge of those who came before me, the knowledge of those who will come after.
The knowledge that the truth is beautiful.
All of you can resist, fight until you die, as you will die someday. I can't look to any of you for permanence as you're in a sense, dead already. But I can, and you can look at the truth—beautiful, permanent, absolute.
The discoverers are the forces of Nature, born to a grand tradition, tasked with the seeming impossible, fighting throughout time, throughout history, to make history itself.
The discoverers are the living embodiment of the forces of Good against Evil. The people who push the limits of knowledge for the benefit of all, and the love of Truth.
We are the greatest warriors for Good, of all time, through all time, through all worlds and realities where sentient beings live.
We are the discoverers.
[A reply to someone who said that “History is also full of people that proposed ridiculous ideas which now are dead and buried”.]
Most people show worth through effort. For instance, if you value your job, one would assume that you put a lot of effort into it. Similarly, if you value a relationship, you're willing to work to keep it going.
People like Ullrich value trying to attack the new, as evidenced by his efforts in that regard.
There are a lot of people out there I probably would suppose are nutty, who have ideas that I don't care at all about, and then again there are people who are supposedly very competent, or expert, who have ideas that I don't care about, and you know what?
I don't spend time on them.
Evil is the pursuit of ignorance, challenging new truths to hold on to an old, and comfortable point of view. It's also stupid as change is inevitable, and the energy of the evil person is wasted.
However, in spite of its stupidity evil poses challenges, which the discoverer is ever tasked with fighting through, including handling those who fight for evil in their attempts to maintain their own comfort against knowledge.
That fight is one of the continuing burdens of the Universe's first, greatest, and last fighting force.
[Another reply to David C. Ullrich.]
Fuck off!!! You fucking stupid idiot!!! How many times do I have to tell you that your attention is not wanted David Ullrich?
When will you fucking get the goddamn fucking message to fuck off?
FUCK OFF David Ullrich!!!!!!! Fuck off you stupid fucking idiot!!!!
FUCK OFF!!!!!!!!!!!!
Now then, again here's my discovery:
(5 a_1(x) + 7)(5 a_2(x) + 7)(5 b_3(x) + 22) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)
where b_3(x) = a_3(x) - 3 and the a's are roots of
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)
and when x=0, a_1(0) = a_2(0) = b_3(0) = 0.
In that form it's hard to understand what follows next unless you pay attention to what you have, specifically that cubic defining the a's.
I can get it because of the symmetry of
(5 a_1(x) + 7)(5 a_2(x) + 7)(5 a_3(x) + 7) = 49(300125 x^3 - 18375 x^2 - 360 x + 22)
where I've gone ahead and substituted a_3(x) back in to replace b_3(x), and it's important that you focus on that symmetry.
It's that symmetry which allows the cubic
a^3 + 3(-1 + 49x)a^2 - 49(2401 x^3 - 147 x^2 + 3x)
to define ALL the a's, but something happens when I divide by 49.
Then the symmetry is broken. Without that symmetry it's impossible to find a SINGLE cubic to handle what results when you divide both sides by 49.
That's important because it's why the functions are NOT algebraic integer functions!!!
Now then, I'll recap. Symmetry allows the a's to be defined by a cubic, which shows them to be algebraic integer functions, but dividing by 49 *breaks* that symmetry, taking away the ability to find some cubic to define the results, which proves that the resulting functions are not algebraic integer functions.
(5 b_1(x) + 1)(5 b_2(x) + 1)(5 b_3(x) + 22) = 300125 x^3 - 18375 x^2 - 360 x + 22
where the b's are roots of
b^3 + ? b^2 + ? b - (2401 x^3 - 147 x^2 + 3x)
and when x=0, b_1(0) = b_2(0) = b_3(0) = 0.
My point is that the second and third coefficients are impossible to define in general.
You may find them for some particular x, but in general, they are forever hidden from you.
Notice that doing that substitution with a_3(x) for b_3(x) gives me
(5 b_1(x) + 1)(5 b_2(x) + 1)(5 a_3(x) + 7) = 300125 x^3 - 18375 x^2 - 360 x + 22
but you have broken symmetry since the other constant terms are 1 and 1, so you're still stuck.
Now by emphasizing what happens after 49 is divided from both sides I'm trying to get at least some of you to face the mathematical realities here, and I've made other posts pointing it out as well.
Courage in mathematics might sound new to you as an idea, but those who came before you had courage. That's how they learned of sqrt(2), and sqrt(-1), and much other mathematics that some refused, fought, and bitterly attacked as they lacked that courage.
Of course, the mathematics should win, assuming that humanity survives long enough, but regardless each of you now faces an individual test of your own courage. It's not yet in the history books this time, where you can read and just imagine that you'd never have fought over sqrt(2) or refused to accept sqrt(-1) as an "imaginary" number.
Here it's the present, and your test of courage is now.
Oh, and be sure to check my blog archives:
http://mathforprofit.blogspot.com/2003_10_01_mathforprofit_archive.html
And hey, if you're moping and miserable because mathematics tests you, then maybe, if you think you're a mathematician, you might want to try a different field.
[A reply to David C. Ullrich.]
Well sure, delete out the argument which settles the issue, and then come back and claim that it's not settled.
Now math history is full of people like David Ullrich, a guy with a title, fighting against some new idea. In the past it was sqrt(-1), as mathematicians fought against an idea they thought of as silly.
The hallmark of such people is a refusal to follow mathematical logic. You give a detailed mathematical position, and they just delete it all out, and claim it's not what it is.
Yet the facts speak for themselves:
My work depends on advanced mathematical tools for factoring polynomials, where I use one polynomial to factor another polynomial into non-polynomial factors.
I'd be curious if anyone can find anything like it in the recorded history of mathematics.
Along with that work, I have my prime number research, where I have a partial difference equation used to count prime numbers—a first in math history.
People like David Ullrich are the anti-knowledge crowd. Those people who know just enough to feel comfortable in a complex world. The kind of people who say, "no more knowledge".
Ullrich probably feels very comfortable with whatever mathematical knowledge he has. He can pay his bills using the money given him by taxpayers, as he's a professor at Oklahoma State University. He probably has lots of feedback from various people in society to make him feel good about himself.
What I represent is the unknowable future. The power that comes no matter how comfortable people are, no matter how satisfied they are with themselves or their positions—the power of change.
In that sense I'm a force of Nature, a force of the Universe, a living emodiment of change itself, so I understand the fear I can induce in people like David Ullrich.
But the spirit of mathematics is ultimately about what's not known, what's beyond what's known today, as that defines the work of the future.
I stand here part of a great tradition of discoverers, people who got past the Ullrich's to push forward knowledge so that today you have sqrt(2), i, e, e^x, planes, trains, automobiles, spacecraft, and computers.
It's too easy to look at the past as if it were easy. As if new ideas were just accepted because they were right, and could be proven logically, mathematically, or practically, so that you stand by today, forcing yet another discovery to break through.
But that is your loss, as my strength comes from the knowledge I have, the knowledge of those who came before me, the knowledge of those who will come after.
The knowledge that the truth is beautiful.
All of you can resist, fight until you die, as you will die someday. I can't look to any of you for permanence as you're in a sense, dead already. But I can, and you can look at the truth—beautiful, permanent, absolute.
The discoverers are the forces of Nature, born to a grand tradition, tasked with the seeming impossible, fighting throughout time, throughout history, to make history itself.
The discoverers are the living embodiment of the forces of Good against Evil. The people who push the limits of knowledge for the benefit of all, and the love of Truth.
We are the greatest warriors for Good, of all time, through all time, through all worlds and realities where sentient beings live.
We are the discoverers.
[A reply to someone who said that “History is also full of people that proposed ridiculous ideas which now are dead and buried”.]
Most people show worth through effort. For instance, if you value your job, one would assume that you put a lot of effort into it. Similarly, if you value a relationship, you're willing to work to keep it going.
People like Ullrich value trying to attack the new, as evidenced by his efforts in that regard.
There are a lot of people out there I probably would suppose are nutty, who have ideas that I don't care at all about, and then again there are people who are supposedly very competent, or expert, who have ideas that I don't care about, and you know what?
I don't spend time on them.
Evil is the pursuit of ignorance, challenging new truths to hold on to an old, and comfortable point of view. It's also stupid as change is inevitable, and the energy of the evil person is wasted.
However, in spite of its stupidity evil poses challenges, which the discoverer is ever tasked with fighting through, including handling those who fight for evil in their attempts to maintain their own comfort against knowledge.
That fight is one of the continuing burdens of the Universe's first, greatest, and last fighting force.
[Another reply to David C. Ullrich.]
Fuck off!!! You fucking stupid idiot!!! How many times do I have to tell you that your attention is not wanted David Ullrich?
When will you fucking get the goddamn fucking message to fuck off?
FUCK OFF David Ullrich!!!!!!! Fuck off you stupid fucking idiot!!!!
FUCK OFF!!!!!!!!!!!!
# posted by James Harris @ 19:46 
