Re: JSH: Psychology of the denial of "core error"

James,

I realize that this is an exercise in futility, but I
am going to try
to demonstrate to you why the mathematical community
does not take
anything you say seriously. Many of us do, however,
find you
marvelously entertaining. As I remarked several
times already, you
write very well, and you spin an absorbing, if silly,
science fiction
conspiracy involving mathematicians and, in
particular, number
theorists. Never before have we been considered so
powerful. We
certainly don't feel that way.

I will intersperse my specific comments withing your
text, where
appropriate, to try to show you how mathematicians
view the
mathematics of what you do. I and others have done
this before to
little effect, but I happen to be in the mood.
Beethoven's fifth is
playing in my speakers, so I guess I just feel
empowered or something.

On Jul 4, 6:41 pm, JSH <jst...@xxxxxxxxx> wrote:
Turns out that if you follow rigorous mathematics
it is trivial to
show a problem with the use of the ring of
algebraic integers with a
quadratic factorization that I've often given
before:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

This appears to be a monic polynomial of degree 2
with coefficients
in Z[x]. As such, its roots would be in some sort of
integral
extension (this is a technical term you may look up)
of Z[x] living
within a field that is an extension of degree 2 over
Z(x). It is not
an algebraic integer. I am unfamiliar with the
factorization
properties of this particular extension of Z[x], but
I have no reason
to suppose that it is a unique factorization domain,
nor do I have any
idea what the structure of its unit group might be.
These are
interesting questions, but I don't have the time and
energy right no,
if ever, to try and figure this out.

The primary problem here is that if you let the
ring be the ring of
algebraic integers, you get something never before
seen in human
mathematics which is a constraint on a constant
factor revealed on the
right hand side of

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

which is obviously not there on the left hand side.

I simply don't know what you mean by this. As
previously mentioned,
a_1 and a_2 are NOT algebraic integers. If you do
out the solution of
the quadratic equation you presented, getting a bunch
of square roots
of polynomial expressions involving x, you will
probably, with perhaps
some effort, be able to see that the two sides are
obviously equal,
assuming you have done the calculation correctly. My
real point is
that, since you are not dealing in the ring of
algebraic integers when
you use this expression, you can't pull out anything
very direct that
you can say about the algebraic integers. The fact
that you are
getting confused here and have done so now for a very
long time does
not increase the respect you get from the
mathematical community.

To understand what I mean--I'm sure some posters
would reply that they
find it incomprehensible--consider a simpler
example in the ring of
integers:

7(x^2 + 3x + 2) = (7x + 7)(x + 2)

where the 7 is still unconstrained, and you can in
fact move it around
at will, so


I have no idea what an unconstrained 7 might be. I
don't even know
what a constrained 7 might be. I thought it was just
a number. What
are you talking about?

7(x^2 + 3x + 2) = (7x + 7)( x + 2).

Now the mathematically astute of you may notice
that the first example
IS fixable so that you have equivalence on both
sides of the equals
and the 7 is unconstrained if you use normalized
functions:

7(175x^2 - 15x + 2) = (5(7)b_1(x) + 7)(5b_2(x)+ 2)

where

7b_1(x) = a_1(x) and b_2(x) = a_2(x) + 1

and the a's are still roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

and the b's are normalized because they both equal
0 when x=0.

Now, apparently, the b's are functions of x, since
they take values
when x = 0. I have never heard of an algebraic
integer taking a
value when x = 0. So what is going on here?

NOW notice that algebra still works and the 7 is
unconstrained again
and you can move it around, like before:

7(175x^2 - 15x + 2) = (5b_1(x) + 1)(5(7)b_2(x)+ 14)

or even divide it off completely:

(175x^2 - 15x + 2) = (5b_1(x) + 1)(5b_2(x)+ 2)

But now the ring of algebraic integers itself
becomes the problem as
unlike any other known ring in mathematics it will
not allow you to do
the above!!!  The b's are not in general in that
ring!

Not only are they not in general in that ring, but,
since you have
defined them in terms of the a's which are in some
integral extension
of a polynomial ring, and so also not in the
algebraic integers, this
does not require 2 exlamation points.

The issue with the exclamation points can be explained:
while you were listening to Beethoven's fifth, James
was watching Seinfeld's "Muffin Tops" episode, where
Elaine repeatedly overused exclamation points:
Top of the Muffin to You!!!





It does not
even require a
single exclamation point. It is commpletely obvious.
What, perhaps,
requires an exclamation point is that you seem to
think that they
should be.

That is because

7b_1(x) = a_1(x)

means that a_1(x) has 7 as a factor which requires
that one of the
roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0

have 7 as a factor, and you can get a contradiction
with x=1, which
gives

Again you are setting x to a value within an
algebraic integer(!!).
There are other, less esoteric why the a's and b's
are not algebraic
integers.

a^2 - 6a + 35 = 0

as provably in the ring of algebraic integers
NEITHER of the roots of
that quadratic can have 7 as a factor.

Now this polynomial, where a simply represents a
variable, does
actually have roots that are algebraic integers. And
the two roots
are both divisible by the prime that lies over 7. If
you studiend a
little bit of algebraic number theory, a truly
fascinating subject,
you would know what I am talking about and why it is
true. But it is
totally unclear what point you are trying to make by
saying this.

That's the easy algebra and it reveals that the
ring of algebraic
integers is mathematically distinct from ALL other
known rings,
including fields of course, as the problem does not
emerge in anything
else!

The ring of algebraic integers actually blocks
algebra itself, by not
allowing certain algebraic operations to occur, and
that is only
revealed by this remarkable construct:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0.

What happens there is that you get the
unconstrained 7 on the left--
math students are normally taught to divide such
factors off--which
cannot in general be divided off from the right
hand side ONLY in the
ring of algebraic integers, which I call
entanglement.

The 7 is somehow trapped in place, entangled, on
the right hand side!

Fascinating. You have gathered together several
nonsensical
conceptions into one neat package here. You present
2 a's that are in
a ring that most us never have studied, although many
surely have, and
which are surely not algebraic integers, and yet you
seem to think
that they should have the properties of algebraic
integers.

By the way, most rings do not have roots to all
polynomials defined
over them. The only rings with that property are
algebraically closed
fields. So, again, what are you on about?

Then, somehow, you get this unconstrained 7, a term
you don't define,
which I have never heard of, and which sounds
ridiculous, and tell us
that it is trapped in place and actually entangled on
the other side.
Nice use of language, certainly, but meaningless
nonsense
mathemtically in this context.

I've proven that this problem leads to the
*appearance* of
contradiction and allows a math practitioner to
appear to prove things
that are not true, or are not actually proven.

With errors in mathematics you can get this odd
ability to appear to
prove just about anything.

Ok, the math is easy.  I've simplified explaining
over the years and
explained over and over and over again in many
different ways and even
had a paper published in this area with slightly
more complicated
cubics before I simplified to quadratics.

Now that story is interesting as it reveals an
amazing psychology
here: publication supposedly means something in the
mathematical
community but when I got published sci.math'er said
it meant nothing,
some of them conspired on the newsgroup to attack
the paper by emails
and did so, panicked the editors who pulled my
paper and the journal
died after managing one more edition!

Now rational people knowing the math, knowing the
error allows people
to do fake math, would get very suspicious with
that story, but this
particular error appears to run so deep that the
bulk of the
mathematical community might rather hide it than
deal with it.

That's because the ring of algebraic integers was
introduced over one
hundred years ago.  The sheer volume of erroneous
publications
revolving around the error can defy imagination and
may represent the
BULK of number theory work over the last century
plus encompassing the
ENTIRE 20th century!!!

One dead math journal under remarkable circumstance
is minor with such
a situation.

Notice here also that practitioners in number
theory with decades in
the field and lots of awards or prizes have the
most to lose from the
revelation of the error.

Graduate students as well can have serious
investment in the error as
imagine being one who considers his graduate thesis
and realizes it's
junk!!!

So for years I've included math undergrads who are
really in an
awkward position here, as the people they rely on
the most may be the
most compromised but unlike professors and grads
they don't have as
many years or as many "successes" invested in wrong
mathematics!

And remember, for someone who believed they were
brilliant or had
great accomplishments this error really can cut
deep emotionally as
well as in many other ways.  It's like it can rip
their entire world
apart.

I have been careful in trying to figure out the
best way to hopefully
get some resolution here as I understand just how
high emotions can
run, and some of the replies I get can give readers
some perspective,
but make no mistake, the fight is to keep doing
wrong math, and to
teach it to others.

So from one perspective, the professors who didn't
have a chance and
the grad students who didn't have a chance are
victims, yes, but they
are also abusers when they are teaching to new
students who DO have a
chance to not be taught this error, and to do
number theory and start
advancing it again after, oh, about a hundred years
of stagnation.


No stagnation. Incredible advancement in so many
ways that I could go
on for quite some while. The best resolution here
would be for you to
realize how wrong you've been about all this and
start to learn what
algebraic number theory is really about. You
wouldn't have the great
joy of being the only human being who is onto a
terrible conspiracy
that must be stopped, who is being thwarted at every
turn by an
alliance of crazed, power-mad number theorists, but
you would have the
joy of learning a beautiful subject, one of many that
could become
accessible to you, and that joy would be a lot more
grounded in
reality.

Regards,
Achava

James Harris

.

Relevant Pages

  • Re: JSH: A simple error
    ... algebraic integers routinely move outside the ring. ... the tautology ... Extreme Mathematics Google group, as is my latest paper which is more ...
    (sci.math)
  • Re: New paper, algebraic integers, Galois Theory
    ... >>You're suggesting that it is possible that one of the roots of the ... existence of common factors requires the context of a ring. ... can find the common factors whose existence you're apparently disputing. ... You have maintained that the ring of algebraic integers is somehow ...
    (sci.math)
  • Re: Math rules and publication
    ... > even faced formal peer review. ... > out of the ring of algebraic integers has to be determined separately. ... a piece of purported mathematics is accepted or not, ...
    (sci.math)
  • Re: JSH: A simple error
    ... algebraic integers routinely move outside the ring. ... I start with a modular tautology in terms of variables r, s, ... Extreme Mathematics Google group, as is my latest paper which is more ...
    (sci.math)
  • Re: JSH: Math journals do not just die
    ... Yup, the Southwest Journal of Pure and Applied Mathematics, a minor ... And what electronic journals are very old? ... DECLARED to be in the ring of algebraic integers definitely were not at ... simply enough are roots of monic polynomials with integer coefficients. ...
    (sci.skeptic)