What I've found is a problem at the heart of mathematics, which can be
fixed if attention is paid, which allows supposed proof of two
contradictory things.

That's why it's important.

The hallmark of an error at the heart of mathematics is that you can
have two camps, each with their "proofs", and in fact, what I have is
simple enough to show extremely quickly.

See http://mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=782

I point you there as their allowal of LaTeX makes for a better looking
and easier to read presentation than posting text.

If you're wondering how an error in mathematics is possible, well in
this case there's this definition for algebraic integers as the roots
of monic polynomials with integer coefficients.

Well it turns out that some roots of non-monic polynomials should also
be included, so it's an error of arbitrary exclusion.

That is, some people are using a definition that just so happens to
keep some numbers out, which leads to a situation where you can seem
to prove two different and contradictory things.

Fixing the problem is rather straightforward as instead of using that
definition for algebraic integers, the following definition is needed
for the holder of a group of numbers I call objects:

The Object Ring is a commutative ring that includes all numbers such
that -1 and 1 are the only members that are both a unit and an
integer, where no non-unit member is a factor of any two integers that
are coprime.

Now the problem is that while I can rather easily give the mathematics
that explains the problem, I'm facing what appears to be an
institutional refusal from mathematicians to acknowledge it.

But remember, an error in core mathematics is serious as it allows
supposed proof of things which are false. And worse, if one is in
core, it now must be considered that other errors are there as well,
and the current refusal by mathematicians to acknowledge this one, is
not comforting.

So some of the things you may believe in physics based on mathematical
"proofs" could be suspect.

Physicists have a strong interest in keeping mathematics healthy, so I
ask for your help here. Checking me is easy. Challenging the
mathematicians, I know, is harder.

My question, is any one of you brave enough?


James Harris

James Harris wrote:

What I've found is a problem at the heart of mathematics,
[snip]

If you're wondering how an error in mathematics is possible, well in
this case there's this definition for algebraic integers as the roots
of monic polynomials with integer coefficients.

Scores of kilobytes have been donated by your betters punctiliously
outlining your egregious ignorance of the subject. Like "****Head"
Donald Shead being inebriated by English units and an inability to do
arithmetic, you are also an ineducable boring dysfunctional psychotic
incapable of climbing out of your hole.

[snip]

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!

In sci.physics, James Harris

wrote
on 3 Oct 2003 08:00:15 -0700
:
What I've found is a problem at the heart of mathematics, which can be
fixed if attention is paid, which allows supposed proof of two
contradictory things.

That's why it's important.

The hallmark of an error at the heart of mathematics is that you can
have two camps, each with their "proofs", and in fact, what I have is
simple enough to show extremely quickly.

See http://mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=782

I point you there as their allowal of LaTeX makes for a better looking
and easier to read presentation than posting text.

If you're wondering how an error in mathematics is possible, well in
this case there's this definition for algebraic integers as the roots
of monic polynomials with integer coefficients.

Well it turns out that some roots of non-monic polynomials should also
be included, so it's an error of arbitrary exclusion.

That is, some people are using a definition that just so happens to
keep some numbers out, which leads to a situation where you can seem
to prove two different and contradictory things.

Fixing the problem is rather straightforward as instead of using that
definition for algebraic integers, the following definition is needed
for the holder of a group of numbers I call objects:

The Object Ring is a commutative ring that includes all numbers such
that -1 and 1 are the only members that are both a unit and an
integer, where no non-unit member is a factor of any two integers that
are coprime.

Now the problem is that while I can rather easily give the mathematics
that explains the problem, I'm facing what appears to be an
institutional refusal from mathematicians to acknowledge it.

But remember, an error in core mathematics is serious as it allows
supposed proof of things which are false. And worse, if one is in
core, it now must be considered that other errors are there as well,
and the current refusal by mathematicians to acknowledge this one, is
not comforting.

So some of the things you may believe in physics based on mathematical
"proofs" could be suspect.

Physicists have a strong interest in keeping mathematics healthy, so I
ask for your help here. Checking me is easy. Challenging the
mathematicians, I know, is harder.

My question, is any one of you brave enough?


James Harris

I wonder why you don't flip the problem, ultimately yielding
something like

Q(m) = (y + 1/a_1)(y + 1/a_2)(y + 1/a_3)

where y = uf/x. (I'd have to work out what it does to your other form.
Note that P(m) = x^3Q(m). Might make it slightly easier to analyze.)

As it is, one can have the following situations, as far as I can tell.
(Remember that the a_i are algebraic integers.)

[1] An a can have f^2.
[2] Two a's can have f. (This is your conclusion.)
[3] All three a's can divide various issues of f; recall, for instance,
that sqrt(f), f being any nonzero integer (including negative),
is an algebraic integer, solving the equation z^2 - f = 0.
In fact f^(m/n) is an algebraic integer, m,n 0. So one possibility
here is that each a has f^(2/3). Another is that a_1 has f^(1/4),
a_2 has f^(2/3), and a_3 has f^(13/12). a_1a_2a_3 therefore has
a factor of f^2 as required by P(0).

Therefore your conclusion that g_1 = a_1x + uf and g_2 = a_2x + uf
both have f as a factor does not seem to follow.

As for keeping numbers out, is Z[1/2] = Q, R or C, and why?

--
#191,
It's still legal to go .sigless.

(James Harris) wrote in message om...
What I've found is a problem at the heart of mathematics, which can be
fixed if attention is paid, which allows supposed proof of two
contradictory things.

That's why it's important.

The hallmark of an error at the heart of mathematics is that you can
have two camps, each with their "proofs", and in fact, what I have is
simple enough to show extremely quickly.

See http://mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=782

I point you there as their allowal of LaTeX makes for a better looking
and easier to read presentation than posting text.

It's worth pointing out how *short* the presentation is, and some may
be confused or wondering what the issue is.

The issue is that mathematicians have been avoiding or arguing against
the conclusion that only *two* of the g's have a factor that is f,
which is revealed from the constant terms.

The contrary position is that instead the g's *all* have some factor
in common with f, with the specific factor varying dependent on the
value of m, which is why you see my emphasis on constant terms being
constant.

So the contrary position flies in the face of basic algebra, and those
who doubt that mathematicians would ever do such a thing as argue
something so wacky, need only go to sci.math and see the continuing
arguments.

So why would they argue, or try to ignore the result?

Apparently it's a small thing in one way as I give the fix in my
original post, and a big thing in another, as they have to change a
LOT of textbooks.

Maybe mathematicians are just terrified contemplating all the
writing!!!

In any event, it's easy to verify I'm right. It's up to you to show
how much you truly value civilization. After all, it's well and good
to claim you're an intellectual or "scientist" when you *think* you
have back-up from a large community, including mathematicians, and
another to actually prove your ideals when mathematicians have gone
their own way.


James Harris

James Harris wrote:
(James Harris) wrote in message om...

What I've found is a problem at the heart of mathematics, which can be
fixed if attention is paid, which allows supposed proof of two
contradictory things.

That's why it's important.


The fact of the matter is that most people (including all competent
people, BTW) take the appearance of a contradiction of this nature
to be *prima facie* evidence of a flaw in the novel argument.

JSH has produced an argument that has, as a direct conclusion, the
following assertion:

ASSERTION (JSH):
There are algebraic integers u,v,w,x,y for which

uv = x
uw = y

where u is not a unit, and x and y are coprime.

JSH claims a proof that has the above ASSERTION as a corollary. To see
that the corollary is manifestly false, note that the two algebraic
integers x and y are coprime iff there exist algebraic integers m and n
for which

mx + ny = 1.

Substituting (via the above factorizations), one has:

m(uv) + n(uw) = 1.

After rearranging factors, this is

u(mv) + u(nw) = 1,

or

u(mv + nw) = 1.

Note that u was not a unit earlier, yet we have explicitly provided
the multiplicative inverse of u as (mv + nw).

It is *this* situation that JSH claims shows the alleged flaw in the
ring of algebraic integers, rather than showing the fact that his method
is flawed.

Just think. Put yourself in this position: you have been working for
years, proposing one thing after another, only to have all your claims
shot down. You stumble across an argument that seems to make sense to
you, and it looks solid, the only problem being that it produces some
anomalous results: factorizations you claim are impossible, numbers that
you claim cannot be algebraic integers, yet others using standard
methods produce the impossible factorizations and proofs of integrality.

What should a person do? Reject this new method because it gives results
that are at variance with fact, or accept it and claim that the whole of
mathematics is flawed? Which side should win in such a clash:

(a) the body of science that has withstood over a century
of rigorous analysis, and been the subject of constant
scrutiny by many thousands of students, researchers, and
developers of applications,

or

(b) the new argument that appears solid in that you can't
identify a step that is in error, yet which produces
results that prove to be at variance with the methods
and results of (a) above?

JSH is saying that the proper route is (b). The fact that he *sees* his
method to be correct, although no one else does, means (to him) that all
these contradictions are contradictions in mathematics itself.

Rather than announce that mathematics is flawed, he rephrases this claim
as being a problem of definition: somehow the mere act of defining the
algebraic integers as the set of roots of monic polynomials over the
integers, and nothing else, has produced an error. To him, it doesn't
matter that the early researchers (Dedekind, possibly?) proved that this
set met the criteria of being a ring. Simply recognizing that some set
has a certain property, and codifying that knowledge in a definition is
enough to cause the edifice of mathematics to crumble! What a house of
cards we must be building, as mathematicians!

To JSH, this ring is "incomplete", a term that he has refused to define;
instead, he says certain numbers "should" be in the ring but aren't. (By
"should", I assume he means that, for his analysis to hold, these values
need to be algebraic integers. However, that's my assumption, and absent
a definition by JSH, we are only left to guess.) Recently, JSH held to
the untenable position that there were algebraic integers m and n for
which mn was not an algebraic integer. It was only after much derision
that he came to the recognition that his position meant a rejection of
the fact that the algebraic integers form a ring. That is, JSH was
showing that he failed to comprehend the very definition of "ring" by
holding to his assertion. Since "one of the top number theorists in the
world today" can't be seen to be a know-nothing in the area of algebra,
he recanted that position.

The hallmark of an error at the heart of mathematics is that you can
have two camps, each with their "proofs", and in fact, what I have is
simple enough to show extremely quickly.

See http://mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=782

I point you there as their allowal of LaTeX makes for a better looking
and easier to read presentation than posting text.


It's worth pointing out how *short* the presentation is, and some may
be confused or wondering what the issue is.


It's also worth pointing out that the presentation is totally symbolic.
His argument uses a polynomial in four (4) variables, and presents it as
though it were a polynomial in a single variable. Setting m = 0, he
believes he has found "the constant term", ignoring that this constant
term is a polynomial in three variables. This sort of sleight of hand
could not be done with specific numbers, or with a polynomial of a
single variable, despite JSH's many efforts: recall all the pretend
examples he has come up with, claiming that a certain factorization
didn't exist, or that when a factorization existed, it had to have some
specific properties, only to have others display his errors explicitly.

Yet, from this argument, we are to believe that mathematics is a house
of cards. It shouldn't matter that the person making this claim has a
long history of making false claims, denying any and all evidence of the
falsity of his claims, while displaying abysmal ignorance of the very
area that he claims is deeply flawed.

The issue is that mathematicians have been avoiding or arguing against
the conclusion that only *two* of the g's have a factor that is f,
which is revealed from the constant terms.


OK, you'll let f be a parameter. Then, your constant term is merely a
polynomial in *two* variables, parametrized by f. You insist that it is
constant!

The contrary position is that instead the g's *all* have some factor
in common with f, with the specific factor varying dependent on the
value of m, which is why you see my emphasis on constant terms being
constant.


The contrary position has shown, in *explicit* calculations, that your
claims with respect to *specific* factorizations are simply mistaken. If
your method were correct, then it would specialize to correct assertions
with regard to the factorizations you've been trumpeting.

So the contrary position flies in the face of basic algebra, and those
who doubt that mathematicians would ever do such a thing as argue
something so wacky, need only go to sci.math and see the continuing
arguments.


Note how the argument requires going to JSH's pasting to some Hong Kong
web site. Note how one needs to ignore the argument I gave towards the
top of this article, that takes only the ASSERTION that JSH makes, along
with the simplest of algebraic manipulation, to reach a contradiction.

Beyond JSH's ASSERTION, what assumptions were made?

1. Algebraic integers form a ring.
2. Definition of coprime.

So, one could have (a) the ASSERTION is wrong, or (b) one or both of the
above assumptions is incorrect.

Which is it?

So why would they argue, or try to ignore the result?


Why would you ignore my demonstration that, STARTING WITH YOUR OWN
ASSERTION, and making no fancy assumptions, one gets a contradiction?

Your answer that "there is a wacky problem in math" is no response. It
amounts to admitting that you don't know what you're talking about.

Apparently it's a small thing in one way as I give the fix in my
original post, and a big thing in another, as they have to change a
LOT of textbooks.


Right. If I drive on the wrong side of the street, there will be lots
of wacky driving, or even collisions. Looks like driving is just full
of lots of wacky errors, and all the traffic laws will need to be
changed!

Maybe mathematicians are just terrified contemplating all the
writing!!!


Oooh, I'm trembling as I type this. No, wait, that's from laughing so
hard that I can hardly see the monitor.

In any event, it's easy to verify I'm right. It's up to you to show
how much you truly value civilization. After all, it's well and good
to claim you're an intellectual or "scientist" when you *think* you
have back-up from a large community, including mathematicians, and
another to actually prove your ideals when mathematicians have gone
their own way.


And, never forget the primary axiom

"JSH is never mistaken"


James Harris

Dale

James Harris wrote:

What I've found is a problem at the heart of mathematics,

Namely that's a cipher you'll never figure out.

John VanSickle wrote in message ...
James Harris wrote:

What I've found is a problem at the heart of mathematics,

Namely that's a cipher you'll never figure out.

Hmmm...my guess is that you believe I've said something that's
impossible, which depends on your definition of "mathematics".

At other times I've said it's a problem in taught mathematics in an
attempt to help, but that get tedious.

Basically mathematics is a body of what are hoped to be truths for
many reasons which have been compiled over a period of many years with
effort from lots of people.

What I've found is a rather small error that crept into that process
based possibly on too much assumption. The problem comes from a
definition which focuses on *roots* of monic polynoials with integer
coefficients.

Just by reflex some of you might wonder why there can't be some roots
from non-monic polynomials with integer coefficients included, and in
fact, it turns out that not including them creates a situation where
you can *appear* to prove two different and contradictory things,
which is a bane of mathematics.

See if you can figure it out, and look at the simple argument at

http://mathdb.math.cuhk.edu.hk/forum/e_show.php?msg=782

where the allowal of the use of LaTeX makes for a *much* nicer
presentation.

Remember, my point is that the definition problem allows the
appearance of two mathematical proofs that are contradicting each
other--but proofs can't contradict each other--and the contrary
position is that what I have shown is wrong.

So then, look at it, and see if you can find a mistake.

But you see, my point is that you will not be able to find an error in
the reasoning and in fact the most important point is that a constant
term IS constant.


James Harris