Saturday, July 04, 2009


JSH: Psychology of the denial of "core error"

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.

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.

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

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)


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 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!

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

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.

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!

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.

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