Harristotelian Logic

I have a peer reviewed and published result, which has been argued out in extreme detail for years, and there is no room for error.

That result shows a coverage problem with the ring of algebraic integers, so for instance,

x^2 - 12x + 65 = 0

has one root that can be shown to be coprime to 5 but it can also be proven that none of the roots are coprime to 5 in the ring of algebraic integers.

An analogy to understand what happens can easily be seen by considering evens, and noting that if you have evens as a ring, then 6 is coprime to 2. Wow!!! How can that be??!!!!

It's true because 3 is not even, that's how.

So, with x^2 - 12x + 65 = 0, two of the roots have sqrt(5) as a factor—but not in the ring of algebraic integers—just like 6 has 2 as a factor, but not in evens as a ring.

It's not even hard to understand, so why all the arguing, and how is it possible?

Well, it turns out that it's the definition of algebraic integers as roots of monic polynomials with integer coefficients that is the culprit.

That highly specific definition blocks certain numbers, which still behave more like integers than like fractions, that is, more like 2 than 1/2, which is why I abstracted out the key properties of the ring of integers, to get the object ring:

The object ring is defined by two conditions, and includes all numbers such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.
2. Given a member m of the ring there must exist a non-zero member n such that mn is an integer, and if mn is not a factor of m, then n cannot be a unit in the ring.

And that's it. Turns out that you only need two key properties and you have numbers that are integer-like, or are in fact, integers. I call such numbers objects.

Even that concept isn't terribly hard as consider 1/2.

Do you ever get 1/2 or do you get 1/2 of something?

You always get 1/2 of something.

So 1/2 is like an instruction—a set instruction—telling you to take one of two equal objects from a set of two.

Hey folks, this isn't rocket science. But it is overturning, as if you do the mathematics right, wow, out the window goes—<gasp>—ideal theory!!!

And there you have it. Why mathematicians would ignore this result—they're not real mathematicians!

Now Wiles has nothing, neither does Ribet, nor Taylor.

Suddenly, they're just ordinary people who join the ranks of people who thought they had something proven mathematically, but did not because they missed something small, that nevertheless overturns their results.

Would you tell the truth if you were them? Knowing that your ENTIRE CAREER could go up in smoke?

So there, people with Ph.D's in mathematics whose thesis would go away, have lots of motivation to keep arguing over this until they keel over and freaking die, because the alternative is to admit being an ordinary person—not a brilliant mathematician after all.

Just another person who tried and failed.

But make no mistake, Andrew Wiles, Ribet, Taylor, Granville and any number of other supposedly top mathematicians are not.

They are failures, who today are simply succeeding at hiding the reality.

Well, at least they have that.