Harristotelian Logic

One of my most powerful research results is also one of the easiest to understand, when you break it down to simplest concepts, which I'll do in this post.

First consider

7*P(x) = (f(x) + 7)*(g(x) + 2)

where P(x) is a polynomial, f(x) and g(x) are functions and

f(0) = g(0) = 0.

To make it a little easier, I'll also give P(x) explicitly:

7*P(x) = (49x^2 - 14x)5^2 + (7x-1)(7)(5) + 49

And yes, that is a polynomial just written a little oddly, as you can verify by multiplying out and simplifying.

But now back to

7*P(x) = (f(x) + 7)*(g(x) + 2)

as the simple question, the million dollar question is, where did the 7 multiply through?

Now that "multiply through" is a phrase that some posters in reply to me have seized upon, mocking it, going on and on about it as if it is just so unimaginably bizarre as to be nonsensical, but they NEED to attack at the simplest concepts.

Now then, where did the 7 go through?

Let's back up a bit, and instead I'll show

P(x) = 7*(h(x) + 1)*(g(x) + 2)

where I've introduced yet another function, h(x), where h(0) = 0, and some may cry, FOUL!!!

I'm loading the question, right? Clearly I divided 7 out through f(x) + 7, which is the result I WANT, but is there any other way to do it?

Is there? Given

7*P(x) = (f(x) + 7)*(g(x) + 2)

where f(0) = g(0) = 0, is there any other way the 7 could have multiplied through, as notice it's there on the left hand side multiplying times P(x).

7*P(x) = (49x^2 - 14x)5^2 + (7x-1)(7)(5) + 49

so its just a polynomial, which you can write out more simply in a more familiar form if you wish, which has 7 as a factor for all x.

Is the any other way that 7 could have multiplied through with

7*P(x) = (f(x) + 7)*(g(x) + 2)

than in such a way that

P(x) = 7*(h(x) + 1)*(g(x) + 2)

is the proper result if you take it back out?

Is that really so complicated? So bizarre? Made up math? So nonsensical as to be insane?

Harristotelian?

Why is it such a big deal then?

Because I say the distributive property does NOT care what value f(x) has, so if the 7 multiplied in that simple way indicated then it did so for every x.

BUT, now non-polynomial factorization steps in as a very clever idea, as I can now go to something only slightly different:

7*P(x) = (5a_1(x) + 7)(5a_2(x)+ 7)

where the a's are roots of a monic polynomial, and at x=0, one of them is 0, while the other is -1.

But which one is which?

Don't know. The math doesn't say.

But they are roots of

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

and I can use my earlier result about how the 7 multiplied through.

But what about f(x) and g(x), versus a_1(x) and a_2(x)?

Well, f(x) equals whichever of the a's goes to 0 at x=0, while g(x) equals the other one plus 1, as f(0) = g(0) = 0.

Why can't I say WHICH of the a's is 0 and which is -1 at x=0?

Because of the ambiguity of the square root:

a = ((7x-1) +/- sqrt(1 - 3(14)x - 3(49)x^2))/2

That's the complicated looking expression you get if you solve

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

using the quadratic formula.

So why would people argue with me over these ideas, why would a journal that published a paper over these ideas, pull it under social pressure from the sci.math newsgroup, and later die—yup the freaking math journal freaking DIED—and that supposedly NOT be a big deal?

How could a supposed crackpot get published in a math journal, the journal cave under Usenet pressure, and die a little later and that supposedly just be the most casual thing as if it happens everyday and is barely worth mentioning?

Because these results run counter to what mathematicians currently teach.

For mathematicians who are professors at universities the discussions on these newsgroups can be comforting, if they are working to deny the results, as the ridicule heaped on non-polynomial factorization by dedicated posters is a sign to them that they can keep teaching what they have taught before, and get away with it with their students and the public.

So the posters who argue with me, serve a vital function to those mathematicians who need to keep teaching the wrong theory to their students, as they can sit back and watch how the wind is blowing on the newsgroups, and determine if they can keep getting away with teaching the flawed ideas.

That's why those posters have to argue and argue and argue and can never stop, no matter how many threads I put up, as day after day they play a vital function for the current mathematical community: protecting.