Harristotelian Logic

Since I can go from talking about FLT, to discussing an esoteric error in the ring of algebraic integers, to my prime counting function, to partial differential equations, to what happens when you add 1/2 to the ring of integers, some might get a little lost, or not realize that all of it is connected by a rigid, logical framework based
primarily on modern ideas.

And in fact the modern ideas I use have a lot to do with your reading this post as object-oriented thinking is quite important in computer programming.

What I want to do in this post is give a roadmap, connect all the dots, so that the big picture people understand how it all relates.

First off, while object-oriented thinking permeates modern computer programming, and has always been in the sciences, mathematicians seem to have missed the train. If you've picked up a textbook on abstract algebra, you might notice there are all these little, dry rules. Being someone who focuses on the concrete, when I picked up a text on abstract algebra, back when I was fourteen at Duke University, I put it back down after a few minutes, and instead went back to playing with calculus.

To me it's like when Ptolemy was a big wheel. For those who forgot their science history, Ptolemy worked out a system for figuring out where objects in the sky would be at various times using spheres and circles. Those were used because the heavens were considered the abode of God, and spheres and circles were considered perfection, so people figured God would only use perfect stuff.

The problem is that left little errors, which were fixed with, guess what, little circles called epicycles. It was a kludgy system that involved a lot of calculation and still would be off.

Well along comes Copernicus and Kepler, and Kepler drops the circle business and uses ellipses. Being himself religious Kepler came up with his own reasons for why God would use the supposedly less perfect ellipse, rather than circles.

When I decided to try and use basic algebra to find a short proof of Fermat's Last Theorem, which I really, really, really hoped existed, I made a conscious decision not to bother with overdone approaches. I like simple.

What happened is that I came upon a rather basic, straightforward approach which boils down to factoring x^p + y^p - z^p indirectly. However, that approach revealed that mathematicians hadn't discovered enough mathematical infrastructure to handle factorizations at that level.

So I was forced to work out that infrastructure myself, which I call object mathematics.

While thinking about such things, I found myself chatting about simple polynomials like x+1, and (x+1)(x+3), which got me to thinking about prime numbers, and a few weeks later I had a way to count them that mathematicians got close too, but never quite got the full thing.

I know they didn't because I can look at their work where they got close, and see how close they came to what I have. And also I can see what my discovery does that what they have cannot do.

I played with prime counting for a while, including working out a partial differential equation that follows from my functional way to count prime numbers, and then went back to thinking about my FLT work.

For a while I was convinced by others that I needed algebraic integers, which are numbers defined to be the roots of monic polynomials with integer coefficients. You know, like x^2 + 2x + 2, as the polynomial is monic because the first coefficient is 1.

So I put object mathematics to the side, hoping that maybe mathematicians had indeed built up the infrastructure needed for my FLT work, but then a little later I found out that no, they hadn't, and in fact there was this intriguing little problem with algebraic integers.

That forced me to go back and blow the dust off of my work on object mathematics, and I finally worked it out thoroughly within the last few weeks, as part of the polishing process.

Then I was surprised to find that mathematicians seemed to not know basic things about their own work, which thinking back to Ptolemy, doesn't surprise me now, as when you have a lot of excess, based on unnecessary rules, people can learn things by rote, and not understand.

So mathematicians apparently don't understand that including fractions like 1/2 with numbers like integers gives you the field of reals. Their belief comes from arbitrary rules where they exclude infinite sums on an ad hoc basis.

Seeing that is easy. Consider that

1/(k-1) = 1/k + 1/k^2 +…

when k is a nonzero integer other than 1 or -1, which is easy to prove in the classic way using S.

S = 1/k + 1/k^2 +… = 1/k(1 + 1/k + 1/k^2+…) = 1/k(1+ S), so

kS = 1 + S, S = 1/(k-1).

So if you add 1/2 in with integers, you have 1/4 = 1/2(1/2), so you get 1/3, and now you can have 1/12 = 1/4(1/3), which gives you 1/11, both from the formula above, and that process leads you on and on until you have the field of reals.

I say that such infinite sums are decidable since you can get an answer, and there's no reason to exclude them. Mathematicians want them excluded so they yelp, and start tossing out arbitrary rules. And I think back to Ptolemy.

So that's an overview of my work and for LOTS of mathematics you can check

http://groups.msn.com/AmateurMath

where I go into a lot of detail, giving a short proof of FLT, my prime counting function and its associated partial differential equation, and I have a paper outlining the problem with algebraic integers.

Also I have the framework object mathematics with discussion on why I unearthed it, and I even connect back to Gauss's Fundamental Theorem of Algebra.

Basically I do a lot in a few pages which is what you can do with concise and potent mathematics. If you can make it through and understand it all, you are at least a hundred years ahead of current mathematicians. But don't tell them that as they seem to get upset very easily.