Friday, December 18, 2009


JSH: Consideration of idea worth

Having gone through getting at least an undergraduate degree in physics I can appreciate how annoying it can be when some person thinks he has VERY important ideas, which challenge orthodoxy, while now I can appreciate that feeling that you have something important and someone should listen.

So what's the resolution? Well for me I've tossed things out there and do my arguing for my own purposes or posts like this one as well, and figure, things will work out.

But I have that luxury.

Ten years ago this month I had the privilege of finding a simple extension of mathematical ideas pioneered by Gauss himself, which I called tautological spaces (I even got to do a cool name, bonus).

A simple tautological space: x+y+vz = x+y+vz

So yeah, this post is partly to commemorate that ten year anniversary.

Identities are so dismissed that I also have the bonus of being someone who found a surprisingly simple use for them—probing equations for analysis.

The 'v' variable is free, so you always have one extra degree of freedom and I like to say it was designated by me 'v' for Victory.

My innovation moving forward from Gauss besides using special identities was to use "mod".

x+y+vz = 0(mod x+y+vz), which is the equivalent of x+y+vz = x+y+vz

Here's using tautological spaces with x^2 + y^2 = z^2.

x+y+vz = 0(mod x+y+vz), so x+y = -vz(mod x+y+vz), squaring both sides:

x^2 + 2xy + y^2 = v^2 z^2 (mod x + y + vz), now subtract out x^2 + y^2 = z^2, giving

2xy = (v^2 - 1) z^2 (mod x+y+vz), which is (v^2 - 1) z^2 = 2xy mod (x+y+vz), so:

(v^2 - 1)z^2 - 2xy = 0 (mod x+y+vz)

and I can get rid of x, with x = -y-vz (mod x+y+vz), so I have:

(v^2 - 1)z^2 + 2vyz + 2y^2 = 0 (mod x+y+vz)

which just says that for ANY v you choose, you have that x+y+vz, will be a factor of what's on the left hand side which is the residue. And that is not really ring specific, as if you don't use "mod" you can do it all explicitly. But it's most meaningful in rings where "factor" is meaningful.

So like with v=1, I have 2yz + 2y^2 = 0 (mod x+y+z), and trivially with x=3, y=4, z=5, you have:

2(4)(5) + 2(16) = 72 has 12 as a factor.

So you get the result that if x^2 + y^2 = z^2, then 2y(z+y) has x+y+z as a factor.

So why should physics people care?

Because you can put ANY equation from the canons of physics or any other area of human endeavor through a tautological space, subtracting it from an identity, and poke at it by setting v to whatever value you wish. (Higher order tautological spaces are available depending on the number of variables or other aspects of the equation. For instance, I used x^2 + y^2 + vz^2 = 0(mod x^2 + y^2 + vz^2) to prove Fermat's Last Theorem.)

v is yours entirely, no matter what the equation.

It is quite simply, a mathematical probing tool.

Which means I can say that tautological spaces encapsulate all prior human mathematical knowledge.

ALL prior human mathematical knowledge.

For the theoretical physicist looking for an edge it can't hurt to be able to grab the essence of any mathematical equation in your field, get a residue, and poke at it, at will.

Given that I've taken identities and turned them into a powerful analysis technique and named the field of study, the resistance against the ideas is fierce.

Fierce resistance is a challenge. Overcoming it is a thrill.

But that just means I can point out that people using these techniques have an edge.

You now know they have an edge.

They lose nothing from what was currently known, but may find out surprising things with a super-powered technique, so I'm thinking it probably is already secretly used by people wanting that edge.

Just for fun I handled binary quadratic Diophantine equations. The entire field.

Ten years. Years ago I was desperate for answers. Today I celebrate my successes and ponder how fascinating the journey has been. Yes, I do dominate discussions in odd ways, while still supposedly being fringe. But you do not know what advances may secretly have depended on my ideas. You do not know if you're behind today because that guy in front of you, was smarter in a way you didn't realize.

Which is how it should be, I guess. In any event that's how it is.

I've noted that Google searches around mathematical fields related to this research are dominated by my research.

So yeah, you can get backup to my having handled the ENTIRE FIELD of binary quadratic Diophantine equations, just with a Google search.

The entire field.

Weirdly enough, I covered 2000 years worth of research in a few days, re-discovering some things along the way. 2000 years, in a few days.

2000 years of human effort. With tautological spaces, I could match that by myself in a few days. Wow.

Happy Birthday Tautological Spaces! Now ten years old. And what a brilliant child of mine you are!

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