So back December 1999 out of desperation I came up with using identities to help analyze mathematical equations, where after some thought I first focused on the identity:
x+y+vz = x+y+vz
where I made it part of modular arithmetic very simply:
x+y+vz = 0(mod x+y+vz)
which is equivalent, and I decided to call mod x+y+vz a tautological space, as it represents a region where that is true, and I liked the sound of it. I've also considered logic and mathematics, so it was easy enough for me to meld them a bit with that term.
And I can do things with tautological spaces like I came up with a more refined and general way to reduce binary quadratic Diophantine equations, among other things.
And I've advanced tautological spaces methodology. The general form is actually:
a1h + a2h + v1a3h +...+vn-2anh = 0(mod a1h + a2h +v1a3h +...+vn-2anh )
where h is what I call the hyperdimension and n is what I'll now call the tautological freedom.
So with my first tautological space you have a hyperdimension of 1 and a tautological freedom of 1, which means three variables, which are x, y and z.
The generalization to the full form is needed for more dimensions than x, y and z, as well as for the calculus as I'd started pondering how to bring it in with my idea, which I realized required that you have at least two degrees of tautological freedom. That is, n=4 or greater is required for the calculus.
For instance if you differentiate against the tautological space itself, you do a partial differentiation, for example with h = 1 and n = 4, you have:
a1 + a2 + v1a3 + v2a4 = 0(mod a1 + a2 +v1a3 +v2a4)
And one way to differentiate is:
a3 + a4dv2/dv1 = 0(mod a3 + a4dv2/dv1)
That's curious. I'll call it a tautological differentiated space or a tautological D-space, or a TDS.
What's interesting is, you do not in general differentiate from within the tautological space, so you'd differentiate a conditional outside of it, as when you differentiate inside the tautological space you lose variables!
The problem is that differentiation is one thing that must impact the modulus, and it's only possible to do it against the v's as otherwise it's trivial.
And I don't know the utility of that expression, but will keep it up.
But there is another way to introduce the calculus in a possibly more useful way, since the a's are so free, as consider the following tautological space:
x + y + v1yx' + v2z = 0(mod x + y + v1yx' + v2z)
which allows you to introduce a conditional with differentials, as I can just use as many of the a's as needed, and you can use partial differentials or full differentials as the a's do not care.
The general principle is that the shape of the tautological space is driven by the conditional.
There are an infinity of tautological spaces. You simply pick from that infinity to suit your needs.
Pulling from the Absolute.
Because it's an identity it MUST be true, and I do wonder what great things can now be done.
Excellent, so you CAN use the calculus, and my generalization was needed for that effort, where now I see where differentials can be added in a way that makes more sense to me.
And I haven't even reached for integration yet.
That's enough of that effort. Thinking at foundation level is just SO hard.
Makes you really hungry too, but that's a benefit, as food tastes so much better.