Throughout human experience with mathematics since the arrival of algebra, there has been algebraic manipulation which is a very human activity. Sure computers can also do it in our times, but that can be debatable as to efficacy in discovery.
So you can have mathematical arguments which are lots about manipulating algebraically. Well what if you let the math do it instead?
My own math discipline around tautological spaces is actually having the math do the heavy lifting of most difficult algebraic manipulations. Which is great. I find all that work tedious, and why bother when the math can do it, and can do it with perfect ability?
The math cannot even make a mistake! Where yes, started talking about the math in a different way when considered this reality. Where greatly enhanced further math development may depend on relying on the math as simply does algebraic manipulation far better than any humans or machines.
Which seems to be a natural progression in the development of mathematics, and possibly as needed for certain mathematical advancements as algebra itself was needed.
Like ran into the complexity of:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants, which is so great has never been as simply and generally reduced as possible until my methods.
The advance was:
x+y+vz = 0(mod x+y+vz)
You can expand upon with basic algebraic manipulations and subtract some expression which has x,y and z from the result to get what I call a conditional residue. Then you can probe that residue with v, which is a free variable.
One of the basic things I have done is for some prime p use: v = -(x+y)z-1 mod p
That would allow me to probe for a particular prime or find that I had a result true for any, which then is forced to be a general result. And then is forced to be an algebraic manipulation.
Where the math did all the work.
And great thing? The math does not care. What is telling about us humans is how much we can miss when WE try to just manipulate algebraic expressions ourselves.
And my favorite example like to call the BQD Iterator:
It must be that if you have:
u2 + Dv2 = F
then it must also be true that
(u-Dv)2 + D(u+v)2 = F(D+1)
-------------------------------------------------------------
Easily demonstrated:
1. x2 - 2y2 = 1
2. (x+2y)2 - 2(x+y)2 = -1
3. (3x+4y)2 - 2(2x + 3y)2 = 1
4. (7x + 10y)2 - 2(5x + 7y)2 = -1
5. (17x + 24y)2 - 2(12x + 17y)2 = 1
and you can keep going out to infinity.
Have looked at that simplicity for years wondering, why new? Why didn't human beings find that just playing around?
Reality is that algebraic manipulation can have an art to it, which maybe people take too much for granted. But when the math does it, is logically perfect.
The math does not have human limitations in that regard.
Am confident that how mathematics was done, will go away. And in time will seem amazing to mathematicians that human beings ever tried to discover any mathematical truth by themselves trying to manipulate algebraic expressions. Which is really cool.
So use of tautological spaces is actually an uber-discipline as use the German word for over, and encapsulates almost all prior mathematics. There are exceptions which are of mathematical interest.
Where yeah if absorb that and can accept? Then you're facing what have considered for YEARS now trying to process. That math can simply do better what humans have tried to do for so long is not really surprising I guess.
That the math can do math itself is something I write just to read it, over and over again.
Then with tautological spaces you are more like a researcher asking the math questions.
Which means which questions you ask become more definitive in terms of what you can learn, but the primary thing is, the math answers.
Resistance to progress in this area is something I welcome to a large extent, while feeling a sense of duty and responsibility hard to explain as well, so is a mixed feeling. From my perspective MASSIVE resistance is best scenario. However also feel like must never forget consequences for other human beings and especially other human lives.
We just have to be sure as a species. And reality is, in that quest for certainty, there are going to be consequences.
James Harris
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