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Saturday, July 05, 2014

More modular algebra symbology

The advent of algebra itself was one of the greatest things in human history, as considering unknowns with numbers vastly increased the analytical powers of the human species.

An equation like x+y = z may not seem like much to us today, but clearly its flexibility is far greater than 2+3 = 5.

With modular arithmetic I think there has long been a focus on explicit numbers versus heavy use of symbols, so it parallels a focus before on numbers as that's a more practical concern.

While a shift to the greater purity of symbols also gives more flexibility but can be difficult as an increase in abstraction which is my guess at how I got to be the person who introduced tautological spaces.

So I got to be the guy who pushed the symbology of modular arithmetic more into a modular algebra with something as simple as:

x+y+vz = 0(mod x+y+vz)

And the extent to which that was NOT intuitive to the mathematical world can be seen at the reticence in increasing the symbology of modular arithmetic into more of a modular algebra which exists to this day.

Talking about it on my blog helps then to increase the spread of human knowledge.

So you can see a HUGE increase in modular algebra symbology in my research, which has an extra level of abstraction which is clearly not evident in the more simpler approaches you see from the traditional, where for instance you may see a lot of things like: x = a mod b. So yeah there are still symbols, where actually a lot of times you'll see numbers, but nothing like the heavier symbology in my mod x+y+vz.

But is that a necessary level of abstraction like algebra was to simple counting or an unnecessary layer of complexity?

Well consider, using my approach I was able to simplify reducing binary quadratic Diophantine equations!

That achievement in and of itself I would think should be kind of big, right? I mean we're talking about approaches that were handed down from ancient times one could say.

Before my research I'm sure none of the top number theorists in human history even knew that such an improvement could even exist. They thought I'm sure, wrongly we now know, they had the best possible in mathematics already.

Yet I improved upon them? Oh yeah, that could make it one of the greatest in mathematics, but that's just an opinion. But what makes it kind of weird even to me is that it was so easy.

But then again the increase in abstraction with the symbols of algebra turned a lot of problems into far easier to approach ones as well! So I'm looking at a justification for the necessity of this approach.

A good introduction to these things including links to the full mathematical work and proof can be found through my post:

somemath.blogspot.com/2013/09/leveraging-asymmetrical-forms.html

The shift in thinking necessary to embracing greater symbology in modular algebra appears to be kind of big, as I've had this research for quite a while now.

But that's not a surprise. Each major advance in human thought requires a paradigm shift. And while I can think it's an advance things move slowly until more and more people agree with me! That's ok. It's to be expected.

And quite reasonably people need a LOT of justification and time to accept such things, as once that shift is made, a TREMENDOUS amount of human effort will be involved from then on in that direction.

Which we've seen with algebra--one of the most important subjects, ever.

It makes sense that people take their time and are careful.

For me it gets to just be exciting.


James Harris
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