Most of my mathematical research has huge implications which makes checking me on it easy. Which may seem counter-intuitive, so thought would talk a bit about how that works.
Better math should have clear indicators.
1. There should be things never before doable from it which we can now do--as a species.
Mathematical advance MUST mean humanity can now do more. That means you can check me on things human beings just could not do before something of mine came along.
2. In general, people tend to use things that are useful and to move to things more useful than prior.
So yeah is just common sense that if there ARE better mathematical tools then people will go for them. That is not hard to find if is happening.
3. Human emotion is telling by behavior.
Which is something I quietly check routinely. And it is just human emotional reality--why put your efforts in an area where you know you will NOT get a benefit?
With plenty of things now doable that were not, am hesitant in laying out things in THIS post I think are cool. Is not really about my opinion. Human interest of others is more interesting I think with regard to certain things. The math does not care, but people may.
Checking how people react? Why not? Some people may over-estimate their ability to hide their emotions. To me is kind of funny.
Where started checking in that way years ago. Web makes it easy.
So yeah, I have discovered things that allow the human species to do things it could not do before. Have looked at indicators showing global interest and also global use, because there are things that can now be done easily which were either difficult, or impossible before.
And have noted human emotional reality by doing some web searches on certain areas to try and get a handle on current research in number theory.
Is fun. Passes the time. Though admit for me also is kind of weird. People can be so wacky.
Math does not care. Mathematics tells the truth regardless of social consequence.
James Harris
Saturday, September 22, 2018
Saturday, September 15, 2018
To what audience then?
Have wanted to move from posts which to me were trying to push certain people to acknowledge what am certain are important mathematical truths. But it can be hard not to be emotional, which part of me thinks is a path to being more convincing. However philosophically prefer to stick to the value of simply expositing the truth.
By the rules established my most compelling results for others should be around a published result. Where feel important to emphasize how wacky that got back in 2004. And recently emphasized was demonstrating how a correct argument under all the established rules of mathematical rigor could nonetheless lead to a wrong conclusion!
That result is as valid today as was back then, of course. And is simply a demonstration of how declaring the ring to be the ring of algebraic integers can lead to a result outside that ring despite starting with expressions valid in the ring and only using ring operations.
More recently defending that result found myself fascinated with a general factorization where have repeated over and over but also I like the math.
In the complex plane: P(x) = (g1(x) + 1)(g2(x) + 2)
where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x. The simplest example is: P(x) = x2 + 3x + 2
Can solve with some simple things: g1(x) = f1(x)/k and g2(x) = f2(x) + k-2
Multiply both sides by k, and substitute for the g's--to force symmetry:
k*P(x) = (f1(x) + k)(f2(x) + k)
Forcing symmetry lets me solve after introducing a handle function: f1(x) + f2(x) = H(x)
Then have a defining quadratic:
f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0
Can now solve using quadratic formula but main thing is that with integer k, have algebraic integer solutions because is a monic quadratic with integer coefficients then.
And with k = 1 or -1, you are forced to have algebraic integer solutions for the f's and the g's. Ok well that's cool.
But move to other integers like k = 2, and g1(x) = f1(x)/k blocks one of the g's from being an algebraic integer if P(x) is irreducible as then have: g1(x) = f1(x)/2
(Oh, if get confused can use simplest P(x) which IS reducible to get a handle on what is happening.)
That is so devastating to so much of prior thinking in mathematics.
But with such elementary methods, backing up a published proof.
So why does nothing happen now to officially address and fix the problem? How can a mathematical community continue in error against such simple proof?
Will leave as rhetorical. At least is a pure math result, so had no impact on applied mathematics. Which means doesn't matter for our science and technology at THIS point at least. There may be applied mathematics down the line, yet to be discovered though from the corrected mathematics.
The wild thing then is that we now are looking at one of the g's which is still integer-like but cannot be an algebraic integer, so what is it?
Years ago I pondered. And pondered, and came up with something. That something is talked about in the first post for this blog.
These new to us numbers have intriguing properties. Or I say new to us, as in new to humanity in general, while now have been contemplating them for...how long? Guess really since 2003 before my paper.
Were the reason really for that paper. New numbers previously not catalogued. And a human species still not quite fully coming to grips with them.
But the math does not care. To the extent these numbers rule? They do not care what we think.
Will still work at shifting the tone as ponder what audience finds the math to be of interest, and push myself to give up on others who do not.
You cannot force true curiosity and why bother?
Mathematics can demand the most the human mind can bring to barely understand. Without interest?
There is no way the math will make sense at the outer limits of human knowledge without working hard for that understanding.
James Harris
By the rules established my most compelling results for others should be around a published result. Where feel important to emphasize how wacky that got back in 2004. And recently emphasized was demonstrating how a correct argument under all the established rules of mathematical rigor could nonetheless lead to a wrong conclusion!
That result is as valid today as was back then, of course. And is simply a demonstration of how declaring the ring to be the ring of algebraic integers can lead to a result outside that ring despite starting with expressions valid in the ring and only using ring operations.
More recently defending that result found myself fascinated with a general factorization where have repeated over and over but also I like the math.
In the complex plane: P(x) = (g1(x) + 1)(g2(x) + 2)
where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x. The simplest example is: P(x) = x2 + 3x + 2
Can solve with some simple things: g1(x) = f1(x)/k and g2(x) = f2(x) + k-2
Multiply both sides by k, and substitute for the g's--to force symmetry:
k*P(x) = (f1(x) + k)(f2(x) + k)
Forcing symmetry lets me solve after introducing a handle function: f1(x) + f2(x) = H(x)
Then have a defining quadratic:
f12(x) - H(x)f1(x) - kH(x) - k2 + k*P(x) = 0
Can now solve using quadratic formula but main thing is that with integer k, have algebraic integer solutions because is a monic quadratic with integer coefficients then.
And with k = 1 or -1, you are forced to have algebraic integer solutions for the f's and the g's. Ok well that's cool.
But move to other integers like k = 2, and g1(x) = f1(x)/k blocks one of the g's from being an algebraic integer if P(x) is irreducible as then have: g1(x) = f1(x)/2
(Oh, if get confused can use simplest P(x) which IS reducible to get a handle on what is happening.)
That is so devastating to so much of prior thinking in mathematics.
But with such elementary methods, backing up a published proof.
So why does nothing happen now to officially address and fix the problem? How can a mathematical community continue in error against such simple proof?
Will leave as rhetorical. At least is a pure math result, so had no impact on applied mathematics. Which means doesn't matter for our science and technology at THIS point at least. There may be applied mathematics down the line, yet to be discovered though from the corrected mathematics.
The wild thing then is that we now are looking at one of the g's which is still integer-like but cannot be an algebraic integer, so what is it?
Years ago I pondered. And pondered, and came up with something. That something is talked about in the first post for this blog.
These new to us numbers have intriguing properties. Or I say new to us, as in new to humanity in general, while now have been contemplating them for...how long? Guess really since 2003 before my paper.
Were the reason really for that paper. New numbers previously not catalogued. And a human species still not quite fully coming to grips with them.
But the math does not care. To the extent these numbers rule? They do not care what we think.
Will still work at shifting the tone as ponder what audience finds the math to be of interest, and push myself to give up on others who do not.
You cannot force true curiosity and why bother?
Mathematics can demand the most the human mind can bring to barely understand. Without interest?
There is no way the math will make sense at the outer limits of human knowledge without working hard for that understanding.
James Harris
Wednesday, September 05, 2018
Have been surprised so some math reality
Should admit if also for myself as DO read through my posts wondering at times, years later, have been surprised at how things have gone with some recent results.
Like last year was back almost to holding my breath with the modular inverse find. Luckily am smart enough to realize...ok, was VERY surprised. So to recap though have talked relentlessly back May of last year discovered there was a new way, MY way, to calculate the modular inverse. Explain modular inverse in overview here.
Yet here we are.
Is clear to me that mathematicians are going to play the ignore game--again.
Which is very weird. Also FINALLY trotted out a general method for reducing certain types of three variable quadratic equations. And figured hey with a UNIQUE result showing something number theorists couldn't do, maybe something would happen. Yet here we are.
Guess mathematicians are going to play the ignore game there too!
So what happens? People use the results. I can watch evidence on web search but also there is other evidence. There are things you can do with my math research that humanity as a species struggled with, without it.
Like in this case strongly suspect my modular inverse method can be made VERY much faster than anything prior. And being able to generally reduce even a certain type of three variable quadratics for the FIRST time in human history?
Well, we live in a three dimensional spatial reality, you know?
Oh well. Our species can be so wacky. What can I do about reaction of OTHER people to major mathematical finds? Not much. But have explained that plenty of times.
My primary job is just to make the information available, for the good of humanity. It's a really cool job too. However so far? Doesn't pay very well, monetarily.
You cannot make any particular humans beings be interested in math, in reality. Fantasy though?
Well lots of people are all over that. Real math can be VERY hard, to discover.
So I get to be the major discoverer who gets to deal with a variation on past themes.
Ok. There was a slot available. To me? Is actually maybe more interesting this way.
Am working on not being surprised is all. Now have covered enough ways to be a major math discoverer guess that's it.
Well eventually will get over my surprise. Actually doesn't change things much for me--as long as I'm not surprised!!! You'd think I'd know by now though. Like have over 16 years with my explanation for the prime distribution thingy. Still part of me was clinging to something which I find hard to let go.
Was a view of the human species that I held but must relinquish. We're just more emotional as a species than moved by fact.
You have to appeal to people's FEELINGS whether you like it or not. Oh.
Ok. Yeah needed to put that down. So can remind myself.
Maybe will get around to that--eventually. Is not like it's hard to do. Still part of me is like, really?
Are you sure?
Just bothers me on principle I think. More and more am into logic.
Part of me thinks I simply need to expand the problem space. But what's my motivation? Eh?
Knew that was coming.
Maybe my motivation is the same as always: I just want to know--more.
James Harris
Like last year was back almost to holding my breath with the modular inverse find. Luckily am smart enough to realize...ok, was VERY surprised. So to recap though have talked relentlessly back May of last year discovered there was a new way, MY way, to calculate the modular inverse. Explain modular inverse in overview here.
Yet here we are.
Is clear to me that mathematicians are going to play the ignore game--again.
Which is very weird. Also FINALLY trotted out a general method for reducing certain types of three variable quadratic equations. And figured hey with a UNIQUE result showing something number theorists couldn't do, maybe something would happen. Yet here we are.
Guess mathematicians are going to play the ignore game there too!
So what happens? People use the results. I can watch evidence on web search but also there is other evidence. There are things you can do with my math research that humanity as a species struggled with, without it.
Like in this case strongly suspect my modular inverse method can be made VERY much faster than anything prior. And being able to generally reduce even a certain type of three variable quadratics for the FIRST time in human history?
Well, we live in a three dimensional spatial reality, you know?
Oh well. Our species can be so wacky. What can I do about reaction of OTHER people to major mathematical finds? Not much. But have explained that plenty of times.
My primary job is just to make the information available, for the good of humanity. It's a really cool job too. However so far? Doesn't pay very well, monetarily.
You cannot make any particular humans beings be interested in math, in reality. Fantasy though?
Well lots of people are all over that. Real math can be VERY hard, to discover.
So I get to be the major discoverer who gets to deal with a variation on past themes.
Ok. There was a slot available. To me? Is actually maybe more interesting this way.
Am working on not being surprised is all. Now have covered enough ways to be a major math discoverer guess that's it.
Well eventually will get over my surprise. Actually doesn't change things much for me--as long as I'm not surprised!!! You'd think I'd know by now though. Like have over 16 years with my explanation for the prime distribution thingy. Still part of me was clinging to something which I find hard to let go.
Was a view of the human species that I held but must relinquish. We're just more emotional as a species than moved by fact.
You have to appeal to people's FEELINGS whether you like it or not. Oh.
Ok. Yeah needed to put that down. So can remind myself.
Maybe will get around to that--eventually. Is not like it's hard to do. Still part of me is like, really?
Are you sure?
Just bothers me on principle I think. More and more am into logic.
Part of me thinks I simply need to expand the problem space. But what's my motivation? Eh?
Knew that was coming.
Maybe my motivation is the same as always: I just want to know--more.
James Harris
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