Knew I probably had a long journey ahead back I guess in 2003. Had been working at trying to prove Fermat's Last Theorem which was exasperating but fun--with TONS of errors. But had my approach to the packing of spheres from 1996, which hadn't made it past an editor, and in my mind seemed too simple like he had said.
Now I realize I'd used a spatial modular approach, and modular is the most powerful...how best to describe? Now of course I have lots more modular but still can be hard to grasp how changes entire field of mathematics.
Oh, so yeah, was chasing Fermat's Last Theorem with lots of frustration, but had also found my own way to count primes with a clever thing. I now know that thing is:
ΔS(x,pj) = [x/pj] - 1 - (j-1) - S(x/pj, pj-1)
That is the count of composites for a particular prime excluding composites that are products of lesser primes.
Which I copied from this post which I noticed was trending highly. Is so weird how something so short, and simply presented can change so much. That is the key to questions that Gauss, and Euler and so many others had about prime numbers. And we've had the answers now for over 16 years. But yeah I knew I had a long haul ahead, but not because of THAT result.
My heart sunk when I realized what I would call for a bit the coverage problem. When came across a way to present what looked like a perfect mathematical contradiction--within accepted rules. I could declare the ring as algebraic integers, use expressions valid in that ring, and valid ring operations, and nonetheless get to a conclusion not true in THAT ring.
Is kind of funny now. I just felt like the result was too big. Couldn't comprehend why was mine, and back then also hadn't yet resolved fully in my mind that the foundations of mathematics weren't maybe flawed. So I did try, and did my best to write a paper as close as I could to some standard format. And even shopped journals which I hadn't bothered to do with my spherical packing paper.
Even visited my alma mater Vanderbilt University to pitch the paper in person to a math professor who was an editor of some journal.
But it was later found a journal that would publish as have talked so much. And yeah was in conversation by email with editors at that journal for nine months before got word my paper would be published. My first correction to them was to stop them from giving me the title of Dr. Harris in correspondence. Emphasized was not a mathematician and I don't have a doctorate degree.
For me 2004 was not a year when felt well much of that year. Went from that elation and preparation for the world press when knew my paper would be published to the consternation and disappointment as drama played out instead. And then calm resignation as yeah, would be the long haul I feared.
Now appreciate how lucky I was. I DID get a publication. And then was left to myself to pursue mathematical truth. Eventually figuring out all the details which intrigued me so much. Back then I had inklings and intuition. Figured there had to be these other numbers, but how could I be sure?
Without help from the mathematical establishment, realized I needed a functional definition of mathematical proof. So I could know. And built so many others things. Until the activity was my new normal. And later would also accept being a global figure in my own right, where have puzzled there as well.
Now an authority in my own right on my math and I ponder what I owe my world or my species. I think about what I want for mathematics itself globally, and I wonder.
Is better to know. That I firmly believe. Have been so blessed I wonder why. But then again, so much I know now because have been--always with the questions.
And actually got some really cool answers thank God.
James Harris
Monday, February 25, 2019
Sunday, February 24, 2019
Why new numbers, really?
Feel is important to continue to emphasize what is actually I guess my most dramatic discovery, which is easy to prove with elementary methods. So yeah back in 2003, realized that there were more important numbers than had previously been catalogued.
Eventually, thankfully, came across a very simple way to show their existence, with quadratics.
In the complex plane, consider:
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) is not equal to 0 for ALL x.
The simplest example is: P(x) = x2 + 3x + 2
And that is the end of things just casually making sense. As by the time I knew to use this expression around 2010, also knew I NEEDED to multiply it by an additional, added factor I call k.
So, introduce k, where k is a nonzero, and new functions f1(x), and f2(x), where:
g1(x) = f1(x)/k and g2(x) = f2(x) + k-2
Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:
k*P(x) = (f1(x) + k)(f2(x) + k)
Getting to symmetry lets me solve for the f's, and do so as roots of a monic polynomial with integer coefficients, so I can make them be algebraic integers!
So yeah it is all a way to flip things cleverly. Is like building backwards. Then I get the crucial result which follows.
But for one of the f's: f1(x) = kg1(x)
Where now is built front and notice result NOW will apply in a ring.
And we have a problem if the f's are not rational. We do but the math doesn't. The algebra is straightforward. Is weird how easy it is really.
Of course indices are arbitrary so we simply know that one of the solutions has k as a factor. If you wonder why that is a problem, consider 3+sqrt(-26), as it can be shown that one of its two solutions has 7 as a factor!
Confused? Consider 1+sqrt(4) = 3 or -1. Tend to gloss over that quickly but it is bizarre that people might be taught any other way. It is actually simply mathematically incorrect to say that sqrt(4) is JUST 2, as -2 is a solution. You can't just wish away.
Oh yeah, so of course then 1 - sqrt(4) = -1 or 3. So has 3 as a factor for one of its two values.
But it can be proven that 3+sqrt(-26) also does NOT have 7 as a factor in the ring of algebraic integers.
Well yeah one of the g's cannot be an algebraic integer if k is some nonzero integer other than 1 or -1, when they are NOT rational. (Let them be integers like with simple example noted to notice how easy the math actually is.)
So yeah, my trick is forcing the f's to be algebraic integers, when I multiplied times some prior numbers, and that blows up LOTS of prior number theory when not rational. And then reveals new numbers! Yay!
The situation is like, if people knew of evens and did NOT know of odds, so you caught someone exasperated by the notion that 6 and 2 share 2 as a factor, because they don't know 3 exists.
Well there isn't a unit there which is a big difference. But situation is exactly the same with ring of algebraic integers except there are units.
But if you don't know there are those other numbers, yeah can get apparent contradictions which I exploited in my pivotal paper like to talk so much.
Like with this post: Publication does matter
Are these numbers a big deal? Of course. We're talking about number theory where MOST of the numbers, like an infinity of them, were previously unknown to exist! And I found them. Is cool.
(Oh my God is just so freaking amazing. Why me? How me? Oh my God, oh my God, oh my God.)
Am much better at handling that information now than in the past. See? I'm maturing as a human being.
James Harris
Eventually, thankfully, came across a very simple way to show their existence, with quadratics.
In the complex plane, consider:
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) is not equal to 0 for ALL x.
The simplest example is: P(x) = x2 + 3x + 2
And that is the end of things just casually making sense. As by the time I knew to use this expression around 2010, also knew I NEEDED to multiply it by an additional, added factor I call k.
So, introduce k, where k is a nonzero, and new functions f1(x), and f2(x), where:
g1(x) = f1(x)/k and g2(x) = f2(x) + k-2
Multiply both sides by k, and substitute for the g's, which gives me the now symmetrical form:
k*P(x) = (f1(x) + k)(f2(x) + k)
Getting to symmetry lets me solve for the f's, and do so as roots of a monic polynomial with integer coefficients, so I can make them be algebraic integers!
So yeah it is all a way to flip things cleverly. Is like building backwards. Then I get the crucial result which follows.
But for one of the f's: f1(x) = kg1(x)
Where now is built front and notice result NOW will apply in a ring.
And we have a problem if the f's are not rational. We do but the math doesn't. The algebra is straightforward. Is weird how easy it is really.
Of course indices are arbitrary so we simply know that one of the solutions has k as a factor. If you wonder why that is a problem, consider 3+sqrt(-26), as it can be shown that one of its two solutions has 7 as a factor!
Confused? Consider 1+sqrt(4) = 3 or -1. Tend to gloss over that quickly but it is bizarre that people might be taught any other way. It is actually simply mathematically incorrect to say that sqrt(4) is JUST 2, as -2 is a solution. You can't just wish away.
Oh yeah, so of course then 1 - sqrt(4) = -1 or 3. So has 3 as a factor for one of its two values.
But it can be proven that 3+sqrt(-26) also does NOT have 7 as a factor in the ring of algebraic integers.
Well yeah one of the g's cannot be an algebraic integer if k is some nonzero integer other than 1 or -1, when they are NOT rational. (Let them be integers like with simple example noted to notice how easy the math actually is.)
So yeah, my trick is forcing the f's to be algebraic integers, when I multiplied times some prior numbers, and that blows up LOTS of prior number theory when not rational. And then reveals new numbers! Yay!
The situation is like, if people knew of evens and did NOT know of odds, so you caught someone exasperated by the notion that 6 and 2 share 2 as a factor, because they don't know 3 exists.
Well there isn't a unit there which is a big difference. But situation is exactly the same with ring of algebraic integers except there are units.
But if you don't know there are those other numbers, yeah can get apparent contradictions which I exploited in my pivotal paper like to talk so much.
Like with this post: Publication does matter
Are these numbers a big deal? Of course. We're talking about number theory where MOST of the numbers, like an infinity of them, were previously unknown to exist! And I found them. Is cool.
(Oh my God is just so freaking amazing. Why me? How me? Oh my God, oh my God, oh my God.)
Am much better at handling that information now than in the past. See? I'm maturing as a human being.
James Harris
Monday, February 11, 2019
Better defining what I call the social problem
Back last year noted in a post was turning away from what I call the social problem until 2028. By that phrase mean working to get the acceptance by the established mathematical community of my research.
And that's it.
For years tried various things from working to getting another paper published after my pivotal one got through, and then all that weird happened, to direct contacts with mathematicians like by email. And would do posts at times I felt were targeted to various levels of the mathematical community, to no avail.
Which raises the question then of, to whom are my posts directed now?
Which to me is where the fun is. But my resolve to not focus on the social problem till 2028 was primarily to get me out of a rut, and besides it's kind of embarrassing. Makes me look desperate. Now I just don't bother.
James Harris
And that's it.
For years tried various things from working to getting another paper published after my pivotal one got through, and then all that weird happened, to direct contacts with mathematicians like by email. And would do posts at times I felt were targeted to various levels of the mathematical community, to no avail.
Which raises the question then of, to whom are my posts directed now?
Which to me is where the fun is. But my resolve to not focus on the social problem till 2028 was primarily to get me out of a rut, and besides it's kind of embarrassing. Makes me look desperate. Now I just don't bother.
James Harris
Saturday, February 02, 2019
Abstract reductionism and simpler rules
Debate with myself often my responsibility in making certain things clear. This post is meant to help clarity while also showing the logical framework around my work, especially with regards to what I like to call abstract reductionism.
The most dramatic discovery I made was over 15 years ago, when realized could create a mathematical argument correct by the accepted rules of the mathematical community which nonetheless lead to an incorrect result. So I wrote a paper which was published demonstrating.
So yeah you have this HUGE result which brings into question mathematical consistency. Where the paper demonstrates inconsistency--so stands on its own regardless of anything else.
When the mathematical community did not behave as expected, I realized there was serious corruption as no real mathematicians would leave something like that undiscussed at very high levels and especially not, unresolved.
Which was an opportunity for me as have admitted as discovered problem was JUST with ring of algebraic integers, so a proper ring had to exist--for mathematical consistency. To be sure I also studied logic and concluded that mathematics is a subset of logic. That forced me to resolve the oxymoron of a "logical contradiction" and did that early on this blog.
From the logic axioms linked there established what I called: Three Valued Logic
That was HUGE for me and necessary for later when defined mathematical proof.
And I had mathematical consistency back. Had mathematics as a subset of logic. And had begun abstracting to simpler things, in a way I now call abstract reductionism. Where also focused modular often, and one of my most powerful tools had lead me to the problem with algebraic integers, when focused on x+y+vz = 0(mod x+y+vz). (And I don't use congruence symbol.)
So fifteen years ago, by this time knew that my paper could pass formal peer review and was published but things got weird after. And realized mathematicians as a community were corrupted. And began a distancing which has continued. As of course people with a title of mathematician who don't behave as mathematicians are useless to me.
But abstract reductionism is a logical tool. And importantly went on to use it to functionally define science, and also to functionally define entertainment. That latter was less than four years ago. And world is already in flux as global entertainment has changed rapidly with the US leading those changes with dramatic impact for example with movies with the US box office.
I think it is all SO COOL where have defined cool, of course. Not linking to too much here and trying to keep this post short. The gist of it for those interested in mathematics is to establish much is certain while the world is just taking its time to handle certain things.
To the human species, fifteen years is barely a blink.
While I'm enjoying the ride more and more.
So this post covered logic from axioms to actual Three Valued Logic. Established how mathematical inconsistency appeared with use of ring of algebraic integers, demonstrated with my pivotal paper. And talked abstract reductionism as a logical tool and process which handles mathematics--and beyond.
James Harris
The most dramatic discovery I made was over 15 years ago, when realized could create a mathematical argument correct by the accepted rules of the mathematical community which nonetheless lead to an incorrect result. So I wrote a paper which was published demonstrating.
So yeah you have this HUGE result which brings into question mathematical consistency. Where the paper demonstrates inconsistency--so stands on its own regardless of anything else.
When the mathematical community did not behave as expected, I realized there was serious corruption as no real mathematicians would leave something like that undiscussed at very high levels and especially not, unresolved.
Which was an opportunity for me as have admitted as discovered problem was JUST with ring of algebraic integers, so a proper ring had to exist--for mathematical consistency. To be sure I also studied logic and concluded that mathematics is a subset of logic. That forced me to resolve the oxymoron of a "logical contradiction" and did that early on this blog.
From the logic axioms linked there established what I called: Three Valued Logic
That was HUGE for me and necessary for later when defined mathematical proof.
And I had mathematical consistency back. Had mathematics as a subset of logic. And had begun abstracting to simpler things, in a way I now call abstract reductionism. Where also focused modular often, and one of my most powerful tools had lead me to the problem with algebraic integers, when focused on x+y+vz = 0(mod x+y+vz). (And I don't use congruence symbol.)
So fifteen years ago, by this time knew that my paper could pass formal peer review and was published but things got weird after. And realized mathematicians as a community were corrupted. And began a distancing which has continued. As of course people with a title of mathematician who don't behave as mathematicians are useless to me.
But abstract reductionism is a logical tool. And importantly went on to use it to functionally define science, and also to functionally define entertainment. That latter was less than four years ago. And world is already in flux as global entertainment has changed rapidly with the US leading those changes with dramatic impact for example with movies with the US box office.
I think it is all SO COOL where have defined cool, of course. Not linking to too much here and trying to keep this post short. The gist of it for those interested in mathematics is to establish much is certain while the world is just taking its time to handle certain things.
To the human species, fifteen years is barely a blink.
While I'm enjoying the ride more and more.
So this post covered logic from axioms to actual Three Valued Logic. Established how mathematical inconsistency appeared with use of ring of algebraic integers, demonstrated with my pivotal paper. And talked abstract reductionism as a logical tool and process which handles mathematics--and beyond.
James Harris
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