Sunday, August 31, 2014

For mathematical scientists

There have been problematic aspects to my discoveries where my ability to easily prove that an accepted mathematical framework allowed one to appear to prove two different and conflicting things should have been picked up and widely discussed by others, leading to changes to address the problem. If that happened I missed it. And since it has been over a decade since my 2003 paper I can assume something went wrong.

My reaction was to figure out exactly what, as I had a functional need which is reminiscent of when I found I needed a definition of mathematical proof which could tell me when something was a proof or not.

There is something weird about finding something wrong on such a scale. It shook me to my core. So I began to search earnestly for a firm foundation again. Needed it badly. There were such strange times when I questioned everything for a bit. My skepticism went total.

Looking at behavior that didn't fit my ideas of a mathematician, I had to figure out what I thought one was. Also I needed to understand how the problem had not impacted science, as well as why I didn't seem to be able to fix it by going to scientists. And then I wondered what relation there was between math and science and whether mathematicians were scientists.

Struggling with accepted definitions I realized I needed a functional definition of science in order to figure it out, and along with it came a definition of scientific theory:

science (noun): the art of prediction using methodologies and tools to expand zones of certainty by discovery of a predictive framework.

scientific theory (noun): predictive framework found by using science.

Those are functional definitions that allow me to determine when science is being done.

For instance I was able to determine that many mathematicians are actually mathematical scholars, and do not behave as scientists. Their emphasis is on gathering knowledge, not advancing predictive frameworks related to numbers.

My analysis also indicated that the people who would help me would need to be mathematical scientists, people who use scientific methods to advance mathematical theory.

That allowed me to finally know the most important audience for this blog, and that audience is mathematical scientists.

Now I can easily explain the behavior of mathematicians, including those I consider to be mathematical scientists and those who are not.

Turns out it's not complicated. Modern mathematics as a discipline for the most part was established in the 1800's. At that time European ideas were dominant and included the now completely out-dated notion of the gentleman scholar, the idea that certain men of distinguished breeding could figure things out because they had superior intellects, and working for a living was a mean activity which they should not face.

Well the error I discovered came into the discipline right around its inception which isn't surprising to me. Some of these hero-worship decisions were fascinating like the decision to hand Euler's zeta function to Riemann. The notion that certain human beings are imbued with extra special superpowers is rampant across fan decisions in the math field of the 20th century. Those ideas were completely discredited in other areas and the cult of celebrity around them was mostly extinguished. But in modern mathematics these heroes of the discipline were deemed infallible, and mathematics was thought to just be a building on the foundations they laid.

But they laid an egg.

Thankfully a lot of important mathematics got done regardless, but most of that was outside of pure mathematics where the gentleman scholars held most sway. Today most of what they did is junk.

One of my favorite oxymorons from this bad phase: "logical paradoxes"

Didn't take me long to handle all of those.

The amount of social structure built around these bad ways of thinking is immense and is resistant to change.

But never fear, it will crumble into dust, eventually.

I can use scientific methods easily. Presumably I might use scientific methods against the social problem of mathematical scholars preferring to continue in error in mathematics. My one concern though was in doing so before I fully understood the problem. Science is the most powerful intellectual tool ever found by humanity.

Just because you can do something doesn't necessarily mean you should.

Science gives certainty, but more importantly, predictive certainty. It's not about just believing something, but about knowing what you can make happen. Like flipping a light switch. For so much of human history such a thing would have been extraordinary, but today it's very predictable. Science has a seductive power.

My analysis is complete and my conclusion is I should rely on natural social processes rather than force anything. That analysis indicates that force would do more harm than good.

Analysis indicates a timeframe for change which is acceptable to me. Humanity will be ok.

In the meantime, mathematical scientists who are the future of mathematics can do their own research regardless.

And have all the real fun.

James Harris

Friday, August 15, 2014

Trends in mathematics

Fantasy is one thing. Anyone can dream of having major discoveries, but a necessary reality if you actually manage to have major discoveries is pushing trends in mathematics, which is kind of intimidating. But at least it's something you can check! People are weird. They may not acknowledge things publicly, but major math discoveries demand attention. And then it's hard to act against them, or not to follow in the direction they lead.

So trends in mathematics are a reality check.

Necessarily it comes with the discoveries--if you have them. So forget the fantasy with math, just wait for it. If you found something then have no fear, human nature will do the rest, and your results will push trends. To me it is one of the more definitive reality tests from the social arena, and now I consider that years ago "pure math" in esoteric abstract areas relying heavily on Galois Theory with very little demonstration with actual numbers was all the rage.

Now looking at recent Fields Medal winners including finally a woman winner I could find none of it evident. Yeah!!!

Was happily surprised when I read about the research of the 4 winners.

For those who wonder what are current mathematical trends I suggest you look it over.

And I find it hard to believe there weren't women before who should have won. The youth of the modern mathematical field, which is less than two hundred years old, keeps showing. People can get fooled I think by mathematicians from thousands of years ago like Euclid and Archimedes, who didn't just do mathematics! The notion that people could specialize in just mathematics is what's new. And it's still unproven I think as a hypothesis. Match research of any "pure mathematician" against things I now definitively have, for instance if you wish to debate. And I'm not a mathematician. Yeah Hilbert is cool, but really, how cool? And was he just a mathematician? No, he did physics too. I believe the real world anchors a mathematical understanding and challenges it in ways nothing else does.

Scarier thing is if you move trends when you're wrong. I will admit now that one of my concerns over a decade ago was where my story might lead things, and mostly that dropped away as I was in the early web arguing on the dying medium Usenet where things were misleading.

I likened it to hanging out in swamps with some very odd and loud people with opinions that didn't matter, and still don't, but the web can elevate that kind of thing. Even now when I do searches on the web around me and my math I see mostly the loudmouth opinions of nobodies.

One of my favorite little tests years ago was when I asked a poster who just kept saying no to me, some really basic math questions, which he could not answer. And I mean it was some of the most basic math I could try without getting down to simple arithmetic.

But the reality is mathematical discoveries can't help but draw attention: who is the only person to have advanced Gauss in the modern era with a better way to reduce binary quadratic Diophantine equations?

Who is the one person who used old ideas for prime counting to getter a better simple prime counting function which leads to a partial differential equation?

Who is the only person to come up with an entire new mathematical discipline which has advanced modular algebra symbology?

Part of me wants to pull back from the truth here, but why? The answer to those questions is unfortunately definitive. I didn't make this truth. I've had to live it.

And those things and more draw attention from the planet which I noticed years ago. Comes with the territory. But the intimidation of it all takes a long time to ease. After all, who would wish to lead their world wrong? And how do you know that you do not?

People have this thing, where a person being highly successful at very rare so presumably difficult things gets you noticed.

Which means I have the biggest soapbox. My opinions on mathematics and on the academic world, including funding or tenure have already impacted the world and cannot help but impact the world as I get heard. Guaranteed.

I've never been paid for any math research. Real researchers will work whether they are paid or not. I do believe that people should be able to make a living though. However in certain areas I wonder why there is any public funding. That will not impact research that is real as the best people will do it anyway. Money means nothing to them. I should know.

I've had these positions for years.

And there is a purpose to my positions and I also like to explain so people don't think it's about being antagonistic as it's not. But the reality of mathematical discovery does not fit into convenient boxes about who figures things out, or how the results get received. The fight I've had has necessarily impacted how I look at things, and I have a duty to the discipline.

There are necessary changes. The goal is to maximize discovery.

Certain discoveries will get you attention, whether others like it or not, and then your opinions can have an impact. So you end up in a certain position, whether you feel like you should be there or not, and whether others like it or not. These kinds of positions have a steep learning curve, and my fear for a long time was that I might push people in the wrong direction. But now I feel better.

James Harris

Monday, August 11, 2014

Coverage problem

Over a decade ago I uncovered a coverage problem with the ring of algebraic integers and had a paper published in a now dead math journal, which died a few years after publishing and withdrawing my paper, despite having existed over a decade prior.

Continuing the research path of the paper I have since been able to explain the coverage problem far more simply, using elementary methods:

So what are its consequences?

For fields, it has none because the field of algebraic numbers as a ratio of algebraic integers does not have this problem, though I have shifted to thinking of it as a ratio of objects, from my object ring, which I discovered to fix the coverage issue.

For the ring of algebraic integers the coverage problem allows you to appear to prove two different and contradictory mathematical statements! Which is what my original paper covered.

James Harris

Sunday, August 10, 2014

Is mathematics a science?

Finding functional definitions intrigues me as I try to understand my reality, and recently I came back to ideas about science, where I posted about it on one of my other blogs. For here I'll note I've seized upon the following simple statement:

Science is the art of prediction.

That is a functional definition in my opinion in that you can use it to determine if something is a science or not. For instance, medical science allows doctors to predictably set simple fractures. Geological science allows us to predictably tell the age of rocks. Bureaucratic science allows executives to predictably manage companies of a certain size.

Well where might one predict with mathematics?

How about with the classic equation: x2 - Dy2 = 1?

Its fundamental solution, which is the smallest set of positive nonzero integer values for x and y with positive integer D greater than 1, can vary in wild ways.

The equation is commonly called Pell's Equation by mathematicians, and you can quickly find a paper considering the problem with a search on: Pell's equation size

Doing that search now I found the following paper with open access at #2:

The size of the fundamental solutions of consecutive Pell equations

Now then that has a predictive aspect which should be readily understood, so I say that paper represents a mathematical science inquiry.

Reliably predicting the size of the fundamental solution would be a scientific triumph, and the 'art' to my statement above can be understood with this example! As how would you approach getting this answer?

That is, if you're a math person and I suggest to you the scientific investigation into the 'why' of the fundamental solution, how would you proceed?

If you're quite brave I suggest you try, as you may notice if you read that paper. I scanned through it for this post and noted something I can explain as the authors are puzzled by the very large size of the fundamental solution for D=1621 versus D = 1620.

I can explain it has to do with D being prime, and with D-1 having a lot of small primes as factors, as well as 4, and you can check: 1620 = 4*3*3*5

My full explanation which gives you the full predictive science behind the fundamental solution and connects to derivations is on this blog:

My predictive ability can be compared to the prior paper, which I think helps explain science for those who wonder, as the certainty given can be used to DO something.

So then, to the extent that mathematics gives you predictive ability, you have a science, but when it does not, you do not.

Practical usage of mathematics is often about prediction, and mathematics is often used in science for that purpose. But my example above shows how even with a "pure math" result you can have prediction as well, though it's also possible the method is practical.

What about a counter-example? Well I also have a technique for reliably reducing binary quadratic Diophantine equations! Hmmm...that sounds like it's still a science.

I also have my own way to count prime numbers. That predictably counts prime numbers.

Interesting. Kind of hard to remove prediction here, or am I missing something?

Oh yeah, I have a prime residue axiom which allow you to...why bother? It allows you to reliably predict prime gaps, like the occurrence of twin primes for example.

Well it looks like functionally I'm looking at example after example of prediction given by mathematics, and have an example of a mathematical scientific investigation where you can even challenge me! As you can go forth and try to make predictions about fundamental solutions to x2 - Dy2 = 1.

Starting this post I wasn't sure, as the post itself is, for me, a scientific inquiry, or maybe a metascience one.

My own conclusion is that from what I can tell mathematics is indeed a science, which allows one to gain certainty about numbers. And there is the 'art' in the science which is figuring out how to get to a solution!

You can make a hypothesis, test your hypothesis, and come to a conclusion.

But even better you can have mathematical proof to give absolute certainty, which then is gratifying in supporting of the statement by Gauss that mathematics is the "Queen of the Sciences".

Of course Gauss was a preeminent scientist as well as a mathematician, so no surprise there. Nice to be in agreement with him, of course.

So am I saying that mathematicians are scientists? I guess so, though I wonder how often they act like I tend to think they should! To me scientists are excited by expanding zones of prediction. I'm not sure mathematicians in general behave in that way, as the predictive aspect of mathematics is not emphasized.

So to me, scientists would be intrigued by the greater prediction inherent in my approaches, but I'm still laboring for mainstream acceptance, which indicates to me that the power of prediction is not a driving force with the mathematicians of today.

My conclusion then is: as currently trained mathematicians de-emphasize the importance of prediction in mathematics and so do not behave like scientists not so trained.

But I need to be more careful here, and remind myself that the need is to explain why mathematicians have not behaved as I originally expected.

In my opinion mathematicians don't seem to think like scientists though mathematics itself clearly is a science. That is I'm guessing about training, which emphasizes building on the known, while science emphasizes expanding zones of prediction.

Building on the known can expand zones of prediction, but is not the primary goal, and without the power of prediction to tell you when you err, there isn't an important failsafe.

Prediction has power even in "pure math" areas, as it allows you to say how numbers should behave, and see what happens.

Instead mathematicians built on what came before trusting that no mistakes were in it, which I've found left room for error.

My guess is that mathematics will evolve as a discipline, and that future mathematicians will be scientists like all the others, probably discounting the value of established positions like physicists who tend to try and attack them.

My ability to conclusively demonstrate with classic number theory that prior work was insufficient to explain, may be considered a watershed moment in mathematics.


The current refusal to properly acknowledge a problem is not surprising to me though, as without a true scientific perspective mathematicians today have reverted to typical human behavior valuing belief over scientific reality.

They would prefer to be confidently wrong, rather than predictably right, showing that a scientific perspective is necessary here.

James Harris

Friday, August 01, 2014

Easier for coders

For me playing with numbers is a lot of fun and one of my frustrations it seems like a long time ago was reading through "number theory" without any numbers in it! So there would be this long dense abstruse passage of complex mathematics that lead to some conclusion and nary a number in sight! My research is NOT that way.

Here I'll put up links to some posts with some of what I think is the easiest research to write a simple computer program to watch the numbers.

That doesn't mean I've coded all these myself as most of my effort in that area was with counting prime numbers over a decade ago.

Here's something I noticed again recently--what I think is my own axiom:

It links to my prime gap equation on it as well. I introduced the prime gap equation back in 2006 so there's a post where that is done, but I think the link above helps explain things better, before you get to the prime gap equations, as it's more up to date with my more matured thinking.

Oh yeah, so what are you able to code? Well the prime gap equation lets you predict how many primes in an interval will have a certain gap, like 2 or 4 or 6. You can then go look in those intervals and see how accurate it is. If that's fun for you, then you can do it with that math.

One of my results lets you play with some cubic Diophantine equations:

I haven't seen a lot on this subject with web searches. Coding it should be easy enough then it's just a matter of playing with the numbers and watching them follow the rules exactly.

Of course there's also my prime counting function, also known as a prime number counter, where I'll link to one of my other blogs:

There I use the sieve form and I think it's more accessible to coders as I wrote the post with that in mind, which is why it's on my other blog too.

There are LOTS of examples of math that you can code up with my research as I LOVE playing with numbers. I find it odd that you can have "number theory" which isn't shown with numbers. A fascinating area where you can consider these issues is with my explanation of what was an unsolved problem:

But I think that shows how people can get to an anti-thesis, where they feel they're doing mathematics even when it doesn't actually work with actual numbers! Huh?

Which to me is kind of a clue, you know?

I don't care how established what you're doing is, or how many mathematicians will line up and say it's great and "beautiful" mathematics: if you can't plug in some actual numbers and get a correct answer, what good is it really?

James Harris