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Friday, March 30, 2018

BQD Iterator is a very surprising tool

Realized that one of the more profound results for me both practically and philosophically is a mathematical tool I call the Binary Quadratic Diophantine iterator, or BQD Iterator for short.

Where have this post as my established reference, posted November 28, 2014, which talks in-depth. For here will just give my favorite form again.

Given: u2 + Dv2 = F

then it must also be true that

(u-Dv)2 + D(u+v)2 = F(D+1)

-------------------------------------------------------------

Where do minor variations on that theme here and there as of course can shift with symbols used and signs, and what is weird about that thing is, seems to connect to just about every property of integers.

Wouldn't that be one of the weirdest things ever? Like, ONE thing being key to so much where for thousands of years so many searched and found pieces of the puzzle, when the biggest thing was SO simple.

And yeah I know, may not look like much. Have stared at it often for years now, pondering.

The BQD Iterator quite simply may control, or in so way be related to the controls for all integers. It is the one tool that the math may use, just about everywhere with integers. I find that remarkable, and wonder if is true.

Like MAY lead to THE template for ALL integer factorizations: But does it factor?

Where is one of those areas where I do NOT check thoroughly, as delves into too much scary. (And will collect here another post where talked scary areas with this post with regard to implications from other things.)

The AMAZING thing about the BQD Iterator is that no human found it directly, which includes me, as I used tautological spaces. What is it about it that not even just playing around no human just stumbled across it? Or maybe someone did, but is just not big in the historical record? Have wondered.

And I LOVE to play with it now. Though took a few years, where helped out much when named it for some reason. Like sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 1351xn + 2340yn

yn+1 = 780xn + 1351yn

and x0 = 1, and y0 = 0;

So get x1 = 1351, and y1 = 780. Which is a decent approximation, and next is:

x2 = 3650401 and y2 = 2107560

And 3650401/2107560 is approximately:  1.73205080757

Have used to explain result where Euler and Ramanujan played around, along with lots of other things. And emphasized for me just how much simple we may never know, as how do we know what we do not know?

So much of being human is about aspects of our design with which we are simply born, which must impact much what we can discover. Which is kind of depressing actually.

And yes, talked in this post where mused about my recent find of a third primary way to calculate the modular inverse. Where have been trumpeting that one for a bit now.

For me it is interesting then to have a result where yeah, was me there. And the math didn't hand me THAT result, but then again took me years to notice something that followed from something clever. Where kind of wonder, why did I think of it?

But then again, have been as I say, talking to the math now for quite some time. And has dawned on me that I DO think differently now. Maybe there does happen that better ability to look outside the box as the saying goes, when have watched a process give answers WAY outside of what humanity found on its own.

Yeah that does make sense. Watching the math DO math, could conceivably have shifted how I look at mathematical problems. Like if had watched some human teacher working on math, but am watching an infinite intelligence instead. Wow. These are the kind of posts where am glad just start typing and find out where will go.

Like, what greater teacher of mathematics could there be to watch? Than the math? As can watch the math DOING mathematics, and just be like a student, learning?

And later, me, the student could figure out something cool entirely on my own. Yeah. Maybe.

So yeah, watching the math, do math, can help teach a human, me, how to do math better? Wow. What a concept. Can explain much if true. And I do so like to hedge, do notice. If true.

Yeah but did have the packing of spheres thing, before. Though still bothers me for some reason, how simple that modular approach is. And of course, did figure out myself how to use tautological spaces. Still guess do love the idea of the math itself as one of my greatest teachers. And why wouldn't I?

Makes sense to me.


James Harris

Wednesday, March 28, 2018

More playing with square root of three

Number theory is focused on integers. And mainly am focused on number theory. But there is math that I find myself just doing for the fun of it, which goes into approximation. But really only because is just another demonstration of some of my math discovery. Also though is kind of a reminder of why x2 - Dy2 = 1 had historical importance. As solutions could be used to approximate square roots.

With my research noticed special cases with my BQD Iterator for two and three, where could use to get easy approximations. And is just kind of fun for me and even relaxing to advance. Last post was this one where ended up with sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 362xn + 627yn

yn+1 = 209xn + 362yn

and x0 = 1, and y0 = 0;

And my first post was this one for my reference.

So need the BQD Iterator, with D = -3, to advance some more:

u2 - 3v2 = 1

then it must also be true that

(u+3v)2 - 3(u+v)2 = -2

-------------------------------------------------------------

(362x + 627y + 3(209x + 362y))2 - 3(362x + 627y + 209x + 362y)2= -2

Which is: (989x + 1713y)2 - 3(571x + 989y)2= -2

And one more iteration to be able to divide off some factors:

(989x + 1713y + 3(571x + 989y))2 - 3(989x + 1713y + 571x + 989y)2= 4

Which is: (989x + 1713y + 3(571x + 989y))2 - 3(989x + 1713y + 571x + 989y)2= 4

Which is: (2702x + 4680y)2 - 3(1560x + 2702y)2= 4

So now can divide off that 4, where happens because 2 is the only prime factor of D+1, where is every two iterations, which is why is a special case. Wonder if will always get a cycle if D+1 only has 2 as a prime factor? Sounds like something to check with a computer. But might be mathematically provable also. Checking with computer first could test the hypothesis. Ok, so dividing off the 4.

And finally: (1351x + 2340y)2 - 3(780x + 1351y)2= 1

From which I have, sqrt(3) approximately equals xn+1/yn+1, where:

xn+1 = 1351xn + 2340yn

yn+1 = 780xn + 1351yn

and x0 = 1, and y0 = 0;

So get x1 = 1351, and y1 = 780. Which is a decent approximation, and next is:

x2 = 3650401 and y2 = 2107560

And 3650401/2107560 is approximately:  1.73205080757

Just using how much a web calculation showed. Another trick is to square, and look at what that gives. Where looking at prior post got more digits before.

And 36504012/21075602 approximately equals 3 is what my pc things are saying, so is too close for more with precision of the defaults. Have had LOTS more digits before but just playing around so not interested in fiddling with settings to find out why so few now.

Well guess at limits then of just being able to check with usual pc things. Oh well.

Well that was simple fun. Find it kind of relaxing. Just playing with numbers.


James Harris

Wednesday, March 21, 2018

What math knows

So yeah, much of my own discovery is simply advancing modular approaches. And is surprising to be sure now that modular was underdeveloped in mathematics, as for instance, I use modular algebra where when look elsewhere just see modular arithmetic.

And found out around a decade ago that math could itself do math, when promptly and easily found a way which I now know leads to a general way to reduce a three variable quadratic Diophantine equation. But the modular algebra does not care about Diophantine or not.

Gives a different take on question of whether mathematics is discovered or created by humans, and I think clearly mathematics follows from axioms and logical tools of mathematics. Where can just check at what the math can find that we did not without its help.

So can look at something the math found like three variable quadratic reduction and human beings might NEVER have found without, where was just the human who used the tools that a modular approach allows. Am a person more in awe of that process that can handle algebra vastly better than I can.

Why would the math field not grab such information with gusto? Have theories. Do not care really. Consider that the social problem which may address directly again in 2028. And I knew the math could work math problems itself back in 2008.

Reality is: we are not that important to mathematics. No human is. Is the aggregate of human knowledge that matters. And is like worrying about human beings who do not believe in electricity, to consider those who refuse to advance.

Would we?

Where yeah, have been concerned with influence someone like myself can wield. And maybe should admit long ago realized that I did not need any of the then current mathematicians. And figured that out about a decade ago I guess. With an advanced tool that can do algebra better than any human? Of course I don't. Can replace every single mathematician that decides not to use.

But then again, should I? My decision is, well you're living it now.

Actually do not even need to talk. But more and more think am talking these things now in this way, for my own mental health. Also yes, to help others who are curious and I think deserve answers.

Have often noted am NOT a mathematician, which distances me from people with a label established, who may not be behaving in accordance with it. Am a major mathematical discoverer.

Of course my main tool advance focuses on algebra, but I started with a geometric modular approach, used upon packing of spheres.

For me my own research was very unsettling as was very surprising. And depended so much on the web which helped me figure things out, and also enabled me to experience becoming a global figure. And had to learn lessons there as well. These are not facts meant to convince. These are simply facts.

Mathematics is a zone of truth really. And the better have gained an understanding and appreciating the ability to find truth, more have learned that focusing on facts is best.

The math knows infinity. That's what the math knows. Human beings can talk infinity. We can think we understand in some way. Or do whatever it is that human beings do while we are here.

The math does not need us. That is what have processed through my own discovery.

We discover math. And math can help us discover more math. The math is the greater.

Is comforting actually.  Well it is to me.


James Harris

Talking sense of personal disruption

Have benefited by shifting perspective from talking things one way to more matter-of-fact in terms of explaining. Like carefully walking through how tautological spaces can do algebra for you. Thing is, of course, have known since 2008, when found what I decided to call the Quadratic Diophantine Theorem.

Thing is, years ago when looked to try and find my own math discovery was looking to build on what had already been done. Checking to see if maybe simplifying approaches were available to established discoveries. And definitely was not looking to find some disruptive math technology as way can describe, which could do math itself.

The feeling when realized was not one of awe, but this really hard to explain feeling which can say, is yeah of being emotionally disrupted, and from then on, a search to again find that feeling of solid ground. And also admitted probably had a fight on my hands, which I had never expected or wanted. The only good thing about that though is, time to process.

You do wonder though about simple explanation almost a decade after you have it.

Took me quite some time to explain my prime counting approach simply as well.

May as well note more. Of course I knew had something solid with prime counting in summer 2002. There was no way to doubt. And then in 2004 had a published result, even if things got wacky after. But also knew that results implications: could appear to prove something you could show false with a separate argument. Contradiction. No doubt then, and no doubt now. Resolving lead me to my find of the object ring.

Over and over again just found more as a result of challenging things. Yet also the feeling was not helpful will say. Understand better now.

Realize now that repeatedly would find my sense of reality severely disrupted and maybe the sense of fighting with these results helped in that way. There is a debate have had with myself though as to how much resistance to my research was ever real or imagined by me.

There is a greater sense of stability now though just from the emotions settling down. But still not completely, and yeah, why I like to say, math and emotion do not mix well.

People can think you're arguing with them, when you're trying to find some gap, or silliness or SOMETHING wrong with your own ideas. Or that you're desperate for them to believe, when you're trying to calm your emotions down. And repeat, and repeat and repeat like maybe will feel different with the repetition.

And yeah, eventually it does, I guess. Am talking things better now I think.

The math does not change, of course. And yeah, when you know you have techniques that can help just about anyone do math better than the greatest mathematicians of the past? Really worried about it being picked up, eventually? Nope.

Thing that fascinates me is how we feel can shift how we perceive things. Like how many never just considered that yeah, was giving a better approach in a high profile area than Carl Gauss himself found. So many times have muttered to myself: binary quadratic Diophantine equations. World now has the best way to generally reduce possible, finally.

Yeah because the math found it for me.

Where technically that is, analyzing with tautological spaces was able to complete complex algebraic manipulations easily, using a simple modular algebra approach, which automates much of that process.

And yeah, you learn to talk more technical through the years as well. Much more settled now.

For years though was one human who could not escape that feeling of disruption. Could only run away for so long, or distract myself for so long with other things. Are my ideas.

They are my ideas.


James Harris

Tuesday, March 20, 2018

Math innovation, disruption and explanation

Working through explaining things carefully, while also considering other areas where web disruption has been evident, am better able to explain things in my own story.

So now can better put a context around noting figured out a better way to reduce two variable or also called binary quadratic Diophantine equations than Carl Gauss.

As actually used what I decided to call tautological spaces--getting to make up terminology is fun and a responsibility--with a process have explained quite a bit now.

The math doing the work, is as good as any mathematician can be, and demonstrably better.

The math actually does math better than any human.

Is a surprising, and very cool innovation that math can do math, so have carefully explained and demonstrated with a simple quadratic.

Given that I've had this approach since December 1999, is instructive to consider music industry and taxi industry with web disruption.

In music industry took the late Steve Jobs, through iTunes to allow consumers to purchase music for download. With taxi industry, fights are still ongoing, where key innovations are simple: booking and paying rides over a smartphone.

In mathematics which has great prestige, am sure is hard enough for any outsider who might be considered an interloper. But also to learn the math can DO mathematics? Maybe then the math becomes the greatest threat to mathematicians who first have to adjust to this reality, as of course the math will be better than them.

Greater ability with mathematical manipulation easily in the hands of ANYONE willing to learn some basic modular algebra techniques? More perspective on why there could be fights like elsewhere with disruptive innovation.

Also means people can go back over prior research if tautological spaces apply which in essence I did with Gauss himself, and found better.

And that's just one thing talking my disruption discovery with tautological spaces. I disrupted so many areas. Like even disrupted story around prime numbers. And also importantly disrupted story around packing of spheres with a modular approach so obvious I still puzzle over being first to find that one.

One thing that jumps out at me though about current mathematical institutional reality? Applied mathematics which of course HAS to work in the real world, is tiny in terms of researchers, compared to what is called pure math, where my own research raises doubts. But am wary of digging too deeply there, lately.

Have decided not to focus on what I call the social problem, which is the fight over acceptance until 2028, which is to allow the tone of my posts to be different. And also figure should be over by then, but if necessary then will switch back to addressing directly. Is weird to realize that only ten years might not be enough time. But at least I better understand why.

And at least for others who wonder, and also to put down what I just finally fully realized, am just part of the web disruptions occurring all over, while with me is with math. And yes, web has been a cool part of the discovery. Web has helped me figure things out rapidly, and share widely.


James Harris

Monday, March 19, 2018

Advanced algebraic symbol manipulation

What I like to call tautological spaces are complex identities that can be used to do algebraic manipulation. The benefit of allowing the math to do such manipulations is easier perfection, and insights which seem demonstrably to elude human beings.

My basic demonstration for simplicity and familiarity talks analyzing: x2 + y2 = z2

That can be pulled from a complex identity:

x2 + 2xy + y2 = v2z2 - 2vz(x+y+v) + (x+y+vz)2

Leaving a residue:

(v2-1)z2 = 2xy + 2vz(x+y+vz) - (x+y+vz)2

And since v is still a free variable, you can do equivalent of symbol manipulation by choice of v, for instance, with v = 1 find easily:

2xy = (x+y+z)(x+y-z)

Where notice with our original expression which I like to call the conditional, we can add 2xy to both sides. That simple move is to get a key perfect square, allowing a factorization:

x2 + 2xy + y2 - z2 = 2xy, and then easily: (x+y-z)(x+y+z)  = 2xy

Easy example for demonstration to help with grasping the basic concepts.

You can shift through every possible way to manipulate algebraically with tautological spaces using v as your tool to probe an expression. As you shift v, the algebra moves everything around for you. And again used a simple quadratic for simplicity and familiarity, but this approach does not care.

Importantly, you can use modular to greatly simplify complexity that can build rapidly with the complex identities--where is to its proper purpose as you do not need all of that information.

So the actual tautological space for analysis: x+y+vz = 0(mod x+y+vz)

Where the result then is: (v2-1)z2 = 2xy (mod x+y+vz)

Which I call the conditional residue, which is true when: x2 + y2 = z2

You can rapidly zip through EVERY way an expression can be algebraically manipulated, just looking for things, and asking questions.

The greater ease with which algebraic manipulation can be done then is astonishing, and is interesting to me that it took me finding this approach. And found it December 1999 when looking to get a better handle.

That's strange, huh? And have wondered as what do I care really, just glad to have the tools now! Yet yeah, must be the original discoverer as once you have these ways, how do you go back?

Fumbling with equations and trying a few things here or there which you can do indefinitely, when the math can give you huge answers in minutes? There's no way to go back for me, at least. Would be kind of like choosing to walk a thousand miles when you can just fly there in an airplane.

So I talk to the math now, and ask the math questions, where when tautological spaces are available as an option can just get answers so much more quickly than any other way.


James Harris

Friday, March 16, 2018

Working with tautological spaces

Reaching for more handling with mathematical expressions found out there was another level to what the math could do, which involved using complex identities I decided to call tautological spaces. Have had this approach since December 1999.

For example the simplest tautological space: x+y+vz = x+y+vz

Where usually show the modular algebra form, but will demonstrate like did long ago, where will show explicit. And use a demonstrative case where here is reference from 2014 for the modular algebra form.

With a bit of algebraic manipulation, have:

x+y = -vz + (x+y+vz), and squaring:

x2 + 2xy + y2 = v2z2 - 2vz(x+y+v) + (x+y+vz)2

Where can see all the growing complexity. Now that is STILL an identity of course, so x, y, z and v are completely free variables. But let's put some conditions on them.

Let: x2 + y2 = z2

Then can subtract to get: 2xy= (v2-1)z2 - 2vz(x+y+vz) + (x+y+vz)2

Where now can get to:

(v2-1)z2 = 2xy + 2vz(x+y+vz) - (x+y+vz)2

With some simple algebraic manipulation. And have the result that (v2 - 1)z2 - 2xy has x+y+vz as a factor, when x2 + y2 = z2, and can also easily realize that v is still free.

From the full explicit form can also just look at expressions that result from setting v, like v = 1, gives:

0 = 2xy + 2z(x+y+z) - (x+y+z)2

And of course then: -2xy = (x+y+z)(z-x-y)

Notice also are not of course limited to integers, as is a trigonometric result if z = 1.

That is: -2cos(x)*sine(x) = (cos(x)+ sine(x) + 1)(1 - cos(x) - sine(x))

And get a better feel for how is a true modular algebra when get to modular form, as does not care about Diophantine or not. Learned then that with an actual modular algebra is not just with integers. Should make sure to emphasize which is why am so doing here.

The modular algebra form of course is: (v2-1)z2 - 2xy = 0 mod (x+y+vz)

So what the modular algebra form gives is compactness, and is MUCH easier to manipulate.

Where can also notice a sense of how you get algebraic manipulations by how you set v, which can be a HUGE benefit. With explicit is just given, but with modular form you can do equivalent, as MAY find in pieces that way. Just then stitch together. But can also get an explicit result too.

The freedom to change v in ways that help analysis, was the point of the discovery for me, wanted something I could control. And that result of also giving algebraic manipulation of the conditional expression was really cool to finally realize. Was a big surprise for me when came across.

Here demonstrated the full complex identity where notice raised to a form where could meaningfully subtract the conditional expression which leaves a residue, which can be analyzed. With the full modular algebra form I call that the conditional residue.

That actually is an important limitation on this approach. Like is meaningless to subtract that conditional from x+y+vz as you do not eliminate any part of the conditional expression. So had to square to get to something useful here. And depends on the conditions being set on the variables.

In my experience, tautological spaces are effective when you maintain roughly the same number of terms or less than the conditional expression, and are ineffective if you end up with more.

The tautological space had to be an asymmetric form I learned in order for it to work. And have generalized methodology based on my analysis of what seemed to be necessary.

And you can do fewer variables by simply setting for example y or z, where can consider by setting z = 1, with full case of three variable quadratics, which recently posted on, as yeah for years that's what I did. Which probably helped me figure things out with a bit less complexity.

So highlighting a bit I learned just on my own. And do wonder how much more others could figure out, as am certain only scratched the surface.


James Harris

Tuesday, March 13, 2018

Attention reality and math usefulness

Maybe should address possibly vague notions about value of math discovery and how changed things for me. For one thing do believe there are people who think a committee of some kind of expert human beings MUST accept and promote you, or you have nothing.

That is of course not the case and actually has not been in the past. Where history is full of instances where ideas won out over time, and can say the web speeds that process up.

Math actually spreads on usefulness have witnessed with my own ideas.

However, humans can simply decide to not talk about certain things in certain ways. The math does not care. Still works and humans are attracted to things that work well for them. And some will even notice who made that possible.

So the discoverer gets attention. And in my case have not often handled it well, while decrying it not being of a certain type. I was very picky about what I wanted. And had not gotten it, and threw some tantrums in the past. Am over that now. Where also better appreciate what I have.

Today am potentially known like to say in over 150 countries where relied on something else for baseline. That something else being downloads of an open source project wrote for Java developers and put on SourceForge, an open source repository, back in 2004. Was a reaction to a lot of emotions. Wanted something concrete in a different way. Which helped me with confidence in my math too.

This blog with its old name though had visits according to Google Analytics from 125 countries at its peak that I noticed. Where is just what registered by however Google does those stats. And that was over a decade ago.

That messes with your mind. Do not behave the same trying to process. So I like to say POTENTIALLY known in over 100 countries, where rely then on two sources with data from others, so is objective. And just for my work, with no departments or faculty or other researchers or any academics actually, with any piece of it. Is all mine, alone. Which also messes with your mind.

There is a comfort in having other humans with you with certain things. So much harder when have to go it alone in many ways. While the shared value is still that of humanity with more useful information.

So yeah, alone in the role of discoverer, thankfully with a world of others in the benefits of that knowledge.

The math pulls attention in and of itself. And with good reason. Say some person wants to do something needed with some mathematical tool to solve an important problem, will pick best available if know of, and can, or not?

If you think not, I disagree, as from what I've seen--will pick best available.

Your discovery is not about a committee.

It is not up to other humans to decide the value of your work with their opinions, but for your fellow humans to show they value with use. We do not talk about calculus or algebra as about an opinion.

Obvious to most actually I think, but to the money obsessed? May believe money matters which is so much about celebrity in cultures where celebrity rules. Yet how many track those celebrities over decades? Their moments in the intensity of attention are often barely a few years, or often less.

In mathematics we know people over centuries and even millennia. There is no comparison from modern celebrity. And often am a huge fan, but also do watch modern celebrities come, and modern celebrities go. Audiences endlessly wanting the new. And why not? Our expectations keep rising as learn what is possible.

Was actually such a HUGE advantage for me escaping that noise. And learned what is actually necessary, and turns out it is not. Which has helped me as learned my role. My global position results from my discovery but did not come with an instruction manual. Have worked at figuring out.

No person who understands math, or pursues for the right reasons gets lost on those details for long. Math is not a zone for instant celebrity, and soon thereafter flame-outs. It is an arena where you can help people to know, for as long as there are humans.

Which is comforting really. And I think is important for me to emphasize. Human systems are what change with the new. Like when algebra arrived, mathematics changed rapidly for the better as our species learned more with greater efficiency. The humans back then? Even if their opinion was recorded do we care?

The great thing: great mathematical tools with which can know more. Discovery rules.


James Harris

Sunday, March 11, 2018

Some science speculation

Have tried to refrain from talking physics things much, especially while working at being certain of ideas. But now am more like, why not? But still will emphasize as speculation.

So yeah, was impossible not to think of quarks when considering the entanglement have talked quite a bit, including some recently may as well give a link there.

So we know algebraically there is an entanglement which I found first with cubics, which would maybe mean related equations help determine quark behavior.

But ended up focusing on quadratics as EASIER but realized might mean that there is a quadratic case in physics--undiscovered.

Best guess as to where? Electrons.

Could explain much about electron behavior, if we think is a single particle but is actually two VERY tightly coupled particles, where is just vastly closer than with bosons. Talking as if you have physics knowledge as easier for me, but also still dealing with basics most folks should have.

Then again, problem there is now maybe can develop better quantum chromodynamics where have been problems. Is more difficult than quantum mechanics, but of course is the physics of the nucleus.

That means could get everything from nuclear fusion, to more advanced nuclear weapons.

Like how casually I drop that one, where have pondered for over a decade. And for me, quarks making more sense!--Would be so HUGE is hard to really express. But does the math actually apply there? Takes the real deal physicists to be sure. Just have an undergraduate degree and from two decades ago.

Where I emphasize is speculation, but if true, and some other countries take me seriously, and work it, the United States could wake up to a world where other nations have better nuclear weapons. Where sky is the limit there.

Would change the global order of things, immediately. And as an American? Am not exactly excited at that possibility, of a humbled US having to change how it does all its international politics.

But science does not care. And technology does not care. Where hesitate to note, but my emotion in wanting my situation resolved is also not relevant.

History is full of such stupid stories. I do not want to wake up to learning should have sounded the alarm, when a nuclear explosion in my country reveals things have changed. Where may not even know who set it off. The potential technology may be there.

We think nuclear arms need uranium, like to get to plutonium, but what if they do not?

Could be a simple innovation maybe, if have the math you need, to get the physics, and entire world different as a result. And then the United States no longer the dominant world power.

Since am speculating may as well give scarier scenario where wouldn't have to worry about delivery, if weapons didn't give off radiation. Then could just use a global shipping company.

Could get REALLY wild speculating as hey, I like to write and could be a fiction scenario some day, but here feel like made my point, so will stop myself there, as could make it scarier.

Where think is understandable would want to be VERY sure about my math before putting such speculation out. Think have said some before and pulled back. But do have these innovations with math which could simplify quite a bit. History shows such things cannot be simply ignored.

Why did I spend so much space on negatives though? Just mentioned maybe nuclear fusion, which would just be cool, I think, and went on and on about other. Guess is too easy to focus on negatives for me? But maybe is natural here.

Such things have pondered for years. The dragging from mathematicians I now believe is just typical against such stunning innovation as have found where talk example with rideshare on my other blog.

That the math I found might be used by our reality though? Of course as someone with a physics degree find that intriguing, but also note, is speculation. May as well put down though, like with the electron, if true?

Then THAT would be yet another thing for me in the history books. And one of the greatest as well.

Yeah definitely need that in writing, just in case am right.


James Harris

Saturday, March 03, 2018

When proof rules emotion

One benefit appreciate now more than could in the past was the questioning of certainty. Before did have situations where thought that proof was about a certainty of having checked everything. And could go over mathematical equations over and over again, thinking were correct. But thankfully had that collapse more than once.

People can talk mathematical proof, yet how do you know something is a mathematical proof? Answering that question for me became much about identities.

But still there is that fear of being a person who can look at something believing is true, and mind can play tricks on you. Which I think is how the mathematical discipline has turned to other eyes and checking by others. But human fallibility remains possible. Especially with complex arguments where I DO think computers will handle all proof checking, eventually.

For me though practical reality after looked into computerized proof checking, years ago, even contacting an expert in the area and getting nowhere, pushed me first with functional ideas, and then to number authority.

So defined mathematical proof functionally. But also now like to look at things like:

(462 + 482 + 722)(1722 + 258+ 430+ 6022 + 17622) = 

            615+ 30752 + 141452 + 159902  + 1884972   =  774*210

Which is my favorite one lately. It was found using the BQD Iterator. Where was easy to figure out, but the cool thing is the numerical perfection. Once calculated you know must be perfect. Checking is easy enough with modern systems.

Compare though with:

(x2 + 2y2)(u2 + 3v2)(x'2 + 4y'2)(u'2 + 5v'2) = p2 + 359q2

And finding integer solutions for all the variables is easy.

x = 1, y = 2, u = 2, v = 2, x' = 3, y' = 2, u' = 4, v' = 2, p = 358, q = 2

First iteration is easiest.

Both examples also should say demonstrate modular in that am using BQD Iterator with basic form u2 + Dv2 = F.

So that module is used twice for first, where expanded with a technique for having as many squares as wanted, and four times for second. So went less fancy with second example and left in variables versus putting in solutions like with prior one.

But feels so different to me regardless. There is logic and there is the feeling around looking at things. Now a step backwards to symbols, and is not so easy for me at least to feel certainty. Though mathematical proof IS there. And is interesting I think the emotion can feel despite the proof. Which in my experience?

Is great! The proof does not care about my emotion. When I do the math, it works.

Contrast with emotion the other way, where for instance trusting humans, you find that trust disappointed. Would rather be skeptical and find my skepticism overturned than go that route, as have done before.

I prefer to talk to the math.

The math will never make a mistake. The math will never be wrong.

The math is always right.

To me, when proof rules emotion you know as you go upside down. Trusting not your faith in your ability to look over the mathematics. Which is weird, huh?

So I go through the checks. Like use my definition of mathematical proof to check, every aspect from beginning to end. So I know have a proof, intellectually. Logically it is perfect. Yet part of me still is hesitant, until the relief, when run some numbers.

That burst of positive have found works vastly better to continue, than the other ever did.

Numbers rule.


James Harris