One of the more common refrains with someone claiming important mathematics is: I have mathematical proof!
And a great draw of mathematics is the stated ability to be absolutely correct, though there is a problem, how do you know when something is actually proof?
That question drove me to a simple answer: begin with a truth, and proceed by logical steps, then whatever you have at each step must be true.
That is, I concluded that truth follows from truth. Which revolutionized my thinking, but I had a problem: how did I know that I could really trust logic?
My problem was with the oxymoron "logical paradox", and so I went to look the so-called ones over, and concluded that there were form issues and a failure to appreciate exceptions. For instance, consider the following set:
Consider a set that includes all and only sets that exclude themselves, except itself.
And I found I could remove "paradox" from every so-called example questioning the foundations of logic, as it was a necessity! By linking logic to mathematical proof, I had to resolve any issue bringing the consistency of logic into question. That effort is shown in several posts on this blog including this one:
somemath.blogspot.com/2005/06/three-valued-logic.html
Ultimately I concluded that mathematics is provably a subset of logic, and came up with a rather simple definition of truth itself. That may sound pretentious but it was a necessity of my proof definition. I had to know what "truth" meant in it. By claiming "logical steps" were necessary I had to consider logic and conclude that mathematics depended on logic and not the other way around. From those foundations I found I could move forward, and importantly knew what a mathematical proof was, and how to check a claim of one.
So I went around using it! And found my definition worked perfectly. I could even scan through complex mathematical arguments where much of the math was unfamiliar to me, by keeping up with linkages--individual steps in the argument. Checking linkages can be easier than checking every detail of the mathematical argument. Any break in any one of the linkages mean the conclusion is not proven by the argument given.
It's actually really cool. And the idea that mathematical proofs simply link truth to truth is one I pondered over for quite some time, and it's kind of weird! But it also helped me turn to identities with what I call tautological spaces.
Of course one person can claim mathematical proof, and you can have a perfect way to check one, but what if people disagree anyway? Is there nothing left but endless argument? Nope.
Correct mathematical arguments are not refutable, right?
So far beyond what you may think is possible, the mathematics will work! While incorrect mathematical arguments are limited within perception, the idea that they work operates within a limited zone, with a refutation out there somewhere.
Mathematics is an infinite subject.
Like with my mathematical research, it builds forward I like to say, where there is more that follows from it than I will ever know! Like with algebra. There is no end to it.
In contrast, flawed mathematical arguments are error constrained--even the appearance of success must operate in a special zone.
For instance, I innovated by factoring polynomials into non-polynomial factorizations, which turned out to be the thing which broke mathematical ideas which seemed ok as long as, yup, you didn't factor polynomials into anything other than polynomials!
And it IS remarkable when you see where people have stayed within the lines in such a way that they can't see something that jumps out at you if you just try something different.
One of the troubling things I've seen at times from the mathematical community at large is an emphasis on personal opinion, where things are believed to be true because a bunch of mathematicians say they are true, which is a running away from certainty.
Which is why I emphasize checks which do not require expertise, like asking: what does your research predict about what actual numbers will do?
The innovations I introduced allow you to do more. And the mathematics that follows from the truth will always exceed our finite human imaginations. It is the promise that has been fulfilled with advances like algebra and calculus--there is always an infinite more out there than we now know, and is the promise that makes mathematics such a power in all our lives.
James Harris
Friday, September 26, 2014
Tuesday, September 16, 2014
Power of prediction put to the test
If number theory means knowing what numbers will do then it has predictive power, right? Well there is an ancient equation which puts predictive power to the test.
Years ago I found myself at a critical point with the equation x2 - Dy2 = 1.
Some of my mathematical theory told me that there should be a large fundamental solution, the smallest positive integer solution with y not equal to 0, if D is prime and D-1 had 4 as a factor and small primes as factors, as long as neither D-1, D+1, D-2 nor D+2 is a square.
The most famous case is D=61, with a very large fundamental solution, which I already knew behaved as my research predicted:
61-1 = 60 = 4*3*5
And mathematical science is the art of prediction about the behavior of numbers. So in front of me was a simple test. What about 4*3*5*7 + 1?
I could just go to the next prime and see what happened. If my mathematical research was correct then the numbers had no choice. They would have to behave as predicted. But what if they didn't?
Well, if they didn't, I was wrong.
Turns out that 4*3*5*7 + 1 is 421 and I found the fundamental solution for D=421 on the web, with a quick search, why? Because it's a HUGE solution, so it was listed out there, by people who probably had no clue why it was so huge. Actually I'm sure they had no clue, any more than ancient people understood why they have weight before Sir Isaac Newton. But you see, they didn't have the mathematical science I had.
I felt relief.
If you're curious you can keep checking like check D = 4*3*3*5 + 1 = 181. Did some more web searching and found this useful table and yup, it's on it!
That's with just some of the rules of the mathematical predictive framework.
I put all the rules in a blog post:
somemath.blogspot.com/2011/09/cool-rules-renaming-pells-equation.html
My research tells you predictably how the fundamental solution will behave, so is mathematical science research. Science has a predictive power which people may fail to realize, but it gives predictive certainty.
Mathematical science gives predictive certainty about the behavior of numbers.
So mathematics is a science when it gives you predictive ability as that is a key component of science in general.
Now we can contrast with mathematical scholarship.
Mathematical scholarship has an emphasis on history and other facts of interest to scholars, so for instance you might learn of a guy named John Pell, and find that irrelevant to understanding the ancient equation. I don't care about such things. But my scientific perspective looks for predictive power.
Notice mathematical scholars can give you a way to solve the equation with integers, which does work. But ask them to explain why it works. (I think I can but the probable answer I have in mind is not very exciting to me.) Or also, ask them to tell you when solutions will be larger or not, which is the predictive power mathematical science offers.
This example is a great way to see how traditional mathematicians differ from scientists. Do a web search on Pell's Equation size, read through some published papers, and ask yourself, is that what mathematics is to you?
From those papers, it is clear that for a math scholar it isn't necessary for a mathematical technique to actually tell you much that is useful about how actual numbers should behave! Those papers reveal to me more of an exercise in style.
The scientific itch can drive discovery: what if mathematicians had been irritated by the lack of ability to do something as basic as figure out when the fundamental solution would be large or not?
Well then, the answer might not have been left out there for me to discover.
Predictive power can appear to tell you why, when it actually tells you what will happen. Knowing what will happen with electricity in circuits and through a filament can give you light at the flip of a switch.
James Harris
Years ago I found myself at a critical point with the equation x2 - Dy2 = 1.
Some of my mathematical theory told me that there should be a large fundamental solution, the smallest positive integer solution with y not equal to 0, if D is prime and D-1 had 4 as a factor and small primes as factors, as long as neither D-1, D+1, D-2 nor D+2 is a square.
The most famous case is D=61, with a very large fundamental solution, which I already knew behaved as my research predicted:
61-1 = 60 = 4*3*5
And mathematical science is the art of prediction about the behavior of numbers. So in front of me was a simple test. What about 4*3*5*7 + 1?
I could just go to the next prime and see what happened. If my mathematical research was correct then the numbers had no choice. They would have to behave as predicted. But what if they didn't?
Well, if they didn't, I was wrong.
Turns out that 4*3*5*7 + 1 is 421 and I found the fundamental solution for D=421 on the web, with a quick search, why? Because it's a HUGE solution, so it was listed out there, by people who probably had no clue why it was so huge. Actually I'm sure they had no clue, any more than ancient people understood why they have weight before Sir Isaac Newton. But you see, they didn't have the mathematical science I had.
I felt relief.
If you're curious you can keep checking like check D = 4*3*3*5 + 1 = 181. Did some more web searching and found this useful table and yup, it's on it!
That's with just some of the rules of the mathematical predictive framework.
I put all the rules in a blog post:
somemath.blogspot.com/2011/09/cool-rules-renaming-pells-equation.html
My research tells you predictably how the fundamental solution will behave, so is mathematical science research. Science has a predictive power which people may fail to realize, but it gives predictive certainty.
Mathematical science gives predictive certainty about the behavior of numbers.
So mathematics is a science when it gives you predictive ability as that is a key component of science in general.
Now we can contrast with mathematical scholarship.
Mathematical scholarship has an emphasis on history and other facts of interest to scholars, so for instance you might learn of a guy named John Pell, and find that irrelevant to understanding the ancient equation. I don't care about such things. But my scientific perspective looks for predictive power.
Notice mathematical scholars can give you a way to solve the equation with integers, which does work. But ask them to explain why it works. (I think I can but the probable answer I have in mind is not very exciting to me.) Or also, ask them to tell you when solutions will be larger or not, which is the predictive power mathematical science offers.
This example is a great way to see how traditional mathematicians differ from scientists. Do a web search on Pell's Equation size, read through some published papers, and ask yourself, is that what mathematics is to you?
From those papers, it is clear that for a math scholar it isn't necessary for a mathematical technique to actually tell you much that is useful about how actual numbers should behave! Those papers reveal to me more of an exercise in style.
The scientific itch can drive discovery: what if mathematicians had been irritated by the lack of ability to do something as basic as figure out when the fundamental solution would be large or not?
Well then, the answer might not have been left out there for me to discover.
Predictive power can appear to tell you why, when it actually tells you what will happen. Knowing what will happen with electricity in circuits and through a filament can give you light at the flip of a switch.
James Harris
Saturday, September 06, 2014
Why predict with numbers?
If your math is wrong, one may assume it does not work, as it does not, but how do you know, if you don't DO anything with it with numbers?
Surprisingly you can find mathematical papers which never demonstrate with actual numbers, where in number theory that amazes me to no end.
What good is a math paper in number theory where at the end you cannot see with some number what was supposedly proven?
How is that even satisfying to anyone?
I find that when I look over math papers now (yes I do at times) I skim to the end, and see if they ever show anything with actual numbers.
Seeing mathematicians at their best as mathematical scientists puts the pressure on them to be correct, and numbers are brutally efficient at weeding out charlatans: the math of charlatans does not work!!!
The issue of certainty is one I'm aiming to put front and center across the academic world. That will make funding easy, as policymakers have the tools to tell when researchers are doing real work or not, regardless of the complexity of that research.
I covered that issue on one of my other blogs:
beyondmund.blogspot.com/2014/08/functional-certainty.html
Simply forcing mathematicians to demonstrate what you can predict with their research about numbers and then see if that prediction is verified--where in mathematics 100% is usually required--removes human error in a functional way.
That does not replace mathematical proof. And since I defined mathematical proof I am very well aware of how it works.
However, mathematicians have the skills to evaluate proof. But even a college administrator with no math skills whatsoever can evaluate a predictive test.
If you said one thing would happen and it didn't, then your math is wrong.
James Harris
Surprisingly you can find mathematical papers which never demonstrate with actual numbers, where in number theory that amazes me to no end.
What good is a math paper in number theory where at the end you cannot see with some number what was supposedly proven?
How is that even satisfying to anyone?
I find that when I look over math papers now (yes I do at times) I skim to the end, and see if they ever show anything with actual numbers.
Seeing mathematicians at their best as mathematical scientists puts the pressure on them to be correct, and numbers are brutally efficient at weeding out charlatans: the math of charlatans does not work!!!
The issue of certainty is one I'm aiming to put front and center across the academic world. That will make funding easy, as policymakers have the tools to tell when researchers are doing real work or not, regardless of the complexity of that research.
I covered that issue on one of my other blogs:
beyondmund.blogspot.com/2014/08/functional-certainty.html
Simply forcing mathematicians to demonstrate what you can predict with their research about numbers and then see if that prediction is verified--where in mathematics 100% is usually required--removes human error in a functional way.
That does not replace mathematical proof. And since I defined mathematical proof I am very well aware of how it works.
However, mathematicians have the skills to evaluate proof. But even a college administrator with no math skills whatsoever can evaluate a predictive test.
If you said one thing would happen and it didn't, then your math is wrong.
James Harris
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