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:
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.