One of the greatest things about mathematics is the ability to determine things to be absolutely true. And that is extremely important and is an aspect of mathematical thinking that is absent in so many areas of intellectual thought.
Being someone who accepts that mathematics allows us to get to absolute correctness I was put in a bizarre and very uncomfortable position years ago when I came across something which seemed to defy the above. So for years I've hammered at it, trying to find something wrong, and finally have decided I must accept the result. But now I guess I should address the problem, and try to figure out, did mathematical rigour fail?
How do I reconcile my own results with my own beliefs about mathematical certainty?
The answer may be a bit convoluted and is hard without some math, where I will put a particular result of mine as proven.
Consider: (3 + sqrt(-26))*(3 - sqrt(-26)) = 35
I found I could prove that 7 is a factor of one of the solutions to (3 + sqrt(-26)), which meant logically it is a factor of just one of the solutions to (3 - sqrt(-26)). (In what ring will be explained later.)
If that seems odd consider: (5 ± 2)*(5 ∓ 2) = 21, which is equivalent to: (5 + sqrt(4))*(5 - sqrt(4)) because sqrt(4) is 2 or -2.
My proof has withstood every test of mathematical rigour and the underlying approach has actually been published and pulled by journal editors in a rather remarkable story, which I've discussed elsewhere. One post that can give an overview is:
So what's the big deal?
Why is it important if one of the two solutions to (3 + sqrt(-26)) and (3 - sqrt(-26)), have 7 as a factor, when they multiply to give 35?
The answer requires some math history.
For those who know the history, good. The gist of it though is that the ring of algebraic integers indicates a different conclusion. And I had to figure out where things went wrong with that use of established techniques of mathematical rigour with that ring!
The answer is surprisingly simple. First of all, standard tests of the ring's validity were valid. For instance an algebraic integer is infinitely decomposable into algebraic integers. Given two algebraic integers, their sum and their product is an algebraic integer. The ring has unit factors.
However, those tests do not prove that my result above is impossible! It takes more.
There may be a couple of ways to understand the result, but it was told to me through proof that in the ring of algebraic integers, no non-rational root of a primitive monic polynomial with integer coefficients can have an integer other than 1 or -1 as a factor of just one of its roots.
So the weird thing there is that though 3 + sqrt(-26) is an algebraic integer, and I can prove one of its solutions has 7 as a factor, neither of those solutions are permitted to have 7 as a factor in the ring of algebraic integers!
And that is absolutely correct.
So I went to the logical step of concluding that the ring of algebraic integers was leaving out some numbers, and after pondering things for some time, I came up with a more inclusive ring:
My object ring passes all the tests as well, and includes the ring of algebraic integers, which allowed me to go back and see what was going on in the ring of algebraic integers!!!
And I've talked about it on this blog:
The weird thing I discovered is that despite the tests of the ring, a convoluted path was still available to the math, which lay outside detection.
What my Wrapper Theorem does is explain that route.
So could I be wrong? Well the mathematical argument involves easily checkable algebra and is pleasingly short. That's important with such startling conclusions!
In essence, what I found was that there was a gap where the algebra could do something unexpected, which meant that if you assumed that gap did not exist, you could come to conclusions which were wrong.
But how do you know, what you don't know? Without my methods for proving the result about 3 + sqrt(-26) how would you ever know you needed to worry about wrappers with the ring of algebraic integers, or consider the possibility of units in a more inclusive ring which were somehow NOT units in the ring of algebraic integers?
So the answer is that mathematical rigour did NOT fail in the case of algebraic integers, but people using it simply didn't know enough tests!
That's kind of scary as it makes you wonder if there is always possibly something else out there that we don't know we don't know which could invalidate a particular mathematical argument, and I think the answer is, no.
If you look at the reasoning around algebraic integers it was kind of circular:
1. Run tests thought to determine if the ring was valid.
2. Make assumptions about what is valid from the ring presumably validated.
So I say it was human error.