Faced some important things over a decade ago, which forced me to question how I thought I knew certain things, and as a result I worked out several solutions where I'll talk about an important one now.
When Gauss considered what were eventually called gaussian integers he challenged conventional thinking about numbers not integers.
For instance 1+sqrt(2) behaves a lot like an integer. It could be said to not have certain numbers as factors, for instance 3 is coprime to it in the ring of gaussian integers. Multiplied times another number 1-sqrt(2), it gave an integer. And mathematicians pondered integer-like numbers as something that needed to be categorized.
And they concluded that algebraic integers did the trick. Algebraic integers were simple too! They were just roots of monic polynomials with integer coefficients.
Various tests were done which seemed to indicate these were the solution and included all integer-like numbers including integers themselves. Tests like determining that they were infinitely decomposable into algebraic integers and that the product of two algebraic integers was always an algebraic integer gave confidence to those who believed in them.
However, I came across mathematics that forced me to question if those tests were enough, and began to ponder the issue of "integer like" with the need to resolve what looked like contradictions.
Eventually I focused on units. With integers, of course 1 and -1, are the only units, but there are an infinity of units, in the gaussian integers, and 1+sqrt(2) is one of them. In fields, of course, units are meaningless, but I began to realize that for numbers that behaved like integers when themselves not integers, units might be the key.
So in the month of December 1999, I came up with what I call the object ring. Which was a result so important to me, as it underpins everything else that I find it appropriate that talking about it is the very first post on this blog:
That definition is a lot about units! for instance, it established 1 and -1 as the only rational units, which I realized was one key element that gaussian integers had helped reveal.
In contrast in the field, everything is a unit. For instance 2 is a unit because 2(1/2) = 1.
By focusing on properties I could include the algebraic integers, but other numbers missed! Where I found I could prove the existence of those other numbers and that they were in my object ring.
I consider more of these issue in my post where I questioned if mathematical rigour had failed:
What's interesting to me is that the concept of testing the algebraic integers failed so horribly. But human imagination can only go so far while mathematics is an infinite subject. Those mathematicians couldn't prove with tests that the ring of algebraic integers was complete, as in fact it wasn't!
While by focusing on the intrinsic properties of integers and integer like numbers I could flesh out the rules by which a ring would have to be complete. That is the power of mathematical proof! In essence the rules found preclude the object ring from NOT being complete as to be incomplete it must not be following those rules.
The rules are intrinsic proof. That is, they encapsulate proof within themselves.
That can be a hard concept and the simplest way to see it is to try and come up with numbers that do not follow those rules. For instance 1/2 is blocked because it is rational and a unit.
That is so weird. If a number is a member of the object ring it must follow those rules for membership. By definition then, if it does not follow any one of those rules it is not in the object ring. But those rules encapsulate everything that makes a number either an integer or like an integer; therefore, they must be complete.
What's interesting then is that simply studying gaussian integers is enough to figure out what the rules are, as it must follow all of them! So there cannot be any other rules beyond it, because they must include it.
But you need ALL the rules, and just those actually required.
So the problem with the ring of algebraic integers then is that it includes an unnecessary rule, which is being the root of a monic polynomial with integer coefficients, while leaving out the correct ones.
Rigorous logic allowed me to prove the validity of my own techniques, so mathematical rigour does remain, but the lesson is: when human imagination is used instead of it, you can't really trust the result.
On the face of it, it probably seemed reasonable to those mathematicians to "test" the ring of algebraic integers with everything they could imagine. But mathematics was far too subtle for that to work!
In the end mathematical proof was required even for the question of what number can be considered integer-like when not an integer, and with mathematical proof, total rigour was achieved, and once against mathematical certainty could appropriately be had.