The nature of number

[Categories+Sheaves]

OK Let’s play.

Yay!

1+1=2 lets express this geometrically
We have a line of an unknown length at one end a number at the other a result and between these points a place at which changes one to the other

Never seen anyone arrange this stuff quite like that. Even the syntax of the equation is given a shape. Pretty neat... Your spatial intuition doesn't ever take a break, does it?

For the sake of clarity let’s name these three parts. The origin/‘O’, the function/’F’, and the result/’R’.

I'm so going to try to visualize stuff this way the next time I come across an endofunctor on a category of manifolds.

Ok lets play with this can we now do anything with it?

Depends on what you mean by anything, but the stuff you've brought up all seems to work.

Well I have found you are not restricted to just comparing one number at a time, you can compare a series of numbers to a second series...
...So to draw this up we only need the outer parameters of the two series and a function point...
...You will notice that the direction of the resultant series is in the opposite direction to the original series...
...Any two series with a common factor or point of comparison can be matched...
...For every position in the origin has a corresponding position in the result. No matter where the line going through the function is in the origin it also connects with a corresponding place in the result. So can we can draw the conclusion from that every series with a common function contains an equality of positions?

1. From what I can tell, the class of functions you're looking at (or at least the functions that can be represented like this) are continuous and strictly monatonic (in case you needed some jargon dumped on you )
2. I'm not sure what you mean by 'equality of positions'... are you talking about the correspondence 'y=f(x)' for each number, or the existence of some 'x' that satisfies 'x=f(x)'?
3. If your responses to #1 and #2 are something along the lines of "yes" and "the former", then this all looks good.

Is this actually is a problem? Pi is an irrational number yet it comes directly out of the series. We may not be able to place it exactly in the line of fractions or decimals, because it has no exact decimal or fractional position, but if we have a minimal maximal range where it could sit in the origin then that directly translates to the same area in the resultant series.

Nope, not a problem. This sort of approach is exactly how we're able to define stuff like "2 to the power of sqrt(2)". Because we're able to interpret exponentiation with rational numbers quite easily, we can calculate exponentiation with irrational numbers by (more or less) approximating it with rational numbers (but there's a whole bunch of delta-epsilon math-guts I'm refusing to spill here ).

Actually it seems we only need to place three positions and their values, in the series and we automatically know where all the other positions in the coherent series are, and what their values are. It does not matter which method of counting numbers we use. It seems once you have sufficient parts, the series will generate all the rest.
Given this the supposition could be there is a single template for all series. Is this then is a problem for it seems to be in direct opposition to the conclusions academic mathematicians have drawn?

Well... now it looks like you're looking at an even smaller class of functions (by 'smaller' I mean 'strictly contained in the last classification', so all the stuff I said previously still applies). My tentative answer to this is 'yes, this isn't really challenging anything in modern math' but only with the caveat that your notions of functions and series don't quite match the way these things are thought of in modern math. Depending on how you iron out the specifics of this stuff, I might have more objections later on...