The nature of number

[Categories+Sheaves]

...When maths or any discipline will only converse in its own terms is it actually useful any more?
It seems to me the logic of maths is- does this work, if it works then it becomes part of the subject. I do not wish to know whether this or that formula works, but I am seeking if there is an understanding of what the relationship of numbers to the series are.

The numbers themselves are usually defined by how they function. e.g. isn't not entirely clear prima facie what it would mean for one infinite set to be 'bigger' than another, (I'm echoing Wittgenstein here) so this business with the cardinalities of sets has to be grounded in the behavior of functions between sets (esp. with bijections). These structures and the ways in which the work co-determine each other, so it's a bit wonky to talk about the 'nature' of numbers sans the way they work.

I simply want to know is there a mathematical understanding of the nature of number because I want to make a comparison with classical arts understanding of a similar area.

Well, arguably almost all math is investigating the nature of number (from one angle or another).
There is no singular concepts of 'number'. Remember how I dropped that laundry list of "number" systems on your earlier? You can't discuss 'the nature of number' without some provisional statement of what 'number' means. If you pick out a system of numbers (reals, gaussian integers, whatever) I can (in most cases!) talk intelligently about their structure and what motivates their study. If you give me a clear description of what properties you think numbers should have, I can (usually) talk intelligently about the structural implications of that. Right now I'm trying to figure out the thinking behind these pictures (I can extrapolate beyond them, but that's not where you are) and it's hard to get traction here .

I do not want to have to learn your language, So outside that language is there anything you can say which is useful or is it just 'academic latin', great for priests to talk to each other, about how many angles can sit on the head of a needle, and that ratifies their position as being priests, but that does not contribute any other field of knowledge?

-I don't stand a very good chance of saying anything useful if I can't understand what you're trying to use this stuff for (I currently don't).
-String theorists need the axiom of choice to well-order the real numbers and make sense of their Feynman Integrals. Other folks use this voodoo too

So can a broken series be seen as a single entity? If so in what curcumstances would a series be a unit and not a unit?

-What do you mean by 'series' again? You called these projections of numbers/lines you made 'coherent series' but that's all the leverage I have on this term.
-What do you mean by 'broken series'? Even in the worst possible interpretation of 'broken series' I'm sure the answer is "yes, you can see it as a single entity" (sets have to get way weirder before they're considered proper classes/not suitable as sets) but since I don't see this as problematic (and you're still hesitant about this) I'm not entirely sure you interpretation of "see as a single entity" is the same as mine. What sort of single entity?
-unit = single entity?