The nature of number

[Categories+Sheaves]

Dear Categories+Sheaves
Thank you for continuing with me. My aproach I know is clumsy, as I am coming at this subject from a wildly different angle...
...My problem is that what I am seeing from my end does not seem to tally up with how maths is explained to me.

Nothing wrong with being clumsy, as long as we're all able and willing to think about this stuff.

But if the '0' in what I have termed the result is the projection from the origin through the function, does that not satisfy you that we can call it '0'?

Well, what do we mean by zero?
For the most part, we insist that an element is "zero" when we have some sort of addition operation on a set such that 0 + a = a =a+ 0 for all a our addition operation can take as an input. The origin takes this role in vector addition, and the 'zero function' (f(x)= 0 for all inputs x) fills this role when we're adding functions. In this sense, there can only be one 'zero element' for a given addition operation (if we have 0 and 0' both acting as zeros, 0 = 0 + 0' = 0').

Not to say that you have to approach things that way... but if you aren't doing it that way, you may need to go into more detail/think harder about what you mean by '0'.

Your next explanation is excellent and I think I understand. So I would like to walk away from all the twaddle I was talking about that. But I would like to take up the conversation from this part, because it is central to my theme..

...This is where my problem is: I would contend that Euclid's books are not about maths. Rather they are help books for Artists etc, on how to use maths. As such the definitions are to enable us to make constructions, not definitions that should be used to define the subject itself. I might use points to map out a shape in space, but vectored space has no points.
As such the point is a tool we use not a thing itself. Where I am going with this is that it seems to me a number is the same it has no integrity, but it is created out of the series, any one number is only given its value by the other numbers in the set or series.

The whole "Object X doesn't genuinely exist, we just use it as a tool" angle irks me a little. Here are my gripes expressed by a wiser man.

Also: Euclid pioneered the axiomatic approach to mathematics, so calling his geometry 'not the math itself (but applications of the math)' as a general statement is a little iffy (although I think I can feel where you're going...)

I do not know if this is standard thinking or not. When as an outsider to maths you get an entire hour long BBC programme talking about Cantor and infinity, and declaring he said you cannot map one to one fractions and decimals, without saying that is an analogy. And I try to look up definitions of number and find:-

Can I take where I am with this is a reasonable starting point?

Hrm... I'll try to find a juicy bone to throw your way...

I don't think I understand what you mean by "lying across this hole". Do you mean points on either side of the hole?

Yeah... I would have used 'antipodes' but that can also mean a different pairing of points on the torus... (since it's a direct product of circles...)