The nature of number

[jonb]

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. Also I am not as farmiliar with using the computer as you would expect, so I have fcuked up how to show what you have said and give my replies but I hope my solution to that works for you. My problem is that what I am seeing from my end does not seem to tally up with how maths is explained to me.




Only in the sense that the preimage of 0 is a set with those properties. Or that you've taken something with these properties and called each of its constituents "0". It seems like we could call each point on the line "fish" or "paisley" and achieve the same result. Is there some concrete sense in which these points actually walk or talk like 0?

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'?

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

Geometric shapes are made up of points, which effectively have no dimension.

This is absolutely correct, but it's still worth noting that the "Geometric object A is a collection of points in space" perspective is surprisingly modern. It isn't too hard to learn/imagine what a straight line is, and we can intuitively understand what "a point on a line" is. But do these points themselves constitute the line, or does the line just happen to have points lying along it? That issue isn't quite so clear. While the former line of thinking is far superior when you have a robust theory of sets + calculus/analysis lying around, Euclid (and a whole bunch of other dead mathematicians) originally approached geometry from the other angle.

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.

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?