The nature of number

[Categories+Sheaves]

The link( http://www.thebigquestions.com/2012/02/08/rock-on/), yes that is what I think I may have found a pointer towards.

Well, if you're hungry for more...
This is pretty relevant: it's a discussion of how the natural numbers work, and the axioms we prefer to take concerning them.

Here he spells out a 'strong' realist position concerning mathematics. (I remember throwing this at a currently-absent forumgoer, houseofcantor... I miss that guy)

This collection is a little less relevant to the precise nature of numbers, but still good:
First post in a series on whether math is invented or discovered.
Second post in that series, where he argues explicitly for taking natural numbers as real.
Third in that series where... hey I was taking that stance he's arguing against (in opposition to CliveStaples) in a previous thread...
Here he spells out a 'strong' realist position concerning mathematics.

It will take me a little while to draw up some new graphs to try to make it clearer what I think I have found.

No rush.

When I mentioned Euclid it was not to say the maths was not real, but it seems evident to me that the definitions he was using were designed to apply it rather than examine the thing itself.

Hrm...

By the same token then, Set Theory is designed to apply sets to other problems, rather than examine sets themselves. Euclid's axioms could produce novel statements about lines and how they behaved, but only in regard to the properties declared by those axioms in their application. The difference in cardinalities between the naturals and the reals revealed a great amount of information about how sets can behave and how they can be applied but any questions about what we mean by 'set' (beyond an intuitive "the collection/class of such-and-such stuff") remained in the dark--it's the paradoxes we run into (Cantor's, Russell's, "Skolem's", etc.) that actually struck the foundations. Similarly, the alternative geometries that arose from the investigation of alternatives to Euclid's fifth postulate were the ones that changed the common interpretation of "line".

I may have gone a little off-track with the paradoxes, so to return to my original point: the applications* are the math. A formal treatment of geometry didn't exist prior to Euclid, and new things (about how lines/angles behave) were discovered via the foundations he laid. Exploring the results or applications of axiomatic system A isn't just an integral part of determining what we mean by A and finding a paradox in A. This is why we axiomatized A in the first place.

*I'm trying to use this in the broadest sense possible: e.g. K-theory is applicable to determining whether two topological spaces are homeomorphic or not. Perhaps the issue, jonb is that in those 'early' stages of math, there wasn't much math to discuss beyond the direct, real-world applications.
-------Update-------
I'll try to respond to your new post later..