The nature of number

[Categories+Sheaves]

Ok. To avoid confusion, I'm going to use italics to distinguish expressions occurring in the rationals from expressions occurring in the integers. Now lets's make our rationals pretend they're integers.

So now when I write some (italicized) expression in the rationals, interpreting it requires you to:
0. Look at what I wrote
1. Replace the rationals with integers, following the rule above
2. Evaluate this integer expression in the normal way.
3. Translate the result back into a rational number, reversing the earlier rule.
So, an example
0. 1/2 + 2
1. 2 + 3
2. 5
3. 3/2
Or,
0. 2/3 + 4
1. 9 + 10
2. 19
3. 5/4
Italicized rationals then behave in exactly the same way as regular old integers--they just go by very different names that bear no apparent relation to their "real" value. This weird redefinition of addition, multiplication, etc. on italicized rationals was defined using our normal, intuitive operations on the integers--but surely we can say these "italicized addition" and "italicized multiplication" exist as binary operations on the rationals in their own right. So while rational numbers and integers exist as distinct sets, the existence of a bijection between them (by the axiom of choice, this definitely exists between any two sets with the same cardinality) we can take any structure on one, and made the other set mimic that structure perfectly (but all the names will be wildly different). So which is the "original" and which is the "duplicate"? (they're functionally equivalent, so this distinction is useless)

So uh--in terms of the "different types of sets" thing... Yes, the relations we put on sets is the stuff that's actually interesting, but the semantics of modern math insists that those relations come after the existence of the sets (and hence, within that framework, these structures can't say very much about the nature of sets themselves). The rationals look very different from the perspectives of regular and italicized additions, but it's the same set (the same collection of symbols, objects, whatever) at the end of the day, right?

Afterthought: damn, that was a long rant.