The nature of number

[Categories+Sheaves]

Long but good

but the semantics of modern math insists that those relations come after the existence of the sets

Ah This is where I am having my problem. It seems to me one form of set has those relations inherently built into it where as the other form those relations are not originally expressed.

The first time I read this, I thought you were talking about different concepts of what sets are, and I was going to drop this quote from The Concept of Model,

After the explicit appeal to the mathematics of sets, we have here an appeal, more or less implicit, to the mathematics of integers, namely to the axiom of induction that characterizes. To speak of a model is to presuppose the 'truth' (the existence) of these mathematical practices. We establish ourselves within science from the start. We do not reconstitute it from scratch. We do not found it.

Blah blah phenomenology of math. Although now I'm realizing that you were talking about the two different sets/binary operations.
And... yes, some forms of notation are more intuitive/suggestive than others. I don't know if you've ever used a reverse polish calculator, but when you write addition as ((2,3)Plus,5)Plus the associativity of addition seems a lot less intuitive--this is clearly expressed when we use our infix notation to write 2+3+5 instead of (2+3)+5 or 2+(3+5) This doesn't work with minus signs, but you get the idea.
I guess my short answer is mathematicians don't consider the clarity of the notation to be particularly important. Obviously, the connotations of an expression are important if we're not particularly clear on what the objects are--and this has given philosophers a lot of things to chew on. But given the popularity of the axiomatic method (One mathematician says, "I spy with my little eye a system that satisfies propositions A, B, C, .... M" and then the mathematician says "Aha! You must be talking about the real numbers!") the way we interpret that notation is usually supposed to be defined explicitly enough to eliminate any disagreements on whether certain types of manipulations are valid or not. Isomorphism, invariance, etc. suggest that there's an essential structure that doesn't change when we rename everything, or fiddle with things in some other way. Given your apparent interest in absurdism, this seems like something that ought to have been your natural attitude on this, but why do we have to insist that the italicized additive structure on the rationals is less "inherent" than the usual additive structure on the rationals? Sure, sure, the collection of objects/symbols referring to objects we call the 'rational numbers' was originally developed to express a certain sort of structure in an intuitive way, and you're going to piss people off if you use the italicized number system to fill out your tax forms, but those are issues with the way we choose to express certain sorts of ideas, not the structure of the ideas we're trying to express.