Quote:The numbers themselves are usually defined by how they function. e.g. isn't not entirely clear prima facie what it would mean for one infinite set to be 'bigger' than another, (I'm echoing Wittgenstein here) so this business with the cardinalities of sets has to be grounded in the behavior of functions between sets (esp. with bijections). These structures and the ways in which the work co-determine each other, so it's a bit wonky to talk about the 'nature' of numbers sans the way they work.
In one of the more abstract interpretations of mathematics, statements about 'numbers' (N, Q, R, C, etc.) or about any system in general are actually statements about every system that satisfies the relevant axioms.
So when we say something like "1+1=2", we're not referring to some Platonic "1" and "2". Those are just structural placeholders for any system that satisfies the Peano axioms and which therefore has an analog of "1" and "2".
So the notion of a "number" isn't even necessarily coherent, depending on how you interpret mathematical statements.
“The truth of our faith becomes a matter of ridicule among the infidels if any Catholic, not gifted with the necessary scientific learning, presents as dogma what scientific scrutiny shows to be false.”
