> Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition. ... 1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N).

That's all nice and dandy for mathematicians, but such is still meaningless (from an overall perspective) because ultimately you end up with recursive definitions. "The successor" of "0" means nothing because "successor" isn't defined, unless you define it via dimensions (or something else). Once you try to define dimension or otherwise, you have to define dimension, which leads to defining counting, which leads back to defining 1. Appealing to experience is the only way to end the cycle.

That said, it's perfectly fine to talk about a system within itself, but it's applicability to anything real (which is what I'm concerned with - my apologies if that's not where you're going) is limited to whatever can be described in terms of reality.