No, I haven't "constructed something, labelled it the natural numbers and derived axioms from its properties". For me, the negative numbers are an aspect of nat

No, I haven't "constructed something, labelled it the natural numbers and derived axioms from its properties". For me, the negative numbers are an aspect of nature and I came across them by observation (well, other people did this and someone told me about them when I was a kid, but I can now observe them). Similarly to how I might observe trees and see that there's a property of greenness to their leaves, and then come up with a word to describe what I see. I think we're working on a different ontology of what mathematics is. Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers. I think you're talking about something like a non-standard model of the Peano axioms? My understanding is that this can only put in new 'numbers' c such that n < c for all natural numbers n. So, sure I've chosen a specific model of these axioms, but the model agrees with all other models for every finite number. Either way I don't really see why making a choice here detracts from my argument; numbers exist as a part of nature, and we can describe them with Peano axioms and a choice of model. I didn't assume any (specific mathematical) axioms in the first place, I assumed that reality exists and that anything that can be conceived of is a part of reality. I personally confirm for myself that this is true by observation, although you may not have done so for yourself. But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property. I don't really get what point you’re making here