I know my original argument can be easily defeated via currency. If you have $10 and you owe $3 of your original $10, you are now $-3 (negative 3) and simultaneously $7 (positive 7). The point being the $-3 is a fact. It exists because it was used to deduct a certain determined amount that the $-3 represents precisely and exactly.
What I find so interesting is that our brains also use negativity, as an inhibitor. I want to find out more about this process, in due time I'm sure.
So who cares about any of this? Well, me. I like to understand and understanding must sometimes come from mistakes, sometime uninhibited play and freedom of thought. The case here is uninhibited freedom of thought. Will thoughts such as these lead anywhere? Who knows, but I do know one thing, if I were to be asked to determine the validity of negative integers as their own number, I now have an answer.