If counting/positive numbers exist, do they imply the existence of negative numbers? I'd say yes, because there's always a bijection that maps the lowest counting number of the set to the highest, then the second lowest to the second highest, etc. This reversal of order/mirroring is possible for any set with a strict order. The negative numbers are then some sort of dual space of the positive numbers. This bijection is a mirror permutation, can the idea of permutation groups be applied more generally to integers/sets? I apologize for the borderline vague statements.