Suppose that in the real world we have $n$ finite boxes and we are thinking about what arangements are there, so we abstract it to something that mathematics can tell us about, in this case, the permutations of $n$ elements, and most of the times the abstraction is considered true no matter what. My question then is what makes this abstraction valid.
The set of those $n$ boxes has finite cardinality, so there exists a bijection between the boxes and some finite subset $X$ of $\mathbb N$. But $x_1+x_2=x_3,\ x_1,x_2,x_3\in X$ has no sense in this context, so the assumption that each box can be represented by a number is not totally true. Neither false because there are parts of mathematics that assigning boxes to numbers makes sense. So, what makes the act of assigning abstract concepts to real world objects valid in certain situations, and not in others? In this example, what makes permutations valid for the boxes, but not the addition operator?
