I've talked too much with mathematicians

You are talking to one right now :) (not sure if a bachelors degree is enough to call yourself one)

you can't have 0 apples

You can actually. In fact, right at this moment I have 0 apples. If 0 is not natural, then you have no way of describing the number of apples I have.

There are a lot of concepts (degree of a polynomial, dimension of a space, cardinality of a set, in a graph) where 0 is a natural possibility.

So I think 1 indexing is fine, I use it all the time, but to me 0 belongs with the natutals. I will say tho, that 0 does not make sense to me as an ordinal. "He finished the race in 0-th place"????