There is no universal agreed upon convention whether the natural numbers include zero or not. There is a lot of tradition of it not including zero, because non-computer scientists find counting from one more intuitive. On the other hand, there are good reasons for starting at zero even within pure mathematics itself, e.g. so that natural numbers can be used to represent the cardinalities of finite sets. Zero is quite natural in this context ;-)
Another reason some mathematicians like to start them with 1 is because then you can write 1/n for some natural number n without needing to handle the possibility of a division by 0. As far as I've seen in classes so far, we tend to use |N = 0,1,2,... in algebra and |N = 1,2,3,... in analysis, where one uses 1/n a lot.
Interesting. Having division defined (for a set) can certainly be useful. So division and indexing (as seen in the article here, indexing starts at 1) are two arguments for positive integers, and ranges and empty sets are arguments for non-negative integers.
