Counting a physical process, so you have to allocate space. 0 is that allocation, and as you move along the number line, you update that space (using "add 1"). Note that you only update the space at discrete intervals, which has surprisingly deep implications.
Having a kid I've been thinking a lot about counting, and what it really is. It seems totally wrapped up in repetition, and so I'm wondering if teaching counting as a function of, say, circular motion doesn't give a better intuition than the usual "count this clump of things" approach. (Counting clumps requires the person to simultaneously introduce an ordering and then implement a kind of internalized repetition as they point and count rhythmically. My kid seems to struggle with the ordering part, and no wonder: N objects have N! orderings.)
Having a kid I've been thinking a lot about counting, and what it really is. It seems totally wrapped up in repetition, and so I'm wondering if teaching counting as a function of, say, circular motion doesn't give a better intuition than the usual "count this clump of things" approach. (Counting clumps requires the person to simultaneously introduce an ordering and then implement a kind of internalized repetition as they point and count rhythmically. My kid seems to struggle with the ordering part, and no wonder: N objects have N! orderings.)