Pointing directly to the objects that you are counting is not natural, it is just a bad habit.
People who are interrupted while they are counting and who have to restart the counting later, frequently forget whether they have already counted the pointed object, or not.
Pointing between the objects avoids these errors.
When I was young, I also had the impression that counting from 1 is natural. When I first encountered 0-based languages, I was annoyed by them, especially because I knew how Fortran compilers are implemented and I was familiar with all the tricks that are used by compilers to minimize the extra operations that are required to work with 1-based arrays.
So my initial naive thought was that maybe the implementers of C were not competent enough. However, after many extra years of experience, I believe that all the tricks for the efficient implementation of 1-based arrays are not worthwhile, because even for programmers it is more efficient to think using 0-based indices.
So now I really consider 0, 1, 2, 3 ... as a more "natural" counting sequence, unlike when I was a child.
You don't need to start at 0 to count intervals between objects, it's just easier to say "one" and point your finger to the space to the right of the leftmost item.
> … and point your finger to the space to the right of the leftmost item.
What if there is no leftmost item? Because you skipped the first step, essentially assuming that there was at least one item and including it in your initial condition, that process fails when the set of objects to be counted is empty. Humans are good at dealing with special cases like this; they'll notice the discrepancy and report "zero" instead of following through with the predetermined process. Computers, not so much. So instead we start with a count of zero and all items in the as-yet-uncounted set as our initial condition, and increment the count by one as we move each item from the "uncounted" set to the "counted" set, which correctly handles the case of the empty set.
This whole subthread was about what comes as a natural procedure to humans. Arguing that computers may need a different process is moot.
Anyway, to handle the exception, you could start the algorithm with a "check for the empty collection" base case; that would make it a natural definition for a computational recursive definition as well.
My point was that the natural procedure for computers is also more natural for humans, since it involves fewer special cases. Humans are better than computers at dealing with ad-hoc special cases but that doesn't make them natural.
People who are interrupted while they are counting and who have to restart the counting later, frequently forget whether they have already counted the pointed object, or not.
Pointing between the objects avoids these errors.
When I was young, I also had the impression that counting from 1 is natural. When I first encountered 0-based languages, I was annoyed by them, especially because I knew how Fortran compilers are implemented and I was familiar with all the tricks that are used by compilers to minimize the extra operations that are required to work with 1-based arrays.
So my initial naive thought was that maybe the implementers of C were not competent enough. However, after many extra years of experience, I believe that all the tricks for the efficient implementation of 1-based arrays are not worthwhile, because even for programmers it is more efficient to think using 0-based indices.
So now I really consider 0, 1, 2, 3 ... as a more "natural" counting sequence, unlike when I was a child.