You use zero and one-based counting for different things. Mixing them up is where fencepost errors come from.
Say you're counting the length of a fence, so you are numbering the posts. Of course it's preferable to number the posts starting at zero, because after having numbered N posts, you know the length of the fence is N.
On the other hand, if you are counting the posts themselves, you would want to say "one!" after the first post, as you have just counted one post. In this case, the number you assign the post is the number of posts counted after encountering it. Otherwise you end up having to +1 at the end.
Which is all to say: I've always felt that the disagreement between whether to start at one or zero is due to people not making clear what it is they're counting.
If you're numbering the points between spans (as you tend to be when giving names to dates, memory locations, and fence posts), always start at zero. If you're giving numbers to the spans themselves, it's often convenient to use a physically-based number such as the location of the center of the span (when writing graphics applications I will often refer to the pixel in the upper-left corner as 0.5,0.5, rounding down to get its address).
Those are great insights, thanks for laying it out like that.
This is also why rulers start with 0 (even if you can't see the 0) - because a ruler isn't about counting the marks, it's about measuring the space between the marks.
There's an excellent discussion and illustration of this in the original Inside Macintosh - a few us us were kicking it around a couple of years ago here:
You made me realize that we use 0-indexing because we confused the torsor[1] for the group. What I mean is that we may take differences between labels (numbers on the posts) to see how many posts to the left or the right a post is relative to another, and then we do this all relative to the first post. It is convenient in, say, C to do 0-based indexing because then you can add the index (the difference) to the pointer to the first element to move over that many posts, but that is different from the concept of which ordinal the post is labeled with.
[1] Quickly stated, a torsor over a group is a collection of things which support taking differences, and the differences are elements of the group. One example is pointers vs integers, since the difference between two pointers (to things in the same array in C) is an integer, and the sum of a pointer and an integer is another pointer. Another example is energy vs work, or Euclidean space vs vectors.
Regarding the initial argument, I'd opt for C or D; since here we're being consistent with our signs and therefore the statement's more readable. To me this is more apparent than the number of elements being the upper bound minus the lower, or natural numbers shrinking to unnatural (which I confess I don't understand given here i is by definition an integer and thus natural).
Of these two I'd likely opt for C since this lends itself to natural language (e.g. if I ask someone to choose a number between 1 and 10 I mean the choice is inclusive of 1 and 10), so again is likely more readable.
Also "pernicious: having a harmful effect, especially in a gradual or subtle way". I don't understand this statement either.
A language not mentioned, PowerShell, uses two dots to great effect (ellipses vs two dots is another debate; but I can't see how one would be considered harmful and the other not). `1..10 | %{"we're at number {0}" -f $_}`; simple, clear and obvious.
The whole article reads as if someone's trying to show off their intellect, rather than present an honest analysis of the benefits of the merits of the different indexing/numbering systems.
@TOGoS's comment on this thread, on the other hand gives a compelling and well reasoned argument of when to use each system.
Say you're counting the length of a fence, so you are numbering the posts. Of course it's preferable to number the posts starting at zero, because after having numbered N posts, you know the length of the fence is N.
On the other hand, if you are counting the posts themselves, you would want to say "one!" after the first post, as you have just counted one post. In this case, the number you assign the post is the number of posts counted after encountering it. Otherwise you end up having to +1 at the end.
Which is all to say: I've always felt that the disagreement between whether to start at one or zero is due to people not making clear what it is they're counting.
If you're numbering the points between spans (as you tend to be when giving names to dates, memory locations, and fence posts), always start at zero. If you're giving numbers to the spans themselves, it's often convenient to use a physically-based number such as the location of the center of the span (when writing graphics applications I will often refer to the pixel in the upper-left corner as 0.5,0.5, rounding down to get its address).