Related is Dijkstra explaining the pros and cons of different styles of indexing and slicing: http://www.cs.utexas.edu/users/EWD/transcriptions/EWD08xx/EW...
Given that, his arguments support 1-indexing better.
The equivalent statement in probability theory says that a probability is uniquely determined by its cumulative distribution function, which are sometimes nicer to take limits of. You can also get a probability from a cumulative distribution function, provided your function is right-continuous [1] (which is directly related to half-open intervals). For more examples of half-open intervals in probability, you can look at stochastic processes. See, for example, càdlàgs [2] and Skorohod spaces; they capture the notion that processes that "decide to jump" (think of Markov chains changing states if you like) are right-continuous.
IMO, half-open intervals are just nicer whenever you have to union them or intersect them, and are no worse in other aspects when compared to closed intervals. Also, I think the author makes a big cultural blunder when he dismisses "mathematical aesthetics" as a valid reason. A significant number of mathematicians think of elegance as the ultimate goal in mathematics; as Hardy famously said, "There is no permanent place in the world for ugly mathematics".
[0] http://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension...
[1] Well, we also need the obvious conditions: Correct limits at infinity and monotonicity.
Yes, but we are talking about indexing discrete arrays. All of your examples are continuous, and are not analogous.
As to the "and [thus] are not analogous" part, I recommend the Concrete Mathematics book. You'll certainly see at least a hundred examples of continuous examples alongside with their discrete counterparts, as you can imagine by looking at the book title. Half-open intervals feature prominently when doing discrete calculus in the Sums chapter.
The crux of the issue is treating indices as offsets rather than identifiers, which is perfectly reasonable if and only if one is directly working with sequential memory.
I consider the idea that we should follow human intuition to itself "not rise to the level of wrong". There is nothing intuitive about programming, it is probably the second least intuitive activity known to man behind only pure mathematics. Your intuition needs to be trained from what is basically scratch, not blinding copied from much friendly, human domains. We know what that leads to, we've been doing it for decades; some people seem to argue "Oh, if only we'd try more human-friendliness it would all be better" as if we've never tried it, but we have, repeatedly. We get Applescript and COBOL and other languages that strive to be "human friendly" and produce an impenetrable morass of "human friendliness" that become incredibly hard to use as the size of the project exceeds trivial. As counter-intuitive as it may be, the track record is pretty clear at this point that the best languages are those that do not try to cloak themselves in false human skin.
(Which is of course not to say that we should all be programming assembler. Don't strawman me there. But the task of a language should be seen as a bridge between the human world and the computer world; it is not the job of the language to actually be in the human world itself.)
For the love of avoiding mutation, let's start pitching indexing in the dustbin of history when it comes to programming languages for general use. The fact that the justifications are coming from pointers and arrays ought to be a warning flag that we are looking backward rather than designing the future.
for i = 0, i <= myarray.length - 1, i++
do foo(myarray[i])
end
myarray.get(6) //return the sixth element
myarray.foreach('foo 6 19) //operate on part of the array
We are always going to need this stuff for certain tasks. For example, I'd love to see a simply 3x3 filter operation on an image without using indices. I am also a bit biased; I'm a systems programmer, I don't write UI's and web apps. I tend to work more in the trenches, where abstraction gets in your way almost as often as it helps you get things done more quickly.