You'll typically see it in computer algebra systems whose syntax is intended to be typeset as traditional mathematical notation. Mathematica, for example, wants an array to look like what's being typeset, and it would be confusing (moreso than the potential off-by-one errors) to display `x[3]` while the actual code said `x[2]`.
I think other languages aimed at mathematicians, like Julia[1], do this for similar reasons but, honestly, unless you're writing a CAS, I think it's a bad idea.
Having lived with Dijkstra's ideas for a while, I will say that people find closed-open sequences highly counter-intuitive.
I work on a financial simulator, so we use dates quite a bit. Representing a year as 1-Jan-X up to but not including 1-Jan-(X + 1) is perfectly natural, and really works well with date-time libraries. It still works if you're using dates and decide to extend it to handle datetimes, and even months 1-M-X to nextmonth(1-M-X) do what you'd expect. So he's right, the math works beautifully. You also avoid writing a successor function when you split closed-open ranges; easy with integers, but gets tricky with almost anything else.
But I've had to nag people many times to not use Dec 31. I think the intuition for a closed-closed ranage is deeply ingrained and that extra "Jan 1" seems to stick out and bother people.
From that discussion, here's an interesting argument in favor of 1-based indexing:
> Let me try to briefly convey how I learned to stop worrying and love 1-based indexing. Basically it comes down to the fact that there are three significant numbers you have to think about when it comes to an array: it's initial index, final index, and length. In 0-based indexing, these are all different: 0, n-1, and n. In 1-based indexing, the final index and length are the same: 0, n, n. That's one less thing to think about and keep track of when coding. It seems trivial, but when you're doing something tricky and juggling a lot of complicated things in your head, even the tiniest alleviation of cognitive load helps, I find.
[1]: https://groups.google.com/forum/?hl=en#!topic/julia-dev/tNN7...