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> Dijkstra's paper makes the case from first-principles that the closed, open interval of [0,n) for sequences is the most appropriate

He argues that it is the most appropriate when indexing into an array of the natural numbers starting with 0. If the array in question started with 1, one-based indexing would be most appropriate following exactly the same logic!

[kbd]

I don't think that's a correct reading. You're talking about this section, correct?

> When dealing with a sequence of length N, the elements of which we wish to distinguish by subscript, the next vexing question is what subscript value to assign to its starting element. Adhering to convention a) yields, when starting with subscript 1, the subscript range 1 ≤ i < N+1; starting with 0, however, gives the nicer range 0 ≤ i < N.

He never specifies the contents of the array, he's talking about subscripts (i.e. indexes).

[goto11]

You may be right, but in that case, why is the second range "nicer"? AFACT it is because he explicitly adds 1 in the first example but implicitly subtract 1 in the second, which makes it looks cleaner. If you want the N'th element of an array, the range in the first example is N ≤ i < N+1 while in the second it is N-1 ≤ i < N. Written out like that, I don't see how the second is obviously nicer.

[kbd]

Sure. Something is generally considered more elegant ("nicer") when it has fewer terms. (I'm really happy with my own argument above why Python's indexing is "optimal".) But I'd put what I think you're saying this way: Dijkstra asserts that '1 ≤ i < N+1' is worse than '0 ≤ i < N', but he's choosing his conditions carefully. '1 ≤ i ≤ N' also has no extra terms. (Though in fairness to Dijkstra, this section was after he'd already argued why closed, open ranges are better.)

I will say this though: in all the noise in this sub-thread, I never got any example in answer to my original question asking for any algorithm or formula that works out better with 1-based indexing (my original response was to its parent's claim that 1-based indexing results in fewer ±1s in practice). Except for maybe the "stride" examples[1] for which I still don't understand why the starting index is important. I say this not in victory (ha ha! zero-based indexes are clearly superior!) but in disappointment because I was hoping to gain understanding of why Julia/Matlab and others (which are more geared towards math/stats which is outside of my experience) made their indexing choice. Particularly because it's against the norm, they must have good reasons.

[1] https://news.ycombinator.com/item?id=20425500

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I'm actually arguing the opposite, that 0 ≤ i < N does have an extra term - it is just hidden! Where does the zero come from? The generalized form for picking the N'th number (with zero-based indexing) is N-1 ≤ i < N. So the 0 is N-1 which have been simplified because you know N is 1. But in that case couldn't you also just reduce the terms in the example with 1-based indexing similarly? Reducing the term in the one example to make it simpler is just cheating.

I'm not disputing the choice of closed-open rage, but I'm arguing the expressions are equally simple with either 1 and 0 based indexing.

As for an example where 1-based indexing is better, I had such an example in a another subthread: Translating between the numbers and names of months. Or indeed anywhere where you have a list of things and you want to pick them by ordinal numbers. E.g if you want the N'th president, it is simper to do presidents[N] than presidents[N-1].