To understand his reasoning, I think it is important to keep in mind that Dijkstra was a mathematician and that these people have a very special/peculiar sense of aesthetics. Let me explain what I think Dijkstra did not like about b), c) and d).

Assume you begin counting at 0, so the natural numbers are the set {0,1,2,3,...} and you want to denote the set {0,1}. If you go with b) you have to write -1 < i <= 1. Note that you have to use the number -1, which is not part of the natural numbers, to describe a subset of the natural numbers. As a mathematician, this might feel wrong or at the very least not very elegant.

This does not change if you begin counting at 1, i.e. if you define the natural numbers as the set {1,2,3,4,...}. Assume you want denote the set {1,2}. If you go with b) you have to write 0 < i <= 2. Here you have to use 0 as the lower bound, but we just said that the natural numbers start with 1, so again we need a non-natural number to describe a subset of the natural numbers.

For the empty set... well, say the natural numbers start with 1 and consider the sequence of subsets {1,2,3}, {1,2}, {1}, {}. If you include the upper bound and go with c), you have to describe them as 1 <= i <= 3 (looks good) 1 <= i <= 2 (still ok) 1 <= i <= 1 (ah... I see where this is going...) 1 <= i <= 0 (zero out of nowhere!)

I agree that most people may find it useless to talk about the empty set, but as I said, mathematicians are probably different here.

  • Artir Artir 2017-09-25T17:25:12Z

    Yes, you are right: I got that reasoning wrong about half-closed intervals and his underlying motives.

  • Rational Feed – deluks917 2017-09-16T22:00:20Z

    […] Why Numbering Should Start At One by Artir – the author responds to many well known arguments in favor of 0-indexing. […]

  • kaleberg kaleberg 2017-09-27T01:09:42Z

    APL used to let you choose zero or one based arrays, but the choice applied to the entire workspace. PL/I let you specify the lower bound as any integer you chose as long it was less than the upper bound. The default was one. I get the impression that most modern languages start at zero. It really seems to be a matter of style and taste. The book 'Gravitation' does general relativity with subscripts 0,1,2,3 where 0 is the time dimension, as opposed to a spatial dimension. This actually made sense until physicists started thinking about spaces with more than one temporal dimension.

Citation

In academic work, please cite this essay as:

Ricón, José Luis, “Why numbering should start at one”, Nintil (2017-09-16), available at https://nintil.com/why-numbering-should-start-at-one/.