Dijkstra's argument (https://www.cs.utexas.edu/users/EWD/ewd08xx/EWD831.PDF) is that we count the number of predecessors. This fits well with the mathematical usage, at least among mathematicians who care to dig into the order-theoretic foundations: the von Neumann construction of ordinals (and of cardinals as least ordinals of a fixed, well, cardinality), each finite ordinal is equal to its cardinality: 0 = \emptyset is a set with 0 elements, 1 = {0} is a set with 1 element, 2 = {0, 1} is a set with 2 elements, etc.
If you allow 0 as a valid ordinal, it unifies ordination and measure. That's the best reason to do so.
There. Djikstra as haiku.
Just to be clear, I wasn't making some sort of bizarre appeal to authority; whether something makes sense is a matter of taste, not of mathematics, and can't be proven or dis-. I meant only to respond to:
> In maths they always talk about the first element in a vector, not the zeroth. It is a bad argument to point to mathematics for using zero based indices.
I don't know for sure which convention is more common, though I suspect you're right that it's the 1-based convention, but I do know that it's false that 'they'—meaning, among others, we!—always talk about the first element/component in a vector. I probably will do that in a first Linear Algebra class, because the textbook often does and I don't want to introduce unnecessary confusion; but usually in my own work, and sometimes when I teach upper-level math classes, I use 0-based indexing when it's necessary to choose.