And yes, perhaps one based indexing introduces a class of bugs when you need to call into c. But zero based indexing has the same problem if you need to call into fortran, and calling fortran is really common for numerical code.
Most of the classic linear algebra algorithms seem to be described using 1-based indexing and the last time I needed to use one of these algorithms, I stubbornly tried to translate everything into zero-based indexing which was more difficult than you'd imagine and it was difficult to have confidence that I had correctly captured the algorithm.
I switched to using a 1-based matrix implementation and everything became trivial. It's not that hard to switch between looping from 0 to n (exclusive) and looping from 1 to n (inclusive).
I suspect the main reason 1-based seems convenient for some things is because of convention. Note that 0 wasn't even really used in mathematics until hundreds of years after our system of counting years was created.
Since our year counting is 1-based, we end up with odd things like "2019" being the "19th year" of the "21st century" as opposed to "2018" being "year 18" of "century 20". I suspect it's also fairly unintuitive that the current century began in the year "2001" rather than "2000".
Wow that's the best argument in favor of 0-based counting I've ever heard :)
And since pointer arithmetic is based on offset, it wouldn't make sense for C to use anything other than 0-index. But mathematical languages aren't focusing on mapping the hardware in any way, but to map the mathematics which already uses 1-index for vector/matrix indexing. You can see the relation of languages in [1].
If you want to write generic code for arrays in Julia, you shouldn't use direct indexing anyway, but iterators [2] which allows you to use arrays with any offset you want according to your problem, and for stuff that is tricky to do with 1-indexing like circular buffer the base library already provides solutions (such as mod1()).
[1] https://en.wikipedia.org/wiki/Comparison_of_programming_lang...
This is again just begging the question. When you want to refer to the initial element as the "1st", it is due to the established convention of starting to count from 1. The point is that the reasonining for starting from 1 might only be that: conventional, not based on some inherent logic.
But then I agree that there is no inherent logic, math is invented and not discovered, and you could define it any way you want. If we all had 8 fingers we would probably use base 8 instead of 10 after all.
It just so happens that this edge case of 0 things doesn't occur when we actually need to count something. Starting from 1 is kinda like head is a partial function (bad!) in some functional programming languages. Practicality beats purity.
f = v0 + x
f = v1 + x
Those are the same equations right? Sure, but when I see v1 I'm not really sure what it is or could be, vs if I saw v0 I can assume it may be the initial velocity when I can look up.