> Array indexing is such a core thing and I don't understand why anything mathematical or scientific would start with 1.
Because that's how maths work? Literally everywhere in maths you count from 1, except in software engineering. That's why. I hope that clarified your confusion.
My hot take is that 1-based indexing is often a mistake in math too. It's also not universal, even within math. And linear algebra doesn't need 1-based indexing either, and some operations are even more easily expressed with 0-based indexing.
What index to start with only strongly matters when the indexes have semantics. Otherwise you should just treat it as an opaque index, i.e. eachindex(), keys(), etc. In math when there are semantics, the indices usually include 0. When not (vector components, matrix indices, etc), they usually (but not uniformly) don't.
The one nice side-benefit of Julia's mistake in adopting 1-based indexing is that it provided an extra impetus to build machinery to handle arbitrary indexing, though too much code still doesn't work correctly, and code still gets written to only handle 1-based arrays.
> What index to start with only strongly matters when the indexes have semantics.
Which in everyday computing (as opposed to mathematics) they often do, and those cases are (most?) often, in human terms, much more natural to start from 1: "I have an array of N elements. The first of a bunch of things is thing number one, and the last of N things is thing number N." Hence:
N_things: Array[1-N] of Thing;
for i := 1 to N do begin
Whatever := Whatever + Whateverize(N_things[i]);
end; // for i := 1 to N...
Yeah, that's how old I am: That's Pascal. (With some declarations skipped, and I may have misremembered some syntax.) The canonical example is of course the original Wirth-style max-255-ASCII-characters fixed-length[1] String type: In a string of length N, the nth character is at position n in the string. Character number N is the last one.
> The one nice side-benefit of Julia's mistake in adopting 1-based indexing is that it provided an extra impetus to build machinery to handle arbitrary indexing
1) Arguably, as per the above, not a mistake.
2) Muahaha, "build machinery"? No need to build anything new; that's already existed since the early 70s. (Yeah, that's how old I am: Not adding 19 in front. There was only one "the seventies".) It's not like starting at 1 was mandatory; you could well declare
My_fifty_things: Array[19-68] of Thing;
And then "for i := 19 to 68 do ..." whatever with it, if those specific numbers happened to be somehow essential to your code.
(At least in Turbo, but AFAICR also in original Wirth Pascal. Though probably with the max-255-ASCII-elements limitation in Wirth, and possibly also in Turbo up to v. 2 or 3 or so.)
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[1]: Though from at least Turbo Pascal 3 (probably earlier; also think I saw it on some minicomputer implementation) with the backdoor of changing the length by directly manipulating -- surprise, surprise, it exists! String was a built-in type with its own implementation -- the length bit at index [0]. Better start out with your string declared as length 255, though, so you don't accidentally try to grow it beyond what's allocated.
> often, in human terms, much more natural to start from 1
This meaning of natural is highly cultural dependent. It took the Greeks a startlingly long time to accept that one was a number (because it's a singleton), much less zero. I do not e.g. want arrays that can't have length one, because they have to be containing a number of things.
> No need to build anything new;
Well, no, not "new". Arrays with arbitrary bounds is a well-trod path. But they still had to make it work in Julia: CartesianIndices, LinearIndices, and overloading of "begin", and "end" keywords, etc. And the radical dependence on multimethod dispatch meant they couldn't quite just reuse existing work from other languages.
I'm not "explaining your reasoning" because I don't agree with a single work you just said.
> In math when there are semantics, the indices usually include 0
Complete nonsense. For example, in a Laurent expansion you start in the negatives and go up. Now you're gonna say "but that's an exception, I said _usually_". But it's not, this is the general case.
Because that's how maths work? Literally everywhere in maths you count from 1, except in software engineering. That's why. I hope that clarified your confusion.