this post was submitted on 24 Jul 2022
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Those are valid points and make some practical sense, but I've talked too much with mathematicians about this so let me give you another point of view.
First of all, we do modular arithmetic with integers, not natural numbers, same with all those objects you listed.
On the first point, we are not talking about 0 as a digit but as a number. The main argument against 0 being in N is more a philosophical one. What are we looking at when we study N? What is this set? "The integers starting from 0" seems a bit of a weird definition. Historically, the natural numbers always were the counting numbers, and that doesn't include 0 because you can't have 0 apples, so when we talk about N we're talking about the counting numbers. That's just the consensus where I'm from, if it's more practical to include 0 in whatever you are doing, you use N~0~. Also the axiomatization of N is more natural that way IMO.
You are talking to one right now :) (not sure if a bachelors degree is enough to call yourself one)
You can actually. In fact, right at this moment I have 0 apples. If 0 is not natural, then you have no way of describing the number of apples I have.
There are a lot of concepts (degree of a polynomial, dimension of a space, cardinality of a set, in a graph) where 0 is a natural possibility.
So I think 1 indexing is fine, I use it all the time, but to me 0 belongs with the natutals. I will say tho, that 0 does not make sense to me as an ordinal. "He finished the race in 0-th place"????