If I have five apples and were to divide them among 0 people then nobody gets anything and I can eat them all, so the proper solution would be 5.

Paraphrasing you: "If I have five apples and were to divide them among 0 people, how many does each person get?" This sums up one approach to this problem, and can be thought of in a more intuitive manner than the limit approach. The answer could be zero. Or 1. Or 37. In fact, any number makes as much sense as the question. Which is why either an exception is raised, (or +- Inf is returned for floats, but that's just the limit approach). But perhaps it would be more fun just to return a random number on divide by zero :)

you can't divide the apples among 0 people and then claim to still have them, because in that case you would have divided them among 1 people

Like everything in life, it depends... For example: Storage has 5 items that need to be processed. 5 items need to be split equaly between available processes. There are currently 0 available processes so 5 / 0 = 0 items to be processed is more correct than either 5 or Nan or infinity.

Your example is quite vague (e.g. are we dealing with an integer number of items and processes?) and in general if something looks kinda like a division it doesn't mean it is exactly division. Just like in math, we have the power to simply say: if COND -> divide normally, else -> do something else.

And that's why the answer is 0.

It's more like you had five apples and divided them among zero people, which means not even you get to keep them. They were thrown in the trash instead. The answer is zero.

A stateful expectation of existence is what the denominator describes, but if you forced it to describe people, then you'd phrase it as "how many (ghosts) could possess (any number of apples)?"

Which would be infinite, since ghosts occupy no space and can't interact with physical reality.

As a proportion, compared to nonexistence, any quantity of something is infinitely greater than nothing, so if not n/0, how would you express you expect not the absence of a thing, but its nonexistence?

In other words, the quotient is 0, the reminder is 5?

That's an interesting solution...

I agree wholeheartedly. I think the issue stems from 0 meaning both 0 of "something"/"a concept" and "nil."

If I have 5 apples and divide them in to 0 buckets of apples, that makes sense. If I have 5 apples and divide them into 0 buckets of tractor; that doesn't make sense.

One intuition could be: As you divide 1 by negative numbers of smaller and smaller magnitude, you get negative numbers of increasing magnitude. At 0, the positive infinity of 1/0 is met by the negative infinity of 1/-0 and their average is 0.

why take averages?

Well, it would be simultaneously at both extremes.

The OP uses finite fields and fields are basically harmonic structures (think modular math). Assume the field is the numbers 0..n-1 MOD n. At (n-1) + 1 you will get n MOD n which is equivalent to 0 MOD n which is 0. Now assume n-1 approaches infinity; is 0 not ∞ ?