When there is one (or more) of something, you can observe the thing to say what there is one of. I see an apple, I can say "one apple".

When there is zero of something, what is there zero of? That question becomes a lot more difficult to answer. In theory you could say there is zero of anything not observed, but this isn't a very precise or useful definition and isn't what's done in practice. The things we choose to actively describe there being zero of depends on knowledge of what existed previously, on cultural and linguistics norms, on context etc.

For example, I can say "there are zero apples in the basket", but this requires me to know that there previously existed apples in the basket as opposed to oranges, or that the addressee of my statement was expecting apples to be there. This knowledge wouldn't be required if there was at least one apple.

Using zero fundamentally requires more mental reasoning than using small positive integers.

> "there are zero apples in the basket",

"There are zero apples on that apple tree."

You're inventing an abstraction that is less familiar than the reality. If you know apples go in baskets, you've probably picked them. Or that oranges just don't grow around here.

> When there is zero of something, what is there zero of? That question becomes a lot more difficult to answer.

> Using zero fundamentally requires more mental reasoning than using small positive integers.

I disagree. Positive integers don't exist in a vacuum either.

You always start with a context -- the thing you're counting -- and then you are given a number. Whether 3 or 0.

I see no greater conceptual complexity in either case. I think the flaw is in your sentence:

> This knowledge wouldn't be required if there was at least one apple.

Yes it is required. We're not comparing the statements "0 apples" and "1", we're comparing "0 apples" and "1 apple".

> In the case of an existing object, the prompt is seeing it.

Not necessarily, not at all. It might be, "hmm, do I have any apples left in the sack?" Or "did my daughter retrieve the ears of corn"? Sometimes we randomly come across an object, but in a great many cases (the majority?) we already have the context -- we know what we expect to see (or not see), or what we're looking for, or what we're investigating. Humans are goal-oriented creatures; we're not just responding to current sensory stimuli.

> humans invented the number zero long after the natural numbers

Only in the highly technical sense of a dedicated symbol for balanced financial accounts, or a digit placeholder in a positional number system. Languages all have everyday linguistic equivalents for zero like "none", "no", "aren't any", etc. These mean "zero" in the counting sense, precisely and exactly. There's no evidence at all that these came after something like the number 7.

> and it shows that you must be missing something

Or you yourself, in this case. You are unfortunately looking at an overly limiting definition of context, an overly limiting definition of zero, and you're missing important parts of linguistic history.

Level 1:

There's something or there's nothing

Level 2 if there's something:

There's one or more of this thing and it's not going to transform into same amount of something different.

You seem to have escaped cultural zeros. In Hitchhikers guide... You must prove to Marvin, that you are intelligent, by having tea and no tea, at the same time. A paradox and a cultural reference .

Also with stolen paintings: you have no painting,but you have evidence of it's existence.

And lastly, in song: "can't buy me love."

And of course physics: T(sub) 0. As well as space flight: t-zero.

If I have one apple I still have zero oranges.

>There are no more apples left. There are zero apples.

These are not equivalent statements though. You simply learned at some early age zero represents absence, but that is not a natural concept.

To add to that: After you eat all the apples, the following statements are true: - there are no apples left - there are 0 apples - there are 0 oranges

The following statement does not apply though: - there are no oranges left … as this implies that there have been oranges available at some time before.

In both of your examples, 0 is the result of a counting operation. Only what is counted differs.

"There are 0 apples left." is the answer to the question "How many apples are left?".

"There are 0 oranges." is the answer to the question "How many oranges are on the table?" (or "in the box" or wherever).

Everywhere where 0 appears in speech, it is the result of a counting or measuring operation, which provides the answer to a question, expressed or implied.

That counting or measuring operation could have had any other number as its result, instead of 0, which demonstrates that the nature of 0 is the same as that of any other cardinal number, i.e. it is a quantity (term introduced already by Aristotle, in his "Categories", where the various kinds of concepts and the words that name them were classified by the kinds of questions to which they provide answers).

That is true, but missing the point made by the parent comment. Being familiar with 0, we can clearly see how it follows from counting down. We fail to see why this would be ever difficult, since we’re so used to this concept.

However, this does not match the reality of how we learn about 0. Children learn 0 conceptually way after learning to count, because the concept is trickier to grasp.

What you say about children is true and I have already said in my first comment that the structure of the words meaning "zero" in most old recorded languages indicates that the concept of "zero" must be more recent than the concepts of the other small numbers and of negation.

Nevertheless, even if the concept of zero has been understood later than the concepts of 1, 2, 3 ..., it has still been understood in many places at least four thousand to five thousand years ago, i.e. already by the time of the oldest writings that have been preserved and thousands of years before the invention of the positional system of writing numbers, where the importance of zero has greatly increased and where it has required a dedicated graphic symbol.

But that last statement has more to do with the word "left" than with "zero".

True - the point I am trying to underscore is that the concept of 0 is not the same as counting down and arriving at „nothing“. These are two related, but not equal, concepts.

There are zero horses in my room. The absence of something is a default condition.

> There are zero apples.

"are" and "zero", meaning non-existence/non-"are"-ness, in the same sentence is seen as strange by many, yes, and that include me.

You can have zero of everything in the same place and time.

Which helps make my point, i.e. your Parmenidian view on this problem.

Zero oranges is different from zero apples, because you can have ten apples and zero oranges.

I have ten apples sitting right next to zero apples.