[pflenker]

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.

[adrian_b]

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).

[pflenker]

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.

[adrian_b]

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.

[amelius]

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

[pflenker]

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.