Is it? I ate all the apples. There are no more apples left. There are zero apples. Don’t we come face to face with nothing left all the time? There are zero guests left. There are zero episodes of the show to watch. There are zero days until Christmas. There is zero money in my wallet.
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'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.
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.
> 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".
The problem with your comparison is that you're already starting from too much.
> You always start with a context
And where does that context come from? The true starting point is nothing. The context needs to arise from something. In order to even form a thought about apples and not any of the other thousands of concepts that you're aware of in the first place, you need a prompt.
In the case of an existing object, the prompt is seeing it. In the case of a non-existing object, the prompt is the combination of social and memory factors that I mentioned before. The former is simple, the latter is complex.
Another thing to consider: humans invented the number zero long after the natural numbers. Children learn to understand it after the natural numbers. The brain processes it differently, as mentioned by the article. There is overwhelming evidence that yes, zero is more complicated than the natural numbers for humans to think about.
If your reasoning ends with the conclusion that it's not actually more complicated, then that is in direct conflict with the evidence and it shows that you must be missing something.
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.
"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.
These are not equivalent statements though. You simply learned at some early age zero represents absence, but that is not a natural concept.
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.
"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).
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.
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.
"are" and "zero", meaning non-existence/non-"are"-ness, in the same sentence is seen as strange by many, yes, and that include me.