Are different values of nothing equivalent? Is 'no tigers' the same as 'no zebras'?
It depends on how you are using the term 0. In your question you used it as both a predicate (a property of an object) and as a existential "flag" (non-predicate).
In other words, you could use to it refer to the existence of something, i.e. with 0 essentially representing the existence or non-existence of something:
1 apple = an existing apple 0 apple = a non-existing apple
Or, you could also (in a way) use it as a descriptive characteristic, like a property; in this regard it might be seen as a property all things share. I.E. All existing things share in their non-nothingness; all non-existing things are equal in their nothingness. However in many philosophical contexts, existence is not seen as a property of an object; it is simply viewed a relation between objects (the subject and the object). In other words, it doesn't really say anything about an object's features/characteristics to say that it exists or not.
The overall point here is that you should be careful to avoid linguistic traps with the concept of 0.
- The Devil is greater than nothing.
- Nothing is greater than God.
- Therefore, the Devil is greater than God.
You're missing the point because you're conflating values-with-units and values-without-units.
0 = 0
0 apples = 0 oranges
Okay. But then
1 = 1 = 1 = 1
1 lemon = 1 aircraft carrier = 1 snowman = 1 blog post
Of course all things that you have none of are connected in the most ephemeral sense of you-not-having-any, just like all things that you have one of are connected in that sense. Not very interesting.
But if you are wondering how to turn lemons into aircraft carriers, you need to notice that you can't conflate unitless quantities with unit-containing quantities. Zero lemons is no more like zero aircraft carriers than one lemon is like one aircraft carrier when you care about the units. If you don't care about the units, zero is just as much like zero as one is like one. If you confuse the two--with units and without--you'll be confused.
First, that was not a question, so this is not an answer per se.
If I have an apple zero times, it becomes zero. It is equal to nothing! I never understood that.
It's really not that hard. If you have an apple zero times, you have zero apples. You have no apples. The state of being appleless is really not terribly profound.
In my opinion the 0 is the ultimate connection between everything. The nothingness equals two different things.
Believe it or not, this is actually reasonably close to some interpretations of the primary teaching of the Madhyamaka school of Buddhism. The key term of art used there is Śūnyatā, usually translated by "emptiness", which is also the word for "zero". This, if framed properly, actually is quite profound.
I distinguish between emptiness (Buddhism) and Void (Taoism) as two types of nonduality. they both use the idea of zero, or negativity as their idea of how to point toward nonduality. But I have found in my research that there is a difference between what I call even and odd zero. And my evidence for that is that there is a difference between what comes before the first 1 in the Pascal Triange, and the empty places within the triangle, and this is the difference between emptiness and void.
O 1 101 10201 1030301 etc.
Thus I believe that O does not equal 0.
And in fact there is a theory of Domain Walls that comes from the study of Bose-Einsein Condensates that backs this idea up, that space can have domain walls that differentiate various domains of empty space. So space within a domain wall enclosure is essentially different from space with no domain walls. And this is essentially what we are seeing in the Pascal triangle example.
Zero: The Biography of a Dangerous Idea by Charles Seife
The Nothing that Is: A Natural History of Zero by Robert Kaplan and Ellen Kaplan
Signifying Nothing: The Semiotics of Zero by B. Rotman
Are these statements equivalent?
This basket contains apples and pears, there is 0 apples in this basket. This basket contains apples and pears, there is 0 pears in this basket.
This basket only contains apples, there is 0 apples in this basket. This basket only contains pears, there is 0 pears in this basket.
The first one is definitely different assertion. Asserting that there is 0 apples does not directly imply that there is 0 pears.
On the second version, the statements are equivalent. Asserting that there is 0 apples/pears implies that the basket is empty, and both are asserting that the basket is empty, therefore they are equivalent.
Therefore, whenever we see statements that only asserts
There is 0 apples in this basket. There is 0 pears in this basket.
we had to distinguish by context whether it asserts the first case or the second one.
My personal thought is that 0 oranges != 0 apples.
I much prefer apples and so having 0 apples is of greater concern to me that having 0 oranges, ergo they're not the same in this case. As such, I'd say it depends on what you're talking about and whether you care about the items in question.
Similarly 0 != 0 beers, because 0 beers may indicate that your glass is empty and needs to be refreshed.
Intuitions of correctness can be misleading and these intuitions come to mind easily without explanations of their source and may be substituting an answer to a different but related question. I realized if I think of the problem in terms not having 1 apple when I desire an apple, it is certainly different than not having 1 orange when I have no desire for an orange. It is the wanting that throws me off. But not having something I want is more like having -1 of that thing. I only really have 0 apples when I have 0 desire for apples and in that case it is much easier to equate with 0 of anything I do not desire.
When we have units of something we're counting 'things': 0 apples.
When we have 0 apples, in a way we're saying that we have spaces for apples but those spaces are empty.
By that logic 0 apples != 0 oranges.
0X = 0Y can still follow when X and Y are values of unknown numerical quantities. In that case we don't have any units, just numbers.
Your question is not philosophical; rather is question of logical relation.
An equivalence function inputs a counted set. The type (units) of any counted set is the disjunction of the types in the set; thus the type parameter of a counted set is contravariant. Thus the type of the set of the disjunction of all types (a.k.a. in computer science as ⊥ or bottom) obeys the Liskov Substitution Principle for functions on sets as an input to an equivalence function testing equivalence to a set of type (or even types).
Thus nothing (set without a type, thus the set of the disjunction of all types) is equivalent to any set of no thing (where thing is a type). But no thing(s) is not equivalent to no another thing(s), unless thing(s) and another thing(s) have a common disjunctive relationship.