In the previous post I wrote, that as far as I can see at the moment, semantically zero is none, not nothing. I’ll clarify that a bit. Let’s ask a question: How many? None.
The opposite to “none”, might be everything of some finite amount (in the sense of the set theory). But as I wrote in the previous post, zero might be considered neutral in the way, that zero doesn’t have an opposite.
And I can’t really say, that the opposite to none is everything of some finite amount. If zero is none and the opposite to none is everything of some finite amount, every positive integer could be considered as an opposite to zero, that doesn’t make sense.
Or does it, if we have special cases, where something in some collection or set is some finite amount of elements, for example we have 7 elements in a set. Then, if we have all the elements from the set, we have 7 elements, all of them, instead of having none of the elements from the set.
What about dividing by zero, particularly 0/0? If there is zero amount of something, there isn’t this something at all. So 0/0 could be phrased as “none isn’t divided at all”.
Though I’m using too many negatives in one sentence. Perhaps now it’s better phrased: “None is divided.” Now there’s only one negative in one sentence and it’s better English and we don’t divide anything, particularly not zero. Word “none” prohibits the division.
In case a/0, a ≠ 0, perhaps we could say: “Something is not divided.”
Therefore particularly zero can’t be divided by zero. 🙂
I’ve been thinking about zero, none, nothing and the empty set time to time… And the infinite.
Can the opposite of zero be infinite? No. Why? Zero is a number, infinite is categorically different concept than a number. In the set N (all whole numbers) is infinite amount of numbers, but none of those is infinite.
Therefore, the opposite of zero is not infinite. And as far as I can see at the moment, semantically zero is none, not nothing.
Does zero really in terms of mathematics have an opposite? Is it neutral in a way, that it doesn’t have an opposite?
As to the empty set, it is an empty collection, one could say ”collection of nothing”. Poetically one could ask: Does the empty set exist? The empty set is ”collection of nothing”. If there is a collection of nothing, a collection that consists of nothing, the collection seems non-existent.
Can non-existent exist? Though, to ask, that does some kind of mathematical concept (the empty set) exist, is quite meaningless…
So, perhaps one can say, that the empty set is some kind of nothing… What’s the opposite to nothing, to something completely non-existent? Everything? Everything of what? Everything of everything that exists.
Can one say that there is an opposite to the empty set? If it would be the set of all sets, there is a problem: Also the empty set would be included in the set of all sets — if the empty set exists in same sense than non-empty sets. If the empty set exists, the set of all sets couldn’t be an opposite to the empty set.
The infinite is difficult concept. I’ve read, that Gauss himself objected at first to bring the actual concept of infinite to the mathematics. He would at first wanted to keep it only in philosophy and religion.
As to the infinite, perhaps, to be precise, one really can’t find an opposite to the infinite, not in terms of mathematics nor by the terms of semantics.