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.
This is just humorous thought of mine…
If we have a “collection” where isn’t anything, is it a collection? If someone has got 100 books, the person has a collection of 100 books. But if a person hasn’t got books at all, does the person have a collection of books? No.
So, is the empty set as such a set? A collection where is nothing isn’t a collection.
But is it an empty collection? But is an empty collection a collection at all? 🙂
I’ve been reading e-book ”Introduction to Mathematical Philosophy” originally written by Bertrand Russell and published in the year 1901.
Among other interesting thoughts Russell gives thought to the definition of a number. This is something very interesting; I’ve been thinking myself strange things about number zero. Can zero be considered as a whole number? It doesn’t describe anything existing as whole. If the number of something is zero, this something doesn’t exist at all in somewhere, particularly not as whole.
As to definition of number, Russell discusses about classes. From an old Finnish book that discusses university level algebra, I recently learned the definition of zero as a class. In Russell’s book zero is defined as a class in slightly different way: Russell doesn’t say anything about the empty set, instead he mentions ”null-class”. I think I will read this part of the book over and over again.
This is something fascinating…
Hopefully you got interested in this great book:
This is perhaps funny sounding concept I came up… In sacred geometry everything is defined through geometry; what is post apocalyptic sacred geometry?
With the post apocalyptic sacred geometry I refer to imaginary new world, where geometry defines the ”rules” of this world; what is changed now?
In post apocalyptic world the geometry on which everything is based, makes it impossible to feel pain, get hurt or harm anyone or anything. Some kind of imaginary sacred paradise through geometry that doesn’t allow anything bad or evil; these things don’t exist anymore and every possibility to do anything is good; and from good ”spirit” of every deed arises more good; one good thing is eventually more than just one thing, perhaps fractal-alike thing…
Image courtesy of Danilo Rizzuti at FreeDigitalPhotos.net
In this imaginary world immortality is something considered self-evident; nothing can harm anyone in anyway, because the nature of post apocalyptic sacred geometry.
Through sacred fractals one thing is endlessly interesting, there absolutely doesn’t exist possibility to get bored, which on the other hand is impossible through new post apocalyptic sacred geometry.
Not even smoking can harm anyone’s health in anyway in this imaginary world of post apocalyptic sacred geometry. 🙂
In multiplying one (1) is neutral element: a * 1 = a. For example, 7 * 1 = 7. Number one keeps the identity of a number, which includes a number being even or uneven. But what about zero (0)?
0 * 1 = 0. Does one keep the identity of zero or does zero keep the identity of its own? The property of this identity is ”zeroing” property: a * 0 = 0, were a whatever real number, including one and on the other hand zero.
In case -2 * 0, zero takes the whole identity of number -2: The number being negative and even; as a result we get ”just” zero. Similar happens in 2 * 0 = 0.
“Unique Sphere Shows Standing Out”
Image courtesy of Stuart Miles at FreeDigitalPhotos.net
My two cents: Zero ”zeroes” any number except itself. It ”zeroes” the whole identity – including a number being even or uneven – of any number except from itself; in case 0 * 0 = 0 zero keeps the identity of its own, it doesn’t ”zero” itself, which reflects the identity of zero itself, how it is neutral in a deep sense and meaning.
Mathematical philosophically zero refers to none, there isn’t something at all. Still, zero refers different than nothing. As I’ve written before emptyness (”zero”) can be created, nothing can’t be created; it is from which the creation begins.
Let us assume, that we have two (2) coins. It’s even amount of coins. Let’s give one coin to a poor beggar. Now we have only one coin, uneven amount of coins. We’ll give that coin to a poor beggar too. Now we have no coins at all, the number of coins we have is zero. Do we have still again even number of coins, as we have zero amount of coins? I mean, we don’t have coins left at all!
The coins we had were in a wallet and the two coins were all we had there; now the wallet is empty. Is emptyness even or uneven? Or are we speaking now about different matter?
As far as I can see, if the number of something is different than zero, there must exist something, somehow. This number is even or uneven.
So, number being even or uneven, philosophically would refer to existence; something must somehow exist, that is, the number of something is different than zero. This amount can be negative or positive, even or uneven, but not zero.
But if something doesn’t exist, the amount of this something is zero, that isn’t even or uneven, as stated before. If the “number of something” is even or uneven, something must exist, somehow.
Technically one test to determine, that is a number even, is to divide the number to be tested by 2; if reminder is zero, the number is even. This test is suspicious to zero from two (2) reasons:
- 0 / a = 0 anyway were the number a whatever real number (except zero)
- Two (2) is greater than zero by its absolute value (philosophical mathematical problem)
My two cents: Zero is neutral element in addition and one of its properties is, particularly philosophically, that as to being even or uneven, it is neutral.
(As a sidenote something came into my mind from section 2 above: Is number one (1) somehow fundamentally uneven in natural numbers set?)