Let us examine numbers as distances from some number.
If we choose as starting point the zero (0), we have the usual case.
0 = |0 – 0|
1 = |0 – 1|
2 = |0 – 2|
n = |0 – n|
Each distance can be divided into parts, let’s think the number space as the familiar set R. For example the distance from zero to two can be divided in n parts, we get distances 2 / n between zero and two.
The point from which we started isn’t a distance but — like said — a point, that can’t be divided into smaller parts. Philosophically speaking this point is now in words of Democritus “atomos”, the smallest unit.
In the book “The way to geometry” it is said: “The magnitude of a point is zero.”
When we founded the numbers as distances in relation to zero, the zero point is indivisible; because in question is a point not a distance, dividing zero (a point) by any number, we get zero. Zero doesn’t get smaller by dividing.
Of course zero can’t be divided by itself. In distance interpretation philosophically speaking indivisible would be divided by indivisible.
Now zero is different from other numbers so that it is a point, the other numbers are distances from this point.
As distance interpretation we could start the numbers also in relation for example to number eight (8).
Now eight is a point, we get the other numbers as distances to number eight.
0 = ||8 – 8| – 0||
1 = ||8 – 8| – 1||
2 = ||8 – 8| – 2||
n = ||8 – 8| – n||
Here from number eight is “created new zero”. First we determine the distance of number eight from itself, then the distance of examined number from the determined distance earlier..
We could get the numbers (distances) and the zero point also by denoting:
0 = |8 – (8 + 0)|
1 = |8 – (8 + 1)|
2 = |8 – (8 + 2)|
n = |8 – (8 + n)|
When we see the numbers as distances from some number, we notice, that the distance can be divided into smaller parts, but the point from which we started, we can’t divide into smaller parts.
In our example when we started from eight, we examine first the distance of eight from itself, |8 – 8| = 0, which is and remains as zero.
The distance can be even or uneven, but the point from which the distance is determined can’t be even or uneven, that is the whole point of this post.
If we want also the negative numbers as distance interpretation, we simply put the minus sign before the determined distance, for example -2 = -|0 – 2| and this is how we can get the entire set R so that the distances from zero point can be divided into smaller parts, but zero point is and remains zero and is not a distance but a point. In the words of Euclid that which has no part. In my own words: “Indivisible”.
In this way the specialty of zero can be seen better.
The title is a bit absurd, we could examine the distance of any number from itself..
In the previous post I examined the property of distance being even or uneven. Let’s go back to distance of zero from itself. Formally the distance of zero from itself is |0 – 0| = 0. 0 mod 2 = 0, why isn’t this distance even?
The distance doesn’t exist, instead of distance we have a point. Euclid’s definition to a point is: “A point is that which has no part.” In Euclid words the “distance” of zero from itself, |0 – 0| = 0 has no part. Something like this isn’t even or uneven.
In an e-book called “The Way To Geometry” it says, that magnitude of a point is zero.
Philosophically speaking a point is what Democritus called “atomos” applied to geometry.
In short, there isn’t distance of zero from itself. The meaning of distance |0 – 0| = 0 is geometrically a point, something that has no part.
The main point is, that I see zero as a neutral number. What does that mean?
- Zero isn’t even or uneven
- Zero isn’t negative or positive
- Zero doesn’t have numerical opposite
The meaning of the video clip: “..achievements: Zero.”
Probably zero is my greatest achievement too… 🙂
Semantically zero is none, not nothing.
In multiplication zero overrides the neutral element property of one: 1 * a = a, but if a = 0, zero “zeroes” one. The number one doesn’t keep the identity of zero, but zero “zeroes” number 1’s neutral element property and keeps the identity of its own.
Let us examine the even and uneven property of zero from point of view of distances.
The distance of number 2 from zero is |0 – 2|= 2. 2 mod 2 = 0; 2 is even.
The distance of number 3 from zero is |0 – 3|= 3. 3 mod 2 = 1; 3 is uneven.
The distance of number 4 from zero is |0 – 4|= 4. 4 mod 2 = 0; 4 is even.
But what about our friend zero?
The distance of number zero from itself is |0 – 0| = 0. 0 mod 2 = 0, but now the meaning of zero is, that the distance doesn’t exist. There isn’t distance from zero to zero. The distance that doesn’t exist isn’t even or uneven. Therefore now zero can’t be considered even or uneven.
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.
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: