*To eight goes two four times;*

*To six slips it three times;*

*To four it fits twice;*

*To zero it doesn’t fit;*

*No pairs are there in zero.*

*No trace of two.*

*”Where’s my pair?”*

*Zero asks…*

There are finite number of letters. If we limit the length of a word to finite amount of letters, the number of all possible words is also finite.

If there were infinite number of letters in a word and this word consisted of infinite amount of sounds, the pronunciation of a word would never end.

As there are finite number of words, also the number of consepts formed from these words is finite.

On the other hand, if we would create consepts in mathematical way, we could have infinite number of consepts. Though, all of them couldn’t be explicitly ever written nor said in finite amount of time.

I’ve sometimes wonderd, that how many meaningful natural languages of humans’ from all of these finite number of words could be formed — also how many grammars. What we understand by “meaningful language” might be hard to define…

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.

Let us assume again, that a person has a collection of 100 books. The books are in a shelf in way that between some books there is an empty space. Probably we can say, that the empty space is part of the collection at least visually considering the whole, although that empty space is something different than a book.

In general an empty space can be filled with a book, so that the book is surrounded by an empty space. If no empty space existed, a book couldn’t be taken away from the collection of the books (or in this example put a book to the shelf).

Therefore empty space is part of every collection of books..

I have to correct some things I’ve written about the empty set.

Let’s start. Let A = {0}. Now A \ {0} = {} = Ø.

That is: We get the empty set. Let’s go beyond that.

I’ve written in this blog earlier ”emptyness can be created, ‘nothing’ can’t be created; nothing is from which the creation begins”.

At the beginning of this post, we created emptyness, we got the empty set as a result. By the means of the set theory we can’t get rid of the empty set. It’s as empty space as we can get. Also, in the poetic thought of mine I’ve written, that emptyness can be created, ‘nothing’ can’t be created.

So, we can’t get to this ‘nothing’, we can just see, that it is a ”state” that is ”before” the empty set, for the sake of perfectness. 🙂

A symbol to this ”state” could be

The order: ‘nothing’ → emptyness → something.

This new “collection of nothing” can’t be considered really a set; it actually doesn’t exist, but still, it is there — at least it was… Perhaps it will be… Somewhere…