# Category Archives: Mathematics

## Poem on Two

Two shows what even is;
everywhere where it fits evenly,
it exists;
and so is even that,
where two evenly fits.

So many are there twos in a number,
than the relation possible with two evenly is;
if that possible is.

For if it impossible is,
even number not is.

## The distance of zero from itself

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.

## My view on zero so far

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, but if = 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.

## Fractal of my own

I came up last sleepless night with a fractal of my own. There are two versions of it. The first version has the following rules:

• Build a circle with radius of r
• Draw inside the previous circle a circle of radius of / 2
• Draw circles below, above and sides of the previous circle, the radius of these smaller circles being / 4

Choose for a new circle the circle that’s radius is r / 2. Repeat the previous steps ad infinitum.

Unfortunately on WordPress.com I can’t add the JavaScript-code where you could change the amount of iterations, so let’s just settle for pictures. This kind of fractal might be invented already, but I haven’t seen one ever before.

Version 2 of this fractal: The previous has a bit more complicated rules.

I made two color version of the previous too: The second one: Last night I was a bit playing with these fractal images in JavaScript, below is one more picture: Perhaps this fractal of mine has at least some value of curiosity.

I try to find the time sometime in September to come back to this topic.

## Do we need a new ”set” – going beyond the empty set

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…

## Zero, none and dividing by zero

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. 🙂

## Zero, none, nothing, the empty set and the infinite

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.