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Category Archives: Mathematics

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? 🙂

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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:


In my old article I have implementation for Cantor’s set without recursion. Now I implemented the Cantor’s set with recursion in JavaScript. This is better solution.

The definition of the Cantor’s set in the language of set theory is the following:

If this doesn’t mean anything to you, you might want to check the old post. 🙂

Below is a picture from the output of the program of this post:

 

Below is the JavaScript listing from Notepad++ in full as png-file:

 

cantor listing


Back to the world of fractals…

One of the famous fractals is the Koch curve, described by Helge von Koch in the early 1900s.

The basic idea of this fractal is as follows:

  1. Take an equliteral triangle, build another triangle in the middle of each side of the shape, the new triangle having a base length of 1/3 of the length of the side.
  2. Repeat ad infinitum.

Turtle graphics are very handy in the implementation. The needed functions or methods are Forward: Draw a line in given amount of units forwards to the current drawing angle. The second is Turn: Turns the drawing angle in given amount of degrees.

With recursion we are now ready to go to build a snowflake based on the Koch curve.

Below is two videos of my implemantation:

..and the same with more iterations:

 


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”

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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.

zero2

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:

  1. 0 / = 0 anyway  were the number whatever real number (except zero)
  2. 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?)


Set Z consists of all negative and positive integers and zero:

Z = {…, -3, -2, -1, 0, 1, 2, 3, …}.

Human’s intuition easily make to think that zero (0) is the middle point of set Z. But is it?

In every finite subset of set of Z {-p, -p + 1, –p + 2, …, 0, …, p – 2 , p – 1, p} (p > 3) zero is the middle point on grounds of symmetry:

|(-p) – 0| = |p – 0| = p

(With difference in absolute values, one gets the distance between points.)

But what about the whole set Z? It is an infinite set without beginning or end. Can it have a middle point?

I’ll take a role of a creator. I say: I create the set Z from zero to negative and positive infinity. Someone says: Then zero is the middle point. I answer: No, I ”started” the creation from zero, but after creation of my infinite work, my work doesn’t have a start or an end. Therefore it does not have a middle point.

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Image courtesy of Sira Anamwong at FreeDigitalPhotos.net

Back to the role of a blogger: Because the set Z doesn’t have a start or an end, it doesn’t have a middle point. One can’t measure the distance from “infinity to finite number zero”, at least we humans can’t. The best we can say is that the distance in this case is infinite, but that’s all.

One could also ask: Does set N have a middle point? (consists of all positive integers and is an infinite set) In this case human’s intuition doesn’t make it to think that set would have a middle point.