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? đź™‚

I’ve been reading e-book â€ťIntroduction to Mathematical Philosophyâ€ť 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 â€ťnull-classâ€ť. 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. đź™‚

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:

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:

- 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.
- RepeatÂ
*ad infinitum.*

Turtle graphics are very handy in the implementation. The needed functions or methods areÂ *Forward*: Draw a line in given amount of units forwards to the current drawing angle. The second isÂ *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 â€ťzeroingâ€ť 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 â€ťjustâ€ť zero. Similar happens in 2 * 0 = 0.

“Unique Sphere Shows Standing Out”

*Image courtesy of Stuart Miles at FreeDigitalPhotos.net*

My two cents: Zero â€ťzeroesâ€ť any number except itself. It â€ťzeroesâ€ť the whole identity â€“ including a number being even or uneven â€“ of any number except from itself; in case 0 * 0 = 0 zero keeps the identity of its own, it doesn’t â€ťzeroâ€ť 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 (â€ťzeroâ€ť) 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, somethingÂ *mustÂ *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 /Â
*aÂ*= 0 anywayÂ were the numberÂ*aÂ*whatever real number (except zero) - Two (2) is greater than zero byÂ 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?)

Perhaps particularly in mathematics it is the case that if one meets a paradox, one should get concerned. I myself somehow got alarmed, felt some kind of awakening, when the students including myself were introduced the Russell’s paradox in the course of Eucledian spaces (Euklidiset avaruudet) in the year 1997.

I write now at very general level. If something is interpreted as a set somehow, but on the other hand it isn’t a set according to definition, it may be the case that definition may need to be verified or the philosophy should be developed further from the point of view of semantics. One must see â€ťthat, that isâ€ť.

For example by understanding what the Russell’s paradox is at the level of logic, semantics and philosophy, one new concept is sufficient to see that there is no paradox!

One must deepen one’s view of the given problem and see beyond the known.

*Image courtesy of Stuart Miles at FreeDigitalPhotos.net*

Generally speaking, if one meets a paradox, it may be meaningful to check the definitions, pay attention to the semantics; it may be the case that a new category of concept is needed, if the problem seems to be a paradox, but with deeper view of the problem it may be the case that it in fact isn’t!

As to Russell’s paradox, understanding it something else than a paradox may result in huge development in philosophy, perhaps even new programming languages can be created or new programming techniques â€“ especially: What happens to the set theory?

Perhaps I’m just dreaming…