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.

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

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 ”that, that is”.

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 – especially: What happens to the set theory?

Perhaps I’m just dreaming…

If zero (0) is added to any real number *a*, as a result we get *a*: *a *+ 0 = *a*. What now was added to *a*? Nothing? I would say wrong. In some sense.

Namely from our friend, the set theory, point of view set A = {0} is not empty, there *is *something, namely number zero. If one would say, that 0 is nothing, in set A weren’t anything. In our case there clearly now is something, element 0.

Somehow philosophically 0 isn’t in same extent ”nothing”, that it would lead from view of set theory as the only element in the set to same state as the empty set ({} or ∅), that is so empty, that there simply is nothing; the empty set is more ”nothing” than 0. As a number, zero is considered as neutral element in *some* cases. But it obviously is more… What?

As to empty set, more philosophical question is, does the empty set contain itself – and is it then empty.

Emptyness and nothingness have their differences.

Let us imagine an empty room where there is four walls and a roof. Emptyness gives there space. And also this emptyness, space, has many meanings; if the room has only little space you would probably feel quite uncomfortable there.

In music the fact the there isn’t a note is known as pause. In this case emptyness in the notes gives rhythm to the music, without this non-existence of a note (nothingness from point view of sound?) we would’n have music as we know it.

In speech silence, a pause, can give one some kind of power to the speech itself.

Emptyness and nothingness really are powerful from their beings!

I consider ”nothing” as something that doesn’t exist. Still it does.

**Update! (16/5/2016)**

In order to express all this more precisely zero is interpreted as an positive integer, but not genuinely positive integer; ”nothing” can’t be a positive integer. As to 0 *+ a = a*, to number *a* is added an positive integer – something else than ”nothing”. This makes me consider the empty set being ”more nothing” than number zero. In this particular case to *a *is added a neutral element zero – not ”nothing”, in some sense…

It seems we’re pushing the limits of semantics of ”nothing” to new boundaries… Anyway it is essentially something else than ”emptyness”. See my post on creation.

The question now is: Is the rank of {} zero (0)? Do we need to “divide” zero; do we need a new “zero”?

Many, many years ago I came up with an idea about a fractal that would be based on right triangles and squares. People that had less mathematical experience than I, didn’t take me seriously, when I told about my idea. They thought that this was just some kind of nonsense.

I didn’t visualize my idea and forgot the whole thing… Later, after many years when I started to read e-books about fractals, I found sophisticated ideas how to make nice colored Pythagorean fractal trees. I think I’ve been dealing with wrong people in the past… 🙂

This article discusses only about simplest possible Pythagorean fractal tree, most symmetrical version of it.

We start with a right triangle with two 45 degrees angles and one 90 degrees angle and visually speaking turn the triangle so that hypotenuse is at the bottom:

Next we draw squares against the sides of the triangle so that the length of the side of the square is the length of the side of the triangle:

Next we imagine new right triangles to the farthest side of the ”branch” squares and do the same process as earlier: We draw squares to all the sides of the triangles.

This process is continued infinitely and we have a fractal! …that looks like a tree.

The length of the new hypotenuse *c*‘ = cos(45°) * *c*, where c is the length of the hypotenuse of previous iteration loop’s right triangle(s).

When generating the tree, I recommend using Java with JavaFX package or using JavaScript, because with those one can use the following methods:

- beginPath
- translate
- rotate
- save
- restore

These methods make it quite easy to generate a fractal tree. When drawing the tree into screen, only the squares are drawn.

The e-book below discusses lots of fractal programming and has also an example codes in JavaScript generating these kinds of trees (also colored):

I’ve been fascinated by the mental being of integral since *lukio *(high school in the US). It’s fascinating, for example, to see how it increases the dimension: From length we get into area, from area we get into volume.

Here I deduce a formula of circle’s area starting with length of a circle and the volume of a sphere starting from area of a circle by integrating.

The length of a circle is 2π*r. *By integrating we get into the next dimension: Area.

What is to be integrated is half of the length of the circle due to the way of determining the area. So we will be integrating π*r.*

We will be using the definite integral with –*r* as bottom limit and *r* as the upper limit, so that we get the area of whole circle.

Let’s take a look at the magic of integral:

The area of a circle is π*r*^{2}* , *what you can see as a result above.

We can use the integration formula of even function, because the curve behaves like even function’s curve.

Next to the volume of a sphere from area of a circle. We will be integrating ½π*r*^{2}*; *as half of the area rolls over, we get a sphere, which is what we’re after.

The integration limits are now from 0 to 2*r*. Let’s see the magic:

As we remember, the volume of a sphere is 4/3 π*r*^{3}*. *See the result above.

In this context it is fascinating to try to see the mental being of integral, how it increases the dimension. When you really understand the definition of integral you really can appreciate the mental being of integral.

I apologize that you are forced to read my terrible handwriting. 🙂

People often think that mathematics teaches us only to calculate. However when one goes deeper into mathematics, it teaches us also to think more philosophically whole life, everything.

Mathematics has inspired to views concerning the whole cosmos in its many fields solely from its abstract being that is as such separated from everything else. That is, the philosophical deepness in mathematics is so strong that it kind of stretches views to everything…

Edward Frenkel has written in his book ”Love and Math” as follows: ”Mathematics teaches us to rigorously analyze reality, study the facts, follow them wherever they lead. It liberates us from dogmas and prejudice, nurtures the capacity for innovation. It thus provides tools that transcend the subject itself.”

Copyright: basketman23 / 123RF Stock Photo

Georg Cantor who became famous from his studies on cardinality of the sets has written: ”The essence of mathematics lies in its freedom.” Frenkel has written ”where there is no mathematics, there is no freedom”.

In the own field of mathematics born freedom may inspire to produce very peculiar interpretations. Mathematicians are often misunderstood or they themselves may not seem to understand things that are normal to “normal” people.

Anyway, for the end Edward Frenkel’s beautiful thought from his book ”Love and Math”: ”My dream is that all of us will be able to see, appreciate and marvel at the magic beauty and exquisite harmony of these [mathematical] ideas, formulas and equations, for this will give so much more meaning to our love for this world and for each other.”