Ideas in the Night

Category Archives: Mathematics

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.

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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.ID-100211519

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?

zero

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:

right triangle

 

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:

squares and 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.pytree

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):


Based on the book “Sacred Geometry: Philosophy & Practice”

I have divided this article in short numbered thoughts in Ludwig Wittgenstein’s manner. As an epilogue there’s a bit my own philosophy about forms.

(1) Greek philosopher Plato considered geometry and number as the most reduced and essential and therefore the ideal philosophical language.

But it is only by virtue of functioning at a certain ‘level’ of reality that geometry and number can become a vehicle for philosophic contemplation.

For Plato reality consisted pure essences or archetypal ideas of which the ideas we perceive are only pale reflections. The Greek word “idea” is also translated as form. These ideas cannot be perceived with senses, but with pure reason alone. This is where geometry steps into picture.

Morgens im Wald bei Nebel

Morgens im Wald bei Nebel (c) Fotolia

archs in abandoned farm

© Rafael Laguillo | Dreamstime Stock Photos

(2) Geometry as contemplative practice is personified by elegant, refined woman, for geometry functions as an intuitive, synthesizing, creative yet exact activity of mind associated with feminine principle. But when these geometric laws come to be applied in the technology of daily life they are represented by the rational, masculine principle: contemplative geometry is transformed into practical geometry.

(3) Angle specified the of celestial earthly events.Today the science verifies that the angular position of moon and planets does affect the electromagnetic and cosmic radiation which impact with the Earth and in turn these these field fluctuations affect many biological processes.

Also the word “angle” has same root as “angel”.

In ancient trigonometry an angle is relationship with two whole numbers. This gets us into musical scale system.

(4) Geometry deals with pure form and philosophical geometry re-enacts unfolding of each form out of preceding one. It is a way which the creative essential mystery is rendered visible.

For the end from the book Thomas Taylor’s thought: “All mathematical forms have a primary subsistence in the soul; so that prior to the sensible she contains self-motive numbers; vital figures prior to such as are apparent; harmonic ratios prior to things harmonized; and invisible circles prior to the bodies that are moved in a circle.”

Epilogue

I have come up with some kind of form philosophy myself too. Why do people produce certain kind of forms? Especially the circle form can be found directly in many instances. Typically a glass for drinking has a circle especially at the top of it.

Perhaps it’s easier to drink from this kind of glass. Or are we forced to create these kind of forms because of our “sacred geometry origin”? 🙂

My actual own form philosophy is metaphysical by its nature, where also mental things have geometrical form, that are in interaction with concrete forms creating values together, that have their own forms.

For example it’s not insignificant that from what kind of of “glass” one drinks beer. It is disrespect of beer if one doesn’t drink beer from a pint that retell the mental form of the beer. It’s also of course question of what kind of beer one drinks. Some expensive beers come in a bottle that respect the form of the beer as such, at least almost.

I also have analogical thoughts about wine: It is wrong to drink red wine from a white wine glass. Red wine has its own mental form that the wine glass must retell.

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Image courtesy of Madrolli at FreeDigitalPhotos.net

Well, I have probably written too much about my own form philosophy already, but with cosmetics it gets particularly interesting.

The essence is to see the mental being of things and respect that with right kind of concept by paying attention to right kind of combination of make up, jewelry and clothes and so on. The essence is the respect of the whole, all the mental dimensions of it. Respect in very wide concept. Including self respect.

For end observation let me state that for example in the buildings for different kind of forms were payed a lot more attention in the past that nowadays. At least that is how I see it in Finland.

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Image courtesy of Tuomas_Lehtinen at FreeDigitalPhotos.net

For example the residential districts outside the centers of the towns that are built in the 1970s are especially boring in Finland…

Some relevant links:

Sacred Geometry Kindle e-books on Amazon

Sacred Geometry Jewelry on Amazon


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 2πr. 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 πr.

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:

areaThe area of a circle is πr2 , 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  ½πr2as half of the area rolls over, we get a sphere, which is what we’re after.

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

volume

As we remember, the volume of a sphere is 4/3 πr3See 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. 🙂

On integral calculus


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.”

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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.”