**P Y T H A G O R E A N A R I T H M E T I C S**

**The rules for constructing worlds**

### 1. A parable about Zero and One

### 2. A parable about the Trinity

The world begins with a meeting. Τhe next step of the reasoning requires the introduction of a notion which overcomes separation, and that is the category of interaction.

**Interaction** can be described by the group of three notions: the separated ones and their actual meeting. And now these can be counted. For the first time the category of quantity appears – number three.

** Number** is the category of interaction between the figures.

** Three** is the smallest of numbers and presents the most general, the most concise and powerful of all possible worlds. All worlds stem from the trinity.

Pythagoras said: “*The number rules the world*.” Indeed, the number determines all relationships, properties and rules that can possibly exist in the world.

The world of interaction can be described by the group of three figures: zero and one that we already know and are figures separated and intended to meet. The third figure is the dot, where zero and one meet.

We can say that the dot is the most concise expression of interaction.

To be precise, the **Dot** is the main category of the world of interaction.

In Pythagorean Arithmetics, categories are indestructible, and the relationships between them are dynamic and immortal. Once they are introduced to the reasoning, they continue to act throughout its entire duration. Examining the dot, which is the meeting point of zero and one, we see that it bears qualities of zero, one and, of course, the dot.

It should be noted that non-interacting zero and one are different from the zero and one of the trinity. Now, in zero and in one, there is a dot that overcomes separation and constitutes their first and only similarity. The dot defines the one side of figure one, and ascribes to it new properties: one now acquires an edge and direction towards interaction. Zero loses its isotropy, acquiring a mark at the dot.

Outside the dot, zero and one, as before, maintain a relationship of separation. To be precise, there is absence of any type of relationship between them. The dot is surrounded by space of separation, and the meeting by obscurity. From this point of view, the world of interaction is incomparably smaller than the space of non-interaction.

*The world of Planet Earth, for example, is incomparably smaller than the space surrounding it.*

In this step of the reasoning, for the first time, appear the concepts “large” and “small”, with the maximum possible difference between them. Small is the dot, and large is everything that surrounds it. However, if we stop the reasoning at the dot, the relation between large and small is a relation of separation: either large, or small. Thus, there is the need to expand interaction in the space of non-interaction and examine the surroundings of the dot, where the notions of large and small meet.

The concept of the surroundings plays the role of the third to the concepts of small and large. In the surroundings, there is an increase in the strength of interaction towards the dot, and a decrease away from it.

In the surroundings of interaction, zero looks like a straight line. The part of zero and the part of one together look like the letter “T”.

The dot is unified and cannot be divided. “T” carries inner trinity, thus providing the possibility to distinguish between the parts of zero and one, and the dot. Nevertheless, zero, one and the dot are not separate fragments of “T”, but provide three discreet viewpoints to the category of interaction: the viewpoint of zero, of one and of the dot.

The dot, like the Holy Trinity, is consubstantial and indivisible and expresses the integrity of the world. This allows us to formulate the first rule of constructing worlds: **worlds**** ****are created**** ****only as a whole.**

**The three**** ****qualities of time**

Since “T” gives the possibility to distinguish three different viewpoints to the category of interaction, we can examine the surroundings of the dot accordingly.

From the point of view of one, “T” resembles like three one, three directions that originate from the dot and along them the interaction decreases. This means that from the point of view of one, interaction bares the quality of time, whose basic characteristics are the existence of direction and attenuation while moving further away from the event.

** Time **is essentially threefold, which means that it is possible to distinguish three qualities. Time can not be one, but only three. To the question “WHEN?”there can be only three answers.

The three directions of time are different- two originate from zero and one from one. Those originating from zero can be called right and left, since the part of one distinguishes between right and left. These directions are counterbalancing, a fact that expresses the isotropy of zero. While one, by being anisotropic, it gives only one direction.

Time, as a category, has three qualities – “momentarity” originating from the dot, circularity from zero, and elongation from one.

** Elongation** projects the event towards the direction away from the event.

** Circularity** distinguishes between past and future, which are both as conventional notions, as are right and left. The past and future shrink as we distance from the event: as the distance between them increases, we cannot discriminate one from the other.

** “Momentarity”** is the most categorical notion, since it is indicated by the three directions of “T” and presents maximum condensation. Simultaneity can be understood only by three directions.

The idea of the direction of time derives from the anisotropy of one, and that of examining the surroundings of the dot by the isotropy of zero. Anisotropy is expressed as direction and isotropy as three equipotent directions. The surroundings of the dot can be constructed only outside and around it. The dot is the internal source of time and, for that reason, time is directed outside the dot and is valid only inside the surroundings. Time is an internal quality of the world of interaction and appears exactly at the place of interaction. An answer to the question “WHEN?” can only be given if we answer the question “WHERE?”.

**Trinity of zero**

The surroundings of the dot are located within the space of non interaction and the dot within its surroundings. The space of interaction expresses the relationship between interior and exterior, thus described by notions that describe zero.

In the surroundings of the dot of interaction, the quality of the dot expands along the border. The border of zero becomes the third to the internal and the external and plays the role of interaction proper between the internal and the external. Now, zero, like one, acquires a triadic structure: internal, external, and the boundary between these.

In the world of interaction, **place** is introduced by zero and **time** is introduced by one. Zero as well as one has a triadic structure, for it is only those that bear inner threesomeness, that have the possibility to interact. The concepts of time and place are connected *via* their inner threesomeness, and are introduced as concepts through this similarity.

The inner trinity of zero can be described by the following well-known story, The Tale of Tsar Saltan

*Three fair maidens, late one night*

*Sat and spun by candlelight.*

*«Were our tzar to marry me»,*

*Said the eldest of the three,*

*«I would cook and I would bake –*

*Oh, what royal feasts I’d make.»*

*Said the second of the three:*

*«Were our tzar to marry me,*

*I would weave a cloth of gold*

*Fair and wondrous tho behold.»*

*But the youngest of the three*

*I would give our tzar an heir*

*Handsome, brave, beyond compare.»*

The sister, who is a **weaver**, is engaged with the weaving of the external space, starting from the outer boundary.

The **cook** is engaged with the internal space, from the walls of the stomach towards its interior.

The third and youngest sister, who is going to become a **mother**, occupies just a narrow strip between the internal and the external.

The development of the notion of zero inside the trinity, leads to the idea of the expansion of the border, and it is right there that the third sister exists.

Of course, the recognition that the interior space does not belong to man is rather unusual. This is due to the fact that his senses receive stimuli that derive and belong to the exterior. However, the internal space is present in all objects of the threesome world, with no exceptions.

*The nucleus of the atom does not belong to the atom and the core of the Earth does not belong to the Earth. Yet, the internal space exists and has equal rights to the external space. The world of Earth is a strip of several kilometers, surrounded by internal and external space.*

**The development of interaction**

An increase in the force of interaction leads, on the one hand to an increase in the mutual influence between zero and one, so that properties of one increase in zero and *vice versa*. And, on the other hand, to the expansion of the surroundings of interaction *per se*.

The expansion of the surroundings forces us to take into consideration that zero remains circular, and to expand the influence of interaction to the interaction of the notions of circular and linear. In the expanded surroundings of the dot, zero becomes more anisotropic and acquires direction like one. This looks like a deformation of zero’s boundary in the same way that a stick presses an elastic circle, or *vice versa*.

One acquires rounded wings and resembles an arrow.

All the above depict the category of the trinity, the dot of interaction, which is presented differently according to the force, or scale, of interaction, resembling, accordingly, to a dot, a “T” or an arrow (crown) . Therefore, there are three scales of interaction that correspond to three degrees of manifestation of the trinity.

** Force,** as a category, characterises the tension of the relationship between the three in the group that represent interaction. Change is possible only in the transition from one scale to the other. While, the dynamic relationships between the figures of the group maintain the wholeness of the category at any given scale; they ensure the integrity of the world.

The trinity is the place and the time where the world is constructed. From the viewpoint of space, this world looks like an interior territory, where interaction is possible and is surrounded internally and externally by non-interaction. From the viewpoint of time, the threesome world is about the movement itself that treads on non-interaction, that is to say about the expansion of the world *per se*, where the dynamic relationships between the participants are stable. The participants in the trinity are equivalent, tensed and can be altered only together. **This means that worlds are not only created but, also, alter as a whole**. And this is the second rule of constructing worlds.

**An introduction to the foursome world**

Looking back to the reasoning, we can see that there was only one way to overcome separation: the exercise of a superpower that would make the symbols of separation meet. Thus, separation was surmounted locally.

In the world of interaction, there is only one possible way of existence as a trinity. In the trinity there is complete absence of freedom and autonomy of movement. From the viewpoint of man there can be no movement, since there is no one to move. In essence, this world is empty and resembles a three-directions cross road with no traveler, or knight, to select which way to go. The triadic world looks like the third sister, getting ready to become a mother, yet never becoming pregnant, due to absence of the fourth, of the King. She gives birth to herself by expanding her influence to non-interaction.

Our intention to continue the reasoning from the trinity, places us in the position of the “knight” in front of three directions. Accordingly, we are faced with three possibilities:

- The first concerns the repetition of interaction, resulting in an increase in the quantity of participants, in an increase of figures. In this case we can examine a new way to discern between qualities.
- The second possibility concerns the examination of interaction to the direction of decreasing its strength. In this case, interaction may not result in the creation of new figures-the dot. The space, between the separated zero and one, attains the quality of interaction and becomes a dynamic field of their interrelation. As a result, a new number arises, as well as a new type of interaction between figures; they now know about their mutual placement and affect one another through the field of interaction. The new number has the form of the ten (10) and expresses quality and quantity three.
- The third possibility for the continuation of reasoning has to do with the increase in the interaction to such an extent, so as to cause an irrecoverable alteration of the figures. In the trinity, the boundaries are deformed, but their integrity is maintained for the sake of the integrity of the category. The transgression of this condition results in the rupture of the borders of zero and one. This means that a new category appears, which describes the relation of the part with the whole, and gives the possibility to re-examine all the previous categories.

We chose the first possibility, as the most convenient one.

The repetition of interaction means the appearance of a second dot, which differs from the initial dot of interaction. Between the two dots there is the rupture that guarantees their difference, their lack of connection and their relevant freedom. The fourth figure introduced to the group of three is symbolised by the two dots which define the boundaries of the rupture.

**The rupture is the new category of the reasoning and appears in the foursome world.**

The appearance of the rupture demands the re-examination of the relationship between the previous categories-zero, one and the dot. Now, the rupture appears also between the figures, a fact that allows the examination of each figure separately, while at the same time maintaining the relationships of interaction.

Geometrically, figures zero and one are represented by the circle and the line. Though they left the triadic interaction, they continue to retain its memory.

In zero, remains a mark-the dot, that holds the memory of interaction in the form i.e. of a permanent deformation. One becomes a line with a specific size, since it acquires a second end.

In the trinity, zero and one represented interaction conventionally. In the foursome world, we can, for the first time, talk about the notion of geometrical form. Zero becomes a circle and one becomes a line and, now, it is the circle and the line that can be related. The same applies to the dot, which became a self-existent object, and not a point in zero, existing only due to the fact that one points to it. Now, there are four independent geometrical objects- a circle, a line, a dot and two dots, which maintain their figure relationships, describing the new category-the foursome.

The emergence of the concept of form means that interaction existed and it has stopped, while its memory is not influenced by time. Instead of the triadic time, now appears the idea of tetrad stability. Also on this occasion, we can see how a new category alters the previous one. Interaction in the trinity overcomes separation of zero and one in space, and in the next step of the reasoning, the foursome stability overcomes triadic time.

In the next lecture we will examine in detail the tetrad world and the new category that appeared through the increase in the quantity of the figures and offers the possibility to examine the issue of constructing worlds as inlaid to one another.