## Third Lecture

PYTHAGOREAN ARITHMETICS

The rules for constructing worlds

1. A parable about Zero and One

2. A parable about the Trinity

Essentially, the world of the trinity is empty. There, exists only interaction in space and time. The cause for the continuation of the reasoning is the need to inhabit this world. We must create the one for whom the world was created, the one who inhabits space and time and has the relative freedom to choose his place in space and his point in time. In other words it is about a new world, that from the one hand, exists inside the trinity and, from the other, exists independently of the trinity. It is a world that is able to withstand time and stop its uncontrollable flow. It is a world who aims to win over both time and space and create independence from these. This resembles the creation of non-interaction where interaction has just prevailed.

Let us imagine that zero and one met at the dot and then parted again. This fact does not bring them back to the previous category of non-interaction, since the meeting has already taken place and now its memory exists. The dot as the category of interaction, cannot disappear. In zero, a deformation remains- a mark from the meeting with one. One continues to be confined to its one end. In this way, the dot of interaction was divided, doubled. In the place of the dot of interaction, two dots are formed, separated by a rupture.

Rupture is a new category and is represented by the figure of the two dots. The new category changes the figures zero and one and also the relations between them. Now, in the group that describes the category, belong four figures; zero, one, the dot and the two dots.

With the appearance of the category of rupture the dot, zero and one acquire independence and can be examined separately. Now, for the first time, we can talk about their geometrical form. Zero becomes a circle and one, a straight line (henceforth line). Whereas, it is the circle and the line that can be related. The same stands for the dot that becomes a self existent object and not a mark of zero that exists only because the line points to it.

In order to underline the new properties of the figures, we can ascribe to them new names- circle, line, dot and two dots.

The dots do not differ from each other and their similarity introduces for the first time the notion of repetition. Now interaction, firstly, can be stopped and, secondly, can be, generally speaking, repeated many times.

So, the possibility appeared to examine the identical ones separately and independently from the rest of the notions.

Let us recall that, in the trinity, appeared, for the first time, the category of quantity that is always equal to three. With the appearance of the new figure, a new quantity appears- four. This means that one dot signifies quality and quantity three, and the two dots signify quantity and quality four. The transition from the trinity to the tetrad introduces the notion of counting, as the increase of quantity with the addition of one more unit. Until now, one, together with zero, were referring to quality. In the tetrad, for the first time, one acquires quantitative content. The quantity one expresses the viewpoint of direction as the fundamental quality of counting, while containing the whole of the totality, the integrity of the category of the trinity. Counting with units is like counting with triads, namely dots. In counting, the unit expresses the totality of the dot and the direction of one.

The emphasis on the viewpoint of the unit provides the known form of number four.

Let us examine the surroundings of the triadic dot. From the viewpoint of one, the appearance of a second dot signifies the addition of one more line that points to the new dot. The latter ought to belong to the line, since we give emphasis to the continuation, and this second line ought to appear to the limits of the surroundings of the dot where threesomeness subsides to the point of elimination. The direction of the first line ought to be as less similar to the direction of the first. The picture that arises looks like the known form of number four.

Quantity four is surplus in relation to the category of interaction, in the same way that the notion of counting is surplus to the category of quantity. In the tetrad, the category of quantity is weakened, and in greater extend, so does the qualitative diversity of counting, due to the appearance of similarity to the point of repetition.

Let us remember that space 01 expresses the category of quality and only that. In the trinity, the apparition of quantity violates the purity of separation of qualities. Now in the foursome, the purity of the quantity of the different is being infringed by way of counting the identical.

The sameness of the two dots could mean two different dots from the viewpoint of place, as well as the same dot in two different locations from the viewpoint of time. This signifies the appearance of the possibility of displacement.

On the other side, the two dots are dots of interaction, they are no different and, thus, bear the same power. They introduce the notion of compensating the interaction with interaction and counterbalancing the threesome interaction, ie they stop time. This means that the tetrad created stationarity in time.

Thus, the tetrad introduced both the possibilities of displacement and stationarity. Furthermore, motion is relative, while stationarity is a general property of the foursome world in its totality.

All the laws on preservation of energy, matter, e.t.c, are examples of the foursome organisation of the wholeness.

Nature also exhibits relative change and transformation, while maintaining the equilibrium of the natural environment in its totality.

In which way can we understand the transition from the trinity to the tetrad, from the threesome to the foursome? Let us think about the source of light, the Sun. He is equivalent with the dot of interaction and the light corresponds to the one emitted in all directions. The only thing that the light does is to spread in the space of non-interaction until it reaches an object, for example the Earth. This meeting consists the second dot of interaction and is the source of life on Earth.

Plants originate from the threesome world. The seed- the dot of interaction grows towards different directions- the roots in the soil, the stalk towards the sky. The plant lives as long as it has the possibility to grow, to develop. However, it does not have the possibility of displacement and neither suspects that such a possibility exists. And,then, suddenly, a rabbit passes running in front of the plant. From the viewpoint of the plant, this, of course, is a miracle. Equivalently, for the world of the trinity, the appearance of the second dot constitutes a miraculous unforeseen event.

The relations of the figures in the tetrad

Τhe appearance of the new figure weakened the categorical difference of the figures in the tetrad and altered the properties of zero, one and the dot. Although they continue to represent different aspects of the figure they, now, acquire common properties.

The dot and the two dots show, by their existence, the violation of the continuity of the boarders of non-interaction. Furthermore, they resemble to each other, since they are dots, whose essence is interaction.

In comparison to these, zero and one have a common property- continuity, which hitherto was not possible to detect, since zero and one expressed the absolute dissimilarity, with the exception of their common dot- the dot of interaction. It is impossible to refer to continuity, prior to the emergence of discontinuity. Only with the appearance of the rupture between the two dots, it became evident that the circle and the line symbolise continuity and we can, now, claim that between them there is the agreement of continuity and connection. Now, these appear as graphic objects of continuous lines: zero becomes a circle and one- a line.

In addition, between the circle and the two dots there are also similarities- the presence of inner void, of voidness. From the view point of the two dots, the circle is made of pairs of dots.

The line and the dot also share a similarity: they do not have inner void, they are complete, whole. Furthermore, from the view point of the line, the dot is a line with zero length.

Between the line and the two dots there is a similarity: anisotropy. And the circle and the dot share a common characteristic: isotropy.

So, each one of the four figures share a similarity with the rest under one parameter. Furthermore, each one of the figure differs from the rest under a different parameter.

The dot differs from the two dots, due to its lack of rupture and anisotropy. The line differs from the the circle, due to its anisotropy and openness. And so on.

The similarities and the differences between the four figures appear in pairs. The differences between pairs retain the relation of separation between the figures, while the similarities maintain the properties of the two interactions, as taught by the figure of the “two dots”.

The pair of the circle and the straight line, as continuous lines, have a relation of separation with the pair of the dot and the two dots, where the notion of continuity is totally absent.

Thus, connection and separation in pairs is, now, an intrinsic property of the four figures in the tetrad. The straight line, as a border, separates left from right, whereas a line connects up and down, or reversely depending on its position. The circle continues to separate internal and external, but also connects left and right, up and down. Zero and one exhibit the absolute connectedness along the borders and the absolute separation across the borders.

The dot is the meeting point of the circle and the straight line. On the circle, as well as on the straight line, there is a dot that now belongs, either to the circle, or the line. Therefore, the two dots are the rupture between the circle and the line, and belong both to the circle and the line. Namely, together they constitute the common place for both of them.

The dot and the two dots, as well as the circle and the line, show completely different properties in relation to the viewpoint, to the parameter under consideration. From the aspect of geometry, the viewpoints on every figure can be only two and perpendicular to each other. Thus, perpendicularity expresses the relation between interaction and non-interaction in the tetrad.

Organicity

The emergence of the new quantity introduces a new way to discern qualities. And this means the appearance of a new category. Organicity is the relation between the figures in the category of the tetrad and is analogous to the relation of independence and perpendicularity between the organs in the organism.

The organism exists in the world independently of it and is governed by its own internal rules. It has its own place, its own time and its own will.

There is the triadic aspect on planet Earth. Furthermore, Earth also exists as an organism, thus, constituting an example of the tetradic world. From the viewpoint of the organism, it is not important that the planet moves through time and space following triadic rules. Eventhough it acknowledges the influence of time, the organism is not subject to its rules, since it possesses its own possibilities of internal movement and change.

Focusing on interaction, it corresponds to the second interaction, which counterbalances the first and so, in total, nothing changes. The counterbalancing consists the expression of non- interaction, to which emphasis is given through a different angle, nonetheless equipotential. The organism expresses the balance between independence (separation) and dynamic connection (interaction).

Let us consider a myth by Aesop, where a swan, a fish and a lobster pull a cart towards different directions. Without the cart, they would be moved away from each other, towards the sky, the water and backwards accordingly. The cart connects them in a way of dynamic tetradic stationarity. Thus, it still remains at the same location.

The figurative basis of the organism is composed of organs, whose functions are perpendicular to each other, like the relations of the figures in the tetrad, for the sake of preserving the overall stationarity.

In the human body, even-though the brain may be in a sleeping state, the heart, the liver, the kidneys, etc, continue to function. Furthermore, if one function of the brain is to regulate the movement of the body-parts, the simultaneous effort to understand the underlying processes causes the cancellation of the initial function of movement itself.

If we view Europe as an organism, it ought to have an internal structure. Every country bears discrete properties and plays a particular role, analogous to the organs of the organism. It is, therefore, idiocy to force everyone to conform under identical laws and rules.

Like the figures that describe the tetrad, the organs of an organism play a particular role, have strict mutual placement and discreet functions.

Organicity is depicted by the mythical creature of the Sphinx whose most ancient complete form consists of a human head, a lion torso, eagle wings and the lower part of the bull.

The mutual placement of the figures in the tetrad

As it has already been mentioned, the foursome figures are geometrical objects. On this basis, differentgeometrical shapes can be constructed to depict the possible relations of the four figures. The question appears regarding the relevant position of the figures in the group of geometrical shapes. To be precise, the figures are not aware of their mutual placement, since they are separated. In the trinity, appeared the notion of mutual placement, which exists only in the surroundings of the dot. In the tetrad, the figures acquired geometrical forms, mutually placed in specific positions.

From the viewpoint of the straight line, the figures are mutually placed in pairs, which are connected along it and separated across it. The line can have two positions, perpendicular and horizontal. Analogous to the two dots, there can be only two lines, who ought to be perpendicular to each other. The dot provides the idea of the meeting of the two lines in the dot, and the isotropy of the circle gives an isosceles cross. Thus, the cross is the shape that expresses the relevant positions of the figures in the tetrad, with emphasis to the line. Here, the properties of the line, of the dot and the two dots are strongly expressed. While the properties of the circle appear weak.

If we focus on the circle, a square appears, namely a circle made up of four lines. As geometrical shapes, together the cross and the square express the category of the tetrad in an absolute way.

The basic principles of the organisation of the human life are absolutely described by the cross and the square. The cross does not have a surface area and depicts the relations between the figures- the basic functions that are intrinsic in every organism, and, in particular, in society. It describes a system of dynamic social relations, the structure of the state, since the construction of any state is based on the principles of organicity. The religious and philosophical idea of a common ethic for everyone, is represented by the circle. To this, is opposed political power, the party, that undertakes particular or specific activity, down to the decision making by one and only person, analogous to the single dot. In the horizontal line of the cross, are positioned two dynamic structures, one apparent- the army, and one hidden – the secret security services.

Society itself is organised as a square whose basic characteristics are the closed borders and the existence of internal area. Man conceives himself as a being that exists inside his body, his vesture, his residence and so on, up to his county. Namely, as in a square, inside another square, inside another square, etc. The maximum size of the external square is the Earth. Organicity, as a category, can be achieved only at the scale of the Earth as a whole.

Regarding the cross, the projection of the structure of the state to the scale of the Earth changes completely our understanding of all the four dynamic structures. The army looses its current meaning as an instrument of protection of the external borders of the square. Since, now, the external danger for the total of humanity is the natural environment and not another country. Thus, the external power at the scale of the Earth concerns the activity of transformation of the natural environment. Currently, this means the construction of the world of objects, by exploiting animals, plants and minerals, since the tetradic human being considers nature as an object. The secret internal safety becomes the issue of the civilisation of the relations between people, of the intrinsic knowledge of the specialties of each person and the need of the one for the others. And so on.