**PYTHAGOREAN ARITHMETICS**

*The rules for constructing worlds*

**1. A parable about Zero and One**

**2. A parable about the Trinity**

**4. A parable about the Pentad and the Dyad**, or

**“The emergence of the notion of Man as Creator”**

The category of interaction was introduced for the first time in quantity “three”. The demand for categorical condensation shows that the trinity only signifies the event of interaction, which is expressed in a minimal way. The development of interaction in the context of the trinity allows the deformation of the borders of zero and one, which extends to the surroundings of the dot, but not for the violation of the continuity of the borders of the interacting symbols. This means that interaction in the trinity, categorically bears the characteristic of wholeness. The fundamental idea of the trinity is integrity, the unity of the three in the interaction.

In the tetrad, while constructing the category of organicity, we acquired a new quality-autonomy, due to repetition and not division. The new figure of the “two dots” signifies rapture, which allows the distinction between these. The repetition of interaction bears the new category, without destructing the integrity of the original figures, zero and one, while preserving the continuity of the borders and the minimal expression of the event. Repetition bears the new category of the relation of interactions.

In the tetrad, the evolution of the notion of quantity led to the notion of counting and sequence. If the reasoning continues towards the direction of further increase of quantity, new ways to divide qualities in the category appear. Furthermore, we exhausted the possibilities of discerning the category into figures without inter-crossing. In this sense, we completed the description of the space of separation as the possibility of separate, independent existence of categories. The next step of the reasoning consists of the violation of the very principle that ensures the integrity of the figures themselves, as borders of separation of notions.

Let us notate that every step of the reasoning decreases the separation of notions. This is expressed by the limitation of one at the one end in the trinity, and in both ends in the tetrad. One in itself bears the quality of counting directed to the increase of quantity, the transition from the one integer quantity to the next. The possibility of limitation of one is completed in the tetrad. The continuation of counting brings the need to divide one.

Let us examine the interaction that is so strong as to lead to the rapture of the borders of the interacting zero and one. Let us name **strong **this new interaction and **weak** the triadic one, that leads only to deformation. This is depicted accordingly by the following picture. The triadic interaction resembles to the pressure of one to zero, which leads to the deformation of zero or the bending of one. The new interaction looks like the partitioning of zero by one and results to the rapture of the borders of *both* zero and one.

The places of interaction become places of rapture and, even though in the trinity the interaction means the appearance of a common dot, in the strong interaction it builds two ruptures, two voids in the dots, where one and zero intercross. The two voids destroy the integrity of the initial symbols of space. In the dots of rupture of the borders of zero and one, the distinction between circular and linear disappeared. And here, these notions cannot be discerned anymore.

The two voids on the one hand cut zero and one into parts, and on the other hand connect the notions which, up to now, were separated by zero and one. We can say that strong interaction is the absolute victory over the initial separation, distinct from the trinity, which was the first event of local destruction of separation.

The two voids bear internally the quality of interconnection of everything that was separated before. For the first time in the world of interaction, appears the category which expresses the Unexpressed Whole, before separation, before the appearance of arithmetic worlds. The two constructed voids have the same power with the void before the world of zero and one, with the Unexpressed Whole, with the unexpressed Big Figure. Let us examine what happens in the borders of the interacting symbols during the strong interaction.

The zero is cut in the middle and the line in three parts.

The pieces of zero make a pair and are connected due to the knowledge that they belonged to zero. One, by diving the halves of zero in left and right, does not separate them any longer. Thus we can count them and gain a new quantity, two.

**Two** concerns the two halves. The dyad signifies the pair of connected notions which constituted one whole, were separated and, while being separated, are designated for their meeting. The strong interaction connects all the notions in pairs and distinguishes between the pairs.

The **Half** is a new category which means two notions connected in a pair.

The two halves of zero preserve the memory of the whole, but do not carry all its qualities. Like before, they exhibit curvature and, along the borders, they separate the internal from the external. However, they are not closed, they are anisotropic and are distinguished in left and right. Now the halves of zero bear the qualities of one. For the first time, the inter-crossing of notions which were separated in space is possible.

*For example, man and woman are two halves which are distinguished between them under one parameter and coincide in all the rest. They inter-cross while being different. *

Together, the two halves of zero and the three parts of one express quantity **five**. The one fifth means one out of five parts that arose by the strong interaction. The one fifth does not express a distinct new quality and is not a figure, since in each one of the parts the qualities of figures zero and one inter-cross.

The appearance of the notion of inter-crossing brings forth the problem of counting the figures that describe the category of strong interaction. Based at the previous reasoning, in order to define quantity, it was necessary to introduce a way of discerning the Big Figure in a group of figures , which were so inherently different, as to not allow their inter-crossing. Strong interaction, in essence means the inter-crossing of notions, which, now, does not allow for separate examination of the figures.

Here, the notion of quantity is expressed with two different qualities- **two** and **five**. From the point of view of the constructed voids, they manifest themselves in quantity two. From the point of view of the sum of pieces that result from the interaction, they manifest themselves in quantity five.

This reasoning does not allow for the use of old ways of transiting to the new category. We cannot build the pentad in the same way as the tetrad, by simply adding a new figure. For the first time, the categorical difference is violated- the pentad and the dyad ** together** express one quality, one category-strong interaction. For the first time the

**Pair**appears as a category-the pair 5-2.

The present form of figures five and two results from the image of strong interaction, where the two halves of zero drew away from each other and were attached to some parts of one.

5-2 is a number which demonstrates the new category of interaction of figures 5 and 2. Figure two in this number bares the quality of connecting in pairs, the quality of the inter-crossing itself. Five discerns the whole into parts and numbers all the possible pairs of inter-crossings.

Two appears in the void, in the absence, in the rapture. 5-2 introduced a new quality-presence and absence. The dyad does not express quality, since we cannot count absence, and the pentad does not count qualities, since it cannot discern between them so as to examine them separately. With the appearance of 5-2, the dyad engages the area of the invisible. From the point of view of the figure’s manifestation, the dyad does not exist. The pentad exists as the two halves of zero and the three parts of one.

Thus, appear the pairs: visible-invisible, whole-part.

The appearance of the pair, that is of connectedness, of inter-crossing, is possible under a certain parameter. Consequentially, the rest of the parameters are not taken into consideration. For example in the pair whole-part, there is no way to express the relationship between visible and invisible.

On the other hand, the pair can be only a pair of figures. Five can be only a quantity of numbers – of relationships between figures. Thus, when talking about the dyad, we mean two figures, while the pentad concerns only five numbers.

In the sequence of numbers, the dyad is before the triad, the later being the first number. Namely the dyad is not a number. The pentad is so far away from 01, that the category of figures can be expressed only quantitatively. The dyad exists only in combination with five. In relation with the category of numbers, five is pleonastic and the dyad is deficit.

We can say that, as we approach the pentad, the notion of ‘loosening’ the category of the figure appears, up to the point of excessiveness. Correspondingly, the reasoning may proceed towards compression, towards decrease of the quality in the direction of the dyad. Thus, the reasoning for strong interaction occurs exactly towards two directions.

**The difference between the two dots in the tetrad and the notion of pair**

In the tetrad, the two dots do not consist a pair because they are the same, they are two identically same dots of interactions. The “two dots” are one of the four figures in the group of figures that describe the tetrad. Between this and any other tetradic figure, differences and similarities can be traced- all the figures have equal rights.

In the world 5-2, the pair includes the five and the two, which differ quantitatively and express different aspects of the quality of strong interaction. Thus, the pair consists of the two raptures and the five parts of zero and one. The differences between five and two are absolute: the two voids are invisible while the parts are visible, the former do not exist, the later exist. Nevertheless, the two voids, together with the parts of the borders, contain within them the quality of pair, since the raptures are two and the parts derive from zero and one. Thus, the pair constructs the halves that bare inner “pairness”.

Generally speaking, the fundamental quality of the two dots in the tetrad is rapture, which is one and resists them. The difference between the two dots is their independence, which is the essence of the category of the tetrad. They cannot maintain a pair-like relationship and, generally speaking, we cannot count them, since they are separated. It is not important if they are different dots, or the same dot in different moments in time. The important thing is that they are distinct.

**Five Pairs**

Since the pair creates a connection according to one criterion, in world 5-2, it is possible for five types of pairs to exist. The sensory engagement of the power of interaction between the two halves can only be fivefold. Place and time are tensed and can only be perceived between the two halves. World 5-2 organises the senses, so as that each half can sense the other half in five different ways, with five sensory organs, and there is no “someplace” in which the two halves do not interact.

*One example could be the five senses, each one of which consisting the inter-crossing of perception and the perceived object. In particular, the eye and the object make a pair. During their inter-crossing, a visible image is created, which does not contain any idea on the taste, or the smell, of the object.*

*The two hands, each one of which bears five fingers, are organized on the basis of 5-2.*

*Another example is the word ,that connects the two speakers,the two organs- the mouth and the ears. This word exists only between them and consists the point of inter-crossing of the two different ones. The human being, as an organism, is self-sufficient and separated from other organisms. Nevertheless, between two people that conversate appears the fivefold phenomenon of speech, which emerges outside the organism and is not internal to it.*

*“In the beginning was the word (logos)” said St John about the appearance of the world 5-2. Jesus said: “ I brought fire into the world and I will safeguard it until it grows”. Fire is the word and the growing of the fire means that the word will obtain full categorical force.*

The development of the world 5-2 happens in enumeration of all types of pairs that can construct the whole. This is not an easy task, since pairs must present a maximal difference between them. For example, do the five sensory organs differ sufficiently between them? Furthermore, as we know, they do not record the electrical charge of objects, which they try to feel, to recognise.

The world of the pentad consists of pairs. In order to synthesize wholeness, it is necessary to concentrate five types of pairs.

In the world of non interaction, a pair can be created by anything. In the world of the pentad, it can exist only in one way, which has a fivefold verification.

*The Parable of the Samaritan in the well. *

*“You had five men but none of them was your husband”, said Jesus meaning that each one of the five men of the Samaritan woman was her husband under one parameter: one was handsome, the second smart, the third tender, the fourth rich, the fifth sparing. None of them could satisfy all five parameters.*

*The husband must comply with five different criteria. If even one of these is not satisfied, he cannot be part of the pair. Metaphorically speaking, spouses must match up to their fingerprints.*

*The example about two and the wall of the interrogation office separating them:*

*If the two are a pair, they give answers deriving from the whole. If they are a pair under some parameter only, not all their answers coincide. The result is that the one spouse ends up in jail and the other becomes a grass widow.*

The world of strong interaction contains the separation of the beginning of times, up to the fragmentation into parts and the connection in pairs at the end of times. The absence of coincidence of criteria allows for polygamy. The coincidence leads to the connection in the wholeness and, thus, the real pair cannot be destroyed.

Quinary can only be the worlds, where the pair is a category. There, it is not possible for “non-pairs” to exist. There are no widows and no single men and no reason for adultery.

**Cognition**

In the pentad a new category appears-** Cognition**, which signifies the total victory over ignorance, over non- interaction. It is the knowledge of one for the other, which expresses the highest possible force of interaction.

The quality of strong interaction is expressed in the fact that zero and one acquired the maximum knowledge for one another. Now, they do not exist as figures of interaction. As a result, appear the fragments of the interacting figures, each of which knows the existence of both zero and one. Namely, in every part, the properties of zero and one inter-cross under one parameter. So, there are similarities between the part, and between each part and the whole.

The parts can be similar to the whole and expresses this ability from its side. If we count all the possible similarities between the parts under all parameters, it is possible that the particular expresses the whole. And this is the new category- Cognition.

Or, to put it differently, cognition is called the ability to divide into parts, so that each part and the relations between them express the whole.

*This relation can be expressed quantitatively as the analogy of the whole to the part in the golden mean: the smallest part of the straight line is related to the largest part in the same way that the largest part is related to the whole. If we solve the equation a/b = b / (a + b), we get the number 1,618… or (1 + √5)/2. This number is symbolised by the letter **Φ**. Namely , it coincides with the symbol of strong interaction, with one cutting zero.*

*Another beautiful example for the relation of the part and the whole is the five-pointed star, where all the parts that stem from the inter-crossing of the radiuses are in analogy to number Φ. And the drawing of the star can be made only using diabetes and a ruler, or a circle and a line.*

Cognition is man’s basic characteristic that discerns him from animals, plants, etc. Categorically, man differs from them, in the same way that the pentad differs from the tetrad and so on.

The tetrad won over time and produced a stable world, where everything is given and nothing can change, due to the tensed relationships between the four figures.

The pentad, as the next category, deniesdynamics, as perceived by the triad, *and* stability, as perceived by the tetrad. Here, dynamics is perceived as a gradual transition from one pair to the other, from one stable state of the world to another, amongst which there is a great difference.

*For example, evolution in our world presents a scalar organisation and it is possible to trace four scales-kingdoms: the mineral, the plant, the animal and the human one. Consequentially, man ought to present a categorical difference, that is to be fully discernible from animals, in the same way animals are from plants, and plants from minerals. *

In the pentad, appears the idea of building new worlds and of gradual change of the given worlds. By being constructed, the pentad has equal possibilities of creation with the primordial wholeness, from which everything originates. The pentad acquires the potency of the Big Figure, becomes a source of construction of new categories and constructs its worlds.

The idea of construction resembles the pair “evolution-involution”. Involution is described by internal, paired, intrinsic. Evolution by external, single, constructed. Evolution does not exist without involution, given that they maintain a pair relationship.

Involution is moving backwards, it concentrates. It is oriented towards the previous categories and pursues to connect them in pairs.

Today we, as Europeans, have a thorough knowledge of the notion of evolution. Man constructs worlds in great variety and quantity. Namely, he exhibits preference towards one – the external, the separated, the singular. Obviously, this is the result of a partial, segmental understanding of cognition, which leads to the increase of quantity, to homogeneity, to the world of objects and the possibility of counting the visible, the apparent.

Man has the possibility, not only to adjust to the external circumstances, but to also alter them, by surrounding his existence with buildings, cars etc, and by changing the natural environment, including the creation of new kinds of animal and plant species and also chemical compounds that did not exist before in nature. In the 20^{th} century, man has started to alter nature on a global scale and now has the possibility even to destroy it. The preference towards evolution ignores the laws of connection in pairs and leads to the construction of worlds that are dynamic, nevertheless temporary and finite.

The adherence to evolution gives birth to viewpoints such as that knowledge is infinite and the natural resources are endless. The phrase “Knowing the world is possible, but it is impossible to know everything” concentrates the paradox of modern science, which came about as the result of partial, segmented cognition.

Pythagoras says that the knowledge about the categories is complete and of course man ought to know everything. The literate man, Man in essence, constitutes the last category of Arithmetics. In this sense, by describing the category 5.2 we completed the sequence of lectures on Arithmetics. The study of the following quantities -6,7,8,9 etc – does not reveal new categories but merely describes the relationship between the categories of space, time, organicity and cognition. In order to move further we ought to give to the reasoning the characteristic of activity, namely to construct worlds founded on cognition, from the position of the man who, at last, is literate.

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