First Lecture

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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


With enthusiasm and joy, for the first time in its history, PIFEA Institute publishes the research results of the Pythagorean School.

The essence of the School does not reside in a given set of knowledge, which, being safely kept, survived to the present day, but to reasoning itself as the only way for the existence of the School. Knowledge, unlike reasoning, presents a fixed image of the world, putting a stop to reasoning. Thus, the accumulation of knowledge does not lead to the better understanding of the world, but represents just one study of its various, already sufficiently known, forms.

According to the understanding of modern philosophers, the notion of reasoning originates from Aristotle’s logic, which aims to find cause and effect relationships between phenomena. Contemporary philosophy inherited this linear way of thinking, and remains attached to it. As a result, the schools of thought are characterised by internal dualism, and the relations between schools are characterised by unresolved contradictions. Thus, the total image of the world consists of pairs of contradictory dogmas, like Gnosticism and Agnosticism, science and religion, pragmatism and spiritualism. By studying in the above way, man acquires an image of the world which, rather than being characterized by wholeness, where each part matches harmoniously with the rest, it resembles a mosaic consisting of unfitting pieces.

The method used in the Pythagorean School is the continuous categorical reasoning. Continuity inhibits the appearance of gaps and stops in the reasoning, in order, for example, to apply partial conclusions. The reasoning over categories  inhibits  repetitions and omissions, and provides the distinction between categories, as well as between notions within each category. The absence of omissions guarantees the wholeness of the description of the category.

The Category  prevails over the ‘ocean of notions’ contained within it in the ultimately condensed form.

Attributing a specific definition to the category is not possible, because the definition per se, by connecting a group of notions, constitutes a weak expression of the category. The category is so powerful that it can not be expressed in a finite set of words (concepts), therefore, many parables are chosen as its description, which do not speak directly, but hint.

Pythagorean Arithmetics is a reasoning on numbers, which the Hellene philosopher regarded as manifestations of the divine, independent of man and his knowledge about them. Contemporary man has difficulty in imagining that numbers bear qualities, while his understanding of them is strictly related to measureable figures- for example 10 euro, the 300 Spartans, the 12 months of the year etc.

Consequently, numbers today are used solely to answer the question “how much?”, while few people suspect that they contain their own strength and the whole knowledge on the structure of the cosmos.

The only indication that numbers have a further content resides in the form of the figures, which, though known to every student, no teacher is able to explain why one is pictured with a straight line, nine as a reversed six and the infinite as a sideways eight.

Nevertheless, by thoroughly examining the form of the figures, we can trace the sum of qualities, whereupon distinct worlds, corresponding to the quantity symbolised by each figure, are founded.

In the Pythagorean School, the Figure is a category of Arithmetics. Let’s call this the ‘Big Figure’  category, which within itself contains all the figures and all the categories that describe it, only it, and nothing else.

A Parable about Zero and One

The known figures : 0,1,2,3,4,5,6,7,8 and 9, provide us with the understanding of the notion of ‘figure’. How many notions (figures) are necessary to describe Figure as a category? And with which figures can we do it, so that its content is expressed with the maximum possible compression, while it maintains its absolute wholeness?

It is obvious that, in order to narrate the first parable about the Figure, it is necessary for qualitative and quantitative diversity of notions to exist. While the very notions of “quality” and “quantity” create the beginning of the parable.

Quality, as a category, is the way to separate the Big Figure into a group of figures where each figure offers its own unique point of view on it.

Quantity is ‘the quantity of different qualities’ in which the Big Figure is separated into the different groups

This quantity cannot be smaller or bigger than the quantity of figures in the group of figures. In this sense, quantity, substantially, cannot be decreased, since each figure in the group represents a specific attribute of the Big Figure, which cannot be repeated by any other figure in the same group. A decrease in quantity leads to the loss of the wholeness of the description of the Big Figure. While, an increase of quantity describes the interrelation of figures of the group and does not offer a new viewpoint to the Big Figure. Thus, there is no real increase of quantity. The quantity of qualities is equal to the quantity of figures. The qualities are absolutely defined by their quantity. The way, the power of discriminating the qualities gives the right to the equivalent quantities to exist.

It is obvious, that one “uniform” symbol able to describe all that is, (all the notions), fully cannot exist. Its existence would signify the cancellation of the meaning of the notions themselves. Because one symbol cannot include, for example, existence and non-existence of something at the same time.  In this sense, only conventionally can we name and depict the Big Figure.

The first appearance of the Big Figure occurs through a distinction, that is, by separating or dividing the notions that are contained in the Whole.

For example, the notion “human” is represented as a man and a woman, but a human cannot be found even when searching for one ‘ with a lit lamp in broad day light’.

The force of the division of the category must be maximal, meaning that the viewpoints within the category must be totally different from one another. The Figure category, maintaining its wholeness, is described by the group of figures Zero and One.

Zero separates the notions internal and external. One separates the notions right and left. Together, and not separately, zero and one signify the separation of circular and linear, of closed and open, of isotropic and anisotropic .

Zero and one are described by notions that present maximal separation between them, ignore the existence of each other, and it is impossible for them to meet. In other words, the possibility to interact in any possible way is absent. This is the world governed fully by Non-interaction.

The understanding of the world of non-interaction is simplified by the use of the word “either, or”: “either left, or right”, “either internal, or external”.

The notions “internal” and “external” are not known to one. Likewise the notions “right” and “left” are not known to zero. Thus, the totality of the Big Figure category is expressed by zero and one, and not by either zero, or one. Zero cannot exist without one and vice versa.

Although together, zero and one express the wholeness of the Big Figure, we cannot claim that they signify quantity two. This is because they are completely separated, do not have the potential to meet, and ignore the existence of each other. Consequently, it is impossible to be counted. Thus, the category of quantity is completely absent in the world of non-interaction. Separation is so powerful that it does not allow for the existence of any quantity.

To be precise, simultaneous depiction of zero and one, as well as their very depiction, is opposed to their essence which is absolute separation. We, human beings, live in a complicated world where pure categorical notions do not exist, so we are forced to use this kind of conventions.

One other distinguished feature of the world of non-interaction is the obligatory use of the negative meaning of notions, for example, no-thing, no-body, non-interaction, etc. Without the particle “no”, or the conjunction “either”, it is impossible to define the notions of the world of non-interaction.

However, the negation of notions in the world of non-interaction does not oblige them to coincide with the opposite notions. Rather, it enforces the quality of separation between them. In our world, negation always leads to the opposite. In the world of non-interaction, this, fundamentally, does not happen.

Usually, a “non-thief” signifies an “honest” man. In the world of non-interaction, a “non-thief” might signify anything, for example, a drunken sailor or bacon and eggs.

Thus, the figures zero and one describe the Figure category by means of separation of notions. However, we can not say that zero and one are any number, because they are absolutely separated, never meet and do not know about each other, and therefore can not be counted. This means that in the world of non-interaction there is no category of quantity. The power of separation is so great that it does not allow any quantities to exist.

In Pythagorean Arithmetic, the capacity of a notion increases gradually towards the category, and the most powerful notions have a dynamic impact on the less powerful ones. The degree of this power fundamentally is dependent on the quantity and quality content of the group of notions.

Essentially, the notion of one is incomparably stronger in combination with the qualitative diversity of zero, than with the qualitative diversity of the other nine figures.

Moreover, it is not zero and one that describe the Big Figure, but the Big Figure that dynamically reveals itself in the positions of zero and one. By means of qualitative and quantitative diversity, the concision and brevity are not an idle demand, but a sign of the power relations between them. So those that are less obvious and detailed always present stronger dynamics, and are revealed through the ones that are more obvious and detailed.

Figures zero and one, by describing separation, presuppose the notion of space. The notions “interior-exterior”, “left-right” describe fully the Space where each notion ignores completely all the rest. Moreover, if we want to be precise, we must clarify that ‘the world of non-interaction’ is not really a ‘world’, but pure space that is characterized by the absolute separation of notions. In fact, we can name it ‘a non-world of non-interaction’.

Thus, with the language of the simplest symbols and notions, we described the Space of Non-interaction as the source of infinite power for constructing Worlds of Interaction, where there is victory over separation, and an acquaintance are possible. Because non-interaction means separation, non-meeting and non-acquaintance. So, at this point, we can say farewell to both the non-interacting symbols and to the possibility of not constructing worlds.

  1. An Introduction to the category of interaction

The next step in the reasoning requires the meeting and the interaction of the separated ones, in order to overcome separation. The category of Interaction can be described by the following group of notions: the separated figures and their meeting. And from now on, these notions can be counted.

The Quantity category appears for the first time in the number three.

Whereas,  Number is the category of interaction between the figures.

According to Pythagoras, “the number governs the world”. The appearance of the number marks the beginning of the world, and determines all the relationships, properties and rules that are possibly immanent in it.

The Trinity is the minimum number, and constitutes the most general, condensed and powerful of all the possible worlds. All worlds come from the trinity.

The prayer “Holy God, Holy Mighty, Holy Immortal, have mercy on us” is a prayer for the beginning of everything which exists in time.

Quantity three is described by the group of three figures. Zero and one, which we already know, are separated figures designated for their meeting. The third figure is the dot, where zero and one meet. The dot constitutes the most concise expression of interaction.

The Dot is the basic category of the world of interaction.

In Pythagorean Arithmetics, categories are indestructible, and the relationships between them are dynamic and immortal. As long as categorical notions are introduced in the reasoning, they continue to act throughout its entire duration. Thus, by examining the dot, where zero and one meet, we are able to detect properties of zero and one and the dot.

Zero and one of separation differ from zero and one in the trinity. Now at zero and one there is a dot where separation has violated, and where they have learned about each other and become similar.

Furthermore, the dot ascribes new properties to one by appointing its one side. One now acquires an end and preference towards the direction of interaction. Zero loses its isotropy by acquiring a characteristic at the dot: it becomes a marked circle.

Beyond the dot, zero and one, as before, maintain a relationship of separation. To be precise, there is absence of any kind of relationship between them. The dot is surrounded by space of separation, and meeting is hemmed in complete unknowing. The world of interaction is incomparably smaller than the space of non-interaction.

The world of planet Earth is incomparably smaller than the space that surrounds it.

Up to this point, we have been introduced to the notions and symbols of the phenomenon of interaction. While omitting a few steps which make up the reasoning, to which we are going to be presented during the second lecture, we can briefly draw a few conclusions here.

Specifically, we can say that the triadic relationship of zero and one leads to the deformation of their boundaries. The force of interaction changes the form of zero, as well as one. One’s pressure towards zero results in the deformation of zero. In the vicinity of the dot, the form of interaction resembles a crown, as shown in the following image.

We can focus on zero that appears deformed, on one or the dot, and simultaneously examine the complete picture of interaction which shows that, in reality, the trinity is consubstantial and inseparable. Although the force of distinction between zero, one and the dot gives us the possibility to examine the category of interaction from three different viewpoints, it does not allow for the separate existence of the three figures.

For now, it is enough to say that, from the viewpoint of one, the interaction resembles an arrow pointing towards the dot. From the viewpoint of zero, the interaction resembles the known to us number 3. Yet these are only aspects, in which attributes of zero and one manifest to a greater or lesser extent.


Since the trinity is inseparable, in 3 appears the line belonging to one. The reason why, nowadays, three is written without this line is a topic discussed during the following lectures, where we are going to examine the omissions in the present reasoning, as well as the remaining numbers of the decimal system. We will see that each notion has its own position in the reasoning, and it cannot be introduced to it, neither sooner, nor later.

Second lecture

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