Tuesday, June 6, 2023

Pythagoras and the Theme of Number, by Mário Ferreira dos Santos

“God made the integers, all the rest is the work of men” (Kroneker).

“I am everything that has been, and everything that will be, and no mortal has ever lifted my veil” (Isis, according to the inscription of Sais, reported by Plutarch).

Preface

The publication of this work marks the culmination of an impulse that took shape in me since my youth. When I listened to the first philosophy lessons in school from distinguished and profound teachers, I was always greatly dissatisfied with the way Pythagorean philosophy was presented. And sometimes, addressing my philosophy instructor, I would ask if there was not a gross falsification in the presentation of a philosophy whose role in the entire process of European thought was undeniable. He himself confessed that this was how he had learned and transmitted knowledge, and that he had not found any works capable of presenting a thought other than what was taught in his classes.

These facts have always had a significant influence on me, and I made the decision to seek all means throughout my life to properly study Pythagoreanism and, above all, the figure of Pythagoras, who exerted an extraordinary fascination over me.

Years went by, and I realized how difficult it was to gather material for an in-depth study of this philosophical position. In my travels to various countries, I acquired everything I could find on the subject in order to later organize a framework of trends. It would be lengthy to describe the work and effort I put into it, work that I never spared, for if I have any merit, I consider this the highest: to be a tireless worker for whom every hour of the day is a duty.

Reading various authors put me in a determined position: Pythagoreanism was the most falsified of philosophical currents. This truth became evident to me after reading so many books from all corners of thought, which revealed the presence of old prejudices stemming from Aristotle, who was undoubtedly one of the major culprits in this falsification, as we demonstrate in this book, as well as in our “Aristotle and Metaphysics,” where we discuss the famous work of the Stagirite and demonstrate the gross falsifications based on lesser authors and so-called Pythagoreans, or due to errors of common understanding and even sophistry, which favored a view very remote from the truth of the master of Samos' thought.

I recognized, through all of this, how difficult it was to accomplish a synthesis of the real thought of that great philosopher. Such work would require time and great care, as the elements at my disposal were insufficient. It was necessary to sift through everything and extract what was genuinely Pythagorean from what was not, whether from suspicious authors or scribes. In short, a work of a doxographer was necessary. And I could not fail to recognize the great difficulties I would have to face. What initially appeared to me as indispensable was to gather, from Pythagorean material, what showed the greatest coherence with the statements of the master of Samos, statements about which there was no doubt, at least regarding ideological authenticity.

By following and always employing the methods of Concrete Philosophy, which I created to construct a coherent and solid vision of thought, whose justification I will provide in due course, I was able to perform the work of a doxographer. However, I did not rely solely on subjective criteria of appreciation, which are inherent to all doxa, but rather on objective criteria capable of giving the desired solidity to the premises that my theses can propose. My book is more of an exegesis, but it is based on the method I created for examining philosophical thought, a method that has proven valuable and creative for me, allowing me to confidently navigate the most difficult paths and find solutions and clarity where others have encountered obscurity and confusion. As for the value of this method for those who read my works, their judgment will not be difficult after delving into the study of my works. As for those who do not read me, I cannot say anything to them, nor can they say anything about what I do and undertake.

This book is already an application of my dialectical-concrete method, and the conclusions obtained are based on real and historical foundations sufficient to ensure the correctness of my statements, which I always seek to demonstrate.

Mário Ferreira dos Santos

Quick report on Pythagoras

We do not intend to dwell on the account of the Pythagorean legend, which we do in our annotated edition of “Verses Aurei” by Pythagoras.

Not that we consider only legend what has been written about this marvelous life, because in these descriptions there is undoubtedly much historical and true. However, the difficulty lies in being able to separate what is historical from what is the product of imagination and fictional cooperation of those who have dedicated themselves to describing the life of the famous philosopher from Samos.

The fact that the historicity of Pythagoras is peremptorily denied (as some do) due to the lack of sufficient documentation does not prevent Pythagoreanism from being an exciting reality in the history of philosophy, whose influence spans centuries to our present day.

What happened to Pythagoras is similar to what happened to Shakespeare, whose existence was often denied. If Pythagoras of Samos did not exist, there was certainly someone who constructed that doctrine and coincidentally bore the name Pythagoras. We can paraphrase what was said about Shakespeare. But setting aside these naive scruples of certain authors who prefer to declare him non-existent, as if there were greater value in denying his historicity than affirming it, let us proceed to briefly relate something about this legend that will greatly assist us in the study we are undertaking in this book.

In 1917, near Porta Maggiori, under the railway tracks that connect Rome to Naples, a crypt was discovered, initially believed to be the entrance to an underground Christian chapel. Later, it was determined that it was a construction dating back to the time of Claudius, around 41-54 AD, and that it was nothing more than a temple where the members of a mysterious sect gathered, which was eventually revealed to be Pythagorean. It is now known, based on historical evidence, that even before Caesar’s time, Pythagorean temples were flourishing. If this sect was so combated, it was more because of its secrecy than its ideas. In a work now cherished by Pythagoreans, Carcopino (“La Brasilique pythagoricienne de la Porte Maqeure”) provides a detailed account of this temple. Undoubtedly, this significant discovery prompted new studies on the doctrine of Pythagoras, which aim to demonstrate the great role that this order played in history for twenty-five centuries. The order still exists and has followers, although it is, in our day as it was in the past, irretrievably tainted with foreign ideas that, in our view, distort the genuine thought of Pythagoras of Samos.

It is widely accepted, almost without disagreement among all who have studied his life, that Pythagoras was born in Samos between 592 and 570 BC, in the same century that saw the emergence of many great leaders and creators of religions such as Gautama Buddha, Zoroaster (Zarathustra), Confucius, and Laozi.

There are numerous disagreements about Pythagoras' true nationality, as some claim he was of Egyptian origin, others say he was Syrian or from Tyre.

According to legend, Pythagoras, whose name means “the Announced” or “Pythian,” was the son of Menesarco and Partemis, or Pythais. Once, his mother took him to the Pythia of Delphi, and the priestess prophesied a great role for him, which led his mother to devote herself to his education with utmost care. It is said that Pythagoras, who revealed prodigious abilities from childhood, had Hermodamas of Samos as his first teacher until the age of 18, then Ferécides of Syros, and later became a student of Thales in Miletus and attended the lectures of Anaximander. He then became a disciple of Sonchis, an Egyptian priest, and also met Zarathustra or Zoroaster in Babylon during his stay in that great ancient metropolis.

The legend also tells us that the hierophant Adornai advised him to go to Egypt, recommended to Pharaoh Amun, where he is said to have been initiated into the Egyptian mysteries in the sanctuaries of Memphis, Diospolis, and Heliopolis. It is also claimed that he had a retreat on Mount Carmel and in Chaldea, where he was taken prisoner by the troops of Cambyses and then taken to Babylon. It was during his journey to this ancient metropolis that he became acquainted with the thoughts of ancient Eastern religions and attended lectures given by famous teachers of that time.

For many, we are in the realm of pure legend because there is not enough historical evidence to confirm the truth of these events. But based on similar thinking, little would remain to be asserted as genuinely historical about great figures of the past. In our time, even the historical value of Christ has been denied simply because his contemporaries did not recognize his worth, as if it were not an old custom of those at the forefront of literature and history not to perceive the truly great values that are contemporaneous to them.

Christ was seen by the Pharisees and scholars of his time as a mere wonder-worker who preached unacceptable ideas.

It is not surprising, therefore, especially among the Greeks, whose historical records are so incomplete and deficient, that the historicity of Pythagoras, like that of many other characters, has not been transmitted to us with the utmost certainty. Moreover, there were several Pythagorases in different places, often confused with the one who founded the school of Croton, so it is not surprising that many people become perplexed and skeptical about the accounts usually given of his life. But the truth is that Pythagoreanism did exist and still exists, and it left behind a monumental work that scholars can delve into.

However, in all the sources that relate Pythagoras' life, it is observed that in his youth he undertook numerous journeys and pilgrimages, and he returned to Samos at the age of 56. His teachings attracted many disciples, but they also provoked the enmity of Polycrates, who was the tyrant of Samos at the time. This forced the wise man into exile in Magna Graecia (Italy), where he founded his famous Institute.1

Before settling in Magna Graecia, it is reported that he had contact with the Orphics, who were already in decline, in the Peloponnese, and there he met the famous priestess Theoclea of Delphi.

But it was in Italy that Pythagoras played an extraordinary role because it was there that he founded his famous Institute, which, despite being opposed by the democrats of the time, was ultimately destroyed. According to legend, Pythagoras either perished in the fire along with his most beloved disciples or managed to escape in an unknown direction.2

According to the best sources, Pythagoras must have passed away between 510 and 480 BC. The Pythagorean society continued after his death but disappeared after the famous massacre in Metapontum following the defeat of the Crotonian League.3 From that catastrophe, only Lysis (who is credited as the author of “Verses Aurei”) and Philolaus, one of the most famous Pythagoreans of all time, survived, possibly (and there are sufficient elements in favor of this possibility) without having personally known Pythagoras. Some novices, including Hippocrates of Chios, who later lived in Athens, Hipparchus, and Hippias, were saved along with them. However, they were later considered traitors for revealing certain secrets of the order and were “excommunicated.” Among the notable followers from that period, Archytas of Tarentum should be highlighted.

Philolaus himself was also considered a traitor by many Pythagoreans for publishing works that revealed certain aspects of Pythagoras' philosophy and for reportedly selling three secret books to Dion, the brother of Dionysius the Elder.

For the chosen ones, there was a period of novitiate and an apprenticeship initiation that lasted five years (paraskeiê, preparation), followed by catharsis, purification (catharsis), corresponding to the Masonic fellowcraft degree, and finally, teleiôtes (telos, end), the master’s degree, in which the first and ultimate causes of things were revealed.

As for the causes of the destruction of the Pythagorean order, we examine them in our “Verses Aurei de Pythagoras,” where we discuss the political errors committed by the disciples who failed to follow the rules set forth by the master.

Justifications for the method used in this book

In Concrete Philosophy, we justify the method employed in this work, in which we carry out a mathematization of philosophical thought based on universally valid judgments, that is, in an apodictic manner. In it, we examine the various paths used by human intelligence to achieve demonstration: the Aristotelian-scholastic formal path, primarily deductive, the deductive-inductive and inductive-deductive paths, the demonstration in the manner of geometry, reductio ad absurdum, conversio, the idealistic dialectic, the Socratic-Platonic dialectic, especially inductive in the search for analogous logos, Raimundo Llull’s circular method, and finally, through dialectical analysis, our concrete dialectic, which includes the pentadialectic, the decadialectic, the symbolic dialectic, and the noetic dialectic.

In demonstrating the theses, which are the positivities of philosophical thought, achieved by humans throughout time, we use various demonstrative paths in order to show that through all the paths that humans have constructed to reach the truth accessible to them, they obtain the same result, which is what we have postulated as the thesis.

Concrete Philosophy, as we present it, as seen in the book bearing this title, is a consequence not only deductive but above all dialectical of the ontological truths that humans are capable of postulating. We verified that subsequent theses, some unsuspected in the beginning, had such solid foundations in their favor that they became apodictic and universally valid. Without ever resorting to mystical frameworks, we were able to explore the highest paths of philosophizing by following the methods we had advocated.

We are aware, without any doubt, that many may think that this attempt is contrary to the true nature of philosophy. For them, philosophy is a field of plausible demonstrations, of assertions dominated by doxa, by opinion, which, despite their theoretical basis, are not sufficient to ensure certainty and firm assent of the human mind without fear of error.

However, no matter how considerable and respectable these opinions may be, we are certain that the work we have carried out has demonstrated categorically and indisputably that the search for positivities in philosophy was possible in the manner in which it has been done. Furthermore, we have made every effort to prevent human schematics, in a purely psychological sense, from decisively influencing the examinations to be conducted. Based on the suspicion required by the examination of the epistemological capacity of humans, we were afraid that value judgments would be so powerful as to distort the reasoning itself, leading to affirmations that best align with our most hidden desires. For this reason, we always sought to give the concepts we employed as rigid a content as possible, eminently ontological, disregarding even formal interpretations and those obviously influenced by the schematics derived from our experience. Thus, we established for the concepts not arbitrary meanings or merely suitable ones for our way of philosophizing, but those that necessarily arise when ontologically constructed.

Thus, we discovered that the concept of infinity, independent of human schematics, could only have an ontological content, which was to point to a being absolutely independent of another, whose only reason for existence would be found in itself. In this way, we not only escaped the crude conceptualization of infinity when considered extensively, as in quantitative infinity, but also when considered intensively, as in qualitative infinity. We thus achieved what we could call the unique and true essence of the ontological concept of infinity, and other meanings that might be used would only be analogous to it, never identical. Thus, it became evident to us that only one being could be infinite, as the ontological content of this concept excluded the possibility of dualism and even more so of pluralism. If the construction we carried out, which we call Concrete Philosophy, occasionally repeats positivities of this or that philosophical current, this or that conception, this or that doctrine, our way of philosophizing was not subordinate to such tendencies, but rather, such tendencies participated in positivities inherent to the nature of the structure of concrete philosophy.

In this way, this philosophy is not a synthesis of alien positivities, a syncretism of axiologically chosen positivities, as it may seem to those who have not grasped it in all its reality. The construction of concrete philosophy does not arise like the work of an architect. The architect will preselect the usable and suitable materials for what he intends to accomplish and give them the order that corresponds to what he has previously built in his mind. Concrete philosophy does not arise from a more or less skillful adaptation of scattered positivities, connected in a coherent whole. It arises as a strictly ontological consequence of principles shown and demonstrated apodictically.

And if here and there we find a thesis that is the same as that of this or that doctrine, it is because this or that doctrine has captured such positivities, but concrete philosophy is not a collection of dispersed materials.

Well, at this moment when we undertake the examination of Pythagoras' thought, we cannot fail to follow the methods we have already established. Rarely in the history of human thought has a doctrine been so distorted, so disfigured, and so caricatured, not only by its disciples but also by its exegetes and adversaries, as was the thought of the sage of Samos.

No one can deny that Pythagorean thought played a grand role in the history of human philosophizing. Perhaps no other doctrine can present a list of such great names as Pythagoreanism in general. And it not only had a huge number of disciples but also fertilized the thought of all subsequent philosophy. Everywhere, in all human thought, we always find a persistent presence of one or another Pythagorean thesis. But despite all this, there has never been such a great incomprehension, such great divergences, such disparate evaluations, even among the disciples themselves, as there are regarding Pythagoras' thought.

All religions, all religious thoughts that have emerged over these twenty-five centuries, all doctrines, theories, and even philosophical currents, all without exception, received the influence of this thought, as we will see.

Considering all this, we decided to act differently. Using the same method employed in Concrete Philosophy, we decided to classify as doubtful any statement about Pythagoreanism, including those expressed by disciples considered the most perfect spokesmen of the founder of the Italic school. In this way, we set aside anything that may be subject to doubt and searched within the thought presented thus far for those statements that could not be doubted, that is, those statements that all Pythagoreans recognize as immediately emanating from the founder of this doctrine.

However, this rigorous selection work was not enough. We also needed to start from a thesis about which there was no doubt. This thesis would have an auxiliary role and would act in our examination to the same extent that mathematics acts in scientific research.

This point of utmost importance is what we call the point of coherence, which we need to examine because without fully understanding it, we will not be able to proceed through the construction we intend to present in this work.

Coherence, in its general sense, refers to the more or less close union formed between two beings or the elements that constitute the parts of a whole. Its origin is in the verb “haeo,” whose past participle is “haesum,” from which the word “herdeiro” (heir) comes, which, when combined with the preposition “cum,” gives “coerência” in Portuguese, indicating the united presence of the elements that constitute a totality.

In this way, we can speak of physical, chemical, vital, social coherence, etc. But we also use the term coherence when referring to the set of ideas presented that follow logical or psychological connections that interweave and form them in a way that is suitable to each other, making them congruent with each other. Any thought that deviates from this connection is called incoherent.

If we survey the works of truly great philosophers, it is easily observed that coherence is a norm present in the exposition of their ideas. In the analysis of a thinker’s work, one can find one or another thought that deviates from the connection or asserts something incongruent with the fundamental postulates on which their thought is based. However, many times, such accusations are the product of hasty analyses because a more careful and rigorous examination ultimately shows that this incoherence is not real.

On the other hand, it can also be observed that a philosopher, in the course of their work, is not always consistent with the foundations previously established or, at a given moment, in the face of a specific aspect, proposes postulates that are inconsistent with the fundamental principles. We could call this subjective incoherence, which is found in the work of an author and consists of the famous errors, so proverbial, from which even the greatest have not been exempt (Aristotle, for example, negligence in writing, let us remember).

But there is another coherence that we want to establish now.

Once the fundamental postulates of a philosophical thought have been carefully and clearly set forth, the exegete, as long as they adhere to extreme ontological and dialectical rigor, can develop the thought of an author with coherence, which is indeed possible. And after establishing the strictly ontological framework, suddenly encountering a postulate that is incongruent with what has been previously affirmed or in opposition to the foundations of that thought.

This is a rather easy norm, to a certain extent, and one to which both Aristotle and the Scholastics paid great attention. It consists of the following: the validity and strength of a philosophical thought are judged by whether that thought, when taken to its ultimate consequences, does not reveal absurdities. If a dialectical rigor is followed in the sense we take it, it will easily be seen that few doctrines withstand this examination. And many, sooner than others, reveal absurdity when least expected.

We will call this second coherence objective coherence.

If we want to examine the work of Pythagoras (and we refer specifically to that of Pythagoras), we encounter great difficulties because, in truth, we have to rely on the statements of those who proclaimed themselves his disciples. None of the books that are said to have been written by Pythagoras, if he indeed wrote any, have reached our hands. It is presumable, with good reason, that “The Sacred Word” (“Hieros Logos”), attributed to Pythagoras, was merely a book of symbolically written maxims that could only be properly interpreted by those familiar with his thought.

Furthermore, we know (because this is a constant in the works of the Pythagoreans) that Pythagoras and the order he founded strictly prohibited making exoteric the thoughts and knowledge that were only meant for those initiated at a higher degree. Thus, the true thought of the Pythagorean order was reserved only for those who reached the highest initiatic degrees and could in no way become exoteric at that historical moment.

We want to draw the reader’s attention to Philolaus, who is undoubtedly one of the most prominent disciples and possibly had direct contact with Pythagoras, although we say this with many reservations. However, he was certainly a disciple of other masters who had direct contact with the founder of the Italic school.

Philolaus was accused of being a traitor by the Pythagoreans of his time because he made certain knowledge exoteric, which should have remained guarded and hidden, and which should only be revealed when the opportune moment arrived, that is, when humans deserved it or had already reached such a high degree that they could learn it without causing greater harm.

This is a point of great importance, we repeat, because numerous fragments of Philolaus remain that can guide us to a more accurate understanding of Pythagorean thought, and they can also serve to compare the theses we will present with his statements.4 If Philolaus' exoteric exposition of thought did not align with the most secret teachings of the Pythagorean order, we could not understand why he was labeled as a traitor and revealer of secrets. Unless it came from disciples who were unaware of the true Pythagorean thought and believed that what he had revealed should remain hidden. However, regardless of the case, throughout time, all authors recognized him as an unquestionable authority. In the method we follow in this work, we will not only rely on Philolaus but, above all, on other points that we consider more reliable and irrefutable, without disregarding the contributions of this great disciple whenever his statements are consistent with the foundations that we will have the opportunity to clarify. Moreover, we will see that Nicomachus of Gerasa also expounded Pythagorean thought in certain aspects, with undeniable coherence. Still, others, in certain passages, provide valuable elements for justifying what we will present in due course.

Having established these points regarding subjective and objective coherence in relation to Pythagoras, we are led to say that since we do not have any written work by him that could serve as a reference for analyzing his thought, we cannot assert either coherence or incoherence, subjectively considered. It is not possible to affirm whether Pythagoras was coherent or not. However, it is observed that great philosophers are characterized by coherence in their works, so we can start from the postulate that Pythagoras' work should have been subjectively coherent. But based on the objective coherence mentioned above, we can construct, thanks to the methods of our dialectic, the fundamental theses of his thought and build his conception with rigor and coherence. Therefore, we can establish how Pythagoras must have thought, and since nothing can be asserted regarding subjective coherence or incoherence, we can establish the postulate that his thought must have been, at least, objectively coherent.

With the Pythagorean thought established with rigorous objective coherence, we can examine all the opinions expressed by his disciples and also by those who claimed to be his followers, comparing their statements with those that will constitute what we will call “pythagorikon,” as the Greeks would say.

And what we will proceed to examine, after reproducing and commenting on some of the most important fragments.

Pythagorean Fragments of Philolaus

The following are some fragments from Philolaus, valuable as points of support for the examinations that need to be carried out throughout this work.

Fr. 1: a. The being that belongs to the world (cosmos) is a harmonious compound of unlimited and limited elements: this applies to both the world (cosmos) as a whole and to all the things it contains.

b. All beings are necessarily either limited or unlimited, or both limited and unlimited, but they cannot all be unlimited alone… (This corresponds to Fr. 2 by Diceas).

These theses are distinguished:

    1. There is a being that belongs to the world (cosmos). Consequently, there must be another being that does not belong to the world.
    1. The first being is a compound; therefore, there must be another being that is not compound.
    1. The composition of the first being is constituted by the harmony of opposed elements, some unlimited (apeiron) and others limited (perainónton).
    1. The cosmos as a whole and the parts that compose it are compounds.
    1. All beings (eonta) are necessarily (anánka) either limited or unlimited, or simultaneously limited and unlimited.

However, they cannot all be unlimited.

The fragment continues:

Now, since it is clear that beings cannot be formed from elements that are all limited or all unlimited, it is evident that the world as a whole and the beings within it are a harmonious compound of limited and unlimited elements. This is observed in works of art (those created by humans). Among these, those made from limited elements are themselves limited; those made from limited and unlimited elements are both limited and unlimited, and those made from unlimited elements appear unlimited.

We can deduce the following theses:

    1. Beings in this world (cosmos) cannot be formed from solely limited elements. In this case, where is the atomism of the Pythagoreans, as some claim?
    1. Neither can the cosmos be a compound of unlimited beings.
    1. The cosmos is a harmonious combination of limited and unlimited beings. These beings are harmoniously constituted by limited and unlimited elements. Therefore, in the expression “in go,” only the preposition is used to indicate the material of which something is made, not the verb.
    1. There is a distinction between the world (cosmos) and culture, things made by humans (erga), or in other words, the distinction between Nature and Culture. In Nature, things are as we have said above. However, in Culture, they can be created from only unlimited elements, or from both limited and unlimited elements, or from only unlimited elements, as they appear.
    1. It will be shown in due course that, for Pythagoras, number is the harmonious combination of the even and the odd, the limited and the unlimited, whose apophantic sense we will clarify further on.

The fragments continue:

“And all things, at least those that are known, have number; for it is not possible for any thing to be thought of or known without number. Number has two proper forms: the odd and the even, and a third form resulting from the combination of the other two, the even-odd. Each of these forms is susceptible to numerous “forms” that each individually manifests.”

    1. It is attributed to Philolaus by Nicomachus of Gerasa that “harmony is universally the result of opposites, that it is the unity of the multiple, the agreement of the discordant.” If this passage can be doubted as authored by Philolaus, it cannot be doubted that this is the concept of harmony accepted by the Pythagoreans, as we will see later. Number is a harmonious combination of the even and the odd. The even and odd precede number, as will be seen. And according to the text, the following can be inferred:

Known things are known because they are limited-unlimited. In another fragment, quoted by Jamblichus, Philolaus states that “the object of knowledge cannot be known in advance if it is only unlimited.” That is, if it reveals only oddness, the absence of evenness, of something that can be paired with another, that resembles another, for which there are schemata that allow assimilation.

Therefore, for something to be knowable (gignoskoménon), it requires number, or rather, that it has a number (evenness and oddness). Number has two proper forms (ideas): evenness and oddness.

Each of these forms (ideas) is capable of numerous “forms” (morphai), each individually manifesting. Morphê corresponds better to the German term Gestalt. They would be Gestalten, constitutive and structural forms of the thing.

Philolaus continues:

“Regarding nature and harmony, it should not be believed that philosophers (the Pythagoreans) begin with principles that are, so to speak, opposed: they know the principle that is placed above these two elements… for it is God who hypostatizes the limited and the unlimited.” Philolaus shows that “it is through the limit that every coordinated series of things approaches the One, and it is through infinity that the lower series is produced. Thus, above these two principles, they (the philosophers) placed the unique and separate cause, distinct from everything by its excellence. This cause, which Archytas called the cause before the cause, is affirmed with strength as the principle of everything, and of which Brontinus says that it surpasses in power and dignity all reason and all essence.” (See the fragments of Archytas later on).

    1. The One is the Supreme Being, God. It is above opposites, above the limited and the unlimited. The limit brings things closer to the One but does not reach it, and it is through unlimitedness that the lower series is produced. God is the cause of causes, the first cause of all things, and surpasses them all in dignity.

“The one who commands and governs everything is one God, eternally existing, immutable, motionless, identical to himself, different from all things.”

“God keeps all things as captives and reveals that He is one and superior to matter.”

To keep as captives means to be the master of all things, to govern them, that is: there is nothing outside of Him because He is the foundation and support of all things.

Referring to the world (cosmos), which has always existed through its mutations, Philolaus says, “...since the motor has been acting for all eternity” (ab aeterno, as it is governed by a principle whose nature is similar to that of the world but whose power is all-powerful and sovereign), “and its action continues eternally, and the movable receives its manner of being from the motor that acts upon it, it necessarily follows that one part of the world always imparts motion, which the other part passively receives. One part entirely belongs to reason and soul, while the other part belongs to generation and mutation. One is prior and superior in power, the other is posterior and subordinate. The compound of these two things, the eternally moving divine and the constantly changing generation, is the World. That is why it is reasonable to say that it is the eternal energy of God and becoming, obedient to the laws of changing nature. The One remains eternally in the same state and identical to itself; the rest constitutes the domain of plurality, which is born and perishes. However, the very things that perish preserve their essence and form through generation, which reproduces the form identical to that of the father who begot and shaped them.”

These passages from Philolaus categorically demonstrate that they never considered number (arithmos) to be the ultimate essence and reason for everything that exists, but merely the proximate reason of things, not the ultimate.

And it is reason (but the one developed through the study of mathematics) that is the true faculty of discernment and judgment. However, for Philolaus, as well as for the Pythagoreans, it is not the commonly considered reason but the reason developed through the study of mathematics that is capable of comprehending the nature of everything and that has some essential affinity with it, for “it is in the nature of things that the similar be understood by the similar,” as he affirmed.

Pythagorean Fragments of Archytas

Here are some of the fragments of Archytas that best contribute to the examinations we intend to undertake.

"There are necessarily two principles of beings, one that encompasses the series of ordered and limited beings, and another that encompasses the disordered and unlimited beings. One principle can be expressed by the word, which is capable of being understood by us, encompasses beings, and at the same time determines and orders non-being.

For every time it approaches things in the process of becoming, it places them in order and measure, and makes them participate in the essence and form of the universal. On the other hand, the series of beings, which are absent from the word and reason, attends to ordered things, destroys those that aspire to essence and becoming; for every time it approaches them, it assimilates them to its own nature.

But because there are two principles of things of contrary nature, one principle of good, another principle of evil, there are necessarily also two reasons, one of the benevolent nature, another of the malevolent nature.

That is why both things that owe their birth to art (tekhnê) and those that owe it to nature must first participate in these two principles: form and substance.

Form is the cause of essence; substance is the substrate that receives form. Neither substance can, by itself, participate in form, nor form, by itself, apply to substance; therefore, there must be another cause that moves the substance of things and leads it to form. This cause is the first from the perspective of power and the most excellent of all. The name that suits it is God. There are, therefore, three principles: God, the substance of things, and form. God is the artist, the mover; substance is the matter, the movable; essence is like art, and what substance is moved towards by the mover. But as a movable, it contains forces that are contrary to itself—they are those of simple bodies—and the contraries need a principle that establishes harmony and unity in them, so it is necessary to receive the effective virtues and proportions of numbers, and everything that manifests itself in numbers and geometric forms, virtues and proportions capable of binding and uniting contraries in form, which exist in the substance of things. For substance itself is formless: only when it is moved towards form does it become formed and receive the rational relation of order. Similarly, if movement exists beyond the moved thing, it is necessary that there be a first mover; therefore, there are necessarily three principles: the substance of things, form, and the self-moving principle, which is the first by its power; this principle not only must be intelligent but must be above intelligence, and what is above intelligence we call God, so it is evident that the relation of equality applies to the being that can be defined by language and reason. The relation of inequality applies to the irrational being and cannot be fixed by language: it is substance; that is why all becoming and all destruction occur in substance and do not occur without it."

Comments: There are two principles: one of ordered and limited things, of finite beings, and another of beings that are not yet ordered and unlimited (potential). One principle is proficient, and the other deficient; one pertains to being, and the other to non-being, to what is not yet or what cannot become.

Substance is the substrate that receives form. But how could the conjunction of both be given without another thing that unites them, in short, an efficient cause? Thus, it can be seen that, in contrast to Aristotle, long before him, the material cause, the formal cause, and the efficient cause were already affirmed by the Pythagoreans, and the formulation of the latter was attributed to Aristotle. And what is this efficient cause? It is God, who is the first in power, excellence, and eminence. In order to have the adequacy and harmonization of the contraries, which are unified into a whole, we need arithmos, number, which marks the intrinsic proportions of things, which unites contraries.

The first mover is above human intelligence, and it is God who moves all things. Thus, Archytas laid the foundations for what would later become Aristotle’s thought, which shows how unaware he was of genuine Pythagorean thought, as he could not and should not have ignored the work of Archytas, since they were contemporaries, and Archytas died when Aristotle was still a young man of 18 to 20 years old.

Fragment 2

“The philosophers (Pythagoreans), in summary, began only with the principles that are, so to speak, contrary, but above these elements, they knew of another superior one, as Philolaus attests, who says that God produced (hypostesai), brought about the limited and the unlimited, and showed that the series which has a greater affinity with the One is related to the limit, and to infinity, to what is above. Thus, above the two principles, they placed a unifying and superior cause to everything. This cause, according to Archenetes, is the cause before the cause (aitían pro aitías), which Philolaus calls the universal principle.”

Comments: This fragment clearly reveals that, for the Pythagoreans, there was a transcendent principle beyond the contraries, as we have seen in our discussion of Philolaus, a principle similar to what Krishna promises Arjuna, that he will teach him “to transcend the pairs of opposites” and to reach the One, the principle of all things, the cause of causes; in other words, the first cause. Therefore, the claims that Pythagoras did not reach the transcendent being are unfounded, as we will further demonstrate later.

Fragment 3

“Which unity do you want to speak of? Is it the supreme unity or the infinitely small unity that reveals itself in the parts? In short, the Pythagoreans distinguish between unity and the Monad, as many ancient Pythagoreans spoke, as exemplified by Archytas, who says, 'The One and the Monad have an affinity of nature but differ from each other.’”

Fragment 3-bis

“Archytas and Philolaus indifferently give the name Monad to unity and the name unity to the Monad. Most of the time, they add to the word Monad the determination of the first Monad because there is a Monad that is not the first and that comes after the Monad itself and unity.”

Comments: These two fragments clearly reveal that the Pythagoreans did not confuse the first Monad with the second. There was an affinity of nature, but they differed from each other, distinct from one another.

There is a first Monad, transcendent to all things, which is the Monad itself, the unique being that has its own reason for being; therefore, in it, essence and existence are identical, for it is what it is because it is what it is: God, in short, for the superior thought.

Fragment 4

"The beginning of knowledge of beings lies in the things that occur within them. Of these things that occur within them, some are intelligible, and others are sensible; those that are intelligible are immovable, and the others, which are sensible, are not moved. The criterion for intelligible things is the world (the cosmos); the criterion for sensible things is sensation.

Of the things that do not manifest their own beings, some are knowledge, and others are opinion; knowledge is immovable, and opinion is changeable.

Furthermore, it is necessary to admit these three things: the subject that judges, the object that is judged, and the rule by which the object is judged. The one that judges is the spirit (nous) or sensation; the one that is judged is rational essence (logos); the rule of judgment is the very act that occurs in the being, which is either intelligible or sensible. The spirit is the judge of essence, whether it refers to an intelligible being or a sensible being. When reason seeks intelligible things, it refers to the intelligible element; when it seeks sensible things, it refers to the element of sensible things. Hence arise those false graphical representations in figures and numbers, typical of geometry, those investigations into probable causes and ends, which have as their object the beings subject to becoming and moral acts, and those sought in physiology and politics. When referring to the intelligible element, reason knows that harmony lies in the dual relationship; but this fact, that the dual relationship is consonant, is attested to us only by sensation. In mechanics, science deals with figures, numbers, proportions, in other words, rational elements; the effects are perceived by sensation, for they cannot be studied and known apart from matter and motion. In short, it is impossible to know the reason (dià ti) of an individual thing if one has not previously grasped in the spirit the essence of the individual thing (ta ti enti ekaston). Knowledge of existence (óti énti) and quality (ontôs exei) belongs to reason and sensation; to reason, whenever we subject ourselves to the demonstration of something through a syllogism that necessarily concludes; to sensation, whenever we attest to the essence of something through sensation."

Comments: It is of utmost importance, for a clear understanding of Pythagorean thought, to distinguish between sophia (wisdom) and doxa (opinion), between science and opinion. The former is immutable, and the latter is mutable.

It is not possible to comprehend anything solely through knowledge of the thing in its externality if one does not penetrate the essence of the individual thing, its intrinsic nature, which provides the path to understanding its reason (dià ti). As for the meaning of science (epistéme) and opinion (doxa), we will take care to examine it further.

Fragment 5

"Sensation occurs in the body, reason in the soul. One is the principle of sensible beings; the other, the principle of intelligible beings. For the multitude, number is their measure; length, the foot; weight and balance, the scale; the rule and measure of rectitude, both vertically and longitudinally, is the right angle.

Thus, sensation is the principle and measure of bodies; reason, the principle and measure of intelligible beings. One is the principle of intelligible and first beings by nature; the other, the principle of sensible and second beings by nature. For reason is the principle of our soul; sensation, the principle of our body. Spirit is the judge of the noblest objects; sensation, of the most useful. Sensation was created for the sake of the body and to serve it; reason was created for the sake of the soul and to give rise to wisdom. Reason is the principle of science; sensation, of opinion (doxa). One derives its activity from sensible things, the other from intelligible things. Sensible objects partake of movement and change; intelligible objects partake of immutability and eternity. There is an analogy between sensation and reason: sensation has as its object the sensible, and the sensible moves, changes, and is never identical to itself: also, as can be seen, it becomes more and less, better and worse. Reason has as its object the intelligible; now, the intelligible is essentially immovable; hence, in the intelligible, one cannot conceive of more or less, better or worse; and just as reason sees the first being and the paradigm, in the same way sensation sees the image and the second being. Reason sees man in himself; sensation sees in them not only the circle of the sun, but also the forms of artificial objects. Reason is perfectly simple and indivisible, like unity and the point; the same applies to the intelligible being.

The idea is neither the limit nor the boundary of the body: it is merely the figure of being, the reason why being is, whereas sensation has parts and is divisible.

Among beings, some are perceived by sensation; others, by opinion; a third category by science; and a last one by reason.

Resistant bodies are sensible; opinion knows those that participate in ideas and are like images of them. Thus, a certain man participates in the idea of man, a certain triangle in the idea of a triangle. Science has as its object the necessary accidents of ideas; thus, geometry has as its object the properties of figures; reason knows the ideas themselves and the principles of science and its objects, for example, the circle, the triangle, the sphere itself. There are still four kinds of knowledge in us, in our soul: pure thought (nous), science, opinion, and sensation. Two are principles of knowledge: they are thought and sensation; two of them are their end: they are science and opinion.

It is always the similar that is capable of knowing the similar; reason knows intelligible things; science, cognoscible things; opinion, conjectural things; sensation, sensible things.

That is why thought must rise from sensible things to conjectural things, from conjectural things to cognoscible things, from cognoscible things to intelligible things; and he who wishes to know the truth about these objects must gather all these means and objects of knowledge into a harmonious whole. Once established in this way, they can be represented as a line divided into two equal parts, and each of these parts is in turn divided in the same way: let us thus separate the sensible, and divide it into two parts in the same proportion; these two parts would be distinguished, one by clarity, the other by obscurity. One of the sections of the sensible contains the images of things, and those that are perceived in water, and those that are seen in mirrors; the second section represents plants and animals, of which the first gives the images. The intelligible receives a similar division in which the various kinds of science represent the images: for geometers begin by establishing, by hypothesis, the odd and the even, the figures, the three kinds of angles, and derive their science from these hypotheses; as for the things themselves, they leave them aside, as if they knew them, although they cannot explain them to themselves or to others; they use sensible things as images, but such things are neither the object nor the end they aim for in their searches and reasonings, for they seek only the diameter and the square in themselves. The second section is that of the intelligible, the object of dialectic; it does not truly establish hypotheses: it places principles from which it rises to reach the unconditioned, to the universal principle; and then, by a reverse movement, clinging to this principle, it descends to the term of reasoning, without employing a sensible object and using only pure ideas. These four divisions can also be used to analyze the states of the soul, and the name of Thought can be given to the highest, of Reasoning to the second, of Foot to the third, of Imagination to the fourth."

Comments: We clearly have an overview of the gnoseological conception of Pythagoreanism here. There are four kinds of knowledge. Reason knows intelligible things; science, cognoscible things; opinion, conjectural things; and sensation, sensible things. The path of knowledge starts from sensation (the foundation of Aristotelian empiricist rationalism), through opinion, conjectures, from them to cognoscible things, until reaching intelligences, the eide. And true knowledge is concrete; it is the one that concretizes all sides of the path through a harmonious whole.

We can clearly see the abstractive and concretist paths, as we call them; the one that mentally separates and the one that mentally unites, which allows us to reach the unconditioned, the principle of all things.

And only after traversing these paths is the human spirit able to use pure ideas; that is, to metamathematize philosophical knowledge, as we explain in Concrete Philosophy. In this way, we see that the method employed in that work follows the Pythagorean line, that is, to establish philosophy based on apodictic judgments, under penalty of remaining only in the realm of the cognoscible, which offers the abstractist path or conjectural path, as is the case with the domain of assertoric judgments, of opinion, which unfortunately has done more harm to philosophy than good, as is evident in the historical process of this matter.

Fragment 6

...Man was born, was created to know the essence of universal nature; and the function of wisdom is precisely to possess and contemplate the intelligence that manifests itself in beings.

Wisdom does not have as its object any particular being, but absolutely all beings, and it is necessary that it begin, not by seeking the principles of an individual being, but rather the principles common to all beings. Wisdom has all beings as its object, just as vision has all visible things as its object. To see in its entirety and to know the universal attributes of all beings is characteristic of wisdom, and this is how wisdom discovers the principles of all beings.

He who is capable of analyzing all genera and relating them and uniting them, by an inverse operation, into a single and same principle, seems to me to be the wisest, the one closest to the truth, and who seems to have found this sublime observatory, from the heights of which he will be able to see God and all things belonging to the series of the divine: as the lord of this royal path, his spirit can ascend straight ahead and reach the pinnacle of the race, connecting the principles to the ends of things, and knowing that God is the beginning, the middle, and the end of all things, created according to the rules of justice and right reason."

Comments: Man was created to know the essence of universal nature. Note these words: “he who is capable of analyzing all genera and relating them and uniting them, by an inverse operation” (inverse to the abstractive, which is the concretist path we adopt, which is not merely concretist) “into a single principle, such seems to me to be the wisest, the closest to the truth.”

It is clear that true Pythagorean philosophizing is concrete, not just abstract. We can see unequivocally that Aristotle based his criticisms of Pythagoreanism on the works of lesser Pythagoreans, as he indeed did, and not on the works of its true luminaries. That is why Aristotle’s criticism is subject to reproach, for with a mind of such caliber, it is in no way justified that he would be a victim of an ignoratio elenchi of such proportions.

With these important elements at our disposal, our subsequent analysis and conclusions will be strengthened in what is most solid in Pythagoreanism. It is true that many important works have been lost, which would give us sufficient light on the foundations of this doctrine, which we are now forced, with meager but sufficient elements, to reconstruct in an unequivocal and well-founded manner, thereby contributing to dissipate the deformation that has prevailed for over twenty-three centuries, with serious damage to the progress of Philosophy. This is what we will show in the following chapters.

Pythagoreanism in Greek culture

The Greeks are often accused of imposing a model on the world, of rationalizing the phenomenal world to such an extent that the model they constructed imposed itself as reality. This capacity to surpass the boundaries of appearance is considered the essence of the so-called “Greek miracle.” It is also added that this model was merely an act of faith.

This dual way of viewing the world did not originate with Greek philosophy. It only gave it new contours and justifications. It belongs to the entire religious and psychological manner in which the Greeks considered the world, always fashioned in the image of the gods, where the world of phenomena either copies or participates in the higher reality of the world of forms. Thus, it can be established that the most typical aspect of Greek thought is the visualization of two planes: the realm of pure and immutable ideas, eternal and ungenerated, and the realm of appearance, of phenomena, the world of becoming, of constant change.

It is precisely in Pythagoras that this way of seeing takes on a philosophical form and becomes the foundation of his entire doctrine. For many, this is the great Greek myth, and when they deviate from it, Greece sinks into the vicious forms of sophistry and marks its great finale. One could say, following the example of Spengler, that the essence of Greek culture lies in accepting this myth, sufficient to explain its art, its religion, its philosophy, its politics, its ideals, and also its melancholic ending.

All the efforts of their great philosophers (and the great Greek philosophers were Pythagoras, Socrates, and Plato) were focused on justifying this thesis. Aristotle, with his rationalistic empiricism, would be nothing more than a barbarian, in the noblest sense of the term. He truly came from the borders of Greece, and this explains why he deviated from the great myth, seeking another way to visualize the world. This is also the reason why he had such a significant influence on the West through Scholasticism. His model was not Hellenic.

If we examine Greek cults, from the most primitive to Pythagoreanism, understood here in its religious sense, the two planes are clearly evident. In their decline, the borrowed religions, cults originating from the East, were no longer Greek; they were pseudomorphoses of a culture that only adopted some external forms of Greece, but their content no longer had roots in the soul of that people, as they had lost it.

There is some truth in all of this, but it would be simplistic to think that only this explanation encompasses all Greek philosophy.

A brief examination of Pythagoreanism is enough to distance ourselves from this theory. The Dionysian cults originated from Thrace, and it is impossible to hide the influences that Eastern mysticism and also Egyptian mysticism had on the religious thought of the Greeks in their beginnings. It is undeniable that Orphism, coming from Phrygia, was influenced by the Phoenicians, and the fusion of these two cults undoubtedly occurred through contact with the Egyptians and the East. In fact, Pythagoreanism is not devoid of Orphism since, after Pythagoras, it is difficult to distinguish between the Orphic and Pythagorean authors. Many of their rituals and ceremonies were copies of others of Orphic origin. These facts allow us to consider Pythagoras as a true reformer of Orphism. However, it would be a mistake not to consider the extraordinary innovations he brought to this cult, to the point that his doctrine took on its own distinct character. The entire 5th century and part of the 4th century were influenced by him. Great Pythagoreans of this period were Timaeus, Archytas of Tarentum, Philolaus, and Theodorus. One of their greatest figures was undoubtedly Socrates, whose development culminated in his great disciple Plato, the brightest expression of Pythagoreanism as well as of human thought in general. In Plato, Pythagoreanism reached its grand phase.

Just as the Platonic Academy deviated from its master after his death, the Neo-Pythagorean movements can also be accused of distancing themselves from the master of Crotona. Plato never publicly declared himself a Pythagorean, just as Socrates did not. This was natural because Pythagoreanism was outlawed. Platonic doctrine remains within the realm of the two planes.

This world, the phenomenal world, is fashioned after an eternal and immutable model, the true world, the world of pure forms. Among the critics of Plato and Socrates, there are some who doubt the Pythagorean affiliation of these authors. The affirmations of Aristotle are considered insufficient. However, Xenophon, in his portrayal of Socrates, presents him as a Pythagorean when he says: “He was one of those Pythagoreans seeking redemption.”

All the Platonic terminology of the idea-forms is Pythagorean. Eidos, idea, skhema, morphe are terms used by them.

The doctrine of forms undoubtedly has that origin, and when Plato, in his dialogues, speaks of the “lovers of ideas,” he is referring to them. And today, based on what we know, Eudoxus, who succeeded Plato, his uncle, in the Academy before Xenocrates, wrote a treatise “On Pythagorean Numbers,” indicating that the teaching given in the Academy was Pythagorean.

Pythagoras not only sowed the seeds of Greek philosophical thought with his ideas but also the most fundamental of his doctrines has reached our days, as modern science is more Pythagorean than ever.

By considering number as the foundation of things, he introduced calculation into physics and allied mathematics with science, leading to the great progress it has known. Pythagoreans included Timaeus, who invented algebra, Theodorus of Cyrene, who was the teacher of Theaetetus, Anaxagoras of Clazomenae, who was the teacher of Pericles and studied notions of the infinite, Archytas, Oenopides, and Eudoxus the great astronomer, and finally, Plato, whose mathematical teachings with rational methods paved the way for the advent of the great Pythagorean Euclid. We must also highlight Asclepius, whose role was significant in medicine, Alcmaeon, who was the first to practice dissection, and the greatest of them all, Hippocrates of Kos, the precursor of modern medicine, as well as the great poet Pindar.

The teachings of the sophists were undoubtedly a movement against the doctrine of the Pythagoreans.

It is necessary to distinguish the exoteric doctrine of Pythagoreanism from its esoteric doctrine, which was intended for the profane and reserved only for the initiated. In this book, we aim to reconstruct the esoteric thought, especially regarding numbers (arithmoi), with the few data we have at our disposal, as this has often been distorted by commentators.

When asked, “What is the essence of anything?” the Pythagoreans invariably responded with a dual assertion: “Things consist of numbers” and “Things are formed in imitation of numbers.”

Considering numbers as particles that form reality is a primitive way of thinking.

The true doctrine can only be interpreted thus: things consist of numbers on the eidetic plane, and in the natural plane, they are formed in imitation of numbers due to the mathematical laws that govern them. Materially, things imitate numbers and are therefore also numbers. There is no contradiction there, only apparent contradiction, as we will have the opportunity to demonstrate later.

According to Plato in the Sophist (238b), he says, “According to our way of seeing, number, as a whole, is Being.” Which being he is referring to, we will see in due course.5

Undoubtedly, mathematics had its great epistemic impulse with the Greeks. It is truly with them that demonstration developed. It should not be thought that demonstration, proof, began with them, as it was already employed by the Egyptians. There is a fragment from the work of Democritus that is quite expressive. Describing his travels to Egypt, he refers to it on one occasion: “I traveled to many countries… and conversed with many wise men, but when it comes to the combination of lines with demonstration, no one surpassed me, not even those whom we call harpedonates in Egypt”…

No one surpassed him in demonstration (apodeixos, from which apodictic, demonstration, proof). We know this passage through the Stromata of Clement of Alexandria. Democritus attributed to the harpedonates a demonstrative science that did not surpass his own, which proves that the Egyptians also used demonstration in mathematics, which, moreover, is inherent in the spirit of mathematics.

Long before Democritus, the Pythagoreans dedicated themselves to demonstration. According to the available documents, Pythagoras always emphasized to his disciples the difference that should be established between doxa and episteme. The Pythagorean ideal of Malhe sis Magisthe, Supreme, of supreme instruction, could only be achieved by man through episteme, cultivated knowledge, demonstrated wisdom, which is the path of the knowledge-seeking man, the one who loves knowledge, the philosopher (from philoo, I love, and sophia, wisdom).6

In order for a discipline to become epistemic, it must distance itself from doxa, opinions, from the subject matter on which everyone has opinions and diametrically opposed points of view, to the point that what is confidently affirmed as certain and true by one is considered utterly false by another, as we see happening in the field of the so-called cultural sciences. The evaluation of knowledge can only be obtained epistemically if the criterion used for evaluation is truly based on objective foundations. And how can such foundations be obtained if not through apodictic demonstrations, like those offered by mathematics?

Aesthetics is founded on subjective criteria, while episteme is based on objective criteria. For this reason, modern aesthetics, which still falls within the realm of doxa, allows its scholars to diverge into opposing and even contradictory fields. This is possible because the apodictic foundation of aesthetic postulates has not yet been established, at least among modern aesthetes who are unaware of the work carried out by the Pythagoreans in this field. Therefore, it can be established that in the natural sciences, where objective criteria prevail, the process of mathematization is easier in the proper sense. However, in cultural sciences, due to the persistence of deeply rooted prejudices, mathematization, also using the term in a Pythagorean sense, becomes more difficult but not impossible, as some individuals desire in their natural inclination towards speculation and ungrounded and facile assertions.

“No research deserves to be called science if it does not undergo mathematical demonstration.” This maxim by Leonardo da Vinci is genuinely Pythagorean and genuinely Greek, as it embodies the spirit of Greek science. Demonstration is a characteristic of the Greek logical spirit and its rhetoric. We can observe this in Plato’s dialogues, where demonstrations aim to be as convincing as possible. We can already notice it in the speeches of Demosthenes, where he seeks to destroy the arguments of his opponents and reveal the absurdity contained within them. Undoubtedly, it is with Pythagoras that the method of demonstration develops to reach its peak in Hippocrates of Kos, in Aristotle’s Analytics, and in the astonishing accomplishment of Euclid’s Elements, where it is applied in an extraordinary and definitive manner.

Paul-Henri Michel, in “De Pythagore à Euclide”, page 676, writes these words:

“But the Pythagoreans are not primitive! If, according to them, all things are numbers, it is not only because every sensible object can be considered a ‘collection,’ as the sum of its indivisible parts. The number existing in itself, apart from the plurality of material objects, as well as magnitude, which leads us to another aspect of the theory, to transcendence and the notion of the number as the model of things. The arithmos (which should not be confused, as Aristotle often seems to do, with the plethos) is translated into being through harmony. Firmly convinced of this through their examination of vibrating strings, the Pythagoreans could state in the catechism of acousmatics: 'What is the wisest thing? Number.’”

Furthermore, Paul-Henri Michel (same source, page 680) asserts that the Pythagoreans initially had only a quantitative view of number and later tended to qualify it. “This dual providence,” he claims, “was perhaps never conscious; undoubtedly, it was never explicitly stated by the Pythagoreans, but it was underlying in their conception of number. Only it can justify the kind of fascination that numbers, taken individually, exerted in their thoughts, and moreover, it was not limited to their school.”

This assertion is partly correct because in the study of the first and second degree (degrees of paraskeiê and catharsis), the studied number is quantitative, as an abstraction of quantity. However, the number, in a qualitative, vectorial, modal, etc., sense, is examined later as the initiate delves into higher knowledge. It would be naive to think that all of Euclid’s mathematical thought was exposed in his Elements, which is a work of geometric initiation.

This is the reason why at the Pythagorean Institute, there was a distich at the entrance, later copied by Plato: “Let no one ignorant of geometry enter here.” In other words, initiation is impossible for those who have not acquired the demonstrative habits of geometry.

This was indeed the case, as evidenced by the disdain the Pythagoreans showed for Logistikê, the art of calculation and the number of calculation, demonstrating that they did not confuse the Pythagorean arithmos with the sensible number, the number of accounts, calculations, and measurements alone.

The number (arithmos) is not merely a second-degree abstraction of quantity, as one might think based solely on the works of Pythagoreans of the degree of paraskeiê.

There is no doubt that the constant in Pythagorean exegesis, known throughout the ages, has always conflated the concept of number in its generic aspect with the quantitative number, which is only a kind of number or a “scheme of quantity participation.” This quantitative aspect is the specific difference of the genus arithmos, but it is not everything and only that. This is a viewpoint that we insistently emphasize, although we acknowledge that there were Pythagoreans who never attained any other vision besides the purely quantitative, as can be observed in the modern work of some Pythagoreans and neo-Pythagoreans.

It is evident from the statements of the school that numbers are not the ultimate nature of things, as the school affirms that the number originates from the harmonic combination of the unlimited and the limited (the infinite and the finite, even and odd). Furthermore, the One is not a number.

On the other hand, it cannot be claimed that Pythagoras' conception of the world was atomistic, as in this case, discontinuity would be the ultimate nature of things. This claim is not valid because he affirmed that the ultimate hypokeimenon, the ultimate support of things, is the aether, and it is pure continuity and immutability in its essence.

Paul Kucharski reaches the same conclusion when criticizing A. E. Taylor’s opinions in his “Étude sur la doctrine pythagoricienne de la tétrade,” page 59.

For the Pythagoreans, there undoubtedly existed a transcendent mathematics in contrast to immanent mathematics. The latter corresponds to abstractions of quantity, while the former refers to forms or ideas, following the order we presented regarding ontological mathematization, as we did in Concrete Philosophy. There are several allusions to this transcendent mathematics in Plato’s Laws, which was only known and handled by those initiated into higher degrees, including within the Platonic school. Léon Robin, in his book Platon (Paris, Alcan, 1935, page 238), made valuable comments in this regard.

This general exposition of Pythagoreanism within Greek thought and culture was necessary for a clearer understanding of the theses to be subsequently examined, as well as to strengthen the reasons that we will present at the appropriate time.

The arithmos for Pythagoras, according to common exegesis

This chapter could be longer; however, we would only need to repeat the same statements, as the essential aspects of what has been said about Pythagoreanism regarding numbers are summarized here. It can even be said that this is the official doctrine of its critics and exegetes, as authors throughout time have done nothing more than monotonously repeat the same affirmations, with many so-called Pythagoreans not exempt from this error.

Due to the lack of sufficient writings from the early Pythagoreans and the distortions caused by minor disciples, who not only altered Pythagorean legend but also his ideas, scholars naturally encountered great difficulty in examining this thought. Some, like Zei-ler, claimed that it is difficult to separate what truly belongs to Pythagoras from what belongs to his later disciples. This led some to an extreme position, as is the case with Reinhardt, Frank, and others, who argued that Pythagoreanism in its beginnings was merely a mystic-religious sect, similar to the Orphic-Bacchic Thyades, in which Pythagoras himself was nothing more than a thaumaturge without any scientific character. According to this view, it was through Philolaus and Archytas that Pythagoreanism entered the realm of speculative science. For Dõring, Pythagoreanism penetrated the scientific field through Alcmaeon, or through Philolaus as Covotti believes, or solely through Archytas as Burnett thinks.

Such conceptions suggest that Pythagoras was merely a moral and religious reformer who found a suitable environment in 6th century BC Greece. Others seek to reconcile these extreme opinions, like Mondolfo, presenting Pythagoras not only as a mystic-religious figure but also as a philosopher, drawing on the positive aspects of studies by Bumett, Zeller, Joel, Stenzel, Rey, Jaeger, and others.

The similarities between Pythagoreanism and Orphism allowed these affirmations to be substantiated. Undeniably, there is a religious impulse in Pythagoreanism, and the language of religion is undoubtedly present. However, the symbolic foundations of Pythagoreanism, as seen in the opening verses of the “Golden Verses,” reveal that the language of religions was merely a symbolism of the divine language. When humanity loses the sense of the symbol and the meaning of things, it falls into profane language. Thus, there were three languages belonging to the three initiatory degrees: the profane, the religious, and the divine.7

Regarding numbers, almost all expositors of Pythagoras agree, following more or less the Aristotelian line, that numbers are the essence of things and not merely the substance of things. In this case, things are composed of numbers, and these numbers, which are their elements, constitute a number that is the form. Thus, the form is a number, but so is the primary substance, the matter. “The Pythagoreans conceive things as numbers because they conceive numbers as things,” says Aristotle in Metaphysics. And he continues, “And furthermore (lá méi álla), since Nature seemed to resemble numbers entirely, and since numbers are the first (proton) in Nature, they supposed that the elements of numbers are the elements of things” (Met. I 5:958 b 15). In passage 986 a 15, Aristotle says, “Now in this respect, it seems that they (the Pythagoreans) also consider number to be a principle, both as the matter of beings and as constituting their modifications and states.” In other words, as the material and efficient cause of things. After caricaturing Pythagorean concepts in this way, it was easy to dismantle them with slight blows, as Aristotle intended, although his statements always contain a reservation, as he consistently says “it seems” (hanontai dè…).

For Aristotle, the supreme Unity has extension, and numbers, which for him are always quantitative, are the things themselves. Among the officially recognized scholars of Pythagoreanism, numbers were not the “models” of things, as will be seen later in Plato, but rather, and solely, the things themselves. Thus, Pythagorean mimesis (imitation) would be subsequent to Pythagoras (which, as we will see, is unfounded), and Plato would construct a new Pythagoreanism in this manner. The geometric reproductions of numbers made by the Pythagoreans, solely intended as didactic examples for those initiated in the degree of paras keiê (preparation), become the definitive ones. All manuals and works by academic expositors of Pythagoreanism repetitively restate the same thing, not neglecting to replicate Aristotle’s tone of superiority and regarding Pythagoras as a poor devil of Philosophy, a naive thaumaturge. Thus, the symbolism of numbers found in Pythagorean works, which only served to pave the way for initiation, is no longer the symbol but the symbolized.

Indeed, one characteristic of periods of intellectual decline is the loss of the significance of symbols, which come to be considered as symbolized, as was already evident in the time of Socrates, Plato, and Aristotle when Greece was overwhelmed by inevitable decline, as we observe in Western thought today. Aristotle reproduces these passages without fully comprehending the symbolic meaning, ascribing to them the character of the symbolized. Thus, 1 is the limited-unlimited. But here, the coupling is not understood as a symbol, but as pure existence.

The meaning of Pythagorean krasis was never properly understood. The union of opposites was understood in the most vulgar way, and it was not perceived that there is a transimmanence, for krasis is not merely a reunion of opposites but a formal overcoming that gives rise to a new tension. Thus, krasis, the Pythagorean kosmesein, is considered to be merely a bond that unites the opposing elements of things. Krasis would be only harmony. Hence, what constitutes things are the numbers (as material elements) and the harmony that coordinates them. The universe is nothing more than the harmonization of numbers, a kind of unity of multiples (almost atomized, if not atomized).

For others, the Golden Verses are merely “a set of disconnected and disjointed sentences, compiled by Lysis.” And the Pythagorean symbols are deemed ridiculous maxims or written in an enigmatic language. These critics are unaware that enigmatic maxims are used in all secret orders, intelligible only to the initiated.

The theological thought of Pythagoreanism is presented in the most ridiculous manner. According to these critics, Pythagoreanism never reached the concept of a single, transcendent God. The fragments we presented from Philolaus and Archytas demonstrate the opposite. It is enough to cite Diels 44B20, where Philolaus describes God as the Lord of all things, unique, eternal, immutable, motionless, always identical to itself. How, then, can one conceive that this Monad, through division, generates all other beings, as many claim from their academic pulpits?

Pythagorean fragments on number

Here we present a sequence of Pythagorean fragments on numbers, whose validity is indisputable. These fragments will serve as evidential elements to support the subsequent demonstrations we will propose regarding the theses and principles.

“Arithmou dé te panfepeoiken” (“Everything is arranged (organized, constructed) according to number” - Phrase attributed to Pythagoras, according to Aristoxenus of Tarentum).

“Pythagóras panta ta prágmata apeikathôs tois arithmois” (“For Pythagoras, all things imitate (are modeled, copied by) numbers” - Phrase by Pythagoras, cited by Plato).

These two fragments belonged to the Pythagorean Catechism.

“Pythagôran mathein tà perz tous aríthmous pará Aigyptiôn” (“Pythagoras acquired epistemic knowledge of numbers through the Egyptians” - Affirmed by Aristotle).

“panta tà gignoskómena arithmòn exonti” (“All things become known through numbers” - Fragment 3 by Philolaus, cited by Diels).

“… ay. aiüon íên kai tou emai ôs ôroi oionai stigmat tôn megethôn” (“Numbers are the causes of substances and being, as limits determine magnitudes” - Phrase by Aristotle in Met. 1092b-8, referring to Pythagorean Eurytus of Tarentum, a disciple of Philolaus).

“… pânta là prágmata apeikázôn tois arithmois” (“All things are made in imitation of numbers” - Phrase by an anonymous person, cited by Diels).

“… arithmon stoikeia tôn ontôn stoikeia” (“The principle of number is the principle (element) of being (entity)” - Phrase attributed to Pythagoras).

ORPHIC HYMN DEVOTED TO NUMBER ACCEPTED BY THE PYTHAGOREANS.

“From the immaculate recess of the Monad to the sacred name of the Tetraktys, from which truly emerged the fruitful mother of everything, which, more important than everything, encompasses everything, unshakable, eternal, known as the Pure Decad…” 8

There is also what refers to the Trinity and the Transcendent Unity of the One God, symbolized by the triangle with its three sides, where the fourth side represents the figure given as totality.

Based on the sentences we have just reproduced, extracted from the most reliable documents of the past, we conclude that for Pythagoras there were two numbers: the one that exists in things and the one that things copy, which serve as models for them.

Aristotle, to whom we owe much of the confusion surrounding Pythagorean thought, in Book Alpha of Metaphysics, where he examines them in a general way, but actually referring to the works of authors he knew, who, being lesser, gave him a partial view of Pythagorean thought, concluded in 989 b. 30 that they admitted mathematical entities not belonging to physical things, as they did not classify them among beings with motion.

Mathematical things were thus immovable and immutable. He recognized that the Pythagoreans did not reduce all reality to sensible reality, admitting a higher reality than that of physical things. But he claimed not to understand (990 a. 10) how these mathematical things, which would be numbers, could operate generation and corruption without motion and mutation. These are his words: “The Pythagoreans do not provide us with any clarification, nor do they explain how they can operate generation and corruption, or the revolutions of the celestial bodies that move in the sky.”

They also did not explain the lightness and heaviness of bodies. Nor did they explain the causes of beings and the becoming of the material universe, for he asked, “Is there no other number outside this number from which number is composed?” And further on, he asked, “Is the number that we should understand as representing each of these abstractions the same as the one in the universe, or is it a distinct number from it? Plato asserts that it is another number. However, he also believes that all these beings, as well as their causes, are numbers; only, for him, intelligible numbers are causes, and the others are sensible.”

Aristotle noted that the Pythagoreans confused the immanent numbers of things (sensible numbers) with the numbers transcendent to them. Thus, he faced the following dilemma: how could numbers simultaneously constitute things, be the things themselves, and be the cause of their existence? This question from Aristotle was also posed by Silvester Maurus, and it led to the affirmation that the number would be the cause of itself, which is absurd. It is evident, from the start, the influence of Aristotle’s empirical scheme in understanding numbers. And the dilemma he faces regarding Pythagoreanism is more of a subjective origin than an objective one, as we will show.

Plato, who cannot be denied, is an initiate of Pythagoras, spoke of the distinction between number in an eidetic sense and concrete number, the number in things. And if Plato’s affirmation were not enough to justify this thesis, the previously cited sentences would suffice, as they show that all things are arranged, ordered, constructed according to number, and in another, things of our experience, sensible things, are copies of numbers.

The word “pragma” indicates the things made, the effects, just as “praxis” indicates the fact of action, the exercise of action, the act of doing something. These realized, accomplished things are modeled by numbers, for the word “apeikathos,” which comes from the verb “apeikazo,” means to copy, represent, figure according to a model, and “apetkasia” means image, representation. In this case, sensible things are constructed by numbers and, in turn, copy the numbers. Thus, there is the number that exists in the thing, “in re” (concrete), and the number that precedes the thing, “ante rem,” which the thing copies (“eidos”). Let us call the former concrete number and the latter eidetic number, and we will have perfectly translated the thought of these sentences.

From there, it is clear that Aristotle’s criticism is not justified, for things are not causa sui ipsius (the cause of themselves), as these things, which are arranged, ordered, constructed by numbers, copy the eidetic number, which corresponds to the Platonic form. This number is immutable and eternal, just as the Platonic forms are immutable and eternal.

The other number exists in things that undergo mutations. But even these, which exist in things and are constituent elements of a totality, in turn, copy eidetic numbers. It is easy to conclude from this that there are immutable numbers. Thus, a wooden triangle has its concrete number, the number that exists in the relation of the things that constitute it, but this triangle copies the form (eidetic number) of triangularity.

We have demonstrated in “The One and the Multiple in Plato” and in “Treatise on Symbolics” that Plato’s theory of participation, expressed in different terms, has an identical content to the Pythagorean theory of imitation. For the participant, by participating (metexis), imitates what is participated (mimesis). In those works, we have demonstrated, in an apodictic manner, the predominance of Pythagorean thought in Plato, just as in this work, we want to demonstrate the ontological foundations of that thought, which will further serve to affirm that Plato is one of the most faithful interpreters and disciples of Pythagoras.

By examining the Orphic hymn dedicated to number, from which we have reproduced an important part, we arrive at several conclusions that are fundamental for subsequent analyses.

No one can deny the intimate relations between Pythagoras and Greek Orphism. It is true that upon arriving in the Peloponnese, as the legend tells us, he found Orphism in a state of decline. But this decline, if it had already affected the majority of the followers of Orphism, had not yet destroyed it entirely, and a Orphic center remained immune to the decline taking place in Greece. All biographers of Pythagoras tell us about this famous passage in which he had contact with the great priestess Theoclea and moreover, he was received by the Orphic priests as a great initiate and master, and his advice was heard and followed.

Furthermore, it is known that Pythagoras never distanced himself from the deepest roots of Orphism, and this hymn was always considered indispensable in Pythagorean rituals, and it was constantly repeated. Therefore, we affirm that this hymn is undeniably incorporated into the structure of Pythagorean thought, and from it, we can dialectically deduce consequences that are consistent with its genuine thought.

"From the One (monad) until the attainment of the sacred number of the tetrad, it shows us that the One ontologically precedes the tetrad, that is, the One precedes all arithmoi. This precedence is ontological and not chronological, and we say this because it is the tetrad that is the pure decade (the ten fundamental laws that encompass everything). 9 From the tetrad emerges the fertile mother of all things, she who conceives all things and encompasses all things, unshakable, without undergoing mutations, eternal. From her arose the immortal gods and men, symbolically representing the maximum spirituality and the minimum; and this mother, who conceives all things, is the pure decade, outlined in the ten laws of Pythagoreanism.

This thought demonstrates to us that the One, as the source and principle of all things, transcends the decade itself. This decade derives from it ontologically. This One is being, for if it were not, it would be nothing, and it would affirm the absurdity that all things would have arisen from nothing, and that nothing could have created things, which would automatically affirm it as being, lending it efficiency.

Number for Pythagoras

The word “number” comes from the Latin word “numerus,” which in turn comes from the Greek word “nomos,” meaning law or norm. The corresponding word in Greek is “arithmos.” It comes from the term “rhythmos,” from the root “rhe,” from which “rheo” is derived, from the verb “rhein,” meaning to flow. Therefore, there is a relationship between number and rhythm. There is an analogy in which both are identified. The flow of creation implies number.

“Rhythm is perceived periodicity. It is the measure by which such periodicity deforms in us the habitual flow of time. Thus, every phenomenon perceptible to our senses stands out from the set of irregular phenomena… to act only upon our senses and impress them in a way that is completely disproportionate to the richness of each acting element,” writes Pythagorean Pius Servianus. Matila C. Ghyka synthesizes it with these words: “rhythm is the experience of the ordered flow of movement.”

Thus, rhythm is to time as symmetry is to space, emphasizes Warrain.

Spatial harmony (extensivist) is symmetrical; temporal harmony (intensivist) is rhythm.

Pythagoras said, confirmed by all later Pythagoreans, that arithmos was “posotetos Khyma ex monadon synkeimenon,” that is, the moving series that springs (flows) from the Monad.

Arithmos is thus something of movable things, of things that undergo mutations of any kind; that is, those that undergo the mutations already studied by Aristotle. There is arithmos (number) where there is generation and corruption, where there is increase and decrease, where there is alteration, where there is motion (translation). Therefore, all finite things, which constitute the series of created things, are numbers; they have numbers.

Every finite being is characterized by composition, for the only absolutely simple being, of absolute simplicity, is the Supreme Being.

The One (Hen Proté = a first one) is not a number, just as the Hen-Dyas aóristos (the one-dyad-indeterminate) is not, for this one, being generated by that one (and note that it is generated and not created), is still that one in its outward procession. The generation of the Hen-Dyas occurs through an inward procession, still within the Supreme Being. Plato’s Hen-Dyas (One-multiple) is the One in its creative activity, which creates the indeterminate dyad (the determination, which is the formative act of Aristotle, and the determinability, which is the materiable potency). Number will arise in the opposition between determination and determinability, for it is the moving series that flows from the Monad, the product of the relations between the opposites in universal substance. This is what we will demonstrate shortly, after outlining some essential points, such as those we will reproduce from our “Treatise on Symbolics”:

"Aristotle defined number as multiplicity measured by unity. But in this sense, it is soon noticed that the Aristotelian concept is merely quantitative.

In our “Theory of Knowledge,” we studied, albeit in general terms, the concept of number for the Pythagoreans, who undoubtedly were and are the ones who studied it best.

In the Pythagorean sense of the degree of teleiotes, the degree of perfection for the initiated, number is not only the measure of the quantitative by unity, but it is also form, as the intrinsic proportionality of things, and can be taken, as it really is, in various modalities.

Summarizing what we wrote at that time, we can say the following about Pythagorean thought:

While number is commonly nothing more than an abstract expression of quantity, they judged that within this conception, the conception of Pythagoras was also included.

But if he also saw number in this way, he did not see it only in this way. *1

The word “number” comes from the Greek term “nomos,” which means rule, law, order. However, he used the word “arithmos” for number in a generic sense.

Order is the relationship between a whole and its parts, and if we consider that where this relationship between the whole and the parts exists, there is a certain coherence, we see that the idea of order becomes enriched.

For the Master, number is also this order, this coherence, which gives the physiognomy of tension to a whole.

In later mathematics, in our era, we see that number is not only quantity but also relation, and even relation of relation, that is, function.

For him, number always encompasses the numerous, for it demands a relationship, and in every relationship there is a demand for more than one. The One is not number. The One is the whole. The Absolute is the One. (One should not confuse it with arithmetic one).

“The unity is the opposition between limit and unlimited; the unity serves as a moment of tension and approximation of two kinds of realities.” It is a Pythagorean postulate.

We can form any conception of essence, but in all of them, one note is indispensable: in essence, there is always what is indispensable for a thing to be what it is.

For a thing to be what it is, it must have an order; or rather, a relationship of the parts to the whole, a certain coherence different from the others so that it can be what it is, and not what other things are.

Is this order not number? We can say: all things have their number (arithmos) or their order, their essence; therefore, every concept is a number.

In order to experience his thought, we need to rid ourselves of this superficial conception that number is only what points to the quantitative aspect. No; number points us beyond the quantitative to the qualitative, the relational, the modalities, values, and other categories.

Thus, arithmos (number) is quantity, relation, function, tension, law, order, rule.

“All known things have a number because without it, nothing could be known or understood” (Philolaus, frag. 4).

If we pay attention to the facts that constitute our world, and in this concept, we should include all bodies and psychic phenomena, we see that they do not all constitute a coherence, or, to use our language, static, stagnant, inert tensions, but rather dynamic tensions, undergoing processes, moving from one state to another, taking a direction.

Therefore, number is also process, rhythm, vector, flow.

The facts that constitute the world sometimes appear similar to each other, sometimes different, as they also show us that sometimes they complement each other without repelling each other, sometimes not.

When two opposing facts are placed facing each other and form a relationship, a concordance, an adjustment, as if they constituted something new, they harmonize.

Through music, we all have an experience of harmony.

Pythagoras saw harmony as the ideal point, already revealed by nature itself, for all facts, including those of man.

Harmony is the result of the adjustment of opposing aspects. Harmony can only occur where there are qualitative oppositions. Two equal beings do not harmonize; they only “symmetrize.” In order for harmony to occur, there must be other differences, distinctions other than numerical ones.

Our universe is composed of different units, and when they fit together, they achieve harmony.

In aesthetics, he proposed that we should not only seek the harmony of symmetry but also the harmony of opposites in motion (khiasma). And it was through this great thought that Greek art, by realizing it, succeeded in creating something new in the field of aesthetics, which effectively contributed to the emergence of the so-called “Greek miracle.”

Pythagoras observed, studying harmony, that when certain relationships are obeyed, it is verified. These relationships constitute the so-called “golden numbers,” which play an important role in all arts and their higher periods.

Thus, harmony is the highest ideal of the Pythagoreans, which consists of adjusting the diverse elements of nature.

Arithmos is also harmony.

He also found that certain combinations, obedient to certain numbers and under certain circumstances, are more valuable than others.

Therefore, numbers are also values because they reveal values, possessing, when realized, a power capable of effecting something beneficial or harmful.

Since values can be both positive and negative, and since through numbers we actualize and bring forth powers, numbers are also magical. The word “magic” always encompasses the idea of a greater power that can be awakened.

Supreme instruction, superior knowledge of man and divine things (Mathesis), is an activity; mathema is study, knowledge.

The One (Hen), which is alone (Holos in Greek, alone), is the emanating source of everything. The arithmoi arkhai (supreme numbers) are the supreme principles that emanate from the One. From the cooperation of these arithmoi arkhai, known only to the initiated, and which are the supreme powers, arises the organization of the Kosmos (which means universal order). [Note the influence of the arithmoi arkhai on the Platonic forms (eide), which are nothing more than symbols of the Pythagorean arkhai, esoterically expounded by the author of the "Republic."]

The One, as the supreme source emanating the arithmoi arkhai, generated the One. The One is act, pure effectiveness, absolute simplicity, therefore pure act. Its activity (verbum in Latin) is of its own essence, but it plays a role because in activity it is always itself (ipsum esse of the scholastics), although it plays another role (persona = hypostasis), that of activity, but it is the same substance as the supreme One, to which it is united, fused by love, which unites the One to the One, forming the first Pythagorean triad, which, when studied closely, differs little from the Christian trinity, expounded by Thomas Aquinas.

The One, plus the One generated by it and the love that unites them, form the Pythagorean triad, symbolized by the sacred triangle with equal sides."

In the emanation (procession ad extra, as the previous one between the One and the One and love, the procession is ad intra), the Two, the Dyad, emerges. Being takes on the extreme modes of being, which, being inverse, are identified in being. With the emergence of the Two, which becomes heterogeneous, all numerical combinations (arithmetikai) are possible.10

The arithmos is also a concept; for the concept is an anth-mos of notes (skhema by aphairesis, that is, schema by abstraction).

Therefore, we have:

it is quantity (arithmos posotes) it is quality (arithmos timos) it is relation (arithmos poiâ skesin) it is function (arithmos skesis) it is law, order, rule (arithmos nomos) it is process (arithmos proodos, or kethados), whose inverse movement is conversion (episthrophe), which realizes effective return (anados). These arithmoi arise from the arithmoi archai, and are produced by the emanation of the One, and they return to the One after combining with other arithmoi.

Fluxes (arithmoi khyma) by which the Pythagoreans mathematized studies on emanations and flows of any kind (of light, for example).

The rhythmic number (arithmos rythmos), the periodic number; sets are numbers (arithmoi plethos); and when they become tensions, they are arithmoi tonoi.

Pythagoras was also concerned with the conjunction of numbers that produce transient qualitative aspects, different from the component elements, such as the percussion of different notes forming a new qualitative aspect. Hence the symphonic numbers (arithmoi symphonikoi), which in turn form the numbers of harmony (harmonikoi arithmoi).

Proportions of all kinds lead to the construction of the analogical number (analogikos arithmos).

There were also other numbers belonging to Pythagorean mathematics.

We also have the numbers of punctual growth of the Pythagoreans, which are nothing more than Dedekind segments, the so-called dynarnei symmetroi (commensurable numbers in power) and others like the sympathetikoi arithmoi and antipathetikoi arithmoi, which are completely different from the episthemikos arithmos, the scientific number, the number of profane mathematics.

Only when the number is placed in this true Pythagorean sense can its symbolism be understood, which is, moreover, the subject matter of Aritmosophy, which studies its significance. However, one must not forget that in various religious myths, the number, taken in this sense, may appear at first glance to have a value in itself when, in fact, as we will have the opportunity to appreciate through the analyses we will proceed to carry out, it is not a power in itself, but only an indication of power, which refers to the so-called arithmoi archai, the archetypal numbers, whose study we will undertake from a general perspective.

Natural phenomena and their laws lead us to coefficients that are numbers, and all things in the cosmic world are arithmomically realities that imitate certain numbers. Crystals, plants, humans, stars, sounds, spectra, chemicals reveal numbers and a numerical law, which is the same. Mathematics shows us how number is an extraordinary instrument for our knowledge, to the point that when we cannot reduce a phenomenon to numbers, we feel as if in a vacuum.

As Pascal showed, “there are common properties to all things, whose knowledge opens the greatest wonders of Nature to the mind.” And it is these “common properties” that analogize the facts to one another and allow us to capture the references to numbers, indicating to us the symbolism that emerges through time.

Leibniz recognized that “mathematical language” could communicate many of the secrets of nature, and it was not uncommon in philosophy to repeat that mathematics is the language of God, and that divinity constructed the universe as a perfect mathematician, whose symbolism we see in many religious artistic manifestations, including Christianity.

Numbers have been studied since ancient times, and we find works and references among the Vedas, the Egyptians, the Chaldeans, the Babylonians, the Greeks, and the early fathers of the Church.

In general, for the Pythagoreans, numbers were entities intermediate between the Supreme Being, the One, which is not a number, and the other beings, in which, being created and consequently finite, number is partly a negative limit, as it indicates where this entity is what it is, and positively, what it is, its quiddity, as form, morphê, or eidos or skhema, in the Aristotelian sense, is number, which Aristotle partly understood.

The Aristotelian form corresponds to the Pythagorean form, which is the law of intrinsic proportionality of beings because if this is this and not that, it is because it has a certain intrinsic proportionality, which is its arithmos.

Saint Augustine emphasized that “the unintelligibility of numbers prevented the understanding of many figurative and mystical passages of the Scriptures.”

For genuine Pythagoreanism, we can consider the set of created beings according to two triads, the lower and the upper, which offer us a clear vision of reality.

If we start from sensible things, like beings most directly in contact with our senses, it is easy to perceive that they are constituted of a geometric structure revealed by their dimensions. These geometric structures can be reduced to mathematical numbers (arithmoi mathematikoi), as algebra, algebraic geometry, etc., do, for example. In this way, the lower triad is formed by mathematical numbers, geometric structures, and sensible things that can be schematized by mathematics, as is actually done.

However, the schematic possibilities of knowledge of things are by no means exhausted if we consider them only within this triad. And this becomes clear immediately because things reveal an intrinsic proportionality, a scheme that makes them what they are and not something else, in short: their form.

These forms (commonly called Platonic ideas) constitute the point of connection with the lower triad. The forms are no longer objects of sensible knowledge but of intellectual knowledge since they require an abstracting activity of the mind that separates from the phantasm (phántasma, that which appears, emerges, is seen; phaos, light) the eidetic schema (eidos, morphê) of the thing, that by which (quo) the thing is what it is and not something else, this intrinsic proportionality, this arithmos plethos (this number of proportional set), which reveals an arithmos tonos (a tension, a coherence of its parts with the whole).

It doesn’t matter which plane it is considered on. And it is easy to understand: that painting is a portrait, a human figure, with harmonious colors. If viewed under a microscope, it would only represent granules of various colors on the canvas, and if it did not allow, in that state, the same overall view, the capture of its arithmos plethos, it would not prevent the viewer from seeing it in this set of coordinates as a portrait of a particular person. Its form, in this relation, is this, and in another, it will present a heterogeneity of form. If we see it here as a whole (plethos), in another position, we would see it as a heterogeneous being of other totalities, without excluding that, in this set of coordinates, it constitutes a coherent whole, a tension different from the tensions of the elements that compose it, which, in turn, can form other tensions with heterogeneous elements, and so on.

This point, of fundamental importance in the “General Theory of Tensions,” reveals to us that substantial forms, above all, are the arithmos of tension, which, in turn, is a coherent schema that implies the heterogeneous because, as tension (tonos), it is one and homogeneous, but heterogeneous in its parts, which are transcended by the whole, which forms a unity qualitatively different from the component parts, which, in totality, can only be considered quantitatively.

Thus, form is not a sensible being, it is not a subsisting thing per se, but it is given in the thing because the thing is what it is by the form it has, that is, by the schematics that present the intrinsic proportionality of its parts.

Plato reached this point in his dialogues because this is the exoteric field of Pythagorean thought.

These forms are imitated by things, for they are of this or that. Thus, in a wooden or iron triangle, for example, triangularity is the scheme of the intrinsic proportions of this wooden triangle, which is a triangle not because it is made of wood but because it participates in the proportionality of the angles that constitute its essence.

Thus, the eidetic schema of the triangle is the law of intrinsic proportionality of triangularity, imitated (in the Pythagorean sense) by this or that object or participated in (in the Platonic sense) by the same.

But this or that object is not triangularity itself but only triangles because they participate in triangularity.

This is not a subsistent being per se, like something that occupies a place. Triangularity does not have a where or a when. It does not happen here or there. It is and subsists in being, in the order of being; better yet, in the infinite power of being. Triangularity is a potential-being that sensible things here or there imitate, triangularizing themselves through the intrinsic proportionality they possess. And thus, the noetic-eidetic schema that we construct in the mind is the enunciation of this law of triangular proportionality, in intentional terms, in noetic terms, according to our mind and its capacity for assimilation and schema construction, which captures triangularity in facts. Therefore, for Platonism, as well as for Pythagoreanism, the eidetic schema of the thing belongs to the omnipotence of being; it is, therefore, ante rem. In the thing, we have the concrete schema through imitation (mimesis); that is, in re, and in the human mind, we have the noetic-eidetic schema, post rem (after the thing).11

One cannot properly understand Platonic or Pythagorean thought (since Plato is undoubtedly Pythagorean) without placing them in the terms we have just outlined.

Thus, with the forms, we have the first element of the higher triad. But the forms reveal an ontological structure that corresponds, in the eidetic field, to the geometric structures in the field of the lower triad, that of sensible things.

The intrinsic proportionality of things, the eidetic arithmos, presents an ontological structure, while sensible things present an ontic, singular structure.

This ontological structure reveals the arithmoi archai, the archetypal numbers, which are immediately below the One, the Supreme Being, the Divinity, which is not a number because number belongs to multiplicity, to what is dual, to the dyad, as seen in the esoteric thought of Pythagoreanism, which we will not discuss here for now.

In this way, we would have the two triads arranged as follows:

  • Higher Triad:

    • arithmoi archai (archetypal numbers)
    • ontological structures
    • forms (eidetic arithmoi)

  • Lower Triad:

    • mathematical numbers (mathematical arithmoi)
    • geometric structures
    • sensible things

In the realm of symbolism, we could therefore say that sensible things participate in geometric structures, figures, mathematical numbers, forms, etc. Thus, things can symbolize the highest, even reaching the arithmoi archai.

We can symbolize a sensitive being through figures, which are geometric structures; for example, a cubist expression of Napoleon. We would have an apparent inversion, as the participant would be symbolized by what is participated. But it is not quite so. When we symbolize Napoleon through a cubist figure, there is an association, through the figure of the Corso, reduced to a figurative scheme. This is not a complete symbolization but a copy, an imitation of its geometric structure. The symbol, as we have seen, includes more in its language because it addresses the eidetic, for example, when we symbolize Napoleon with an eagle.

The symbol contains something imitative, as there is no assimilation without a corresponding accommodation, which implies imitation. But if imitation is a co-principle of the symbol, it is not sufficient in itself to indicate its essence because otherwise, we would have to include all imitations in the species of the symbol.

If the figurative can symbolize, as the cylindrical shape symbolizes the phallus, there is not truly the revelation of the hidden here, which is also a characteristic of the symbol it points to. This does not imply that the figurative cannot symbolize, but it only does so partially because it points to the figure of the immediate symbolized. It symbolizes by pointing to the symbolized and by making present a note or notes of it, not contained in the symbol, which are hidden because they belong to the symbolized. The symbol points, through the imitative, to the symbolized, but it does not intend only this, but what belongs to the symbolized, not contained in the symbol. The symbol is thus always less than the symbolized, taken hierarchically, because the symbol participates in something of the symbolized, which is the participated, and participates to a lesser degree than the other has in fullness.

The symbol is a means of making present what is absent. Therefore, it is not only the imitative that must be considered but what is more in the symbolized.

This equality exists, but it implies the presence of what differentiates them. The aesthetic pleasure that the symbolic provokes in art lies in this aspect. In itself, the work of art expresses what it is in its figurative aspect, but as it points beyond and provides enjoyment of a fullness, it offers aesthetic pleasure that goes beyond mere sensory apprehension because otherwise, we would consider art only from the angle of aesthetics, from the angle of the senses, without considering it from the angle of the spirit, which is important.

Aesthetic emotion is complex, not only immediate intuition of what it expresses externally but also apophantic intuition, thus mystical, which allows for an insight into the intrinsic nature of the work of art, which is experienced to varying degrees according to the viewer’s capacity. This is the reason why art can never be exclusively realistic in the abstract sense that term assumes as a copy of reality. In any case, this same reality speaks a symbolic language, and that is why realists are “impossible realists” because, whether they want to or not, they go beyond their conscious intentions. Thus, all art, in its means of expression, is realistic, but it is symbolically transcendent despite the artist’s intentionality. Therefore, it allows for a symbolic interpretation often in disagreement with the artist’s “original intentions,” which do not fail to reveal the “second intentions” that he is not always capable of perceiving. 12

We can now classify numbers (arithmoi) within the sciences that include them as a material object. Thus, we have:

(“arithmói”) Numbers:

  • Pure = “arithmologia”
  • Scientific = “epistemikós arithmós”
  • Sensible = “arithmós logistikós” (ordinary mathematics, calculation number).

Scientific numbers, according to Nicomachus of Gerasa:

  1. Limited multitude (posótes). It is the quantitative number, the abstraction of quantity.

  2. Composition of monads (plethos, tonós). Classes of classes.

  3. Flow (khyma).

Nicomachus gave the following definition of the second species:

“The Pythagoreans considered all the terms of a natural series of numbers as principles, so that three (the triad) is the principle of three among sensible objects, and four (the tetrad) is the principle of all fours, etc.”

This definition is similar to the one offered by some modern logicians of numbers as “classes of classes.”

The pure numbers, which constitute the subject matter of Arithmology, are defined by Nicomachus as follows: “The principles (arkhai), in the sense of origins of Number and of everything and all things, are the ‘Same’ and the ‘Other,’ or 'the quality of being the same thing or being another thing.’”

The relation between two objects or magnitudes is arithmos skesis.

And harmony, according to Philolaus, is “the unification of the diverse and the placing in agreement of the discordant.”

Thus, the essences of things, the Forms, are also numbers. There are those who identify form with essence, and these with number. However, despite everything, there is a need to distinguish. The eidetic form, as an exemplar in the order of the Supreme Being, is ante rem. The forms in things, the concrete forms, in re, are the laws of intrinsic proportionality that constitute the formal structure of sensible things, Plato’s eidola (little forms).

The eidetic noetic forms, which fit into the logical definition, are in intellectu post rem; they are constructed according to human intentionality, which is nothing more than concepts. These can be conceived logically when emptied of all pragmatic content, taken only in their logical structure, following the Aristotelian norms, which fit into the definition, which is equal to the proximate genus and the specific difference, and the historical-social concept, a noetic form in which there is the contribution of human experiences, whose variation is immense, and it is up to Schematology to study.

By placing number (arithmos) in this genuine Pythagorean sense, the countless false interpretations are immediately dispelled; the true thought of the master from Samos is definitively clarified.

It is understood, then, that Mathematics, in the sense of Pythagoras, is not the common mathematics that we study in schools. The latter is included in the former but does not encompass the entirety of mathematical thought.

In order to avoid the common confusions, we prefer to call this Pythagorean theorizing Metamathematics since the term mathematics is definitively compromised due to its vulgar connotation.

In our comments on Aristotle’s “Metaphysics,” we have the opportunity to examine the errors that pervade Aristotelian analysis and that stem from the great Stagirite’s lack of knowledge of legitimate Pythagoreanism, which is acceptable since we understand that Pythagorean thought was, in his time, and still is, forbidden thought that deliberately remains disfigured, distorted, and falsified. It is not surprising, therefore, that many are Pythagoreans without knowing it. This is the reason that leads us to affirm, and we will still demonstrate conclusively, that Pythagoras has fertilized Western thought more than anyone else, and the presence of his theses is found throughout human speculative work.

The foundations of the Socratic-Platonic dialectic, which we study in “Concrete Philosophy,” have their bases in the arithmetic relationship examined by the Pythagoreans.

The arithmetic relationship reveals itself:

  1. through the perception of a functional relationship or a hierarchy of values between two objects of knowledge;

  2. discernment or comparison of values, qualitative or quantitative (a/b). Form of a fraction with properties of a fraction, which is equivalent to the quotient of a divided by b; that is, number.

According to Euclid, proportion is “the equivalence of two relationships.”

In analogy, at least three terms are required (proportional analogy):

a:b :: b:c

And it is in the logos that the analogy takes place because it is the logos that analogizes; just as a is to b in the proportion in which b is to c.

Plato said in “Timaeus” that “it is impossible to combine two things well without a third; there must be a bond between them.” This bond is the logos.

And whenever there is an analogy of proportion, it is possible to derive a resultant for the Socratic-Platonic dialectic between two terms.

Let’s illustrate this with a passage we wrote in “Concrete Philosophy”:

"Between two analogized particular premises, Socrates induces the analogizing logos (for Socratic-Platonic dialectic is predominantly inductive, unlike Aristotelian dialectic).

Let’s consider the classical example: The lion is the king of the desert.

King Manuel is the king of Portugal.

From these two particular premises, nothing can be deduced within the Aristotelian canons.

But within the Socratic canons, it is possible to induce, provided we find the analogizing logos. Aristotle was right when he said, in the “Metaphysics,” that Socrates was the creator of inductive reasons, of inductive logoi.

These two premises can be reduced to a proportion (analogy).

Just as the king rules his kingdom, the lion rules the desert. But if there is similarity between them, we can still highlight the differences, for the reigning of the king is different from the reigning of the lion, but ultimately, through Socratic inductions, we arrive at an analogizing logos, which is this: the relatively more powerful always dominates in their respective field of activities. Now, the lion is the relatively more powerful in the desert, thus dominating in its respective field of activities, just as the king dominates in the kingdom.

This analogizing logos can ultimately be generically reduced to the analogizing logos that “the agent acts proportionately to its nature and proportionately to the field of its activity.” This proportionality, in turn, is generically reduced to the logos that “the agent acts and the patient suffers proportionately to their natures.” This occurs through the law of Being, already induced by the theses we have examined, for if the agent were to act beyond its nature, the supply would come from itself or from another or from nothingness. If from itself, then it would already contain it, it would already be powerful, and therefore its action would be proportionate to its nature; if from another, its action would still be proportionate to its nature and to the supply from another, which would then be the agent. From nothingness, it is absurd. Therefore, the fitting and congruent is for the agent to act proportionately to its nature, that is, action is analogized to its nature, to itself."

According to Stobaeus, cited by Aristoxenus, the first notion that Pythagoras taught his disciples is that of even and odd before delving into the examination of numbers. Current studies on infant noogenesis and the examination of modern primitives reveal that even and odd precede, in a certain way, the formation of the concept of numbers, for if two objects are taken out of a group of 18 objects, the primitive Australian will notice the absence less quickly than if only one is removed. And it is understandable that the idea of parity and oddity always presides over all human actions and is particularly present in the process of connecting stimuli from the external world to one another, as well as through psychological assimilation, which occurs through accommodation-assimilation, that is, the accommodation of schemes to facts and the assimilation of facts to schemes. Parity always presides over all human actions and is particularly present in the formation of the Self, as the individual distinguishes himself more and more from the external world through his consciousness. Oddity arises from the unexpected, the unheard of, the never seen before, that which cannot be immediately compared to something that is, in a certain way, similar. Oddity, or also disparity, is the absence of parity and manifests itself in what lacks correspondence, symmetry, equality, or even similarity.

Pythagoras stated that the even number is the apeiron, the unlimited, because between its two parts there remains nothing, while the odd number is finite, limited, because when divided into two equal parts, there always remains an indivisible unit between them, which is the even-odd.

Even:

. | .

Odd:

  .
. | .

For Pythagoras, the number is a combination, a harmony of the even and the odd, of parity and non-parity. As we will see, the number is the “scheme of participation,” where there is both parity and non-parity between participant and participated. The participation is not a physical composition but merely formal, as we have already demonstrated to be the foundation of Pythagorean mimesis, where the imitator does not physically compose with the imitated but only reproduces it formally, proportionally to the nature of the imitator. In this case, the scheme of participation is a combination of even and odd, to remain within the arithmological language.13

And the number is a harmonization of the unlimited with the limited. Indeed, the first ten numbers are the fundamental ones, as the others are only repetitions of them. Thus, the decad encompasses all numbers with their properties.

According to Philolaus, the decad (tetractys) is great and all-powerful, the source of everything, the beginning and the model of all things. It is the number of the universe, whose symbolism we study in the corresponding chapter.

Without the decad, everything is mysterious, confused, and obscure. It symbolizes the perfect and contains within itself the essence of all numbers. It has an equal number of evens and odds, and the One, which is both even and odd, is the first even, the first odd, and the first square, four. It is constituted by the sum of the first four numbers: 1 + 2 + 3 + 4 = 10.

Now, the decad is the tetractys of the 10 universal laws (logói), which are the revelation of the principles that govern the entire Universe, principles of all things. Chaignet writes: “The first ten numbers, whose decad is the limit, as the Pythagoreans say, explain the infinite variety of things, from the simple herb to the sun, from the most material reality to the attributes, modes, and properties of things, even to the gods themselves.”

We have already examined the tetractys and the various ways of considering it, according to Theon of Smyrna, but in the sense of the ten laws of Being, we will examine it in due course.

Pythagorean arithmetic is geometric, and inversely, their geometry is arithmetic because numbers are distinguished by their geometric characters. However, this distinction was based only on the first-degree vision of mathematics, which was that of the initiate in the paraskeiê degree, the apprentice degree.

Thus, numbers were represented by points and lines tending to form figures, as seen in philosophy manuals.

Aristotle, as we have seen, stated in Metaphysics that the Pythagoreans (presumably those he referred to) considered numbers to be:

  • a) the principle of all things;

  • b) the substance of all things.

But he claimed that, for them, numbers were extensive, as the Monad itself was extensive.

Thus, the number would be, in a twofold sense, the matter and form of things, or rather, the form and matter of things were numbers. And just as there is an opposition between form and matter since both are positivities placed against each other, numbers are also opposed. Even and odd, one limited and the other unlimited, etc.

Aristotle further claimed that, for the Pythagoreans, the One proceeded from the combination of two numbers because it was simultaneously even and odd. But the number proceeds from the One, and the entire Universe is composed of numbers (in Metaphysics, 986 a 19-21).

After stating these theses, it was easy for him to show how filled with absurdities Pythagoreanism was. However, in truth, Aristotle knew little about Pythagoreanism.14 Perhaps he was familiar with the works of some major Pythagoreans, but only fragmentarily, or with lesser sympathizers who were his contemporaries. It is well known that Pythagoreanism in Greece was “outside the law” and, as such, was not immune to the inherent distortions that doctrines “outside the law” undergo. There is no need to search for examples in history when even today is so fertile with them.

In Aristotle’s work, there is a clear intention to devalue all preceding thought. The criticism that Aristotle has endured throughout time is very justifiable because, as an expositor of other people’s thoughts, he was unfaithful and, in this unfaithfulness, there is an unmistakable tendency to distort. It should not be concluded from this that everything Aristotle wrote about Pythagoreanism is false. There are valuable contributions, and in our analysis of his Metaphysics, we have the opportunity to examine those points where the great philosopher vacillates and weakens, distorting other people’s thoughts. However, we must also highlight the great contribution he made to the understanding of Pythagoreanism.

For him, number, starting from unity, proceeds in two ways. Either by adding unity to itself, going from one to two, from two to three by adding a new unity, or by multiplying unity. Now, the number cannot undergo such operations if it does not simultaneously participate in Unity and Multiplicity. Numbers are composed of monads; it is a multiplicity of monads. Formally, it is a singular monad, a unified entity. It is a numbered unity, as part of a number (matter), and a numbering unity, formally unifying the number. Therefore, Aristotle affirms that numbers participate in the One, which is their formal principle, and in the indeterminate Dyad (multiplicity), which is their material principle. And if that is so, the One and the Dyad end up transcending all numbers because numbers participate in them.

While there are many errors regarding genuine Pythagoreanism, as we have already demonstrated, there is also much truth. But the truth is what there is of Platonism in Aristotle.

Strictly speaking, numbers do not arise from the One in an uninterrupted creation. They are ab eterno in the Supreme and infinite One. Multiplicity implies the ontological precedence of the One, just as all finite entities imply the ontological precedence of Being. Two does not arise from addition; it is the two of arithmetic, not the two as an eidetic arithmos. All numbers are already given in the order of being, and therefore, in a certain sense, they are infinite because the human mind could never limit them. The thoughts of the Supreme Being are, in a certain sense, infinite because its power is infinite.

Genuine Pythagoreanism, at a higher initiatory level, says nothing else. Just as we have shown in “The One and the Multiple in Plato” that forms are infinite because they exist before thoughts in the Supreme Being, their powers that know no limits, numbers are also infinite. We present an apodictic proof of this thesis in “Concrete Philosophy.”

As an empiricist, Aristotle could only understand the infinite number as potentially infinite. But since the power of the Supreme Being is an infinitely active potential because it can be everything that can be, and since potentiality can only be limited by absolute nothingness, which is absurd and has been definitively eliminated by the proofs we presented in that work, its power is potentially infinite. However, the infinite power of the Supreme Being is actuality in it, and as the numbers are possible to be realized in creatures, they are, in a certain sense, actually infinite in it because they belong to the infinitude of its actuality.

As a form, the number is not a unified set because the form, taken in itself, is not a multiplicity but a unity. If, to stay within Aristotelianism, we consider man as animality and rationality, the human form is not a unity composed of the multiple animality plus rationality, a composite being in the physical sense. The structures here are ontological, not physical. Animality and rationality are distinguished in man, but only ontologically. In reality, human rationality already includes animality. It is only a degree of perfection that includes the previous one. Thus, triangularity, as such, is not the product of the sum of sides, like this triangle, but it is a formal structure per se. Otherwise, the square would be a triangle to which an extra side has been added. While this square may have actually emerged in this way, the square does not arise from a modification suffered by triangularity. Triangularity continues to be what it is, even though this wooden triangle will now, along with another side, form a square.

We are in full Platonism, but also in full Pythagoreanism since, as we demonstrated in “The One and the Multiple in Plato,” the thought of Aristotle’s master is fundamentally Pythagorean in the true sense that should be given to this doctrine.

If material things are numbers, it does not mean that number is matter in the sense usually given to that term, the physical sense.

The indeterminate Dyad of the Great and the Small (dyas aoristos) is the aptitude for the maximum and the minimum, for more and less, for addition and subtraction.

Aristotle concludes that the One is, as a material principle, prior to the Dyad, but as a formal principle, it is posterior to it.

It is believed that speculations about the indeterminate One-Dyad arise in Pythagoreanism, perhaps since its earliest days. This is Aristotle’s thought.

In the “Pitagoric Memoirs” by Alexander Polyhistor, which dates back to the first century of our era, it is stated that such speculations date back much earlier, and referring to this theme, it is expressed as follows: "The principle (arche) of all things is the Monad. It is from it that the indefinite Dyad derives its existence, as matter for the Monad which is the cause; from the Monad and the indefinite Dyad, numbers obtain their existence."15

Now, if two is ontologically subsequent to one, it is not chronologically so. The numbers were already contained, since eternity, in the infinite power of the One, the Supreme Being, the Supreme Monad. To say, as Eudorus said, that when there was the One there was no two, that it only later emerged, is to confuse the thing that is two with the two as form, and to confuse the two as form (eidos) with the two as plethos, as the concrete form of the thing, which, being one, is constituted by two principles. The indefinite Dyad is fundamentally one, but it is indeterminately two.

If the Supreme Being can do everything that can be, it can do more and it can do less. The power of more and the power of less are indeterminate, for otherwise, they would be determined by another being that would limit them, thus ceasing to be first and infinite, because an infinite being can be one and only one, as we have proven in “Concrete Philosophy,” where dualism was totally and absolutely refuted. Or, it would be limited by nothingness, which is absurd. If the Supreme Being can accomplish, it can accomplish the maximum and the minimum of being. And accomplishment implies what is accomplished, for the infinite active power of the Supreme Being must correspond to an unlimited power of being-able-to-be, for doing is simultaneously being something done, doing implies being done, as we have demonstrated.

However, note that what is accomplished will always be limited, which implies that the potential-to-become does not include infinitude. Or rather, a realized being that has infinitude in an absolute sense is not possible.

The indefinite Dyad is thus one; it is the One-Multiple, to use Platonic language; it is the second One, which is generated by the first one, it is the creator of what Pythagoras called the universal substance, the first category of beings.

This being generated by the Supreme Being is One and is the indefinite Dyad (Hen-dyas aóristos). Therefore, the universal substance arises from the determination of determinability, in more or in less, in the maximum and in the minimum of being this or that.

If we stick to the Aristotelian concept of matter, prime matter is, as such, diadically indeterminate, as it can be informed to the maximum and to the minimum, receive maximum determination and minimum determination. And this is genuinely Pythagoreanism. It is this second Monad that gives rise to number, as is clearly derived from the thought of Pythagoras: the One generates the One, and this generates the two (the indefinite Dyad), and so on.

We do not know if Pythagoras used the term “generate” in the sense that we give to generation, according to the content of our scheme, or if he used it analogically. However, for us, he must have used it in this way, as we will show, as a dialectically rigorous consequence of the future examination of Pythagorean theses. One could translate such a passage, if we desire greater ontological rigor for the terms, by saying that the One generates the One, and this creates the universal substance, which is the indefinite Dyad (the category of opposition, which is the second Pythagorean category), and from this arises the relation that occurs between opposites.

And this happens because what receives a limiting determination is what is, and not what is not. But to determinatively determine something is to separate something from something else because, for a being to undergo such a determination, something must be outside of it, something other than it. And what remains outside is something that is (for the absence of nothing is not absence, and then there would be no such determination). 16

The undetermined dyad precedes ontologically the determination, and what is determinable corresponds to Aristotelian potentiality. And potentiality, when determined by form, is this and not that. However, it is not a reference to a being because, otherwise, it would still remain undetermined. Creatures arise from the determination of the undetermined dyad and arise by excluding what can be, for now that they are what they are, what they are not is excluded, but what they could be. Otherwise, what they lack would be mere nothingness, and there would be no limiting determination either.

The One is therefore transcendent to the second One. There is a transcendent Monad to the second Monad, which is Hen-dyas aôristos (Undetermined One-Dyad). The first is identical to the God of scholasticism and is not a number because it is not numerical, for it is absolutely simple.

Eudoro proves our assertion by showing that for Pythagoreanism, there are two planes: the supreme plane, where he places the One, the universal principle of all things, and the secondary plane, where the undetermined One-Dyad is.

And reason, Eudoro adds, is that, for the Pythagoreans, the second One and the Dyad command only a parallel series of the real and are not universal principles themselves. The double Hen-dyas ontologically requires a One principle, as we demonstrated in “Concrete Philosophy,” for otherwise we would fall into the aporias of dualism, already refuted in that work.

And Proclus corroborates our assertions by saying, “Let us not think that for this reason the principles of things should be seen as opposed terms (diereménas, literally divided). In fact, we say that these two parallel series are classified into a common genus, for above all opposition, there is the One, as the Pythagoreans also declare. Well, in truth, after the First Cause, the Dyad appeared from the number of principles, and among the Principles, the Monad surpasses the Dyad, or if you want to speak like Orpheus, “the ether surpasses chaos,” [precedes] and it is in the same way that oppositions [divisions] are realized” (in Tim. I, p. 176.6D).

From these speculations, which also constituted the subject of study for the Platonists, the Peripatetics, the Stoics, the Gnostics, and the Neo-Pythagoreans, it is concluded that there are three ones. The Supreme One, the first Monad, the One-multiple (Hen-dyas aôristos), and the One-and-multiple, the one (plethos) of composite things. And are we not here in full conception of Plato? And is this fact not further evidence in favor of the predominance that Pythagorean thought exerted over him?

But there are still other proofs. By reading the Platonic work, one can conclude, albeit hastily, that the creator (poietén)■ ordered the agitated mass of immeasurable and disorderly movements, the unordered matter (akôsmetos hylê). But Porphyry and Jamblichus have shown us that such affirmations in the Platonic work are merely didactic, for the world, for him, always existed (utou mèn ontos aei tou kosmou) and did not have a beginning in time (agénetos). Plato’s intention was only to show the value that order has alongside matter. If this were admitted, the benevolent will and creative power of the Supreme Being would be denied. We know that Thomas Aquinas also admits the possibility of a creation ab aeterno and does not consider it contrary to the principles of the Church.

The creation of matter, in this sense that we are taking, is the subject of long controversies in Pythagoreanism. Commenting on these controversies, Proclus writes:

"Aristotle demonstrated by other arguments (De Caelo-A 3,270 a 24 ss) that matter is ungenerated because it is not composed, that it is not taken from another matter, and it is not reduced, in turn, to another matter. But the present discussion, by recognizing that matter is eternal, implies the question of whether it is ungenerated independently of any cause and whether, according to Plato, it is necessary to place these two principles of the Universe, matter and God, neither God creating matter nor matter God, so that matter is absolutely eternal and independent of God, and God absolutely independent of matter, and simple. This is exactly the question; it is one of the most disputed, and I have already dealt with it on another occasion. For now, it is sufficient to show against such critics what Plato’s thought is.

That the demiurge is not the first cause of the existence of matter is evident, according to what Plato will say later (Tim. 52 d3), that in the genesis of the world, these three pre-existed: extension (khora), creation (gênesis), and the created as having sprung from it, extension as a mother. Undoubtedly, it appears from this text that Plato establishes an opposition between matter and the Demiurge, according to the characteristic properties of mother and father, and that he makes the created arise from the Demiurge and matter. But perhaps Plato makes matter depend on a class of beings higher than the Demiurge. This is, at least, what he explicitly writes in the Philebus (23 c): “We said elsewhere that God manifested in beings both the limit and the unlimited (to peras e to ápeiron),” from which results, for bodies, as for all things, their composition. If, therefore, bodies also result from the limit and the unlimited, what is the limit in them? What is the unlimited? Obviously, it is the matter that we will call “unlimited,” and the form “limit.” Therefore, as we had said, if God makes everything unlimited exist, He also makes matter exist, which is the ultimate unlimited. That is the absolutely first and ineffable cause of matter. On the other hand, since sensible properties are related to their intelligible causes, Plato everywhere makes those and these depend on each other, for example, the equal from here with the Equal in itself, and equally for all living beings and plants here, it is clear that, following the same path, he also makes the unlimited from here depend on the First Unlimited, just as the limit from here and the intelligible Limit. Now, I have shown elsewhere that Plato placed this First Unlimited, which comes before the mixtures, at the summit of the intelligibles, and from him the illumination extends from the highest to the lowest steps so that, according to him, matter proceeds from the One and the Unlimited that comes before being one, and if you wish, it also depends on being One, insofar as it, the Unlimited, is a being in potentiality. That is why matter is good in any way and indefinite, a completely obscure and formless being, because of which, for that very reason, it is prior to the forms (of visible forms, he means) and their manifestation".

“The same doctrine,” Proclus continues, “is transmitted by Orpheus (fr. 66 Kem). Just as Plato made two causes come out of the One, the Limit and the Unlimited, likewise, the Theologian made the Ether and Chaos emerge from Time, being everywhere the cause of the limit, and Chaos of the unlimited, and it is from these two principles that he (Orpheus) engenders the divine and visible worlds ... and, finally, the lowest grade of unlimited, which also includes matter”.

Therefore, matter is the unlimited without limits, the “boundless obscurity” of Orpheus. Determination implies the limit and the unlimited, what is and all that is not, for something, being something is what is, and being what is, it is not everything that is not, limitlessly, for being has no limits. The creature, when created, is a composition of limit and unlimited. Here we are in our concept of crisis, which we presented in “Philosophy of Crisis.” Everything that is a creature depends on the Supreme Being and, as such, is limited by dependence, but since everything that is has a form, has a limit, it is in turn not what it is, not being what it is, the unlimited. Therefore, the Dyad is one and an undetermined dyad, it is one in a limited way due to the dependence on the Supreme Being, but it is also unlimited due to its determinability, for it can be everything that can be finite.

In Hermes Trismegistus, the concept of matter is the same because materiality arises from substantiality. The universal substance is the undetermined One-Dyad, which can be one and multiple in what arises from it. And this is also the thought of Plato, as Proclus asserts, adding that he certainly took it from Hermes.

If we consider materiality as the aptitude to receive determined forms, the undetermined One-Dyad, which is substantiality, is not matter. Matter arises from it, as corporeality arises from it, as we demonstrated in “Concrete Philosophy.” Proclus affirms that matter arises from the First Unlimited, thus coming from the Supreme One, but this affirmation lacks a basis in light of what we have already demonstrated, so we reject it completely.

And we also affirm that this was not the Pythagorean thought, which we will discuss shortly, continuing the demonstrations that we had already presented in our “Concrete Philosophy.”

Therefore, for Pythagoreanism, there is the One, which is the principle of all things, and the undetermined One-Dyad, the One-Multiple, which should not be confused with the One-and-Multiple, the second dyad, that of finite things, whose unity is an arithmetic multitude, the unity of multiplicity.

The thesis of crisis in creation is already found in Moderatus of Gades when he says: “The unifying relationship (0 eniaios logos - the One that has a relation, logos) having wanted, as Plato says, to constitute, from itself, the generation of beings, detached from itself, by privation, quantity, after having deprived it of all the relations and forms that are proper to it. And he called this quantity without form, without division and without figure, receiving, however, form, figure, division, quality, and all analogous things.” We do not know where in Plato’s work this passage that Moderatus of Gades mentions can be found.

As for the value of the idea of the demiurge in Plato, we sought to establish its limits in our comments on the Timaeus, without implying that we will not discuss this symbol further in this work, for we are indeed dealing with a myth whose purpose is purely didactic and tendentially exoteric, without implying that Plato accepted the presence of this demiurge as a subjectively considered reality.

It remains to be seen whether, for Pythagoras, the first substance, the universal substance, is something detached from the Supreme Being or is created by it. The solution to this point will come in due course when we examine the Pythagorean theses according to the norms of concrete dialectics, as we have presented in “Concrete Philosophy.” But before reaching this point, it is necessary to examine other aspects that will greatly help us understand Pythagorean thought, as we desire.

The commonly accepted thesis is that materiality is universal substance. Let us assume that it is so, as long as it is capable of receiving formal determinations (of the eidola, the little forms, the forms of sensible things, as Plato explained).

Therefore, materiality would be a “cut-off portion,” extracted from the Supreme Being. This thesis is not Pythagorean, although we may find it in some Pythagoreans. We will prove this later on.

Mathematics was, for the Egyptians, a divine science, therefore religious, of mystical essence, and the numbers and figures that derive from it had a symbolic and divine value, not to be considered only as measurements or mere calculation instruments, as they were for Thales, who according to some, was one of Pythagoras' teachers.

Numbers are not only formal principles. They are also relations between parts, like the laws that order the parts of a totality, the intrinsic law of something (its form), but also the process of becoming of something in the successive exercise of its being (in its dynamism), as well as what is immutable, what remains in itself without translation or mutation from one mode of existence to another. However, this change from one mode of existence or from one state to another has a numerical expression and is also a number.

Things are numbers, but they are also an imitation (mimesis) of numbers because what is finite is a number ontically considered in its singularity and uniqueness, because what is in itself has an order, a law (a logos of its own being), a self-essence, which is its existence, for the essence of the singular existing is the onticity of its own existence, for Pedro is (essentially). But Pedro’s existence imitates an essence (a logos, a form, the human one), which is the general one, that which is common to all men, but also has something that is its own, that characterizes it because it is from the essence of what he has in common with some men (being psychologically this or that type, for example), which reveals a particular essence in relation to the merely human. But this particularization does not end there, for Pedro is also Pedro in his peculiarity, in his individuality, he himself, his own existence. In Pedro, there is something that is always Pedro; a logos of his own individuality, which is unique to himself, despite all the similarities with the particularities of others, and which makes him unique, not only quantitatively, not only in his materiality (this matter of Pedro), but also in his essence (peculiarity), the ultimate determination of form, which is individuality as totality. But this individuality also has an essence, something that endures through the changes that Pedro undergoes; something that is immutable in Pedro, whose logos is unique, and that is Pedro’s soul in all spiritual thoughts.

Pedro is, therefore, the set of all these numbers (arithmoi), and among them, there are those that endure immutably, otherwise, he would cease to be a human being or to be Pedro, and there are those that vary, that undergo mutations, that are merely accidental, for Pedro’s ultimate substance remains unchanged.

As we have seen, the number is, for Pythagoras, substance, essence, logos, law, principle, relation, cause; in short, it is everything, and it is everywhere. This leads us to seek the essence of number (arithmos), which we will address in due course, given the sense that Pythagoras attributed to it, far removed from the primary sense of the sensible number, the number of measurement and calculation, which Aristotle wanted to attribute to it as if it were the only way to conceive it.

Transcendent and immanent, the number is matter, it is form, it is substance, it is quantity, it is quality, all accidents, properties, everything, indeed. And why? Because the number, for Pythagoras, is the “scheme of participation,” the participation in the divine, not only the scheme but also the participation itself, and since all numerical beings are participants, and since participation manifests itself in so many various ways, the number also manifests itself in many various ways. In short, this is “the concrete and schematic expression of participation in all its modalities,” for since everything that exists is a participant in being, everything has a numerical expression that both distinguishes it and homogenizes it with all things and makes it analogous to the Supreme Being.

The lack of clear understanding of Pythagorean arithmetic is due to the false conception of what arithmos meant to him.

If arithmos is prejudiced in a certain way, the theory of Pythagoras becomes prejudiced, and this is the reason why his thought appears so varied in the exegesis of all those who have dedicated themselves to studying it.

Stobaeus, cited by Aristoxenus, says that “Pythagoras seems to have esteemed arithmetic above all else.” But which arithmetic? The one that is commonly considered, or the one that is properly arithmology? Undoubtedly the latter.

To attain the number, as we have seen, he taught his disciples, first of all, what the even and odd were. The even is the unlimited (apeiron) because there is a void between its two parts. The odd is the limited (peras) because if we want to divide something into two equal parts, we find that there always remains an indivisible unity between them, which is the even-odd.

Thus, the number is the combination, the harmony of the even and the odd; that is, the harmony between the unlimited and the limited.

However, to penetrate more deeply into the nature of the number, it was not possible to remain only at this point, without carrying out an ontological analysis of what is evident and latent in these expressions, which undoubtedly conceal much more.

Being the number “the concrete and schematic expression of participation (imitation) in all its modalities,” where there is a number, there is participation.17

One is not a number because if there is only one, there is no participation. There is only a number where there is 2 and 3. But Pythagoras says that the number is the harmonious combination of the even and the odd. And since 2 is the first even and 3 is the first odd, the number arises from the harmonious combination of 2 + 3, that is, participation arises where there is 2 and 3. Now, participation implies a participant and a participated; therefore, 2. This two does not have a limitation because, as such, it is unlimited since there is only one participant and one participated, nothing is limited yet, nothing is delimited. The ultimate participated is undoubtedly the Supreme Being for Pythagoras, consequently, the participant, the one who participates, participates in what is participable of the participated. So, we have three, because there is only participation where the participant participates in something that is participable of the participated. It cannot be said that the participant fully participates in the 4th participation because in this case, it would be included in the other, which is absurd, as the greater would be contained in the lesser, and the entity, which is finite because it participates, would be infinite because it would contain the ultimate and supreme participated. Consequently, the participable of the participated must be participated proportionally to the participant, or rather: the participant participates in the participable of the participated in accordance with its nature. And thus, we have 4. Now, something is this or that to the extent that it imitates (for Pythagoras) the participable of the participated, for then it is this or that. Every number indicates something and is something because where there is a number, there is the participation of something that is participable of the participated by the participant, something that has a structure, a scheme, a logos, a form, in short. Everything that is something, and not absolute nothingness, has a being, has a unity, has a form (eidos).

An accidental relation, which is the simplest of relations, or a purely predicamental relation that does not imply any corruptive mutation of the subject, has a form and can be reduced to a scheme. Motion, which is a real accidental (predicamental) relation, has a unity and a form because it is something, it is a number (arithmós), and it can have a numerical expression.

In the natural sciences, this aspect of Pythagoras' thought is evident. Now, there would be a glaring error (and indeed there has been among its interpreters) if arithmetic were considered only what is expressed in the sensible numbers used for calculation, that is, if Pythagorean arithmology were nothing more than ordinary quantitative mathematics, logistics of the third degree of abstraction.

Absolutely not. In a mere real predicamental relation, there is participation because what relates accidentally to another presents, at least, the following aspects: the relating terms, 2 (the subject of the relation and the terminus ad quem that is referred to); there is also the basis of the relation, 3, which is the relational participable because between two entities that relate to each other, they relate in what is relatable between them and in proportion to their nature. Thus, in a relation of space, such as A being closer to B than C, the relation is formed by the participation of relative proximity; that is, directed towards B by A, which is of a higher degree than C, which also approaches or is close to it, although to a lesser degree than A. Being close to something else allows a schematic expression. And mathematics is full of these schematic expressions.

It is seen, in this way, that for Pythagoras, arithmós is not an accentuator of abstraction but of concretion because things are either like this or they are in another way, depending on the degrees of participation of the participable by them, and the universal heterogeneity is understood within these participation schemes, as all things participate in perfections (eide), some more than others, in varying and variant intensities, which allows for understanding the reason for the immense heterogeneity of all things. The Pythagorean conception of number does not allow, therefore, a pure and simple reduction to numbers as second-degree abstractions of mathematics because if these are numbers, they are also numbers, and it is not only them that are.

Thus, every relation has a form, 5. Now, Pythagoras says that number is the harmonious combination of the unlimited and the limited. Participability is unlimited, but participation, when made effective, realizes an actualization, a limitation. And where there is participation, there is a form because the former is the product of a partem capere of an eidos by something. Therefore, number manifests where there is participation, which clearly justifies what is contained in our definition.

Are forms, therefore, numbers? The question arises, and the answer is not entirely easy. What is participable, and this is formally something because there can be no participation of nothingness, as participation would be annulled. If the participable is a form, it must, in turn, have a number and be a number because it is something, it is a unity. If it is a unity (perfection), it participates in the perfect unity of the Supreme Being. However, this is not enough because the form is this and not that; it is, therefore, eidetically different from the others, which are not it.

Now, to answer this question in a Pythagorean way, it is necessary to delve into what Pythagoras understood at this point, which we will synthesize to subsequently proceed with our analysis, already based on our ontological dialectic, in order to deduce the foundations of this philosophy, which has been so little understood and so falsified and caricatured throughout time.

The first ten numbers were 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, the decade, which Philolaus called the great, all-powerful, and source of everything, the beginning and model of divine and celestial things, as well as of earthly existence. It is the number of the Cosmos and symbolizes perfection because all other numbers are nothing more than repetitions of these ten fundamental numbers.

If all things are numbers, numbers are not the ultimate reality of things. But here, we use the term number in the sense of the numbering number. The first four numbers are called principles (arkhai) and qualified as eide, but eidos is not only the form or structure but also the principle. All things begin with 1 (participated), with 2 (participant and participated), with 3, with the participable of the participated, and with four, the second participable is participated by the participant because the participant participates in the participable in proportion to its nature. Now, we must answer the question we had previously asked, namely: if the participable is a form, is it or is it not a number? In short, are forms numbers? Now, the forms of sensible things are undoubtedly numbers because, in turn, they participate in the Supreme Being. But taken as exemplars or possibilities within it, as done by scholasticism, and done with a solid foundation, forms have a structure, but a formal one, participating in being, in the unity of being, the forms ante rem, and not the forms in things, the forms in re, the intrinsic structure of things, the law (logos) of intrinsic proportionality of things. Although numbers in both cases, they are distinct and should not be confused.

If the forms, in the Supreme Being, participate in it, there is another participable, and that is a form. And what are these participables, if not the divine attributes? But then there would be a participable of the divine attributes, which would lead us to another form, which ultimately would place us in an apparently insoluble aporia. The solution can only exist if we consider well what Pythagoras said: number is the harmonious combination of the unlimited and the limited (finite). The divine attributes are infinite, but the forms ante rem are also infinite, because as such, they are infinitely what they are. However, these are specifically infinite, or rather, they are infinite secundum quid (relatively), because humanity is infinite as humanitas, because as such, it is purely itself. However, such forms are not simpliciter infinite, as the divine attributes are. In the forms ante rem, therefore, there is a combination of formal finitude in relation to other forms because one form is not another, but in itself, it is unlimited, infinite, secundum quid. Therefore, they realize the harmonious combination of the unlimited (infinite) and the limited (finite). The forms participate in the divine attributes, but the participable of these is proportionate to the specifically formal nature of the forms. That is why they are numbers, arithmoi eidetikoi. And it is from their participation, forming 4, that all things arise, not by efficiency but by imitation. Things are made in imitation of the forms. The forms ante rem encompass, therefore, 1, the monadic unity attribute, the 2 of the specific form, the 3 of the participable of the attribute, and 4, which is the form ante rem itself, as a possibility of the omnipotence of the Supreme Being. The forms are thus unlimited-limited, infinite-finite, and in them, there is a harmonization of the unlimited and the limit, as also in things, in re, the form is a harmony of the formal unlimited and the limited, the informed thing.18

Thus, one can say that all things are numbers. But why is it said that the ultimate reality of things is not number? Because the ultimate reality of things is the Supreme Being, its support, and it is not a number because it does not participate in another.

The Pythagorean thought understood in this way, and this is what rigorously follows from a dialectical-ontological analysis, in the mold we advocate, therefore allows for a reconciliation of that with Christian thought, which was considered absolutely impossible by many, an affirmation that we have shown to be unfounded.19

Elements for a concrete foundation of Pythagoreanism

PYTHAGOREAN PRAYER

“Bless us, divine number, you who engendered the gods and humans! O Holy, holy Tetractys, you who contain the root and the source of the eternal flow of creation! For the divine number begins with pure and profound unity and then reaches the sacred four; and it engenders thereafter the mother of everything, who accomplishes everything, the firstborn, the one who never deviates, who never tires, the sacred Ten, who holds the key to all things.”

After examining, as we have done in the previous chapters, the Pythagorean concept of number (arithmôs), and having once and for all excluded the crude way of conceiving it, as we have preconceivedly observed in the exposition of the thought of the Master of Samos, we are now able to establish the fundamental theses of legitimate Pythagoreanism in order to build upon them the concrete construction of Pythagoras' thought, under the criterion of objective coherence, which is the main purpose of this work.

The Pythagorean prayer, which we have reproduced above, has been transmitted through time and preserved by all those who claim to be or consider themselves disciples of the great master. It contains something of the truth of the fundamental conception because there is undoubtedly an influence of symbolic religious language. We have chosen it to serve our analysis and also as a point of reference for examining the thought subsequently expounded by disciples, exegetes, and adversaries, considering it as a document that undoubtedly reveals much of the true thought of Pythagoras.

It was the divine number that engendered humans and gods. But the divine number is not the Supreme One (Hen), for we have already seen that it is not a number. The Sacred Mother, Creation, is the one who engenders everything. She is the Decad, the Sacred Ten, the Holy Tetractys. The divine number begins with pure and profound unity, the Supreme One. Therefore, the divine number is ontologically posterior to the One, to the Supreme Being. And it could not be otherwise.

This is observed in these Pythagorean verses, which are also a fundamental element for our future analyses:

"From the inviolable abyss of the Monad

To the most sacred Tetrad: this one gestated the Mother of all things,

The universal Receptacle, the Venerable, the one that limits all things, The inflexible, the tireless: they call it the pure Decad."

(Pythagorean verses)

The inviolable abyss of the supreme Monad (Monas=Hen) is absolutely simple, for the first Monad is absolutely simple, otherwise another one would be the first. It is the source of everything, for it is from the inviolable abyss of it that we reach the most sacred Tetrad (the Holy Tetractys), the Mother of all Things, Creation.

This thought, attributed to Pythagoras by all Pythagoreans, and preserved as undisputed by the entire Italic school through the ages, also confirms the accuracy of our affirmation:

"For the conductor and ruler of all things is God, One, always existing, monistic (mô-nad), immovable, identical to itself, different from the others (ton allon) [Allós is Nature, for Pythagoras].

“In God, as in a prison, all things are contained.”

The conductor of all things, the conductor and ruler, is God, One (Supreme One — God), always existing, and therefore eternal, monistic, absolutely simple, most simply simple, immovable; hence, not corporeal, for it is different from the others, from corporeal beings [ton Alton, for Allós (another) is Nature, that which is born].

It is identical to itself, self-identical, ipsum esse. It is the source and origin of all things, for all things are in it as in a prison; that is, they are in the Supreme Being and not in nothingness, for nothing exists outside of Being, as it contains all things inflexibly.

The abyss of the Monad is inviolable; therefore, the direct knowledge of divinity is unattainable by man in his current state.

Human knowledge is limited because it is proportionate to its nature. And if, through initiation (from initium, from itere, to go, to walk, to begin, to go on the way, in-ter), man traverses the path that leads to the Supreme Being, his knowledge will present degrees of knowing, but it will not be exhaustive.

These two Pythagorean thoughts confirm this restriction to human knowledge, which perfectly aligns with the epistemological position of scholasticism, without being skeptical, as we have demonstrated in “Theory of Knowledge.”

Philolaus, who is one of the most esteemed Pythagoreans, also expresses himself in this way:

“The essence itself is distant from man. He only knows the things of this world, in which the limited (finite) combines with the unlimited (infinite). And how can he know them? Because there is harmony, analogy, a common principle between him and the things. This principle is given by the One, which provides, with its essence, the measure and intelligibility. This is the common measure between the subject and the object, the reason of things, through which the soul participates in the ultimate reason of the One”20.

Human knowledge is not totaliter; it does not grasp the totality of what exists in its fullness. But it knows the things of this world, in which the limited (finite) combines with the unlimited (infinite). What is finite implies dependence on the infinite, and its being is given by the Supreme and infinite Being. But how can man assimilate what he captures from things to his accommodated schemes, if there were not between them not only an analogy, not only a harmony, but a common principle, an “identity”? This identity is given by Being. We have apodictically proven in “Concrete Philosophy” that there is nothing outside of Being. And Pythagoras affirms that all things are in the Supreme Being as in a prison.

The Pythagorean thought of Philolaus is congruent with that of the Master. How can there be cognitive assimilation without something that analogizes them? And where there is analogy, there is similarity and dissimilarity. Where there is similarity, there is the same and the diverse. Where there is the same, there is sameness, identity as well. And isn’t it the Being that identifies all things? In their being, all things, that are, univocally come together. And since there are no ruptures in Being, since there is neither more nor less in it, as we have seen, it is being that exists in all things that allows harmony, analogy, and identity, which is a common principle, among them21.

We demonstrated, in the same aforementioned work, that between what is and absolute nothingness there would be an infinite distance. Furthermore, it was shown that absolute nothingness is impossible in any aspect. It is no longer relative nothingness. But in this relative nothingness, the positivity it has is given by reference to something that is because, when we say that something is not this or that, this or that are positive, for not being nothing does not deprive a being of anything. To say that nothing is lacking in something is to say that nothing is absent, or in short, it is fully what it is. Being, as being, is the fullness of being, and it lacks nothing to be, for there is no middle ground between being and absolute nothingness because less than being is nothing, and more than being is being.

Therefore, the common link, the principle that unifies all things, is being. It is in this being, as such, that the infinitude of everything that is, while it is, is found. It is not determined by this or that, for while something has this limiting determination, it is deprived of another limiting determination. Not while it is, for there is neither less nor more than being. And since all things, which are, are because they exist, and they exist in and of the Supreme Being (in which they are contained as in a prison), it is through their univocity in it that the intelligent being can know things. However, the content of its knowledge is proportionate to its schematics; therefore, it can know more or less.

It is the One (Supreme Being) that is this principle, and it is he who provides, with his essence, the measure and intelligibility. Now, the Supreme Being is the maximum perfection of being, for it does not suffer any limiting determinations, it is independent, the reason for itself, in which essence and existence are identified because it is absolutely simple (Monas). The perfections that things present are participations in the divine perfection of being, for there is no participation of nothingness, which is not, and furthermore, a perfection that was not contained in being would be in nothingness, which is absurd. Furthermore, we also proved in Concrete Philosophy that perfection in the Supreme Being is absolutely simple, and the beings that arise from it cannot fully enjoy it in their absoluteness because they are limited, finite, dependent beings.

Furthermore, we demonstrated in “Ontology and Cosmology” that the Supreme Being is the measure of qualitative perfections, for qualities are measured by the specific maximum of their perfection, while quantities are measured by a smaller portion of the quantitative. The Supreme Being is, therefore, the qualitative measure because a finite being can have such or such perfection to a high degree, never to the infinite degree of the Supreme Being. And it is in comparison to it that we can speak of a greater or lesser degree of perfection, for we can speak of a greater or lesser goodness, measuring it with the infinite goodness of the Supreme Being; we can speak of a higher or lower value, comparing it to the infinite value of the Supreme Being. And if we do not have actual possession of the infinite perfection of the Supreme Being, we have the virtual possession of that perfection, as we have demonstrated in “Man before the Infinite,” which enables us to make axiological appreciations.

And the perfection of intelligence, which allows for discernment among various things, is a perfection that participates in the perfection of the supreme and absolute intelligibility of the Supreme Being. In short, this is what can be concluded from the thought of Philolaus, which we have mentioned above.

There is another thought of Philolaus, which we are going to reproduce and which clarifies the gnoseological conception of Pythagoreanism.

“The essence of things is an eternal essence; it is a unique and divine nature, whose knowledge does not belong to man. However, it would not be possible for any of the things that exist and are known by us to reach our knowledge if this essence were not the internal foundation of the principles from which the world was formed, that is, the limited and the unlimited elements.”

Accordingly, man attains an ontological and ontic knowledge of the things of his sensory experience, but always proportionate to his schemas. Our eidetic-noetic schematics are given by the intentionality of our “nous,” our spirit, and are constituted by the conceptual contents that man constructs over time. However, such conceptual contents can be examined and rectified according to an ontological criterion, as we show through the use of our dialectic in “Concrete Philosophy.” In any case, ontological rigor, which is achieved through the rigor of anteriority and posteriority—since what is anterior in a certain way precedes what is posterior, which is what follows, regardless of the way in which antecedence and consequence occur, whether chronological, axiological, ontological, etc.—does not give us a frontal vision of them, but only knowledge of what is, without seeing how it is.22

Since vision is the primary organ of our sensory knowledge, we always have the impression that we do not know something well unless we see it. Seeing is the path to belief and certainty. To convince, one must make things visible to the eyes. Thus, all knowledge is strengthened to the extent that it can reduce the known to the visual.

For this reason, vision emerges in mystical thought and in religions as the symbol of perfect knowledge. “Those who are righteous will see God… and in seeing, there will be complete knowledge.” “Man does not see the truth…” Expressions like these are indicative of the influence of optical schemas on knowledge and the formation of secondary schemas, products of the coordination of primary schemas, as examined in Esquematology.

Through symbolic reduction, vision appears in all these thoughts as the complete apprehension of the object by the subject, indicating the fusion of the former into the latter. This is what the Hindus call yoga, and we call it phronesis, giving the Greek term its deepest meaning, which should not be confused with the prudentia of the Romans. Here, there is room for a parenthesis. In Greek concepts, there is always a root that precedes the execution of the process. Thus, phronesis is prudentia, translated only etymologically, as sophia is sapientia, and eudaimonia is felicitas. However, there is a profound distinction between such terms and their translation into Latin, which somewhat betrays them.

Phronesis is intelligent prudence, the virtue of practical intellect. But prudence, for the Romans, is the virtue that is acquired, as it was for Aristotle. But this is undoubtedly a watershed in Greek philosophy, as we prove through the comments we make on his books.

In the Platonic sense, which is more genuinely Greek, phronesis is not only the knowledge acquired by practical intellect or the ever-increasing, or at least expandable, ability to practically know the means to achieve ends, which is the root of wisdom and science. It is also a virtue that comes before, that precedes, that is a priori to experience. One is born intelligent, that is, with the aptitude to be intelligent, to acquire knowledge more and more, and to distinguish.

Phronesis thus has a deeper root in the human soul, while prudence expresses more of a constant acquisition, a habit. The same distinction can be made regarding the eudaimonia of the Greeks and the felicitas of the Latins. Felicitas comes from felix, and this from fenus, from what springs forth, emerges, is obtained. But eudaimonia is the eu daimon, the good spirit within us, which already exists beforehand. Happiness for the Greeks is therefore something for which we already have an inclination, while for the Romans, it is something we can obtain. The same applies to sophia and sapientia, as the sophos is emergently wise, while the sapiens is the one who acquires knowledge in general.

Perhaps because the Romans were conquerors, forming a people that emerged from small tribes, which multiplied through the conquest of others until the city and then the empire were founded, there is a presence of posteriority in Latin schematics, while in Greek schematics, especially those up to Plato, there is a presence of anteriority. In other words, for the Greeks up to him (and we mention Plato not in a chronological sense, but in an ideological sense), emergence is indispensable for subsequent predisposition to act. Thus, an aptitude is needed to undergo one’s own experience, and it is this emergence that can explain the heterogeneity of the contents acquired through experience. For the Latins, predisposition is paramount, as it is from it that experience arises, and thanks to it, the human spirit can shape its schemas.

Now, we know, and we demonstrate in “Concrete Philosophy,” that no being arises from its own emergence because in that case, it would exist before existing, which is absurd.

Only the Supreme Being, the first and origin of all things, does not require a predisposition to be. Thus, all finite beings, being dependent, cannot preexist themselves but can only potentially exist within those that precede them, and which are their cause; that is, those on which they depend. In this case, predisposition precedes the emergence of the finite being. Once it arises, it will act and be subject to the proportionate nature of its being. However, the nature of a being is not only what fits into the Aristotelian definition but also what constitutes individual emergence. In this case, such a being will undergo the action of predisposition in proportion to what it is, according to its nature. Therefore, something precedes experience, and this anteriority is a kharis (from kharis, kharitôs, from which charity, what shines, grace, enchantment, what pleases, what reveals an appetite for something), which moves to grasp the experience in this way and not in another, more perfect than another. In Greek schematics up to Plato, this presence of kharis exists in phronesis, and it is this love that merges the subject with the object, which is why we use this term to express affective intuition, which is fronetic because it involves knowing that arises from a cognizant being who knows a cognitum, which is itself, its own state. These are the reasons that lead us to establish these distinctions in the schematics of Pythagoras and Plato in contrast to the schematics of Aristotle, as we are faced with two ways of visualizing the same path. In our works on both of them, we delve further into this examination, which allows for a better understanding of the divergences between Plato and Aristotle, considering them in terms of their philosophizing, which arises primarily from differences in schematic order.

Aristotle is an empiricist-rationalist, while Plato is a real-idealist. Consequently, their schematics diverge, and they help us understand the reasons why Aristotle deviated from his ancient master in so many points and even distorted his thought, driven, as can be seen, by the fronetic aptitude of his own spirit, which further supports the Platonic position.

Returning to the Pythagorean field, we see that because the essence of things and the very nature of essence, considered here in the twofold Pythagorean aspect of eidetic ante rem and eidetic in re (in the order of eternal Forms and in the order of forms in things), are eternal, they require a cognition that is not human but divine to reach them. For man, in any way, will only know in proportion to his finite nature. Furthermore, knowledge can only satisfy man if it can be reduced to his schematics. To know the eternal essence of things, man would need to see them. However, in seeing them, something would always elude him, something he would need to see, and thus, successively, it would never satisfy him except for fleeting moments. Only a divine mind could grasp the fullness of truth. This Pythagorean thesis is, in other terms, the same that will arise not only in Aristotelian thought but also in the thought of the scholastics up to our present day.

If we can only intellectually and intentionally see the essence of things, it follows that they are formed by something beyond man, something that transcends man, for otherwise, it would not be said that “none of the things that are, and are known by us,” would reach our knowledge if this essence were not the internal foundation of the principles from which the world was formed, that is, limited elements and unlimited elements.

Since our knowledge, which is limited, cannot directly and frontally know the infinite, how could we know things, which constitute the Cosmos? It is because they are both limited and unlimited. However, they cannot be only limited because if they were, it would be possible to know them exhaustively as such. And since something escapes us about them, it is because they are also unlimited. The limit-unlimited category is a Pythagorean category.

This fragment from Philolaus confirms our assertion as genuinely Pythagorean. We reproduce it below:

“The being that belongs to the world (cosmos) is a harmonious compound of unlimited and limited elements. It belongs to the whole world (Cosmos) as well as to all the things it contains. All beings are necessarily either limited or unlimited, or both limited and unlimited. But they cannot all be only unlimited…”

The things that make up our cosmos cannot all be unlimited. Now, the limit marks where something is what it is and where it ceases to be what it is. The limit indicates the boundary of the thing’s being and its non-being. If all things were unlimited, they would always be themselves, without boundaries, and would automatically cease to be finite. In this case, there would be only one thing, an infinite being, and multiplicity would be denied. The Supreme Being, as infinite and as being, is a being without limits because it is only being, as there is nothing outside of being. Only it knows no limits. All dependent things, whose being depends on another, are limited, at least by their dependence. And the creature, because it is dependent, is limited in any way. However, limited things cannot be absolutely limited, simply limited, because the limit is the non-being of the thing, as the thing’s non-being begins where it is limited. And if the limit were absolute in things, they would be forever separated from each other by an abyssal separation, and being would be multiplied, interspersed with absolute nothingness, and pluralism would be affirmed. And no limited thing can, therefore, be absolutely limited because then it would be disconnected from being, it would always be disconnected from it, it would not be dependent on it, and it could not have begun to be. Thus, in the cosmos, all things are neither absolutely unlimited nor absolutely limited. Therefore, the things that compose the cosmos are composed of the limited and the unlimited.

Unlimited-limited is a category of finite things and is a category because it does not reduce to a genus, as is the essence of categories.

Therefore, all things that constitute the cosmos are number and have an arithmos, which is their ontic structure, and they repeat an arithmos, which is their ontological structure, and this is repeated by the ontic structure by imitation of that, as we have shown to be genuine Pythagorean thought.

“What is nature? Another!”

“What is God? One!”

(Pythagoras, according to Aristotle)

Aristotle acknowledges that for Pythagoras, the being of physical things is different (Another) from the being of God, which is One. Another is always nature for him, and nature depends on him. God is therefore transcendent to nature, taken in the sense of cosmos. To be another is to be one before two; it is to be different from One, and since One is primarily the first, Nature (cosmos) is different from him. One is not another than Nature because the ontological structure of otherness, of being another, implies a first antecedent. Being One, prior to any being, it is not another than any being, but any subsequent being is another than it. Because it did not understand this ontological content, Parmenides concludes in the famous dialogue of the same name that One participates in otherness because it is another than the many, the multiples. But the error is already clear, and it allows us to understand the sophistical conclusions that follow from his reasoning.

This warning we make now becomes important to better understand the flaws that arise from Aristotelian analysis, especially when it is clear to us what genuine Pythagorean thought is.

“The Pythagoreans considered unity the principle of all things. From unity arises indeterminate duality (matter and motive cause). From these two elements arise numbers; from numbers, things, etc.”

(Words of Alexander Polyhistor)

Sextus Empiricus also affirmed: “The first unity is not corporeal, nor is it based on mathematical figures. It is only in unity and indeterminate duality (Dyas-aoristos). And all logical categories refer to them. It is from the cooperation of the two principles that numbers and everything else arise.”

Eudorus also stated that for the Pythagoreans, One is the superior divinity. From him derive two principles: the one, and indeterminate duality. The second element is generated by the first.

This is also the thought of Moderatus of Gades. One is the principle; then duality arises, and finally, multiplicity.

Pythagoreans such as Plutarch, Pseudo-Plutarch, Hippolytus, Pseudo-Justinus, etc., also support this thought.

We find it consistently valid and present as a fundamental thesis in all Pythagorean thought throughout the centuries.

According to Archytas (or Pseudo-Archytas), there is the one and duality, the two principles of all things. However, in this, we already find an Aristotelian influence, as both are reduced to Form and Matter. One is, however, above and distinct from both Form and Matter. Numbers are, for him, the links that unite Form and Matter. Thus, One is placed above this opposition.

However, there are Pythagoreans who deny this position, such as Theon of Smyrna, Pseudo-Alexander. And in modern times, Luigi Alessio opposes it because, for him, as for those mentioned above, numbers are the substantial elements of corporeal beings and not forms, as the previously mentioned ones claim, accused of being influenced by Aristotelian thought.

For them, number is the thing, and it is the form of the thing. In this case, both are numbers. The corporeal is thus number, and not number is corporeal. However, all these ideas that arise in opposition can be perfectly transcended by a higher thought of Pythagorean doctrine, as the disagreement arises more from a misplacement of the master’s abstract thought than from a more secure visualization of his ideas. It is enough to pay attention to what we have studied in previous chapters.

One is absolutely necessary, both logically and ontologically because we cannot surpass it. The idea of the multiple imposes it in advance, ontologically and logically. And also the best, the maximum, and the minimum.

If we start from the Aristotelian position, primarily empirical, the multiple is what appears to us in experience, but the multiple always implies the one because we can only know what is one, as what is not one is nothing because it is not one. Having no unity, it has no contours, no limits, and as such, it cannot be an object of knowledge. If we perceive that this book is green, we perceive a green, the green of this book, a book, this one, which is here, and everything we perceive is always one: this curve, one, this triangle, one because every perception is always of a singularity, and singularity is always one. We cannot surpass the one because conceiving more than one is impossible. The one is necessarily demanded by our knowing, both logically and ontologically.

And when the idea of the multiple arises, it automatically implies the one because the multiple is always formed of many ones. And all values imply it, and all knowledge always implies it. It is, therefore, the root of all our knowledge.

"You are One, the root of numbers, but you are not one as an element of numbering because unity does not admit multiplication, form, or change. You are One, and wise men are lost in the secret of your unity because they are ignorant of it. You are One, and your unity never increases, never decreases, nor changes.

You are One, and none of my thoughts can set a limit to you or define you.

You are.

But you are not like a living being because the vision and comprehension of mortals cannot reach your existence or determine your where, how, and why."

(Pythagorean words of Simon Ben Jokai when interpreting the “Johar”).

These words of Ben Jokai are a true Pythagorean prayer and blend perfectly with what we have examined so far because One, the Supreme Being, is infinite and does not admit form (in a figurative sense) or any kind of mutation because it is simple and always itself, the perfect ipsum esse, the being that is always itself. Its unity transcends human knowledge, which can never reach it, and men always remain ignorant of the secret of its unity. No thought can set a limit to it or define it because, for that, it would be necessary to include it within the limits of a definition. It is, and it is regarding it that we can most profoundly say: You are!

“The Pythagoreans called One the idea of identity, unity, equality, concord, and sympathy in the world, and ‘Two’ the idea of ‘otherness,’ discrimination, inequality.” (Moderatus of Gades)

Although we do not consider Moderatus of Gades a Pythagorean of the third degree, his words, however, have great value for the analysis we are conducting. One is primarily identity, the ipsum esse, the being that essentially is its own existence. Thus, implicit in Pythagorean thought is the identity between the essence and existence of the Supreme Being.

And from the examination we have made of the most important sources of Pythagorean thought, when compared to the passages we have cited, which we consider fundamentally Pythagorean, they allow us to construct a concrete understanding of that thought.

Thus, there is a consistently Pythagorean thought that connects what is most solid in this philosophy and shows us that the points that deviate from it are the products of lesser Pythagorean philosophers, who reveal the deficiencies of the philosophizing man, and not the genuine theses of Pythagoreanism, the construction of which is the aim of this work.

“But since the Whole was an unlimited multitude… Order was necessary…; now, it is in the Decad that a natural balance existed between the whole and its elements. This is why, according to reason, the ordering God (literally, ‘the God arranging with art’) used the decad as a canon for the whole… and this is why celestial things to earthly things have sets, and parts have their relations of concordance based on it and ordered according to it.”

These words are from Nicomachus of Gerasa. And from him, the following:

“And the number ten, which according to Pythagorean doctrine, is the most perfect of possible numbers. It is, according to this idea, that ten types of relationships and categories are observed…”

For him, the Whole (Pan) is the Decad, “which serves as a measure for everything like a square and rule in the hand of the Orderer.”

For all Pythagoreans, the “decad” is the “key” to the universe. In the Supreme Being, which is everything that being can be, all possibilities are contained. Among the possibilities, both what can be and what cannot be are potentially present; potentially, possession and privation are contained. In contradiction, there is an incompatibility between possession and privation, as there is when both are simultaneously affirmed. There is contradiction because, at the same time that presence is affirmed, absence is also affirmed. Thus, when it is said that “all men are mortal” and that “some men are not mortal,” there is contradiction because mortality, affirmed in one judgment as applicable to all and therefore also to some, is in the second judgment affirmed as absent from some, which makes these judgments incompatible with each other, as the truth of one judgment implies automatically the falsehood of the other.

However, among the possibilities, privation and possession, before their actualization, are not contradictory, for what can become can also not become.

And when the possible things actualize, the actualization follows an order. The possibilities, considered as such, constitute chaos (Khaos), that which precedes order, that which has not yet been ordered, cosmos (which comes from Khosmein, to order).

That is why Nicomachus of Gerasa wrote:

“The original Chaos, lacking order and form, and everything that distinguishes it according to the categories of quality, quantity, etc., was organized and ordered according to number” (“Theologumena Arithmetica”).

The original chaos lacks the order of existential actualization because possibilities can be contradictory, while actuals cannot be. The order and form are lacking in chaos, but order is the order of being, which we will study through the ten archetypal laws of Pythagoreanism, and form is taken here in the sense of the form in re of the actualized thing, not in the sense of the eternal Forms, which are in the Supreme Being as its thoughts. And this interpretation is justified by the following words because what is lacking in these forms is what distinguishes according to the categories of quality, quantity, and others. The organization and ordering of chaos occur through cosmic actualization, and this, therefore, obeys number (arithmós), because everything cosmic, actualized in the order of Nature, is ordered by number, as we have already seen.

And how do finite beings arise? Plato describes it in the “Timaeus”:

“And it was then that all these kinds thus constituted received their figures from the Organizer, through the action of Forms and Numbers.”

The possibilities, which actualize themselves by repeating the eternal forms, proportionally to their imitability, receive their figures (in the sense of the form in re) thanks to those forms and numbers.

One cannot fail to understand a hierarchy here. There are the possibilities, which are neither general nor singular, those of the eternal Forms. There are the singularizable possibilities, which are those of actualizable things.

This division allows us to reduce these words to our terminology:

There are the eternal forms, which are the ontological eidetic schemas, and the forms in re, the ontic forms of singular beings, which are always possibilities, for otherwise, they would be possibilities in nothingness, which is absurd.

Thus, there is the ontological eidetic form of man, which is a possibility that actualizes itself in the ontic eidetic form of this or that man. The human individual is thus a singular possibility that has been actualized, but it is also something that repeats itself, being what it is, the eidos of humanitas, which is in the order of the Supreme Being.

Thomas Aquinas distinguished these two possibilities, as we have already shown in our other works. And this is still a Pythagorean thought that strictly follows what we have examined so far.

The Whole is Pan, the set of all things, creation, in short. And how was the Whole (all things that are actualized) ordered? These beings that constitute the Whole received their figures (their forms in re) through the action of the eternal Forms and the numbers that order them into singular structures and that copy those Forms.

“Boethius says: 'Everything that, after the origin of things, was engendered by Nature, seems formed according to numerical relations arising from the Wisdom of the Creator… Numbers are in the closest and simplest relations with the ideas of the divine Understanding… The forces that enjoy the Numbers in living nature do not reside in the names of numbers, nor in the numbers used in accounting, but in the numbers of understanding, formal and natural… He who can connect the usual and natural numbers to the divine numbers will perform miracles through Numbers.’”

(Chapter II of the book “Cabbala” by Agrippa).

These words of Agrippa are undoubtedly Pythagorean. The numbers of sensible things (numbered numbers) are symbols of formal numbers (numbering numbers).

In a scholium on Plato’s “Charmides,” Nicomachus of Gerasa writes: “Logistics (calculation) is the theory that deals with countable objects and not with (true) numbers; it does not consider, in fact, the number in the proper sense of the term, but assumes that 1 is unity, and that everything that can be numbered is number (thus, instead of triad, it takes 3; instead of decade, it takes 10) and applies to them the theorems of arithmetic.”

It would be a gross mistake, therefore, to consider that numbers are only the matter of actualized things. Numbers also form the structure of things, the in re structure, the intrinsic law of proportionality of things that are, and this law of proportionality, which is the factual schema of the thing, repeats, by imitation, the ontological eidetic schema (ante rem).

Thus, ante rem, there are two ontological eidetic schemas: that of form, which is the paradigm, the model, and that of the possible form of the singular being. There is in re the eidetic-factual schema, which is in the thing, which is its intrinsic law of proportionality, the number in re, and finally, what the intentionality of our mind captures from things, which is the eidetic-noetic schema, which is post rem.

Pythagoreanism, therefore, concretizes the thought of all the currents that clashed in the great controversy of universals and thus affirms what is positive in all positions. Pythagoreanism, by not excluding the positivities, is a concrete philosophy, in the sense in which we use these words in our works.

The words of Theophrastus, which we quote below, further illustrate our statements:

Theophrastus, in Met. 33,11 to 27, says:

“For Plato and the Pythagoreans, the distance (between the real and sensible beings) is great, but they nevertheless affirm that all things imitate (the real). However, for those who posit a kind of antithesis between the One and the indeterminate Dyad, from which the unlimited, the unorganized arises, in a word, everything that is, so to speak, by itself the absence of form, it is absolutely impossible for the Nature of the Whole to exist without this Dyad. necessarily, there is an equal part between these two principles, or one surpasses the other, so that the principles themselves are contrary to each other. This is why God, since we connect the cause to God, cannot bring all things to the best, but always to the extent possible. And perhaps He would not choose to do so until, if it is true, it would not result in the destruction of being as a whole, for it is made up of contraries and depends on contraries.”

The distance between real beings, singularized by actualization, by their onticity, which is their singular existence, is distinguished from the real, from what gives them reality, which are the Forms they imitate.

We have already examined the Pythagorean Dyad pages ago and how it should be understood. It is unnecessary, therefore, to comment on Theophrastus' words, as they are perfectly congruent with the thought expressed so far.

And this is the reason why finite things, being capable of being better, never attain the axiological perfection of being and can only be better to the extent of the possibilities of dependent beings. And since these beings are always dyadic and depend on the dyad because everything finite participates in the Dyad, as we have seen, they are made of contraries and directly depend on them to be.

And if things change, if they undergo mutations, if there is generation and corruption, the cosmos, the ordered set of singularized beings that constitute the Whole, always constitutes the Whole.

“Therefore, the cosmos remains incorruptible and without deficiencies throughout infinite eternity. No other more powerful cause is found outside or inside that can destroy it. But the cosmos is from eternity and will remain eternal.”

“The One, piloted by the One of the same substance, is omnipotent and unsurpassable. And further, being One and continuous, and breathing the life of nature, and being led to action from eternity, the cosmos contains the principle of movement and transformation. And part of it is immutable, the other part mutable.”

These two Pythagorean thoughts provide us with many more suggestions and deserve to be glossed over, for the Whole (Pan) as such always remains eternally. And the Eon (Aevum) of the Gnostics is what always endures and has always endured, although things within it transform, arise, and perish. Nothing is lost from it; within it, all singular things are contained; it is the Totality, and outside it, there is nothing more powerful that can destroy it because God is not destructive but constructive, for the Supreme Being is life and gives life. Creation is thus ab aeterno. The ab-eternity of creation is admitted by Thomas Aquinas, for it does not contradict in any way the Christian conception of creation. It is in the Cosmos, in the Great Pan, that the principles of movement and transformation are contained, but what ceases to be this becomes that without the Whole suffering any diminution, for what once constituted a being can become nothing of what it was, but not nothingness of being.

And the One is driven to action not by a power outside of it but by its own glory. This was demonstrated in “Concrete Philosophy,” for in the face of the omnipotent power of the Supreme Being, a creation ab aeterno can be admitted as possible without it arising from a necessity imposed upon it, but rather it would necessarily be connected to its immense power. In this case, the Supreme Being would create not out of a necessity to create, but creation would be necessary solely in function of omnipotence and glory.

We can conclude this chapter with this Hermetic prayer, a prayer that, coming from the Egyptians, penetrated Pythagoreanism and has a deep significance in it:

“Of all living beings, man is the most immortal, the one who receives God within himself and lives in union with God. It is only with this living being that God converses, through dreams at night, through signs by day… Everywhere, during your journey, God will come to meet you, and you will see Him everywhere, even where you do not expect to see Him: whether you are awake or asleep, at sea or on the road, at night or during the day, when you speak or keep silent, for there is nothing that God does not know.”

In God, the Supreme Being, His knowledge is Himself, for all things are in Him as in a prison, for there is nothing outside the Supreme Being, the source and origin of all things.

These are thoughts that strictly follow what has been presented here in examining the congruent thought of Pythagoreanism.

The apodictic demonstration, following the dialectical paths of our concrete philosophy, will come in due time, and then it will be the human voice that speaks through its philosophizing, through its possibilities of knowledge, which will serve to humanly corroborate what is expressed divinely in the great book of Nature, which offers the path that the deepest cultivators of Pythagoreanism followed until they reached what is humanly attainable but, in a certain way, surpasses the limits of man himself, just as man surpasses the limits of his animality.

HARMONY

Cosmos, for Pythagoras, who was the first to use this term to indicate the universe, comes from the verb Kosmein, which means to organize, and opposes Khaos, that which has not yet been ordered. Harmony is “the unity of the multiple and the agreement of the discordant,” which is evident everywhere. Thus, the universe is harmonious because in it we see the discordant awakening to a norm that prevails. The universe is not a bundle of absolute perfections secundum quid, but a bundle of disharmonies that harmonize; it is the pre-harmonic multiplicity that harmonizes.

Those who oppose the Pythagorean conception are based on false judgments because they take, as a starting point, that which is not genuinely Pythagorean: they consider that the universe is only the sum of perfect things.

Now, that is not the starting point, for finite things are all deficient, and deficiency implies the limited and the unlimited, for where there is deficiency, there is limit and also that which surpasses it, the unlimited, the non-limit. No thing is perfectly limited in its species; but there is always something that becomes unlimited, that escapes the limit. For this reason, harmonization is a combination of multiplicity, an agreement of the Accordant, which achieves a new, specifically higher unity.

Pitagoras never affirmed the absolute perfection of the universe, but rather the dynamic harmonization, not static, the viably perfecting of the Cosmos; that is, that perfecting is a stage that endures in flow, but which, in turn, flows in a slower succession, but that reveals an agreement among the discordant, a symmetry between opposites, a symmetry that always implies analogous opposites, because harmonization implies something to which pairs harmonize (for the pair is multiplicity: and the many can be considered analogous to pairs of contraries). Thus, many analogous to a logos and in the face of many others, more or less numerically analogous to another logos, can analogize among themselves, like two contrary groups, to a logos that gives them the norm to which they obey, and it is what achieves harmony. Where there is harmony, there are therefore contraries (opposition). In contraries, there is a logos for each that distinguishes them, otherwise they would be identical and there would be no harmony among the identical; there is identification.

Harmony also requires that the contraries have, in addition to a contrary logos, or at least a distinct one, that the component elements of one of the pairs of contraries analogize among themselves or that they have, in a certain aspect, a logos that identifies them, formally, under that aspect, like a group of men who analogize themselves as soldiers of a group, as combatants of a group, and, as such, they identify themselves despite the differences, the heterogeneities that distinguish them from one another. But, functionally, they merge into a logos that points to the functionality of the group. In the contrary group, there is the same functionality and a corresponding analogy.

The combat between the groups analogizes them into a logos, which is the clash between opposing forces that seek to prevail and dominate the other. Thus, in combat, there is harmony. (We chose this example because it seems most dissonant to the eyes and serves well to explain the concept of Pythagorean harmony).

In combat, there is an agreement of the discordant. The parties attack and defend, and despite the variance of positions and attitudes, both analogize themselves in the clash, in the same logos of battle, as warriors, as combatants, with identical ends and identical functions, taken only formally.

The battle between Greeks and Trojans is a harmonization, which is the struggle. Thus, where there is discordance, there is harmony under a certain aspect, and as, in the Cosmos, things, no matter how discordant, analogize themselves to a logos. They are analogous contraries and obedient to a norm (the norm, the rule of battle). There is always, under one aspect, harmony, and under another, disharmony. When the disharmonious aspects are examined, other harmonious aspects are found in them. Thus, in a longer analysis, which is unnecessary to perform, it is found that the universe (Cosmos) reveals an alternation but also the presence of harmony-disharmony, or rather, agreement-discordance, for how can there be agreement without discordance, harmony without opposites? The dialectical concept of harmony, for Pythagoreanism, therefore implies a dynamics, and not only a mere static. Seen merely statically, it leads to the usual confusions, but considered in its dynamics, what it affirmed becomes understandable.

The law of opposites is a universal law (the law of two), a law that governs not only the physical world (alternation, frequency, opposition, contrariness, antinomy, antagonism, etc.) but also the anthropological (philosophical, ethical, social, etc.) world.

The entire Pythagorean conception is based on the cooperation of the opposites, which occur in the Creature! world, the contribution between the limited (peras) and the unlimited (âpeiron), that is, in the theory of oppositions. But these oppositions do not imply an absolute principled dualism, for it is the One, as absolute unity and uniqueness, that precedes all things, but rather a creaturely dualism, for making implies being made, creating implies creative activity and creature, the creans and the creatum, the creator and the being created, for in the act, there is what determines and what is determined, since to be finite implies, to exist, an delimiting act and something that is delimited.

Therefore, Pythagoras always speaks of the Other, which seemed mysterious in the eyes of John Damascene. But this other (alter), whose etymological origin is obscure, implies that which is not before that which is, for in the making of something, parity is implied, and consequently, the two, for how could something be made without something being done?

Making implies being made, creating implies being created, creature. The Supreme One does not imply another beyond and outside of itself to be, but the creature does, for it is not a being that has in itself its ultimate reason for being. And consequently, being dependent, a product of an operation, the operation occurs in it, which is creation. The operated implies the operator, for how could what is made be made without what made it? The creature always implies another. And becoming, to exemplify, implies the other, for how could there be the flow of things without the other, for the flow is always something other than what it was before?

Where there is dynamism, where there is flow, there is always another, but also that which endures, for if, in the flow, there were not the enduring, but only the other, the flowing being would never be in any way, and the flow would vanish into nothingness, it would be only nothing. For something to flow, becoming another, it is necessary for something in itself to remain. Becoming another is, in a certain way, remaining partly what it is and partly ceasing to be what it was to be what it was not yet in a certain way but is now, and so on successively. Becoming is, therefore, dual, and consequently, it implies the dyad of permanence and non-permanence, of limit and unlimited. Becoming is, therefore, a number (arithmos) because it contains parity and oddity.

Thus, there is an immutable part (relatively) and a mutable part (relatively), but the first implies an eternal part, for if immutability were not formed of an eternity and a temporality, it would vanish into nothingness. And dialectically, it is easy to reach this point, for if immutability were only relative, the immutability secundum quid of this being, specifically hoc (in its particularity, only this), being still transient, then what exists would vanish into nothingness. But what ceases to be this (hoc) does not cease to exist absolutely. There is a return to a primordial being, for what ceases to be what it specifically is ceases to be what it is, and does not become an absolute nothing. Consequently, there must be, behind all things, something eternal, something that endures eternally.

The same can be established in the analysis of mutability, which also cannot be absolute, as we have already seen, for then it would vanish into nothingness. As a result, the following classification can be established: creaturely immutability is Immutable-Mutable, and mutability is Mutable-Immutable.

The presence of opposites always imposes itself for a clear vision of things. Therefore, for Pythagoreanism, knowledge always implies a cooperating duality, not only knowledge but also all creaturely existence.

The Supreme One is thus the transcendence of pairs of opposites, and by reaching it, we overcome oppositions. Therefore, the overcoming of opposition can only be obtained, and only exists, transcendentally in the Supreme One, and not in finite beings. In them, there is harmony, the combination of opposites, the agreement of the discordant, the number, for where there is multiplicity, there is number (arithmos).

The failure to understand Pythagoreanism in this way allowed for the various false interpretations that hasty exegetes have made throughout time and that allowed this rich philosophical conception to be undeservedly disparaged, a failure from which we do not exclude some Pythagoreans who did not properly understand the genuine conception of the master from Croton.

Finite beings, when they cease to be what they are, return to the primordiality of the creaturely foundation (hypokeimenon), which is the aether (ether) for Pythagoras, which ensures the continuous cohesion of creaturely being in its source, unlimited-limited, indivisible, permanent, indissoluble. The nature of the soul, for Pythagoras, is eternal. This aether is also the soul of the world, which would be spoken of later by the Platonist-Pythagoreans and the Neo-Pythagoreans.

Pythagorean themes

The possible, as such, can be contradictory because the power to be this or the power to not be this are both possible and, as such, valid. However, actualization already implies order, and the contradictories cannot currently be given, for if this were done, the inability to do this remains in the epimethean state of being. Thus, the cosmos is the actualization of the possible, and while they are taken as such, they are chaos. That is why in the religious ideas of many peoples it arises that God brought order to chaos in the act of creation. Bringing order to chaos is giving the order of existence to the possible, making them actual, which implies, from the outset, the exclusion of the contradictory. The cosmos is the affirmation of the actualized possible, which, as such, affirms an order or, in other words, a connection of necessity with what precedes and follows, and subordination to a totality; otherwise, there would be contradiction. That is why there is order, there is arrangement, there is connection. The Cosmos, as creation, is ordered, and successive facts are connected to previous ones by an order; and that order is the negation of contradiction.

These words of Plato, in the Gorgias, are genuinely Pythagorean:

“The wise, O Callicles, say that friendship, order, reason, and justice together maintain heaven and earth, gods and men; that is why they call this whole the Cosmos, meaning good order.”

If the actualization that follows a fact contradicted the previous one, one of them would be negated because, as we have seen, there is contradiction when both privation and possession are simultaneously affirmed. Something is contradicted when possession is denied, when it is affirmed that something is deprived of what we affirm it possesses. Now, before privation or possession occur, both are possible. But once one of them is actualized, the other cannot occur simultaneously, as it would be contradictory and absurd. And in the sequence of facts, everything that happens obeys this principle. The Cosmos, as something in act, is the negation of Chaos (of the possible).

The Pythagoreans spoke of a Pan-psychê, which is similar to Plato’s World Soul. They affirmed that all things—minerals, plants, animals, and humans—derive from the same reality, and there is a kinship among them (fsyngéneia). There is a real brotherhood among all things, which are connected by a common law to the same being. There is a community of origin that binds them (friendship, love), there is a reason that connects them (logos), and a justice, a distribution to each according to what each deserves by its nature and being.

If we turn our thoughts to what we said earlier, it will be easy to understand what the Pythagoreans meant by these aspects, for since actualization is connected, and facts cannot contradict already verified laws, there is, among all things, the presence of the same law, the presence of non-contradiction, as long as we understand contradiction in the sense we explained above, which is clear and has prevailed throughout time. In modern philosophy today, we find confused thinking because there is confusion between contradiction and distinction, diversity, and difference. In the former, there is a relationship of privation and possession, and the beings of the universe, of the cosmos, are connected by the presence of a logos, of a law, for what exists, what has existed, and what will exist cannot affirm and deny the presence of that law. It is this law that establishes the kinship the Pythagoreans spoke of.

Now the true meaning of the famous phrase of Pythagoras, which he constantly repeated to his disciples, becomes clear:

“You shall know, as far as is permitted to a mortal, that nature is, from every point of view, similar to itself.”

We also find expressions on Pythagorean tombstones that confirm this principle of analogy. We can cite this one found in Turium:

“I come pure from among the pure, O Queen of the underworld… For I also take pride in belonging to your blessed race.”

Here, there is an affirmation of participation, a theme we examined in “The One and the Many in Plato,” the conclusions of which are valid for Pythagoreanism, as Plato’s thought never separates from it, as we have seen in that work and also in this one.

A note by Jamblichus

Jamblichus, in his book “The Mysteries of the Egyptians,” begins with these words: “The God who commands the word, Hermes, was once rightly considered common to all priests; he who presides over the true science of the gods is one and the same in all. It is to him that our ancestors dedicated all the discoveries of their wisdom, giving the name of Hermes to their own writings.”

Hermes thus appears as the author of all profound thoughts, even when expressed by an individual. Just as the initiates in the highest degrees of the Egyptian mysteries were called Hermes, there is a possibility, supported by sufficient evidence, that Pythagoras was also a title given to those who reached the highest initiation in the Pythagorean order. It is common in affiliated organizations to name initiates after famous Pythagoreans like Philolaus, Euclid, Apollonius of Tyana, etc., based on the degrees they attained. This possibility would help us understand the existence of various Pythagoras figures and also explain their presence in different places and times. However, what cannot be doubted is that there must have been an original Pythagoras, just as there must have been an original Hermes preceding all others, just as there was a first Buddha.

It is not difficult to comprehend this possibility when we consider the examples we constantly witness. Those who are great in military conquests are given the name Alexander or Napoleon, just as we speak of a Dante, a Petrarch, or a Leonardo, attributing these titles to those whose works resemble the first bearers of such names.23

The unknown god (Theos agnostos), which we find not only in hermetic writings but also in Greek ones, certainly does not have a Greek origin, as the Greeks never renounced research, and Greek philosophy is characterized by the fearless pursuit of solving the greatest difficulties without ever abandoning it. However, if we turn our attention to the early stages of Greek cultural development—and we refer to the Orphic and Dionysian cults—the unknown god would fit perfectly as a reference to the uninitiated, to those who do not know the way.

Proclus speaks of this unknown god. We also find a reference to the agnostos kath’auten, the unknown in itself, in Aristotle’s Metaphysics, referring to prime matter, or rather, the incognizable, as such. Naturally, prime matter cannot be the object of sensible knowledge because, as such, it is undetermined. But prime matter, in itself, for Aristotle, is an abstraction of our mind because it is always given as determined, although what determines it and its determination are distinct. St. Paul referred to this unknown God and affirmed that he was Christ.

In hermetic doctrines, he is the transcendent God. We also find this thought among the Pythagoreans, which is of utmost importance for understanding the theses we will present in this book. They not only affirmed the transcendence of the One but also the transcendence of God. Sextus Empiricus showed that while all people recognized the existence of the divine and practiced various rituals and customs related to it, they did not have the same conception of the nature of divinity.

For many, the essence of God is unknowable, although it is knowable to others.

Although everyone accepts the existence of God, not everyone can explain His essence. If we turn our attention to the arithmological speculations carried out by the hermetists, we note that whenever they refer to the Supreme Being, they call it the One, the Monad. We often find the assertion that the Lord, Father, and Alone is not the One, but the source of the One. This thought is also found among some Pythagoreans, and it is of great importance.

In the Pythagorean speculations of Philo of Alexandria, we observe the following passage: “For the Being, which is superior to the good, prior to the Monad, simpler and purer than the One, can be seen by no being except itself, for it allows no other to perceive itself” (Praem. 40).

And further, he tells us that just as light is seen by means of light, God, who is His own light, can only be contemplated through Himself. No other being cooperates or can cooperate in the pure perception of the reality of God. Those who depart from created things in order to contemplate the uncreated, which engenders the whole universe, proceed like those who, starting from the dyad, wish to investigate the Monad when it is precisely the opposite, for the dyad must be considered in relation to the Monad because the Monad is the principle.

Here, the Monad is identified with Parmenides' One. And he continues in his other works, telling us that the Monad is in the category of the One God since every number is subsequent to the world, just as God is prior to the world and its creator. The Monad is pure, indivisible, and the path to contemplating being. According to Philo, God is superior to the Monad but is also a Monad. Now, this doctrine is not exclusively Neopythagorean, as Pythagoras said that the One begets the One, which reveals that such a doctrine predates Philo.

This is also the thought expressed by Moderatus of Gades and Theon of Smyrna. Among the Pythagoreans and Neopythagoreans, we consistently find the affirmation of the One above all things, and this One surpasses our knowledge, while there is a second One that is the intelligible, the comprehensible.

However, this Pythagorean doctrine was not always accepted as such, as can be seen from Aristotle’s statements.

Eudorus, before our era, said, “Regarding the Pythagoreans, they considered the One not only as the principle of physical beings but of all things absolutely, placing contraries as secondary and elementary principles, to which, although not being first, they also subordinate the two parallel series.” And he continued, “On the higher plane, it must be said that the Pythagoreans place the One as the principle of all things. On the second plane, there are two principles of reality, the One and the nature opposed to the One. Of all things conceived as opposites, the good ones are subordinate to the One, and the bad ones to the contrary nature. That is why, according to them, these principles are not properly principles: for if one of the two principles is the cause of such things and the other of other things, they are not universal principles of all things, like the One.” And he goes on, “That is why, following another path, they also said that the One is the principle of all things, as it is the principle of both matter and all beings that arise from it; it is the God who is above all.”

The One is a principle, and from the One come the elements (sioi-keia), which they call multiples. These supreme elements are two: the masculine principle and the feminine principle. The knowable (nostón) and the second unknowable (agnostón).

Thus, we have the One and another One opposed to the dyad.

C.J. de Vogei writes in Mnemosyne, Vol. IV, Ser. 2, pp. 205/216:

"Since Aristotle frequently mentions the fact that Plato called matter the Great-and-Small, we must know that Porphyry reports that Derkyllides, in the eleventh book of his ‘Philosophy of Plato,’ where he deals with matter, quotes a passage from Hermodorus, a disciple of Plato, taken from his work on the same subject, in which Hermodorus presents matter as resembling the limitless and the indeterminate, showing it as belonging to things capable of more or less, which also includes the Great-and-Small. Indeed, after saying, ‘Plato says that among beings, some exist by themselves (athéautá)—such as man, horse—and others in relation to other things (prós hétera), and of these, some are relative to contraries (ôs pro enantia)—such as good and bad—others are relative to another term (ôs prós ti), and of all these relatives, some are determined, others indeterminate’ (sg. Hermodorus), he adds: 'Plato also says that everything that is designated as Great-versus-Small (mega prós micron) encompasses more and less, so that through “more,” “greater” and “lesser” extend to infinity; in the same way, “wider,” “narrower,” “heavier,” and “lighter,” and all things designated in this way extend to infinity. But what is designated as the Equal, the Fixed, the Concord, does not encompass more and less, while its opposites do: for there is something more unequal than such unequal, more moved than such moved, more dissonant than such dissonant.

Thus, the two syzygies (combination of sounds), more and less, contain the whole set, except for the term One.

Moreover (Plato says that) the object in such a way (that is, capable of more and less) is presented without fixity, without form, without limit, as non-being by negation of being, and that, on the other hand, this object has nothing in common with either the principle or the essence, but it leads to being dragged into a kind of confusion. Plato shows, in fact, that in the same sense, where the first cause is eminently the efficient cause, in the same sense, it is the principle, and thus matter is not a principle. He also said that for Plato, there is only one principle."

In this way, Vogei argues against the accusation of dualism commonly made against Plato. Matter is not the principle of beings but something that the first cause employs to create corporeal beings. Matter is the determinable indeterminability, to use metamathematical expressions in philosophy. It does not have limitation in itself because it is indeterminate. It includes the Great-and-Small, maximum and minimum determinability. Now, more and less are relative and, as such, indeterminate-determinable, as they can potentially extend to infinity.

Regarding Aristotle’s argument that matter, for Plato, is ungenerated and, if so, was never created, Simplicius responds with the following words:

But if matter is ungenerated, some say, and imperishable, why isn’t it also a kind of First Principle like God? Indeed, if it had been derived from God, it would not be ungenerated. However, what Aristotle designates as ungenerated is not that which depends on a cause but that which does not come into being from a temporal beginning, which he shows at the end of this treatise. He shows that motion is also ungenerated and imperishable, although he has said that every moved thing is moved by a cause. Moreover, in the same way, he would also call ungenerated the properties common to the Forms, thus determining a multiplicity of first principles. And yet, this is what he proclaims: ‘It is not good for there to be many heads.’ In a word, it is as an elementary nature that Aristotle represents matter, which should not be opposed to the efficient or final cause if it is true that it aspires to this visible order ‘like the female to the male, and the ugly to the beautiful’ (Physics 192a23). On the other hand, the heterodox say that matter is evil and make it an opposite principle to the good, and since then, they oppose it to the good as an efficient principle. Consequently, they mention generations originating from matter in their teachings and speak foolishly of the strategic operations of matter, its designs, and its triumphs over the good. But Plato, who, in the Timaeus, teaches the doctrine of the proper causes and the auxiliary causes of the existence of the world, links matter to the auxiliary causes and calls it imperishable, just as he does with the whole world. Moreover, he judges it unfitting to call it the First Principle, as Hermodorus, a disciple of Plato, showed when, in his book on this subject, he set forth, among other Platonic dogmas, those that dealt with matter, as Derkyllides reports.

The Monad and the indeterminate dyad are the principles of the universe, but in any case, the Supreme Being, the Platonic Hen, the Pythagorean One, predates them all.

According to Alexandre of Aphrodisias, Aristotle, in his Pen Tagathou (On the Good), a dialogue that has been lost, presented four arguments that he considered Pythagorean, which we reproduce below, but in a clear manner, offering the criticism that they deserve.

The first argument is as follows: Did the Pythagoreans think that the First and the Simple (to asyntheton) would be the principle of all things? Now, it happens that the surface is prior to bodies because what is simpler and not composed is first by nature. Also, what mathematicians call points and the Pythagoreans call monads are prior to lines (stygmata). Now, points are absolutely uncompounded and therefore are first beings, for compound beings are composed of simple beings, which is why the simple must always precede.

According to Plato, the Forms are prior to beings. It is through them that beings have the being that they are. Plato called these forms Numbers (arithmoi). Since there is nothing prior to Number, Aristotle continues to affirm, in this case, if the Forms are prior, they must necessarily be numbers. That was the reason why Plato said that Numbers were the principles of Forms and that the One is the principle of all that is real.

In this first argument, Aristotle goes on to confuse the Supreme One with the arithmetical one.

Aristotle’s second argument is stated as follows: the Forms are principles of all things, and numbers are principles of the Forms. However, Plato said that the Monad and the Dyad were the principles of numbers. Now, the Dyad is the opposite of the One, and if the One is indivisible, the Dyad is divisible. Here we can also see that Aristotle’s confusion began in his youth, because the arithmoi arkha? and the arithmoi mathemaiikoi are confused as identical.

The third argument is as follows, reproduced verbatim as Alexandre of Aphrodisias presents it:

"Furthermore, thinking to show that the Equal and the Unequal are principles of all beings, both of those that exist in themselves and of opposites — because he sought to include everything in these two, as the simplest principles — he (Plato) connected the Equal to the Monad and the Unequal to Excess-Deficiency: because the Unequal consists of two terms, the Great and the Small, which are the Excess and the Deficiency.

That is why he (Plato) called this form the Indeterminate Dyad, because neither of the two, neither the Excess nor the Exceeded, as such, is determined, but rather indeterminate and unlimited. However, when it was determined by the One, the Indeterminate Dyad became the numerical dyad: because this dyad is formally one thing.";

In summary, the fourth argument alleges that the first number is the numerical dyad, and in it, the One and the Great-and-Small are its elements. In it, there is the double and the half, as double-half is equal to excess-exceeded, but it is not true that excess-exceeded is equal to double-half. “In this way,” Alexandre continues, “excess and exceeded are indeed the elements (stoikheia) of the double. Moreover, excess and exceeded do not become the double and the half until they have been determined — because double and half are no longer indeterminate, just as triple and three, quadruple and four, or any other number is no longer indeterminate, nor is excess determined from now on. It is the nature of the One that produces this determination — because each of these numbers is one and to such an extent that it is something and a definite thing — we place as elements of the numerical dyad, the One and the Great and the Small. Now, the first number is the dyad. Therefore, the elements of the dyad are the One and the Great and the Small” 24.

Aristotle reduces Platonic thought to a sophistical syllogism, as the One, which is an element of the Dyad, is not the One, which is the Supreme Being, but rather the universal substance, as it was for Pythagoras.

Furthermore, all of Aristotle’s reasoning is tainted by the deforming influence of the empiricist schema, as he only admits the surface as prior to bodies and points as prior to lines. If the points are monads, they are not the Monad, which is the One, for the unity (being one) of composite things participates in unity, which, in the Supreme Being, is the highest perfection. Composite beings are one because they are a determined being, but the unity of the Supreme Being is being the Being.

All of this passage by Aristotle regarding the first argument had already been refuted by the authors we previously mentioned, and the remaining arguments deviate from the true meaning of both Pythagoreanism and Platonism. This is what we have the opportunity to demonstrate in this book and in our comments on Plato’s dialogues, where we analyze his famous dialogue “Parmenides” and examine his understanding of the Great and the Small, excess and deficiency, etc.

After this brief analysis of what is most positively Pythagorean throughout the ages, it can be concluded that there is a thought of Pythagoras that we must outline, and a multiple, heterogeneous thought of his disciples, as well as an interpretation that was made based on certain poorly examined statements that constructed what is prejudiced about Pythagoreanism, that is, that set of poorly founded affirmations that served, over time, to almost definitively distort the true philosophy of the great master from Samos, whose doxographical delineation we partially attempt to achieve in this book.

The Hieros Logos

(The sacred discourse)

It is known that the teaching imparted by Pythagoras was oral, for in his time, books were something rare. Furthermore, it is said, there was a fear that by entrusting the deepest knowledge to lifeless letters, they could be used more for evil than for good in the hands of the ill-intentioned.

But this statement must be considered in perspective, for the fear was relative, and so much so that it was affirmed (and there are historical bases to substantiate this claim) that Pythagoras had written a work in verse entitled The Hieros Logos (The Sacred Discourse), in which the foundations of his doctrine were contained in symbolic language. However, the clear understanding of what he wished to convey was thus conditioned by the capacity for symbolic interpretation. And since symbolic interpretation was provided according to the degree of initiation, the extent of the knowledge exposed would be obtained proportionately to the degree. In other words, the reader of such a work would understand it only within the limits of their assimilative abilities, which would prevent the danger of the knowledge obtained falling into inexperienced hands that could use them for purposes other than those genuinely set and marked by the true intent of the Pythagorean order.

According to Diogenes Laertius, Pythagoras also wrote other books on Pedagogy, Politics, and Physics, and he is also attributed with the authorship of a poem called Peri tou holou (On the Whole), whose theme would be the Cosmos, a kind of analysis of all things considered in their unity. It is also said that Plato, at great cost, managed to acquire some of these works from Archytas.

But the authenticity of such books can be doubted, as they may have been written by disciples who attributed them to the master, which was common at that time. Nothing has survived of these works to our days, except for some scattered fragments in the works of Pythagoreans and Pythagorean followers, which, however, have served as elements for the foundation of our exegesis of Pythagoreanism, which we consider to be genuine based on doxographic grounds, and which we justify in this book.

It is said that the Hieros Logos was originally written in verse by Pythagoras and only later translated into Doric prose by him. It is claimed that the publication of this book was carried out by Telauges, husband of Bitale, Pythagoras' granddaughter, based on the notes he had left to his daughter Damo. In this case, this work would never have been published by Pythagoras himself, but only later, long after his death. The belief that this work really existed is reinforced by the fact that the fragments that have come down to us are written in a language that reveals its antiquity, as pointed out by Mr. Delatte.

Hence Rostagni states that the discourse of Ovid in the Metamorphoses is a paraphrase of the original discourse. In the Golden Verses, there are several reproductions of maxims that must have belonged to the Hieros Logos.

The Hieros Logos would be a scientific and philosophical treatise, and from the fragments attributed to it, we will reproduce some that will allow us to have an idea, albeit very vague, of what it consisted of in its reality.

Its first words would be as follows:

“I want to sing to those who can understand; close the doors to the profane.”

“Youth, worship, in respectful silence, all truths.”

Then follows a succinct vision of human life, from the Golden Age, the age of innocence, to the state of fall and sin, original sin, which man attains in his later wisdom.

"O race paralyzed by the fear of death,

Why do you fear the Styx and shadows and vain words?"

Just as wax, on which new figures are imprinted, remains, in a way, the same, although it does not retain the same shape (aspects), so the soul always remains the same, although it migrates, I tell you, through new figures."

Everything changes, everything transforms, but at the same time, everything remains identical to itself through the unified rhythm of numbers.

Everything obeys this law.

"When, having left the body, you depart for the ether.

You will become an immortal god and will never die again.

(From the “Golden Verses”).

To imitate God is the path to the elevation of man. God is within us; we must imitate Him (hepou theou — be a god). And Nó peithou advises, obey the spirit and you will follow God, for to find Him, one must follow the path of wisdom, moving away from passions.

Finally, we shall attain the One (Hena genesthai), generate the one from oneself, become whole. Only then will we achieve inner tranquility and peace among men.

Its main ethical principles are inscribed in the “Golden Verses,” which we reproduce at the end of this book.

Plato’s demiurge and Pythagoreanism

This is one of the themes that have sparked intense debates among scholars and has been used as a basis for unfair criticisms by opponents of Platonism.

Who is this demiurge, this craftsman, this being who brings about the realization of things, with eyes turned towards immutable forms, shaping formless matter into a definite form?

The myth of the demiurge, as it appears in Platonic works, is undoubtedly didactic. However, we should not take this didacticism to the extreme, because thanks to symbolic dialectics and what we have studied so far, it is possible for us to penetrate its true meaning.

But before we present our contribution, it is necessary to start by quoting a passage from Proclus, which provides us with many suggestions and can facilitate a clearer understanding of the myth. Socrates, in “Timaeus,” reveals that it is a “likely myth,” that is, it imitates the truth without being the truth itself, but it is sufficient to provide a didactic understanding to the listeners. Moreover, he adds that very few people are capable of understanding it in its true sense.

Proclus writes in Timaeus (8 c 3) I, p. 300:

“Discovering this Demiurge of the universe is difficult,” says Plato. Indeed, discovery is achieved in two ways: one proceeds from the First Principles through the path of science, and the other proceeds from the Second Principles through the path of recollection. Now, it must be said that the discovery proceeding from the First Principles is difficult because the discovery of the intermediate properties is connected to the highest doctrine. As for the discovery proceeding from the Second Principles, I am almost tempted to say that it is even more difficult. For it is through these Second Principles that we propose to see the essence of the Demiurge and the totality of its properties. It is necessary to consider in their entirety the nature of the beings produced by it, all the visible regions of the world, and all the invisible natural powers that establish the existence of sympathies and antipathies in the universe. Furthermore, we must consider the fixed rules that govern nature and the very natures themselves, both immaterial and material, divine, demonic, and those of mortal beings. We must also consider the different categories of beings that fall under the realm of life—some immortal, others mortal; some untouched by matter, others immersed in matter; some having a complete nature, others consisting of parts; some endowed with reason, others without reason. Additionally, we must consider the most perfect complementary beings through which the entire intermediate region between the gods and mortal nature is closely linked. We must consider the souls of all species, the multitude of gods that vary according to different portions of the universe, the expressible and inexpressible connections that relate the world to the Father. Yes, if we consider all these things, the path that leads to the Demiurge remains quite imperfect for comprehending the Father. For it is not permitted for anything imperfect to have contact with the Perfect Whole (Omniperfect).

Moreover, it is necessary for the soul, having become an intelligent world and having become as similar as possible to the entirety of the intelligible world, to approach the Creator of the Universe. By virtue of this approach, it becomes somewhat acquainted with Him through the continuous application of the spirit—because the activity of thought, interrupted relative to a given object, awakens and enlivens our rational faculties. Through this familiarity, it, having arrived at the gate of the Father, enters into union with Him. This is what discovering God means: to go to meet Him, to become one with Him, to enjoy His presence, alone with Him, to cause Him to reveal Himself in person when the soul is “rapt” away from all activity, and to consider scientific discourse as mere fables. For it is when the soul is united with the Father that it feeds, at the same feast as He, on the truth of being, and in the flash of pure light, it is initiated into perfect visions that never change.

Yes, this is what it means to find God… It is not to discover Him through the path of opinion (for this is uncertain and hardly removed from irrational life) or through the voice of science (for this proceeds through inferences and chains of reasoning, which do not immediately reach the intellectual essence of the Demiurge). It is to find Him through an intuition that allows Him to be seen face to face through contact with the intelligible, through union with the intellect of the Demiurge. And truly, this discovery can rightly be called “pure work” in the proper sense, because it is laborious and unpleasant to obtain, as the object is only revealed to the souls after they have traversed the entire hierarchy of living beings. It is also the true struggle of the souls, for it is after vain pursuits in the created realm, after purification, after the illuminations of science, that they finally ascend to intellectual activity. And the intellect within us, which leads the soul to the harbor in the Father, sets it up far from any blemish in the thoughts of the Demiurge, and adds light to light, not only the light of science but also another, more beautiful light, more intelligent, more similar to the unity than the former. For there is the harbor of the Father, the discovery of the Father, the immaculate union with the Father.

As for the words, “When one finds God, it is impossible to speak of it,” they may well indicate, similar to the Pythagoreans who kept the doctrine of divine things secret and refused to discuss it with anyone, that “the eyes of the Vulgar lack the strength to keep their gaze fixed on the truth,” as the Stranger from Elea says. But it can also be said that these words teach a much more august doctrine, namely, that when one has found God, it is impossible to speak of the things as they were seen. For the discovery would not consist in the soul saying something, but in being initiated into a mystery and being subjected to the influence of divine light… and the soul, maintaining what could be called its silence. In fact, now that the soul is not of a nature to grasp the essence of other realities through naming, definition, or scientific demonstration, and can only be reached by thought, as Plato says in his Letters (VII 342 s), how could it discover the essence of the Demiurge except through purely intellectual means? And having found it this way, how could it convey what it saw through words and verbs and make it known to others? For discursive reasoning, which proceeds through composition, is unable to describe the essentially uniform and simple nature.

But one may ask, isn’t it true what we have been discussing at length about the Demiurge and the other gods, and about the One itself? Undoubtedly. But if we discuss these realities, we do not define any of them in their own essence. We can argue about them, but we cannot express the intuition we have of them: for it is to “encounter” them, as it was said. Now, if the soul “encounters” it only when it is silent, how could the flow of vocal words be sufficient to express the “encountered” object as it is?"

So who is the demiurge then? Let us remember the Pythagorean theses we have already examined. The Hen-Dyas, the One, creates the indeterminate-determinable dyad, and creates determination, the formative act and the informable potential, for to make implies simultaneously what is made, and to create, what is created. Creation does not precede the creature; ontically, creation is to give rise to the creature.

The demiurge is, in short, the formative act, the determinant that determines, and in giving form to matter, it imitates the eternal forms. It is through comparing them that it brings about finite things. The symbolism is easy. Finite beings imitate the perfections of the eternal forms, thus things participate in them. And since things are what they are through the forms that inform them, they require an efficient cause to bring them about. The demiurge is the universal efficient cause: the determinant, the formative act. • í

Pythagoras and man

Id quod inferius úcut quod superius…

(Hermes Trismegistus)

When Pierce says nowadays, “There is something in nature to which the human spirit is analogous; Nature fertilizes the human spirit, suggesting ideas that, as they develop, resemble their Mother, Nature,” he is reproducing Pythagorean words.

"Gnôse dé themis esei physin peri pantós homoien”

These words of Pythagoras, reported to us by Plato, affirm:

“You will know, as much as is possible for a mortal, that Nature is, in everything, similar to itself,” as we see in the Golden Verses.

There we have the fundamental principle of Pythagorean analogy. Man’s knowledge reveals to him that he has an analogy with Nature and with Being, because if the latter is not exhaustively known to him, it is not impermeable to him, and analogically he can know it.

“God, in His providence, has given man two admirable things: the faculty to embrace the truth and the ability to do good to his fellow human beings; both can be compared to the works of God” (Words attributed to Pythagoras by Aelian).

That is where man reaches the supreme for Pythagoreanism. It is in action and knowledge (two genuinely Christian affirmations).

From there, we can infer many Pythagorean theses, implicit in these words, which endure through time as truly from the master of Samos.

The human intellect can know everything that is intelligible.

Being, being intelligible, can be known by man, proportionately to his nature.

Absolute nothingness is unintelligible; consequently, being is intelligible.

Being, taken in its indetermination, is the first, natural and adequate object of human intelligence.

Man does not know being directly, but he knows it abstractively at least.

And man knows the quiddity of things, their intentional form. And since this is an imitation of the immutable and eternal form, he has a knowledge of it proportionate to his nature.

Id quod inferius sicut quod superius… The lower is thus something similar to the higher, and, thanks to analogy, we understand that man imitates the Divinity in some way, as we demonstrate through his works.

There is no doubt that the ancients (pre-Christians) had reached the idea of creation, considering what we have examined through Pythagoreanism. But if we want to look even closer to our time, we will find these words of Epicurus:

“Oudèn gínekú ex tou mè òntos” (no being is generated from non-being).

If no being is generated from non-being, what is generated is generated by being. Epicurus' statement is considered as the denial of creation. And how could he deny creation without having the concept of it?

Some authors usually say that in other thoughts the idea of creation arises, but imperfectly, as the Creator needs to shape something already pre-existing, so that by shaping it, it becomes a creature.

But this explanation should always be considered didactic, as for the uninitiated there is a need to use concepts and representations from the schema of the common man. How can we explain to the common man that the Creator created the creature out of nothing?

Thomas Aquinas explains that this “out of nothing” (ex nihilo) does not mean that the creature was made out of nothing, but that before being, there was nothing of the creature. Or rather, since he accepts the possibility of the eternalness of creation, the creature belongs to temporality, and in eternity, there was nothing of the temporal creature. The Creator, as divine, precedes ontologically and axiologically the creature, which is temporality. - Creation, therefore, did not have a specific day; it has always existed temporally. But Divinity exists eternally; it is transcendent to Creation.

The Christian creationism also does not philosophically solve the theme of creation, and it proclaims itself in the face of a mystery, considering, furthermore, philosophical attempts as respectable but none as dogmatically true. Thus, creation remains open as a question in Christian thought, philosophically speaking, although not in its religious aspect.25

Pythagoras’s dream

The Pythagoreans tell us that when visiting the cave of Proserpina in Greece, Pythagoras fell asleep and had a dream, which he later revealed to his disciples. The dream served as a theme for various interpretations, some misleading, others taken advantage of with the intention of undermining the great contribution of the master from Samos, attempting to interpret it as a confession of the failure of his experience through the path of knowledge to attain the supreme Mathesis, the ultimate goal of all human desire for knowledge, of all human restlessness, for our minds will only be at peace when they find supreme knowledge.

Pitaghoras reported that during the dream, he was analyzing sensible things. From this analysis, he realized that the reality of these things was given by geometrical structures, and these, by mathematical numbers. Upon reaching this state, he realized that what constitutes the reality of things were the forms, the eidetic ciphers, which constitute the intrinsic proportional law of beings. When he had reached the forms, he had reached the great laws of mathematics and the ontological structures of the forms, which placed him face to face with the supreme numbers (archaic numbers). In that instant, he felt as though between Being and Non-Being. In that state of anguish, he seemed to be before the wall of Chaos. There was no longer order, no cosmos. He could touch that wall of Chaos, but suddenly he found himself before an impalpable void, an absolute darkness where nothing could serve as a point of reference, for he touched nothing, felt nothing. An absolute silence surrounded him, a silence of darkness. Nothing.

In an impulse, he wanted to cross that void in search of a limit. It was a desire to find any point of reference. He found nothing, and found nothingness. But suddenly, through the darkness, he seemed to perceive something; it was a figure that gradually took shape. And now he was before a mirror. In that mirror, he saw his own countenance, unreal, static, as if frozen in time, insurmountable, impassable. He felt himself going mad. He thought of his disciples and resorted to the last resource. He appealed to the master of masters, and then he felt himself returning to himself. He hastily left the cave of Proserpina. He did not know what to say. He remained silent for a long time and then recounted to his disciples the dream he had had, whose symbolism needed to be deciphered.

For Pitaghoras' opponents, and among them many who, in truth, are Pythagoreans without knowing it, this dream was a warning of his failure. His search along the paths of knowledge would not reveal to him the immortal Isis, the truth, the Greek aletheia, the Hindu Varuna. No. Just when he thought he had reached the highest point, he found himself in emptiness, having only himself as his contemporary.

At the end of his journey, he returned to the beginning. Everything he had created was only a projection of his spirit. When he thought he had reached the zenith of knowledge, he was at the nadir of man himself. This dream revealed that man, in seeking the transcendent, would ultimately find himself unreal, false. That image was the symbol of our concepts, of our constructions. At the end of all things, man finds only himself, but already stripped of reality, merely a cold and lifeless image of himself. It is the failure of all human science, of all striving, which leads only to the sad and melancholic certainty that man only knows what he knows, and nothing more. In thinking he has encountered the truth, he finds only his truth, only what he constructs, only what he himself is.

Pitaghoras' dream, thus, refuted his own doctrine. Human speculation along the paths of knowledge led him only to a vicious circle: constructing his knowledge with his own schemes, he only arrived at the certainty given to him by the schemes and nothing more. Beyond that, lies the eternally impenetrable, the unknown and the unknowable. Ignoramus et ignorabimus, we do not know and will not know. Pitaghoras thus became an agnostic. In all his striving, he ultimately achieved what he intended to refute. The ultimate certainty was that we know nothing for certain.

But this symbolic interpretation, however well-founded it may seem, completely distorts the meaning of Pythagoreanism.

It is possible that at a certain moment, Pythagoras admitted that it was the duty of man to achieve exhaustive knowledge of the truth. Let us assume this possibility. But this possibility, which is certainty in some Eastern beliefs, is not founded in the essence of Pythagoras' thought. What is certain, as we have seen, is that he never claimed that man’s knowledge could fully and exhaustively grasp the essences themselves. The state of epiphany, of illumination from all sides, which the epopt attains, the theophanic illumination, is only a glimpse and not the direct vision of truth. The entire framework of Pythagoreanism denies this possibility to man. However, it does not deny that man can attain truth proportionate to his nature. This dream is not a proclamation of failure because Pythagoreanism never promised man the impossible. The acquired illumination is always proportionate. The direct vision of divinity, which is truth, is not given to man in his present state. That is what he means. Through the pursuit of science, man can reach even the archaic numbers, comprehend many of the secrets of divinity, but to attain the seventh solitude, the solitude of the Supreme One, would be to reach the state of beatitude of the Christians, the yoga bhakti of the Hindus, and that does not belong to the finitude of our minds.

With this dream, Pythagoras reinforced his position, which is just and logically well-founded. But since his entire philosophy revolves around analogy, and he knew that “quod id inferius sicut quod id superius” (what is below is like what is above), through analogical analysis, which is the foundation of Socratic-Platonic dialectic, man could carefully follow the symbolic path and attain the highest possible enlightenment.

But what about his state of anguish? That anguish was human and arose from the limitations of the body. Between the two worlds, what man achieves and what transcends him, there is the impulse for more, which is the mind’s appetite for transcendental truth, as well as attachment to the body, to our life. In that agonizing moment, anguish had to seize him. To go beyond would require breaking free from what binds him to materiality. Human will is not strong enough to achieve this detachment on its own. We will not discuss here an extremely serious theme, namely, whether human will is sufficient to achieve this detachment. But Pythagoras thought of his disciples: he had a mission to fulfill. He needed to return.

In all religious myths, the initiate, who is the master, goes through these states. It is a myth that is constantly repeated. It is the moment when he feels the limits of his own knowledge. But upon reaching them, he cannot come to the primary conclusion that there is nothing more, because there is nothingness beyond, neither beyond nor on this side of being. He knows this well. The reached limit, which is the affirmation of man, is also the affirmation of what transcends him. The unknown Theos agnostos is unknown to man in his ontic exuberance, in his immanence, but not in his transcendence. Penetrating into the depths of divine immanence is impossible for man in his present state, but he can know it analogically, based on the perfections that his intelligence attains.

The Ten Laws of Pythagoras

The Mother of All Things is the Tetractys (the One, the Two, the Three, and the Four; 1, 2, 3, 4, whose final sum is Ten, the Sacred Decade), and from her all things that are and that may be originate.

In the symbolism of numbers, which we examine in “Treatise on Symbolism,” we delve into the mystical meaning of those numbers, which reflect the so-called ten laws of Pythagoras, constituting the Theíraclys, the Sacred Decade, Mother of all things, because it is from the Ten, from the ten laws, that all things are generated and arise.

The number 1 is the symbol of the One, but also of unity in general, for every unity is 1. In Pythagorean symbolism, it signifies the

The Law of Unity.

It is the law of wholeness, for all things that exist, in whatever manner, constitute a unity. To be, in any way, is unity, to be one. Only nothingness is not unitary, because nothingness is not. The law of oneness governs all beings that participate in the supreme unity of being, to a degree that is intensely lower, proportionate to their nature. The maximum unity is the absolute unity of simple simplicity, of Being, which is simply being and without deficiency; therefore, all being, the Supreme Being, the One.

Because all things are “as if in prison” in the Supreme Being, all participate in this law that governs all things.

Everything that is finite is unitarily what it is and tends to become an integral part of a unity. Nothing happens that is not unitarily so, according to the intensist degrees of unity. This law governs all things.

Thus, the arithmetic number 1 symbolizes Unity and, therefore, can symbolize everything that is and in whatever way it may be.

The Supreme Being, the One, as form, is the Father, generating the One as operation, as action, through an in-intra procession, for the creating One is the Son, begotten by the former. In religions, the Father and the Son appear as symbols of the closest correlation, for the Son is the son of the father and the Father is the father of the son, so the affirmation of one is the affirmation of the other. Translated into philosophical language in the Pythagorean sense, the Hen Prole is existential and essentially itself, unchanging and eternal, because Being, as Being, is absolutely Being. But this Being is active, acts, accomplishes, operates. And operation implies choice, intellection (intellect). The Hen Prole is Will, as wanting, words that can symbolize to us the omnipotence of the Supreme Being, who can do everything that can be. But in accomplishing something, its operation is intellectual, it chooses what will be actualized. The Supreme Being, as operation, is the Hen that generates the indeterminate Dyad, which corresponds to the formative act and the materializable potentiality, to remain, in a certain way, in line with Aristotelianism, or rather, taking advantage of Aristotelian terminology to aid the exposition of Pythagorean thought, naturally preserving the formal structure of Pythagorean thought, for the formative act, the determinant, and the materializable potentiality, the determinability, are only vectors that arise simultaneously from the creative act of the Hen-Dyas aorist, for it is the Son, who is the Creator, because it is Being, when it operates, that creates. But one does not abysmally separate from the other because determination implies determinability. Our abstracting mind separates conceptually what is one in reality but formally distinguishable, for the Logos of the creating One generates, in its acting, the action of the indeterminate Dyad, whose Logos is dual, for action implies the acted upon, as it is inherent in the acted upon and does not separate from it, as Suarez very well showed. Thus, in creation, this belongs to the creature that arises from the Dyad. The Hen (Son) acts by performing the action, but this action is an absolute modality inherent in the acted upon. Thus, closer to us, the movement of a wheel is an absolute inherent action of the wheel. The action is not a modality of the Supreme Being. If it were, it would undergo changes. Its acting consists of accomplishing the action, and the action determines what is determinable. Creation is of the creature and not of the creator. We have already demonstrated this thesis, abundantly supported by evidence, in “Man Facing Infinity” and “Concrete Philosophy,” to which we refer the reader.

It is with the two that finite things arise, and the two, here, symbolize the Dyad. In the indeterminate Dyad, we have, as formally distinct positivities:

indeterminate determination - power (active potency) to determine limitlessly; and

indeterminate determinability - power (passive potency) to be determined limitlessly.

The act can always determine, and potency is always determinable. But absolute determination is impossible because it would be an act, and there would be a contradiction in terms, for the infinite is the endless power to determine, and if everything were already determined, the determined would have reached the limit of its determination. Moreover, a determinant being, as such, fully actualized in the act of determining, would achieve the quantitative in actuality, which is absurd.

Therefore, the act of determining implies a limit, the limit of determination, and it limits the determinable thing. But what is determined is, limitlessly, what is determined; therefore, what has received a determination is, as such, limitlessly itself but limited by what it is not, and also limited by what it is, for it is only what it is until it is what it is. In this way, the creative action, creation, realizes a limited thing that is, as itself, limitlessly itself but limited by itself because it is only what it is until it is what it is, and limited by what it is not, which is what can be, not contained within its nature.

Thus, the indeterminate Dyad is potentially infinite and is everything that can be determined: it is simultaneously the potential infinity of determining and the potential infinity of being determined. In this case, the formative act can endlessly determine everything it can determine, and the materializable potency, which is passive, can be endlessly determined in everything that can be determined.

Here, therefore, the quantitative potential infinity applies, not the actual one. While the latter is absurd, the former is not.

Now, the indeterminate Dyad has no limits in itself; it is itself indeterminate, limitless as such, but it is limiting in its acting. They are not absolutely independent, for they are created by the Hen. They depend on it; therefore, they do not have the absolute simplicity of the Supreme Being, nor its infinitude, which is eternal. They do not have actual infinitude but potential infinitude, the power to be active and passive endlessly.

And here lies the foundation of the ab aeterno creation of the higher Pythagoreans. For the indeterminate Dyad has no beginning in time, for time implies the determination of determined things. Time begins when the formative act shapes the materializable potency. Time belongs to determinate things in a limited way. Thus, the Dyad, which is not eternal since it is not the simultaneous totality of time, because, as we will see, one limits the other, and therefore, there are diverse relationships between them, which we will analyze shortly, and as it is not temporal because time occurs in the succession of determined things generated by it, it belongs to a duration that is not simultaneous. total, not completely simultaneous but that also does not succeed, which includes, as a species, succession, which is time. The duration of the Dyad is eternity, it is aeon.

But both (the formative act and the materializable potency) are positive and not mere nothings. If they are formally distinct, they are also distinct in the realization of the determined entity. They are two positivities, two positions, two theses, they are thetic. One is before the other, “ob” to the other:

position “ob” position

thus, they are opposed.

The Dyad, as itself, is the universal substance, for from it all things are generated. In Aristotelian language, matter is the first substance (ousia prote), and form is the second substance (ousia deutera). A finite being is the composition of these two positivities. For that is the Pythagorean thesis, with the distinction that the substance of things is one, the Dyad in the thing, but formally distinct, that is, the logos of each is distinct from the other.

Thus, everything that is finite is a product of this opposition. And that is why substance is the first Pythagorean category, and opposition is the second, for it is from the conjunction of the two positivities, the formative act and the materializable potency, that any finite being arises.

We cannot avoid an analysis on such a relevant theme as act and potency. In our books “Philosophy and Cosmology” and “Ontology and Cosmology,” we examine the various ways of considering this fundamental theme of Aristotelianism, as well as scholasticism and philosophy itself.

In these works, which precede more comprehensive ones that we intend to undertake, our position is outlined in general terms. Faced with those who affirm the real-real or real-physical distinction between act and potency, we align ourselves with those who deny this diastem, which would aggravate the crisis between the two fundamental modes of being. We know that Thomists affirm the real-real distinction, while Scotists affirm only a formal distinction. The former claim to be based not only on Aristotle but also on Thomas Aquinas. As for the former, we have no reservations, but regarding the latter, there are serious doubts as to whether this was Aquinas’s true thought. In the mentioned works, we presented the fundamental reasons of Scotism against the real-real or physical distinction.

This would cause an immeasurable influx of aporias and would prevent the solution of others that arise from the assertion of the creationist thesis.

In turn, Thomists also offer their arguments. It is impossible here to analyze and criticize these positions, which, as we have already said, will be the subject of future work of ours. However, we want to draw attention for now to an aspect of great importance in philosophizing. Although philosophy tends to achieve greater objectivity and the avoidance of opinionated and evaluative stances, there is undoubtedly, in the face of the theme of act and potency, the presence of a prejudice from doxa, which, in our view, has profoundly influenced the entire philosophical process of the West. This prejudice, of Aristotelian origin, consists of undervaluing potency in the face of act and devaluing it to the point of depriving it of being, turning it into nothingness. This prejudice, whose emerging and predisposing roots would allow us to conduct a lengthy psychological analysis, must be exposed, under the risk of philosophy being unable to reach new stages in its path and consequently solve many of the aporias that previously seemed insoluble. If we glance at Hindu, Egyptian, and Chinese thought, we find that in these civilizations, act and potency are placed on equal axiological and ontological footing.

Among the Greeks, Pythagoras, Socrates, and Plato equally valued act and potency. Look at the definition of being given in the Sophist. Being is fundamentally potency (power). It is being all determinative potency from the highest to the lowest degree, and it is being all determinable potency from the greatest to the smallest degree at any given moment, for the shortest instant of time. Plato was a potentialist, thus following the Pythagorean line.

In Pythagoreanism, the indeterminate Dyad affirms the determining potency (active) and the potency to be determined (passive). In the mentioned books, we defend the thesis that every being, however minimal, is characterized by presence and efficacy. Every being is effective. Act is the realization of that efficacy, and potency is the effectuation of efficacity. Potency is not a non-being but a vectorially inverse mode of what is in act. Potency is virtual and based on efficacy. In “Concrete Philosophy,” we show that to do is to be done because when something is done, something is made. The determining action requires a determinable correspondence, for if this correspondence did not exist, the determining action would annihilate itself because it would act upon nothingness, and to act upon nothingness is to act upon nothing. The idea of determination implies determinability. Thus, the infinite potency of determination must correspond to the infinite potency of determinability. This thought, which we have already presented and which we intend to justify exhaustively and apodictically in a special work, corresponds appropriately to Franciscan thought. The valorization that they made of matter, of potency in sum, led many of their opponents to baselessly accuse St. Francis of being pantheistic and the entire Franciscan school, in philosophy, of achieving a pantheistic work, and therefore, heretical in the eyes of the Church. We do not need to defend the Franciscans against this accusation because they have already defended themselves brilliantly and have shown with sufficient skill that their opponents could deserve the label of pantheists with more reason than they.

This comment we have just made is only intended to show that our interpretation of Pythagoreanism is apodictically well-founded, and that this is also the thought of Plato and Socrates, which still endures in Western thought and represents a victory over one of the prejudicial moments that, in our view, was among the most pernicious for philosophy.

It is not surprising that Aristotelianism, despite its greatness, its vigor, has created prejudices because we know that, psychologically, Aristotle, as his work reveals, was always a man driven by prejudices, by preconceived stances that distorted before his own eyes the work of other authors. Aristotle, despite his genius, falsified and caricatured the thoughts of others, as can be seen in his treatment of the Pythagoreans, Anaxagoras, Empedocles, Heraclitus, and even his own teacher, Plato.

The Law of Opposition.

We have seen that everything finite is the product of this opposition. We are, therefore, facing the Law of opposition, whose symbol is the two. All finite things are composed of two orders of being, at least. And, in the coordination of the elements that compose it, they form oppositional dyads, which are expressed through all pairs of opposites, which constitute the poles, not only of all philosophizing but also of all the most primary classifications and human divisions.

From the opposition between the passive-principle of the determinant and the passive-active of the determinable, all the heterogeneity of finite beings arises. Determination, as we have seen, establishes the limitation of the unlimited, for all things are formally unlimited but materially limited. All things can be visualized as a unity, as a totality, and can be visualized as a bundle of oppositions of opposites, affirms Pythagoreanism. No knowledge is perfect about something unless it examines it as a totality (unity) of opposed aspects, classifiable in a dyadic way.

Everything that is a creature presents this opposition, which governs all things. Two laws were then specified: the law of unity and the law of opposition.

But the opposites are indispensable (the opposites of the formative-act and the power-materializable), as no finite entity excludes them, for they are the fundamental elements. Also, the fundamental opposition, which manifests itself in all beings, is the principle of all finite beings. It is for this reason that opposition is the second category of the Pythagoreans. But the opposites are face to face, one is referred to the other, both correlative in the Pythagorean sense because the formative act is the formative act of the power-materializable, as the power-materializable is the power-materializable of the formative act, both having their base, their kipokeimenon, in the Greek sense, their ultimate subsistence in universal substance.

From the reference that is formed between one and the other, from this relation to another, necessarily, from this reference to another, ad aliquid, arises the relation that constitutes the law of all things, the law of series.

The Law of Relation.

Since the opposites are correlative, indispensable to each other, because the materializable power always has a form, this or that, to be, it requires the formative act, the determination, for the determinant is only such when there is the determinable, for how can something perform determination without something that is determinable to be determined?

The law of relation is therefore fundamental for created beings, for they do not exist without the correlation between opposites. And it is from this correlation that some finite entity arises because it has a form and matter, to use Aristotelian expressions.

But this relation is not like the accidental relations that the entity will later maintain with other beings to which it refers. This relation is primary because, without it, the being does not arise. It is for this reason that relation is the third Pythagorean category. And no being can be properly known if it is not considered from the angle of unity, intrinsic oppositions, and the relations between the oppositions that give it origin and being.

In the relations that are formed between the main opposites, the arithmos in ve emerges because the thing arises from its intrinsic proportionality, from the cooperation of form and matter. The finite thing, considered as a form in re, imitates the eidetic form, which belongs to the power of being, for everything that exists, has existed or will exist, in a certain way repeats a perfection of being. For this reason, created things participate in the perfections of exemplary forms in the order of eternity, of eternal forms.

In the relations that are formed between the main opposites, imbalance and balance arise because when a matter is informed, there are degrees of proportionality that characterize the specific mode of being of the thing in its specific perfection. Balance and imbalance arise as Pythagorean categories, subordinated to opposition, and are therefore sub-categories. The Mega and the Micron (the Great and the Small, of Plato) are also sub-categories of opposition because the great refers to maximum determination and maximum determinability, and the small refers to minimum determination and minimum determinability, as created beings are more or less related to the specific perfection of the exemplary eidos. It is for this reason that Plato spoke of the Great and the Small of the indeterminate Dyad, which is the smaller Dyad, for the great Dyad is that of the Hen-Prote and the Hen-Deuteron, of the Second One, which is the Hen-Dyas aoristos.

In the relations established between the opposites, there is an interaction between them, as the formative act, when informing the materializable power, and playing the role of the Platonic demiurge, as we have seen, is limited by matter, for it can only inform proportionately to its nature as efficient cause, but also proportionately to its nature as efficient cause and proportionately to the capacity of determinability of the materializable power. The materializable power, in turn, undergoes the action of the formative act but also exerts resistance to it. Such resistance is easy to verify, and here it serves as an example when we take the already informed matter, such as clay, which, as the material of the brick, exerts a delimiting action on the form sought to be impressed upon it by the efficient cause.

Thus, there is an interaction between both, which led the Chinese to conceptualize Yang as passive-active and Yin as passive-active. From this interaction, the fourth great Pythagorean law emerges—the law of reciprocity.

The Law of Reciprocity.

In all beings, considered in their intrinsic and extrinsic oppositions, in the relations that are formed between the opposites, there is an interaction, an interactive reciprocity.

We are here in the world of things that make up our cosmos, which is called by many doctrines the sphere of the quaternary, whose symbol is the four.

If all things can be seen unitarily, they can also be seen dyadically, ternarily (as a bundle of relations and also as having a beginning, middle, and end, etc.), and, quaternarily, as a result of the interaction of opposites. If the law of relation governs beings as series, the law of reciprocity governs the primary and fundamental evolution of finite entities. It is also the law of fundamental evolution for Pythagoreanism. For this reason, this interaction of opposites arises not only when the being begins but also during the course of its duration, of its existence, for as long as the entity is, there is a polemos, a constant struggle between the opposites, which determine each other, in a different way, generating the intrinsic heterogeneity of the singular being.

But the reciprocity that occurs between opposites occurs within a law of intrinsic proportionality of being, for its acting and suffering are proportional to its nature. And here is the fifth Pythagorean law, which governs all things—the law of intrinsic proportionality or the law of Concrete Form.

The Law of Form.

All things are determined as such by the form they have. This, together with its matter, is constitutive of the nature of the thing. A thing is its form, but existentially, ontologically, it is the set of the main opposites.

It acts and suffers in proportion to this nature. The reciprocity that occurs between the opposites occurs within established limits, which are the form of the thing, the concrete form, the form in re because, otherwise, the thing would realize or suffer disproportionately to its nature, which is absurd, as we have shown in “Concrete Philosophy.” To properly know a thing, it requires to be considered quintarily according to its law of intrinsic proportionality, for its possibilities as well as its actions are proportional to the concrete form it has.

These five laws, examined so far, contemporaneously govern every being; they govern it simultaneously because any finite being has a form, has reciprocity arising from the interaction of related opposites, which constitute the manifestable aspects of its ultimate subsistence, of its hypokeimenon. Thus, if substance is given by universal substance, which is created by the Hen-Dyas aoristos.

The form is thus the eidetic arithmos in re of the thing, which is symbolized by 5, hence the symbol of the five dots being the symbol of Man because he is capable of grasping the forms of things, albeit intentionally; that is, proportionately to his schematics.

To know a being formally, and the reciprocity that arises from the interaction of the related opposites that constitute its substance, is to have a quinary vision of it and, therefore, a broader one.

Every finite being constitutes a unity formed by its totality, the arithmos plethos, the number of its totality. This totality has a cohesion that harmonizes its parts, the constitutive elements, which are dyadically opposed. As a totality, there is a principal function that belongs to the whole, to which the subsidiary functions of the opposites are subordinated, which are analogous in universal substance, which is the hypokeimenon of the being. The subsidiary functions are subordinate to the principal function, which is obedient to the interest of the whole. When the functioning of all parts, with their respective subsidiaries, is subordinated to the norm given by the totality, then we have harmony in the being.

The Law of Harmony.

The sixth law, symbolized by the hexagram, is the law of Harmony, whose statement we had the opportunity to give above. It is not the result of symmetry between opposites, but the subordination of the subsidiary functions of the analogous opposites to the norm given by the principal function, which is in the interest of the whole.

Not only do beings form harmonic sets in this sense, but they are also, in turn, component elements of totalities, larger structures to which they are subordinated. The law of harmony thus reigns in all things, and when a thing breaks this law, such a breach is only apparent because, properly speaking, it breaks the harmony of one set to integrate into the harmony of another. But the law of harmony, which governs the universe, proclaims that the subsidiary functions of the component elements, ordered within the set of oppositions, function obediently to a norm given by the totality. However, as there are degrees of being among finite things, there are degrees of harmony, and disharmony occurs when there is a rupture or deficiency of the principal norm due to the contrary action of the subsidiary functions. Harmony thus implies disharmony among entities, for they do not always remain within the same totality but go on to integrate others. Thus, there are substantial mutations, mutations of the forms of things, as well as their matter, causing specific qualitative leaps. And the law of seven—the Law of Cosmic Evolution.

The Law of Cosmic Evolution.

Finite beings do not always remain within a norm, for they are constantly, to varying degrees, transmuted from one order to another, from one set to another, from one schematic tension to another. Thus, there is a destiny that corresponds to latent possibilities not actualized when it comes to a formal aspect, and they are predispositions for future information. What a being is currently, in its form, is not all that it is in its virtuality. This set of minerals, which becomes an apple, is not, as an apple, all that it is, for there are, in its being, predispositions to be other forms, not just that of an apple. Once its function is fulfilled, its possibilities, which are constituted in its process, are exhausted by intrinsic dissolution or extrinsic factors, it becomes something else and evolves into another form.

Thus, all things in the cosmic world experience these evolutions, which break the previous adjustment and ordering of opposites (harmony) to undergo qualitative and specific leaps. In the symbolism of all religions, seven is always a symbol of this evolution, as we see in the seven sacraments, the seven mysteries, the seven solitudes, the seven colors, the seven musical notes, the seven days of the week, the seven pure animals of Noah, his seven sons, etc.

Just as there is elementary evolution in the four, in reciprocity, there is a higher evolution in the seven, which is the law we have just indicated. In this way, every unity is the product of a polarization of opposites that, in their relationships, interact, realizing a form that provides the norm for the subsidiary functions of the component elements that tend towards new forms, evolving.

The Law of Higher Evolution.

But evolutions tend towards a higher evolution, which is the eighth law: the Law of Higher Evolution, which is the attainment of a new equilibrium above the previously experienced one. This law is symbolized, in religions, by resurrection because it is the salvation of the being from the cycle of evolution and the attainment of a higher stage, for all things tend towards a good that transcends them, the higher good of the Supreme Being. All are integrated into the grand Whole (Pan). Everything is integrated into the Whole, for there are no ruptures in being. It is the great unifying law of all cosmic beings—the Law of Universal Integration, symbolized by the ship.

The Law of Universal Integration.

But all things, integrated into the Whole, move towards the transcendent Good that is inherent to them, the Transcendental Unity beyond the Cosmic Order, the Whole, which is the Supreme Being, which is the supreme law of the Universe—the Law of Transcendental Unity.

It is the law of participation because all things that are, and in what they are, are so by participating in the infinite power of the one who is the supreme and primeval origin of all things, the Hen-Prote, to whose power all things are as in a prison, in the metaphorical language of Pythagoreanism, and it is the Law of Laws.

Jacob’s Ladder

“Everything that nature systematically arranged in the Universe appears, in its parts as well as in its entirety, to have been determined and harmonized by Number, through the providence and thought of the one who created all things; for the model was established as a preliminary sketch through the dominion of the pre-existing number in the mind of the God who created the world, a number-form, purely immaterial in every aspect, but at the same time, the true and eternal essence, so that according to number, as according to an artistic plan, all things were created, such as Time, motion, heavens, stars, and all cycles of all things.”

(Nicomachus of Gerasa)

The Bible tells us in the book of Genesis (28, v. 12-13):

"And he dreamed: and behold a ladder set up on the earth, and the top of it reached to heaven: and behold the angels of God ascending and descending on it.

And, behold, the Lord stood above it, and said, 'I am the Lord…’"

And this was Jacob’s dream.

This ladder is the symbol of man’s earnest search for truth. This ladder is Philosophy.

Some are on the earth, preparing to climb its steps. Others have already ascended a few, while others, further away, are approaching the top. There, high up, almost disappearing into the clouds, only a few eyes from below can see it. Only those who have climbed a few steps are capable of achieving it.

At the base, the path begins for those who start from sensory experience. It is from there that the empiricists depart, but some remain, like the materialists, sensualists… Not everyone is capable of ascending the steps.

On those steps is Aristotle, who ascends the ladder, starting from empiricism. He wants to reach the top. Further, much further than him, is Plato, looking down towards the earth.

Aristotle seeks to explain the highest, starting from the lowest; Plato explains the lowest, descending from the highest. But the path is the same: the ladder. Only the vectors are different.

There, almost at the top of the Ladder, is Pythagoras.

He does not descend. His eyes turn towards the highest. He seeks the luminous top, which his slightly dazzled eyes can faintly discern.

He does not start from the empirical, nor does he descend the ladder. His doctrine is clear when it can be seen with eyes full of understanding.

At the top, lies the Mathesis Megisthe, the supreme Truth. That path is the path of philosophy, and the striving to climb it is the effort of the philosopher, the lover of supreme knowledge, who overcomes hesitations and faintness, and seeks to rise up to the heights. The higher one climbs, the more difficult it is to conquer the new steps. Perhaps one lifetime is not enough, as many remain on the path. But for the most daring, the ascent does not discourage them.

This symbol facilitates our understanding of these three giants of antiquity: Pythagoras, Plato, and Aristotle. They are three milestones along the way.

Plato descends to bring to men the secrets from above. This great Pythagorean, not always well understood, left a work full of suggestions for future analyses. And Aristotle, despite his convictions, as he climbs the ladder, approaches his old master more and more, much more than he imagined. He was great, as great as the others. And his imperishable work must always be considered as a starting point. This was understood later by Thomas Aquinas and Duns Scotus. They followed in his footsteps, with their eyes turned towards Plato, and from there, towards the heights.

And undoubtedly, they reached the highest steps.

Mathematics and Pythagoreanism

“Things are nothing more than appearances of numbers” (From “Hieros Logos” by Pythagoras).

The great development that mathematics has had in modern science, where its extraordinary results are observed, especially in Physics, allows us to affirm, which is already common, that modern science is under the influence of Pythagoras.

Just as medieval science was predominantly Aristotelian and pre-relativistic science was Democritean, modern science is Pythagorean.

However, there is certainly some exaggeration in these classifications, especially when instead of giving predominance, it ends up considering it as completely dominated by the postulates of one philosophy or another, or by one way of philosophizing.

In “Aristotle and the Mutations,” we demonstrated that the claims that modern physics was predominantly Democritean and deviated from Aristotelian postulates were unfounded. As we saw in that work, modern atomic theory is much more Aristotelian than Democritean. However, what cannot be doubted is that science in general tends more towards Pythagorean postulates than is believed, as long as one has a clear and real vision of the thinking of the great philosopher from Samos.

It can be affirmed that in modern mathematics, we observe the indisputable triumph of abstraction. The theory of groups (sets), stemming from the theory of functions, is an achievement of modern mathematics, which approaches Pythagorean thought, especially the arithmoi plethoi, arithmoi tónoi, and arithmoi khymcú, which the initiates from the second degree onwards had known, with such extraordinary accomplishments that astound the modern mind.

It cannot be denied that Matila Ghyka is one of the most enlightened Pythagoreans of our time. In his book, “Nombre d’Or,” in Volume II, page 112 onwards, he writes these words, which we cannot refrain from reproducing, along with the necessary comments:

“By the operation of our latest mathematical symbols, we highlight an image of the physical world in which only structure is considered a philosophy of pure Form, Form and Rhythm, or at least periodicity; for in this world of physical phenomena (what was once called the world or the material ‘plane’), we will see later that, following the words of Nicomachus, knowledge can only encompass relations and structures; Number, not substance, is the only eternal reality.”

Undeniably, this is the tendency observed in modern science. Atomic theory is reduced to a mathematical expression. And thanks to it, physics has made progress that astonishes the human mind itself.

“The paradoxical subtlety of Cantor’s theory of ‘transfinite numbers’ (the basis of set theory) had caused alarm among some mathematicians, and the controversies between finitists and infinitists regarding the logical possibility of an ‘actual infinity,’ a mathematically realizable infinity (in thought), not just an endlessly approached limit like the exasperating one in classical algebra. Cantor develops, manipulates, enumerates, and partitions in ‘realized’ (conceptual) infinite processions of different orders. At first glance, one does not receive the impression of a hallucinated fantasy, but rather of a discipline that would be worthy of taking part in the classical Temple of Mathesis. The audacity of these concepts, the symbolism in which the Hebrew Aleph of the Zohar and the Tarot becomes the insignia of the cardinals, and the gnostic Omega, that of the transfinite ordinals, brings to mind the Kabbalistic speculation, sephirothic pyramid, tower of white magic, Golem of Symbols of terrible growth, of some disciple of Rabbi Loew on the slopes of Hradschin…”

For the Pythagoreans, the sacred decade is the source of all things, from which numbers and rhythms flow…

As he shows us, rhythm is, and its definition is undoubtedly the best known to us, the experience of the orderly flow of movement. In rhythm, there is time and intensity. It is the connection between the intensivist and the extensionist, the qualitative and the quantitative. The arithmoi rythmoi of the Pythagoreans realized this connection, they materialized it with the arithmoi posotes, the quantitative numbers that Aristotle considered to be the only Pythagorean numbers.

Modern mathematics borders on the qualitative. It is no longer merely an abstract realization of the third degree but also a materializing one, as modern relativity materializes coordinates in a set, and set theory already encompasses materialization. Modern mathematics is not a higher degree of abstraction but an activity that, from extreme abstraction, becomes materializing in turn, as it connects, conjoins, and gathers together what analysis had previously separated. Modern mathematics, in its criticism (in the sense we always give to that word), is more syncretic than diacritical, and it is in this procedure that a new vector, whose fruits have not yet fully ripened, is clearly marked.

Referring to Cantor, he continues: “But these creations of this disquieting magician of the transfinite had become incorporated into our mathematics and logic as a magnificent armor of set theory, which another Cabalist, a tamer and charmer of symbols, using group theory as we have already said, led us, in three transcendental leaps, from paradox to paradox, to the ultra-Pythagorean synthesis stated above the physical universe in number-ideas.” He is referring to Einstein, this Pythagorean of our days, who has given science new directions, undoubtedly synthesizing in his monumental work many of the current achievements in mathematics and physics.

He continues: “If the ‘raw material’ has finally been found, it has also been found that all bodies called material, called solid, including our living bodies, are actually so due to the immense separation of the molecules that make them up, to the apparent fabric, to the gaseous state (the only relatively ‘solid’ bodies known in the universe are three recently discovered stars, including the ‘white dwarf’ or ‘companion of Sirius,’ in which matter is compressed to a density 60,000 times greater than that of water, to the extent that a ton of this matter could fit into a matchbox, and the nuclei of its atoms must be probably close enough to largely suppress the zone of electron orbits and the very planetary electrons). Furthermore, these molecules and atoms, once ‘inseparable,’ forty years ago, are now known to us as small ‘solar systems,’ nearly empty zones in which, in turn, at relatively astronomical distances compared to their dimensions, the last particles of ‘substance’ (no longer matter, as they have lost the only ‘material’ quality: constant mass), particles of pure electricity, negative or positive (electrons or protons), gravitate around each other.”

The concept of matter in philosophy should not be confused with what physics attributes to it, as we have shown in “Concrete Philosophy,” as it still follows, undoubtedly, the prejudices of the 19th century, especially because mass is not the essence of matter.

What is undeniable is that modern physics moves away from the prejudices of the 19th century and the vulgar materialism of that time. Matter becomes pure numbers for physicists like Heisenberg, mere mathematical structures that largely reproduce Pythagoras' famous phrase: “Things are nothing more than appearances of numbers.”

Now we can reproduce these words of Ghyka, which are so significant today: "And if Knowledge also accidentally concerns bodies, supposed material things of incorporeal things, it is nevertheless to these that it will be especially linked. For these immaterial, eternal things constitute the true reality.

But that which is subject to formation and destruction… (matter and bodies) is not currently real by essence. It can be noticed how this conception of the world is similar to the one given to us by Modern Mathematical Physics, in which only structure and invariance matter."

In the Renaissance, Leonardo da Vinci said, “There is no certainty where none of the mathematical sciences can be applied, nor any of those founded on mathematics.” And if we take the concept of mathematics in the genuine Pythagorean sense, and not in the strict sense that prevailed and still prevails in our day, we can understand how true da Vinci’s statement is.

Our mathematics, the one constructed by humans, proportionately repeats the mathematical interpretation that humans are capable of constructing of the great mathematics of the universe, the great Content of Mathesis Megisthe (maihema, mathematos, mathemática). Our mathematical language expresses something of what presided over the great accomplishment of creation. That is why Sir James Jeans, in our day, does not hesitate to say, “The Great Architect of the Universe now appears to us as a pure mathematician.”

Commenting on Greek contributions to our culture, Matila Ghyka recognizes two important ones: one of Egyptian origin, represented by the thought of Pythagoras and the influences they exerted, such as the almost sacred character of geometry, the perfection of form, the importance of secrecy, the magical value of the Word, the magical value of the sign, symbols, rivers, and rhythm, etc., and another contribution of Hyperborean origin, as Herodotus and Heraclides Ponticus called it.

From the Greek spirit, these contributions, which are genuinely Pythagorean, have come to us:

  • A spirit of synthesis and clarity in synthesis;
  • The achievement, in works of art, of perfect formal beauty;
  • The development and completion of Geometry as the ideal model of a synthesis based on axioms and the chain of irrefutable logical deductions (axiomatics);
  • The establishment of the theory of “numbers,” with the entire Universe being “governed by” or “arranged” according to Numbers;
  • Concepts of proportion and rhythm derived from the two aforementioned disciplines (theories of forms and theories of numbers) and applied to the pursuit of Beauty;
  • Theory of musical harmony;
  • Harmonic conception of the Cosmos.

These are the main contributions of Pythagoras to Greek culture, and they are present, as high points, in our culture.

We consider the axiomatization of philosophy to be the most valuable and forgotten of all. There is a place in Philosophy for an axiomatic system, just as there is in mathematics. This axiomatization would allow for the metamathematization of philosophy in the eminent sense of Pythagoras. This is what we have accomplished in “Concrete Philosophy,” whose work, by gathering the positively demonstrated elements, remains under the influence of the great Master of Samos.

We know that Pythagoreanism has given rise to numerous sects, which have arisen more from the deficiency of the disciples than from the greatness of the work of the great initiate that was Pythagoras. Thus, Gnostic sects like the Ophites, Essenes, Cainites, Manicheans, Paulicians, Bogomils, Albigenses, Kabbalists, Rosicrucians, and all the Masonic sects, to name just a few, drew their knowledge from Pythagoreanism, and their heterogeneity arises from the heterogeneous interpretation of the master’s thought.

Our position is clarified from the beginning. We do not intend in this work to judge one sect as more or less correct, truer or less true than others. What interests us, following the method of Cuvier, is to make use of the provisions of our decadialectic and concrete dialectic, which we consider the most skillful means for examining a thought, and to reconstruct the Pythagorean doctrine starting from postulates that are considered unambiguously valid. And through their ontologically rigorous consequences, similar to what we did in “Concrete Philosophy,” to restore the true thought of Pythagoras. And starting from this restoration, whose value is given in itself because it will be proportional to the value of the demonstrations we make, for in Philosophy, the only authority is demonstration, as Thomas Aquinas said, we can then appreciate the value of the various positions and determine which of them can be considered the legitimate heir of the master’s philosophy.

This is what we are going to undertake.

The Philosophy of Pythagoras

Summary of the fundamental theses of Pythagoreanism in light of what has been examined

Based on what we have examined so far, it is possible to establish the fundamental theses of Pythagoreanism, which will serve us for the concrete construction of Pythagorean philosophy and as a starting point for further analysis and critique of the main works of Pythagorean authors, a task that we hope to accomplish one day in special works.

They are as follows:

God is unknown to men, not in all aspects. The agnostic view is not specifically Pythagorean. However, due to the limitations of the human mind, it does not fully grasp divinity. Yet, since all finite things are composed of the limitless-limit, man can surpass himself and comprehend divinity to a certain extent.

All higher knowledge must be initiatory. Initiation is necessary to prevent the unworthy from treading the paths of knowledge and taking misguided directions that may lead more to evil than to good.

As man takes his first steps in the pursuit of the unknown, he attains degrees of enlightenment until he reaches a point where he is capable of seeing things in the divine light of the all-around illumination.

In the highest states, there is always the presence of Khâris, for we only accomplish that for which we have an appetite, an impulse that stimulates us to obtain it. All knowledge, all wisdom (sophia), implies a prior emergence in man.

If God is unknown, man should not renounce the search, because since knowledge is also attainable, he can increase his knowledge to degrees he does not even suspect.

The one who loves knowledge is the philosopher. The supreme knowledge, the supreme instruction, is Mathesis. Philosophy is the striving of man to attain it. Thus, there are many possible paths for those who do not know. He must traverse this path, follow this itinerary, and delve into this initiation. Hence, initium, beginning, and its action, initiation.

Initiation is, therefore, a wholly logical operation (gogic), an action of indicating, guiding, hence pedagogy (leading the young), which indicates the best path to attain Supreme Instruction (Mathesis).

God is the Supreme One, and the Supreme One transcends all finite beings.

Man may not know the nature of God, but all men throughout all times have paid homage to this Supreme Being. Where men differ is in their understanding of the nature of God, giving rise to the diversity of religions.

The Supreme Being, God, is the Supreme Monad. He is also the Lord and Father, and alone, the source of the One (the universal substance, that which gives support and engenders all things, corresponding to creation).

It is from the One that we understand the first Dyad and the second Dyad, and the number three, because in it, and thanks to it, things arise. (“The number three reigns throughout the Universe, and the Monad is its principle,” says an oracle of Zoroaster). It is important not to forget that finite things only arise from relation, and this relation is three. Without the indeterminate Dyad (of actuality and potentiality), there would be no opposition, and without opposition, there would be no source of relation. And without relation, there would be no determinate finite entities.

The Supreme One is infinite and beyond our knowledge; however, the One, the universal substance, is not.

The first One is simply One, the Supreme Being.

The second One is One-multiple since it gives rise to the second Dyad, with the first being formed by the first One and the second One.

(Between them, the supreme trinity is formed. But this thesis can only be proposed later, as it requires further measures to attain and additional evidence to affirm it).

The second indeterminate Dyad arises from this second One. It acts, and its action implies the acted upon. Its action is limitless; consequently, actuality is limitless, which corresponds to what, in Aristotelian terms, is act and potentiality. Act can inform (give form) to everything that can be informed, and potentiality can be informed, correspondingly, by everything that can be informed. Both are thus indeterminate; they have no boundaries or limits in their acting or suffering. That is why it appears as an indeterminate Dyad or limitless to others. In Greek, as we have seen, it is Dyas aóristos (aóristos meaning without limits, from the privative alpha and hórizo, to limit, from hórõs, limit). The determiner has no limits to be determined. Hence, the Megon (the Great) and the Micron (the Small) of Plato, which can increase and diminish.

The determiner (act in the Aristotelian sense) is gnostón, knowable, while potentiality, determinability, is agnóston, unknowable because it contains contradictions within it as possibilities, making it unknowable. One principle is called masculine, the other feminine.

Maximum determinability and minimum determinability imply maximum determination and minimum determination. And since the determiner is determined in proportion to the determinable, there is a Great (determiner-determinability) and the Small (determiner-determinability). The One, which is the universal substance, is thus the source of the Dyad created by it.

Numbers, which are contained in the Supreme Being, in the Supreme One, are eternal forms. The numbers that appear in the determined things of the Dyad are the numbers in things. These can arise from addition; not those that are eternally given, like the thoughts of the Supreme Being. Aristotle did not understand this distinction.

The number in finite things always affirms a schema of participation because it is through the proportionality of this participation that things are what they are, as essence and existence.

The first arithmoi are uncreated.

The second arithmoi eternally arise in creation.

Number is not a unified set but a simple unity. The schema it reveals is a simple unity. Since the possible is infinitely possible, as there is an infinitely passive potentiality, and there is an infinitely active potentiality (because determinability is infinitely determinable, and determination is infinitely determined), numbers are infinitely possible.

If two is ontologically posterior to one, it is not chronologically.

Numbers are already contained, since all eternity, in the infinite power of the Being, of the Monad, from which they flow.

The number that constitutes the schema revealed in the coherence of the structure of a finite thing is the arithmõs plethos. The one, as arithmôs plethos, participates in the One, for every unity, as such, is univocal in its ultimate logos. The logos of one is the indivisum In se, it is that which does not differentiate itself, it is that which is only itself. The Supreme One is absolutely so; the finite one, the plethos, is relatively so. But as an indivisum In se, both are univocal. What distinguishes them is the “self” of each, for one is constituted of finite parts, while the other is absolutely simple. What distinguishes one from the other is their essence and existence, as subjectively considered. Therefore, the plethos, which imitates the Supreme One, imitates it proportionately to its imitating nature.

The indeterminate Dyad is originally one but indeterminately two. The indeterminate Dyad is the One-multiple, according to Plato.

There is the Hen (one) and the Hen-Dyas. The first Hen is transcendent to finite things. These derive from the second One, which gives them existence. They are created by it. In the Christian Trinity, there is the Father, who is the Supreme Being, and there is the Son, who is begotten by the Father. The Son is the Supreme Being who realizes the outward procession of creation. The Holy Spirit unifies both. Thus, three roles represented by the same substance: Being as Form (Father), Being as worker (Son), and the infinite unifying power that identifies them, symbolized by love (the Holy Spirit). According to Pythagoras, the Supreme Hen is the Father (a name that appears among some Pythagoreans). He generates the second One, and this generates the universal substance, which, in its action, is dyadic since action implies being acted upon, just as love implies being loved.

This second One is Hen-Dyas (the Dyadic One, the one-multiple), which will generate all finite things, which are both one and multiple. These finite things constitute the Cosmos, which is One-and-multiple.

Thus, there is the ONE

the One-Multiple, and

the One-and-Multiple

as revealed by Plato.

In the Dyad, there are opposites: determination-determinability. But to determine implies crisis because it is not determined indeterminately but determinately. And where there is determination, there is what belongs to the determination, excluding what does not belong to the determination. Thus, there is a crisis because every determination has a limit and excludes what goes beyond the limit, which is the extent to which the determination is itself. Therefore, to determine is to realize the crisis. And since determining implies being determined, and there would be no determination without the determinable, determining includes in its action the determinable. But what is determined is within the limit established by the determination; consequently, it also reveals the crisis.

Creation is thus a crisis. And because there is the determiner in the act of determining and the determinable, which is determined, the creating One of finite things is One and Two, it is Hen-Dyas.

The crisis reveals the opposition between the determiner and the determinable. From this opposition arises the relation, as one is placed opposite the other. Opposition is the second Pythagorean category, and relation is the third. Quality and quantity arise from the determination of determinability. The determinative limit is quality, and what is included in the realm of the determined, the result of determination, is quantity.

Between the determined being, there is the determiner and the determinable. From this opposition arises a reciprocity, for the determiner acts upon the determinable in proportion to its capacity to act and the capacity of the determinable to be acted upon. Thus, there is a reciprocal determination between both; there is reciprocity. This is the fourth category because from opposition arises relation, and from relation arises reciprocity. This is the manifestation of the Aristotelian categories of active and passive, from which the sub-categories of equilibrium and disequilibrium derive. Just as from determination-determinability, the categories of the Limited and the Unlimited arise, for the determiner, in determining, limits, but the capacity to determine remains limitless, the same applies to the determinable.

Matter is the passive aspect that arises from the indeterminate Hen-Dyas. It is the limitless determinability that can be everything that can be finite without contradictions, every dependent being that does not formally contradict the laws of Being. Plato speaks of an akomestos hylê, an acosmic matter; that is, one that has not yet received form, the ekmaggeion amorphon, the mass not yet shaped. But such expressions are more didactic, as they refer specifically to determinability, which, as such, is akosmetòs but determinable, limitable, by form.

The Creator (Poietén) gives order to chaos, gives actuality to possibilities, arranges (Kosrnein) what is still akosmetos hylê, the potential determinability.

Thus humanity, in this man, is limited to what he is, but humanitas, in him, is unlimited because it is not included within his personal limits; it is thus, in re, the form, limited-unlimited, as are all finite things.

That is why Pythagoreanism affirms that all composite things are composed of the limited and the unlimited.

And we have already seen where the limit and the unlimited come from, which act as two cooperating causes of finite things.

The Hen Prote (The First One) is not substance, and does not undergo accidents. Universal substance belongs to the Hen second, the Hen-Dyas. But this substantiality, here, is in the Pythagorean and also Platonic sense, as it was for Hermes, as the support of all things, including matter, which is the capacity of potentiality to receive determined forms. In this way, universal substance is not matter; matter is posterior to it. Properly, matter is not a separate being, it is only determinability as a capacity to receive corporeal forms. Universal substance is the eternal being-being (the eon of the Gnostics), the first of creation, but it endures eternally, and from it transient and ephemeral things originate.

Thus, the Hen Prote is the superessence, the super-Being, that which is transcendental to beings. The Hen-Dyas is transcendental to finite beings but sustains them. It is eternal as their sustainer, and its presence is constant in them, surpassing them but giving them being, and it is eternal in the Dyaãa Prote as a being generated by the Hen Prote (Father).

Matter depends on a cause, the indeterminate Dyad, but it is ungenerated in the sense that it did not arise in time, for it is ab-eternal, created by universal substance.

Everything that has being, of any kind, is an entity.

Being is always intelligible, but proportionately to the intelligence that apprehends it.

All knowledge is knowledge of a being. If one knows that there is an absence, that absence is of a being or mode of being. To know that there is an absence of nothing is not to know nothing, but to know being in plenitude.

Being is thus the first object of knowledge.

Real being is the object of Mathesis. The being of reason is the being that only exists in the intellect and not in reality. Real being, which exists in itself, is eidetic (like forms) or concrete, which imitates the exemplar model, which is the form in the Hen Prote.

Eidetic being (form) has a formal unity.

The distinction between two forms is formal, real, not physical, however.

The distinction between forms (quidditative distinction for the scholastics) precedes intellectual knowledge. This formal distinction is real and occurs between formally distinct beings.

Since eidetic being (formal being) is not physical, several formal beings can be imitated by a being (unity) of chrono-topic existence, without implying the rupture of the existential unity of the finite entity.

Eidetic being (formal being) has a unity, but this unity is not determined in individuality or universality. Through imitation, form in the thing is determinable, both individually and universally.

The determinations of forms in re do not modify the unity of the eidetic form as an exemplar in the Hen Prote.

The form in re, when taken quidditatively, is univocal to the eidos in the real order of the Hen Prote.

Being, taken quidditatively, is univocal in all beings, when taken only in its logos of being, indeterminately.

The actualization of form in re is always possible when it is not contradictory to the already realized actualizations. Form in re is not the individual or universal actualization of the eidetic form as an exemplar in the Hen Prote.

The individual is an actualization of form in re.

Every actualized being can be the cause of other beings, always proportionately to its quiddity. Causes are hierarchically ordered to quiddities.

Every finite being is affected by another, for it receives its being from another. The First Being (en Prote) is not affected by another, for in Pythagoreanism, being precedes everything, not nothing.

The Uncaused Being by itself and independent of others is the Hen Prote, the supreme Monad, which is called God in religions.

The Supreme Being (Hen Prote) is infinite because it is independent, it is its own reason for being, it is by itself. And because it is absolutely simple, its essence is its existence. Its infinitude is its first and ultimate actuality.

This Platonic thought, removing what is didactic from it, shines in its creative power. Just as the indeterminate Dyad is created ab-eternally, matter is also created, for it belongs to the feminine vector, the passive aspect of the Dyad.

The conception of the Hen-Dyas aôristos in Pythagoreanism is thus equivalent to the conception of the Yang and Yin in Chinese philosophy, whose interaction brings about harmony. Yang is the determining act, and Yin is the determinable potentiality, but there is reciprocity between them because the limiting act gives form to the determinable potentiality, but always proportionately to one another, because the matter, in receiving this form, can be, with the form it has, the matter of another being. Thus, clay is the matter of the brick, but with the form of the brick, it is the matter of the wall. Informed matter receives the new form in proportion to its capacity. If the form limits, it gives limits and contours to matter (determinability), and matter gives limits to form. Thus, gold, being more malleable, better receives the form of the statue than clay.

Matter is the unlimited without limits, but it is limited by form and ends up offering limits to form. The form in re, due to the limitation it offers to matter, imitates the pure form, the eidetic form that belongs to the supreme Hen, but it imitates it deficiently because the pure form is unlimited-limited and perfect, whereas in matter it is limited-unlimited.

The eidos (the Form) of the Supreme Hen (Hentautôs, the One itself, according to Plato) is the Transcendent that generates the One, which is the Poieién, the creator, the one who creates the indeterminate active and passive Dyad. The first One is exterior to the multiple (Hen para tâ pollà), prior and exterior. The second One is the acting of the One, for they are one and the same, although representing two roles, they perform the Dyad, from which numbers and creatures originate. These imitate what is already in the omnipotence of the Hen Proie, the first One.

The form (the eidos, in the Hen) is unlimited because it has no boundaries that separate it from being, but it is formally limited because formally it is this and not that; therefore, it is unlimited-limited. The form in re is limited in things because it has boundaries that separate it, but it is unlimited because in the thing, it is a law of intrinsic proportionality of the circumference in relation to the diameter. If there is no materially considered perfect circumference, it is because the matter that constitutes it cannot perfectly imitate the form. Thus, there is no perfect curve or perfect line materially.

Being infinite and absolutely simple, the Supreme Being is eternal.

What is in the Infinite is infinite. The exemplary forms are therefore infinite in it. The formal distinction of the eide in it does not imply separate actual existences. Their existence is the existence of the Infinite Being, and in it they are identified. Therefore, the distinction among them can only be formal, not physical. It is worth repeating this Scotist thesis, which is profoundly Pythagorean — "The quidditative beings are called Ideas (Forms); as the object of a formally distinct intellection, each divine idea (eidos) possesses a distinct quidditative being, but none of them has a distinct and separate existence; they all exist together by the simple existence of the actually existing Infinite.”

The second One, generated by the Divine and First Being, is its action ad-extra, the cause of all contingent things. The ad-extra action is creation. This power is the omnipotence of the First Being, which can create all possible things immediately, without the need for another cooperating cause. Since creation implies infinitude, only the Supreme Being creates.

The Hen-Dyas is the Supreme Being in its creative action, generating the indeterminate Dyad from which all finite (created) beings flow, dependent on it.

All things are shaped by the number that copies the arithmoi arkhai.

Numbers arise from the indeterminate Dyad, but they copy the exemplary forms of the Hen Proie.

There are eternal and immutable (eidetic) numbers, and there are numbers of physical things (sensible numbers).

The first tetractys is formed by the Trinity and the Supreme Unity of the three.

The Pure Decad (the second tetractys) represents the laws (ten) that govern all creation.

The Pure Decad is the sacred Tetrade because it emanates from the Supreme Being.

The Sacred Decad is the mother of all finite things.

The Sacred Mother is Creation, not the act of creating, but what results from it.

There is no being outside of Being.

The essence of the Supreme Being is inviolable to human eyes and human knowledge, which can apprehend it intentionally, that is, proportionately to the schematic possibilities of man.

Human knowledge of things is totum et non totaliter; it apprehends things as units but does not exhaustively grasp them because human intelligence is limited-unlimited.

Every existing being is unitarily twofold. It has formal unity and existential unity. As an existential being, it contains the simultaneous coexistence of all distinct quidditative entities that constitute its structure. Every existing being, chrono-topically, is an arithmós plethos and also an arithmós tonós.

The Cosmos, as opposed to Chaos, signifies the ordered actualization of possibilities that, in Chaos, are contained in their contradictory possibilities.

Once a possibility is created, subsequent ones cannot contradict it, for that would contradict the law of being, as it would come from nothing, and nothing would then be the creator, which is absurd.

The ex nihilo creation of Christians can only be understood Pythagoreically as the ordering of possibilities into actuality, which, before being such, were considered nothing in actuality, and it does not mean that nothingness created or that the creature was made out of nothing, which is consistent with Christian creationist thought.

Universal substance always remains itself, for the things that arise and disappear do not increase or decrease it because an increase could only come from nothingness and a decrease, being a decrease of being, would be nothingness, which is absurd, as will be seen.

We believe that in these theses we have gathered what is most secure and genuinely Pythagorean, in light of what has passed through the centuries as the fundamental conception of Pythagoras of Samos.

This construction now allows us to compare Pythagorean philosophy with the foundations of Concrete Philosophy, which is our own, and from this comparison, we can highlight what is secure and apodictic in its theses, as well as demonstrate, also apodictically, that this thought has been distorted, disfigured, and falsified throughout time, with serious damage to philosophical progress.

With these premises established, the path ahead will become easier, as it will allow us to establish not only what is, but also what is not genuinely Pythagorean, even though it has dressed itself in the garments, words, and fundamental theses in order to falsify the true doctrine.

Concrete construction of Pythagoras’s thought

Having established the fundamental theses of Pythagorean thought, it is now easy to compare them to the theses of Concrete Philosophy, as presented by us in the book of the same name, which serve as our criterion for evaluating the positivities of any philosophical thought and its vigor.

If we compare what has been established as genuinely Pythagorean, in light of the historical elements we have gathered, we can now juxtapose these theses with the foundations of concrete philosophy, allowing us to construct the concrete nature of this thought.

For Pythagoras, something exists, has existed, and will always exist. Absolute nothingness is impossible and contradictory to the existence of something. However, relative nothingness, the absence of some positivity of being, is allowed.

What exists is being. And something exists that is, that exists.

Something that has always existed, that has always been, that has always existed, still exists.

The heterogeneity of flowing things can be perfectly explained by numbers (arithmoi) in the two senses that we have already examined: the arithmoi arkhai (divine numbers) and the arithmoi mathematikoi, which are present in things.

For Pythagoras, being is presence, positivity, everything to which something positive can be predicated.

Therefore, something has always been. And this Being, which has always been, is the fullness of being, absolute fullness, which is absolute, independent, ungenerated, unprincipled, without limits; in short, infinite, in the rigorous ontological concept of infinitude. It is the Hen-Prote, without which nothing has a reason for being and in which everything that exists finds its first reason for being.

Pythagoras does not admit a middle ground between being and nothingness, as his entire philosophy affirms being.

By admitting relative nothingness, which is nothingness in relation to something positive (since relative nothingness is nothing this or that, and not absolute nothingness), Pythagorean philosophy does not fall into the inevitable absurdities that the Democritean conception and that of the materialists led to.

Being takes priority over relation, and affirmation, ontologically, precedes negation, so that the Being of the Hen-Prote precedes, in any way, every other mode of being.

The absolute Being, the Hen-Prote, is only One, and that it can only be ONE is an inevitable consequence of the Pythagorean theses that we have already examined. There are no two absolute and primary entities that would be the source and origin of all things.

If the Hen-Prote generates the Hen-Dyas aôristos, the correlation symbolized by Father and Son, expressions already used by the Pythagoreans before the Christians, they are two great roles (persons) in one, united by an infinite uniting power, which is given by identity and is equivalent to the Christian concept of the Trinity.

If there were two absolute, independent, and separate beings, both would be deficient, and because the Hen-Prote is the Absolute Being, the admission of another infinite being or any other being outside and independent of it is repugnant. These theses are therefore contained in Pythagoreanism, and there can be no doubt about their apodicticity after what we have done in “Concrete Philosophy”.

The Hen-Prote lacks nothing to be. It is being that is simply being. And just as absolute nothingness in its totality is impossible, it is also impossible and absurd to have absolute partial nothingness surrounding being, like an island of being in an ocean of nothingness.

The absolute simplicity of the Hen-Prote is undeniable, as it is sufficient, proficient, and absolute, unprincipled, ungenerated, and absolutely the first, whose unity is absolutely simple.

The uniqueness of the Hen-Prote is absolute.

All the dialectical theses we presented in Concrete Philosophy are absolutely valid for Pythagoreanism, which does not contradict them in any way.

The Hen-Prote is the first cause of all things, for everything depends on it, and it is a real dependency.

No being can be more than itself, nor can it exist in itself.

Since finite being is dependent, dependent being is necessarily finite. The being that can exist by its own strength has always existed and was not caused. The Hen-Prote has always existed and was not caused, and it has its own reason for being within itself, while finite being does not have this reason for being. All perfections are already contained to a greater degree in the Hen-Prote. And its presence is total, and all power comes from it, which is why it is omnipotent because the power derived from the indeterminate dyad comes from the Second One, which is identical, in essence and existence, to the Hen-Prote.

Nothing can happen that is not already provided for by the Hen-Prote, for all possibilities belong to it and not to nothingness.

An effective something is absolutely first and prior to all beings, and this effective something is the Hen-Prote. And since a first efficient cause is necessary, that is it.

It is undoubtedly infinite because if it were not, it would be composite, and it would be a number, which it is not. And it is infinitely “substantial,” “formal,” and “adverbially”; it is also uncaused. And therefore, it is not effectible, causable, finitizable, materializable, or formable.

All its actions are absolutely free since they do not depend on anything else. It is indivisible because it is simply simple. And because it is indivisible, it is indestructible. It is, in short, absolute immutability. Only it can achieve absolute perfection, which finite beings cannot. It is most actual, most perfect, and everything that is intrinsic to it is so to the highest degree, for it is being intensely in the maximum degree of being.

Every dependent nature is triply dependent. And it is in three that the number of cosmic things arises, as Pythagoras explains, because they occur in a series that arises from the relation of opposites, from which finite beings emerge.

The Hen-Prote is existing in act. And while all things tend towards an end and are moved by another, the Hen-Prote and the Hen-Dyas, as Hen, move by themselves. The Hen-Prote is pure act, and its power is infinite. It is not corporeal and is eternal.

Since every operation implies an operator, nothingness cannot be the goal of an operation. For this reason, the Hen-Dyas aóristos, in its operation, accomplishes dyadic operation, accomplishes, through creation, the aóristos dyad.

Whatever contradicts being is not possible; only what can be real is possible. Possible numbers can be real, as they can find counterparts that imitate them in accordance with their nature. If the numbers (arithmoi eidetikoi), in the order of the Hen-Prote, did not have the possibility of being imitated, they would be possibilities to which no other reality would correspond. Now, since they cannot be contradictory to being, they are possibilities of real imitation. As eidetic numbers, in the order of the Hen-Prote, they are infinite and formally limitless.

The possibilities demand a necessary being, they point to it, for they cannot be founded on nothingness, and the criterion of possibility is given by the cause and the intrinsic reason for being. The reason for the being of finite beings lies in the Hen-Prote, which gives them being through the Hen-Dyas. Finite being cannot depend on nothingness, for it cannot hang from nothingness; it can only depend on the Supreme One.

A magnitude cannot be infinite; therefore, there cannot be an infinite quantity. This is a point of great importance in Pythagoreanism.

Quantitative number (arithmôs posótes) is what modern mathematics considers to be a denumerable finite set. Every magnitude is limited, whether it is continuous (megethos), a set (plethos), or a quantum (poson), etc.

Since quantity is a category, it cannot be infinite because the concept of infinitude cannot be attributed to it (from tribuere, to be attributed, because quantity is not “attributed” to anything, as it is a dependent being, but due to the independent and primary being).

An infinite quantity, an infinite extension, is absurd, as we have demonstrated in “Concrete Philosophy.” According to Pythagoreanism, the number of sensible things can never be infinite, only potentially. We can consider number potentially infinite because we can always add one more unit, just as we can always decrease it by something, without ever reaching absolute nothingness or, on the other hand, reaching quantitative infinitude. This possibility of number, which mathematics presents, confirms the Pythagorean thesis of the indeterminate dyad at its maximum and minimum, allowing for infinitesimal calculus from all aspects.

Therefore, in actuality, quantity is not infinitely large or small, but only potentially. Now, potency is determinable but not yet determined; it is the indeterminate determinable, which confirms the infinitude of the arithmoi in the order of being as possibilities, not as actualized acts, but only subsisting in the act of Being. Mathematics thus proves the infinite potency of Being, which can be everything it can be, and there is never a point of convergence between being and nothingness because, however much the small diminishes without limits, it never reaches nothingness. This statement, in turn, is a way to affirm that there is only Being, and nothingness is only relative, privative, which Aristotle later understood and did not draw all the possible consequences from, as Nicholas of Cusa rightly pointed out when criticizing Aristotle’s insufficient study of privation. Between act and potency, Aristotle placed privation. Every finite being is composed of act and potency, form and matter. The matter tells us what it is made of, and the form tells us how it is, what it is. But it also tells us about privation because, by being this or that determinately, it is not what it is not; for being what it is, it is not what it is not. Determination is always exclusive. And if one possesses what it is, which is its act, it affirms the exclusion of what it is not. And what it is not also limits and characterizes it, just as what it is limits it, being the extent of what it is, pointing to what it is not, for it is by being what it is and not being what it is not that it affirms what it is. Privation is thus the third term of its composition.

Pythagoras said that the ternary is fundamental to finite things, which could be visualized in this way, and one of the ternary ways in which things can be considered is to see them as form, matter, and privation (act, potency, and privation, in short, as possession and privation, as possession includes act and potency). The privation of a finite thing is not absolute nothingness because not having absolute nothingness is not to have, but rather it is a relative nothingness, positive by reference, whose exclusion is positive, for in a being that is not green, the absence of green is positive because green exists. But to say that nothingness is not absent from it is not to proclaim any absence but only to affirm the presence of what is possessed in it.

For this reason, since every quantitative being is a finite being and since it is possession and privation, what it lacks is something positive, and for this reason, no finite, determined, and limited being, because it is in its essence to be such, could be quantitatively infinite.

Finite being is something that is per aliud, by another, for a finite thing is this or that in imitating the form. But a thing, ontically considered, has an intrinsic proportionality, its singular form, in re, for it itself is its number, its arithmós, which corresponds to the haecceity of the Scotists. The essence of its singularity includes the imitated quiddity (form), but it also includes the intrinsic proportionality of its parts that imitate it. Thus, in haecceity, there is the imitated form and the ontic presence of what proportionally constitutes the thing. Thus, in this man, John, there is the repetition, through imitation, of humanity, but there is what John is, the intrinsic proportionality of him that singularizes him. For Aristotelians, it is matter that singularizes beings, as it is matter that singularizes form. For Scotists, it is haecceity, the combination of the formal and the material, that constitutes a whole, which is this (haec) singularity. Scotist thought approaches Pythagorean thought, for singularity has an ontic form, in re. But this form is invariant when it repeats, through mimesis, the eidetic and variant form in what constitutes the matter of this man, which is arranged according to a law of intrinsic proportionality that, despite the arithmetic variances of this being, always remains within the invariance of the law of proportionality, which is the form, as in the case of triangles, which can have different angles without changing their triangularity. If the sides of the triangle increase or decrease, the angles may change, but they always maintain the sum of two right angles, for when angle A decreases, angle B increases, etc. This example facilitates the understanding of Pythagorean thought. Thus, a being like man, although undergoing changes in changeable arithmoi, still remains man. What endures in him is humanity, even though he may be young now, later an adult, and then old. The changes occur in what is changeable and not in what is immutable. But there is also the concrete form of this man (hic), of this one who is here, John. His constant physical transformations do not destroy him as John, and within him, there is the enduring John, and within him, there is the enduring humanity. Thus, through arithmetic changes, there is an unchanging arithmós, that of his singularity (haecceity, for the Scotists), and that of his eidetic form (humanity).

The latter is imitated by him through mimesis, while the second (haecceity, the arithmós of his singularity) is unique to him. He only imitates the possibility of himself because John, who is here and now, was a singularizable possibility in the order of being, for otherwise, he would have come from nothingness, which is absurd.

John imitates himself. His singular eidetic form is uniquely his because only John can imitate it. In this way, among the possible forms, there are those that are generalizable, those that can be imitated by many in common, and there is the one that can be singularized in imitation, the one that only he can imitate, which is uniqueness. Every singular being, as haecceity, is unique. Therefore, uniqueness is the ultimate actualization of a form, but the ultimate and unique, which gives it exceptional and limitless value because only John is infinitely John (haec). He alone, in fullness, is himself, amidst the changes he experiences.

And just as numbers are potentially infinite, unique beings are potentially finite, and just as numerical possibilities could never end, possibilities for the actualization of singularities can never end.

In this way, Pythagoreanism, thanks to its provisions and foundations, manages to unite mathematical thought with ontological and theological thought, with a rigor that always astonishes those who dedicate themselves to studying it in its true roots.

In a singularized being, its existence is its onticity, but also its singular essence is its onticity. In it, in a certain way, essence and existence are identified because John is essentially what John is existentially. Thus, it is in uniqueness that all beings participate in the perfection of divine uniqueness, because every haecceity is limitlessly itself, and only itself.

In John, his essence exists. If he existentially repeats (imitates) the human form, he is his own essence of John. Looked at from this angle, essence and existence only formally distinguish themselves and not physically.

The controversy surrounding the real-physical distinction between essence and existence arises, for Pythagoreanism, only due to a misplacement of the problem. If we consider the existentialized essence, which is haecceity, the uniqueness of a being, essence and existence ontically coincide. However, if we consider the specifically existentialized essence, they do distinguish themselves in a real-physical sense, while the former only formally distinguishes itself (which is the thesis of Scotus and Suarez, while the latter statement is the thesis of the Thomists).

Thus, a law of limitation governs finite beings because a specifically determined being cannot reach infinity since it will always be what it is and not what it is not, which is positive. No being can be more than what it is because if an infinite being could be more than what it is, it would not be infinite, and a finite being cannot be for the reasons already mentioned.

A finite being consists of what it is and the relative non-being that it is not, for it lacks some perfection that is being, for if not, it would not be finite.

It follows, therefore, as a consequence, that for a being to become, it must necessarily be first and antecedently. This antecedence is ontological because becoming implies being, as nothingness does not become. And becoming can only occur in a finite being.

The arithmós plethos, the number of a set, can only occur in totalities formed by parts, which consequently leads to understanding that the infinite being (the Hen) does not have a number of a set, an arithmós plethos because it is not a totality.

Numerically considered, number is always finite. Since the Hen (Infinite Being) is not a totality, it transcends finite beings, and consequently, it is not the All (Pan), which follows from the proofs given in Concrete Philosophy, although any totality, considered as such, is in a certain way transcendent to its parts. The transcendence of the absolute and infinite being of the Hen-Prote is absolute.

Creation for Pythagoras

Based on Pythagorean theses and the concrete construction that we have been presenting of his philosophy, the creationist thought of Pythagoras is clearly explained. We know that our affirmation would provoke the rejection of some who follow the scholastic line, which denies that a clear notion of creation emerged before Christianity among the Greeks.

We also know that Thomas Aquinas made efforts, to some extent, to prove that Aristotle had a creationist thought, but did not give it the importance it deserved. To some extent, we can understand this lack of interest in delineating it in clear terms on the part of Aristotle, because the creationist thesis was opposed to the dominant conception in Greece and threatened to infringe certain norms of religious thought. Now, we all know that in Greece there was always a certain mistrust towards philosophers, who were often fiercely persecuted, as shown by the examples of Empedocles, Anaxagoras, Anaximander, Socrates, Plato, and Aristotle himself, not to mention the great persecutions suffered by the Pythagoreans, one of the most intense and extensive ones, so much so that, after the dissolution of the school of Crotona and the destruction of the institute of Metapontum, the sect of the acousmatics faced serious persecutions, being accused of Pythagoreanism, a forbidden word and persecuted by the polytheism of the time.

The Hen-Prote generates the Hen-Dyas, and this is the creator. Creation belongs, thus, to the Son in Pythagoreanism. The second dyad, dyas-aôristos, the indeterminate dyad, in determining itself, realizes finite things because it is potentially infinite, potentially determinative-determinable since there is an infinite power of determining alongside the infinite power of determinability.

The acting of the Hen accomplishes, in a twofold manner, both the act and the potentiality, in the Aristotelian sense, or the formable act and the materializable potentiality of Pythagoreanism.

Action is thus realized in the acted upon because where there is action, there is the acted upon, for an action without an acted upon is an action suspended in nothingness, which is absurd.

Action is thus inherent in the acted upon and not in the Hen. Pythagoras, as implicitly seen in his thought, places himself before action, visualizing it as a modality of the acted upon thing. In this way, all the Suarezian theses regarding modal theory are perfectly congruent with Pythagorean thought. The distinction between mode and thing is only modal because the mode is absolutely inherent in the modified thing, and this modality of the acted upon is action, which is proportional to it. For a being to be acted upon, it necessarily has to be dependent and finite. Therefore, action is always finite because it is a determination of what is determinable, inherent to the determinable, and thus finite. An infinitely infinite action is absurd, and for this reason, creation cannot be infinitely in act, but only potentially.

Now, finite beings are a product of the operation of the Hen-Dyas, but this operation does not come from nothingness since the finite being, before being, is nothingness of being. It does not originate from nothingness as if the antecedent of creation were nothingness, nor is it a product of nothingness as if nothingness could compose something. The Hen-Prote and the Hen-Dyas precede creation.

The operation of the Supreme Being is twofold, for there is intrinsically the generation of the Hen-Dyas and, extrinsically, creation. Since what results from creation is the finite being, which is always smaller than the Supreme Being, but there is no unbridgeable abyss between them, for otherwise, we would fall into dualism. The created finite being (creature) resembles the infinite being. An effect cannot be infinite because every effect is the effedum, the e-effect, that which is made, dependent on another, which corroborates the thesis of the non-infinitude of the creature.

Thus, in creation, there is a choice because the created being is chosen, while the non-created being is passed over. Temporalization, succession, arises in creation; therefore, creation is dyadic for the Pythagoreans because it involves a choice.

And the Dyas-aôristos is indeterminate; it is potentially infinite because the infinite power of the Supreme Being has the power to determine indeterminately and limitlessly, for otherwise, it would encounter limits to its acting, and such limits could only be opposed by absolute nothingness, which is absurd and impossible, and therefore cannot determine. Now, this potential infinitude of determinability-indeterminability is not nothingness, but rather the reverse vector of determination; therefore, it is positive, but our mind separates it because it is abstracting.

Every act of the agent accomplishes an action, and this action is always selective; in short, there is crisis because being something determined, what constitutes the determination excludes what is not it; otherwise, the determination would not be determination. There is thus a crisis, a fall, the pointing out of privation, which symbolically arises in various creationist religions as the fall of the creature.

The infinite passive potentiality (indeterminate determinability) is not the Hen, but arises from its operating ad extra.

The infinite Being is the first efficient cause of all finite beings. Every dependent being is contingent. Every contingent being is caused. The being that moves is moved because it does not have in itself its reason for being or the reason for its movement, for movement requires an act, and this act, necessarily, has to be the Hen-Prote.

Now, we know that every finite being is composed of act and potentiality, and of privation, and that the series of possibilities is potentially infinite.

Evil, for Pythagoreanism, is deficiency because beings are analogically related in the law of Good, and since evil is deficiency, there can be no absolute evil, for that would be absolute nothingness. That evil is deficiency also follows from the harmony that exists in the subordination of subsidiary functions to the normality of totality. Evil is a break in harmony; therefore, since this is a positivity, disharmony is deficiency.

Pythagoreanism affirms the interaction between opposites and between the opposites that compose beings. The cosmic universe is the universal substance, which ultimately is a unity of simplicity, but the beings that compose it are not completely separated from one another. Since the oppositions are positive, they do not contradict the order of being.

Pythagoreanism does not admit a corporeal infinity. Matter is characterized by specific dimensionality. Thus, matter is a finite and finitizable being; it is determinable. Materiality is the aptitude to receive determinations, and among these, corporeality is properly materiality since potentiality is matter, as corporeality is included among the determinations.

Comparing this doctrine with the theses of Concrete Philosophy, which is ours, it is observed that Pythagoreanism, when the positivities are considered as postulates, is perfectly suited to the philosophy we have presented, allowing us to evaluate how much of positivity there is in that thought, often disfigured throughout time.

Golden Verses of Pythagoras26

PREPARATION:

Render worship to the immortal Gods; Preserve your faith. Give prestige To the sublime Heroes' undying memory And the eternal remembrance of the supreme Spirits.

PURIFICATION:

Be a good son, upright brother, tender husband, and good father; And choose as a friend the friend of virtue, Always yielding to their gentle counsel; Follow the serene path of their lives;

Be sincere and kind, and never abandon it, If it is possible for you: for a severe law Chains Power together with Necessity. It is in your hands to fight and conquer. Your wild passions; learn to control them. Be temperate, active, and chaste; avoid anger. Neither in public nor alone allow Yourself to do evil; and above all, respect yourself. Think before you speak, think before you act: Be just. Remember: an invincible power Commands you to die; and the goods and honors, Easy to acquire, are easy to lose. Regarding the fatal evils that Destiny brings, Judge them for what they are: endure them, seek, As much as possible, to soften their severity: The wise are not delivered to the cruelest Gods. Like Truth, iron counts worshippers: The philosopher approves or gently warns;

And if Error triumphs, he withdraws and waits. Listen, and engrave my words in your heart: Close your eyes and ears to all prejudice;

Fear the example of others, and think for yourself: Consult, deliberate, and choose freely.

Leave aimless and purposeless action to fools; You should contemplate the future in the present.

Do not attempt what you do not know. Learn: everything yields to perseverance and time. Take care of your health: and minister with method To your body nourishment and rest to your spirit. Avoid caring too much or too little; zeal Is equally attached to both extremes. Luxury and avarice have similar effects. In everything, seek the just and good mean.

PERFECTION:

Let not a day pass, friend, without seeking Knowledge: What have I done today? And, today, what did I neglect? If it was wrong, abstain; and if it was good, persevere. Meditate on my advice; cherish and practice it: And it will lead you to divine virtues.

By Him who engraved in our hearts The sacred Tetractys, immense and pure symbol, Source of Nature, and model of the Gods, I swear. However, before your soul, faithful To its duty, invokes with fervor the Gods, Whose immense, valuable, and powerful help Will enable you to complete the works begun, Follow their teachings, and you will not be deceived:

You will probe the strangest essence of beings; You will know the beginning and end of Everything.

And if Heaven permits, you will know that Nature, Uniform in all things, is the same everywhere; Thus understanding all your rights, You will have a heart free from vain desires.

And you will know that the evil that torments men Is the fruit of their own will; and those unhappy Seek outside the goods that spring from within. Those who know how to be happy are very rare. Playthings of passions, swaying on the waves, Blindly roll in an endless and boundless sea, Unable to resist or yield to the storm. Save them, great Zeus, by opening their eyes! But no: it is up to men—yes, the divine race— To discern iron and know the Truth.

Nature serves them. And you, who have penetrated it, Wise and blessed human, peace be with you! Observe my laws, abstain from things That your soul fears, by distinguishing them well; Let Intelligence reign and shine upon your body So that, ascending to the shining Ether, Even among the Immortals, you may become a God!

Translation by Dario Velozo, from the original French by Fabre Olivet.

  1. According to Nicomachus and Iamblichus, on one occasion Pythagoras delivered a famous speech that decisively influenced the foundation of his famous society, in which members committed to practicing communal ownership, devoted to meditation, and thereby achieving the path of knowledge, the Supreme Mathesis (Megisthe), the supreme Sophia, the supreme wisdom. To achieve this, love for knowledge was necessary, and those who were lovers of knowledge became philosophers (from philoo, “I love,” and sophia, “knowledge”), from which Pythagoras coined the name that later became universal: philosophy. Knowledge, gnosis, would enable man to penetrate, following human paths, the path that would lead to Supreme Mathesis, supreme instruction. Only knowledge would bring us happiness, for he affirmed that supreme happiness consists in the true eudaimonia of the soul, in the contemplation of the harmony of the rhythms of the Universe, or, to put it in his famous words, “tes teleiótetos tôn arithmôn,” the perfection of Numbers, number as rhythm and proportion, as recounted by Clement of Alexandria.↩

  2. In the edition of “Verses Aurei de Pythagoras,” we have the opportunity to comment on the historical and legendary passages.↩

  3. The Pythagorean fellowship played a significant historical role in the Crotonian League due to its almost absolute political influence. According to certain information, at that time there were three types of initiates: the contemplative philosophers, who were mathematicians; the nomotetes, responsible for political leadership and social activities; and a third category, the politicians, who had not yet reached the degrees of initiation and served as instruments for carrying out the plans formulated by the leaders.↩

  4. We will reproduce in the next chapter his most significant fragments, as well as some from other authors.↩

  5. The examination of the theme of number will be the subject of the next chapters.↩

  6. It is necessary to consider the deep philosophical roots that Pythagoreanism lays in Greece and Magna Graecia, giving Greek thought a new direction. The lover of knowledge who is satisfied not only with knowing what is and how it occurs, but also the reason for what is.

    However, up to this point, the contribution of Pythagorean thought has not been characterized. We know that there was a strong dose of skepticism among the Greeks, and Pythagoras, who acquired so much knowledge through his travels to the cultural centers of antiquity, brought to the Greeks the great contributions of mathematics, physics, and the arts.

    Faced with these, it is natural that he demonstrated his knowledge and sought to prove his theses. It is not without reason that Pythagoras is attributed with the foundation of geometry based on demonstrated theorems. Not only in mathematics but also in philosophy, he must have presented the reasons for his theses to the initiated, demonstrating them.

    These words of Nicomachus of Gerasa are worth quoting:

    “The ancients, who under the spiritual guidance of Pythagoras, first gave science a systematic form, defined philosophy as the love of Knowledge. Incorporeal things – such as qualities, configurations, equality, relations, arrangements, places, times… are by essence imitable and unchangeable, but can accidentally participate in the vicissitudes of the bodies to which they are connected.”

    And he goes on to say: “And if, accidentally, Knowledge also deals with bodies, material supports of incorporeal things, it is nevertheless to these that it will be especially connected. For these immaterial, eternal things constitute the true reality. But what is subject to formation and destruction… (matter, bodies) is not currently real by essence” (op. cit. p. 22, I).

    The speculative nature of Greek philosophy, the search for apodictic judgments universally valid, in contrast to merely assertoric judgments, which we see emerging in the works of Western philosophy, undoubtedly arises thanks to the great contributions of Pythagoreanism. Mathesis, supreme instruction, is something active that man must strive to conquer. This striving for knowledge is an appetite, a love for the knowledge of Mathesis, it is philosophy. The content of this knowledge is a mathema, whose art in achieving it is mathematika, the art of obtaining the contents of supreme knowledge. In this sense, mathematics is the supreme knowledge of the Pythagoreans, and not in the commonly taken sense of a discipline that studies abstractions of 3.0 degrees of quantity.

    Let’s call it Metamathematics. That is the true philosophy for Pythagoras. And it was he who said that the true lover of knowledge is the one who expresses clearly what he knows and seeks to demonstrate what he knows, following the norms of mathematics, that is, based on apodictic judgments universally valid. When he called himself a lover of knowledge, a philosopher, he did not reveal everything he demanded from a true disciple but only what was possible to say to the uninitiated.

    The true philosophy, for Pythagoras, is Metamathematics, that art which consists in achieving the contents of supreme knowledge and demonstrating its affirmations (theses) through apodictic judgments (universally valid), the true science in summary.↩

  7. The justification of this affirmation and the presentation of the symbolism corresponding to these three languages are matters of such magnitude that they will be the subject of a separate work.↩

  8. According to Theon of Smyrna, there are eleven tetrads (tetractys), of which we reproduce the main ones:

    • 1st) Formed by the first four numbers: .........…

    1 + 2 + 3 + 4 = 10. This was the one sworn by the Pythagoreans.

    • 2nd) Formed by the two geometric progressions of even and odd numbers (1, 2, 4, 8 and 1, 3, 9, 27).
    • 3rd) Which combines, according to the same progression, the nature of all greatness. The point, the line, the surface, the body.

    It is also described by Aristotle in Peri Physeôs (Physics), in a somewhat distinct manner.

    • 4th) Of simple bodies and the corresponding figures: water, air, fire, earth.
    • 5th) It is the classification of engendered things: the seed corresponds to unity and the point, growth in length to the dyad and the line, width to the triad and the surface, and growth in thickness to the tetrad and the solid.
    • 6th) The classification of noetê, which is that of cognitive faculties and the cognizable. Our soul is composed of four parts: intelligence (nous), knowledge (epistéme), opinion (doxa), and sensation (aisthesis), according to the classification of Aecius.

    ↩

  9. We have already examined the main symbolic interpretations of the tetrad (tetractys) proposed by Theon of Smyrna. There are others, however, that we will examine later.↩

  10. The One generates the One, in the intra-procession of the Pythagorean trinity, very similar to the Christian one. In the ad extra procession, which is creation, it generates the one (universal substance), which is a dyad—two—in its functioning.↩

  11. For these reasons, because it does not have a here or a where, the “eide” do not have figure (qualitative determination of quantity), nor any limiting determinations of any kind. Therefore, in order to understand them properly, we cannot reduce them to the schematics of our sensible intuitions (phantasmata), as those who lack sufficient “philosophical mind” claim.↩

  12. The symbolism of numbers is examined by us in “Tratado de Simbólica” (Treatise on Symbolism).↩

  13. The word “combinar” (combine) comes from “çum” and “bini,” and the latter comes from “bis,” the archaic form of “duis,” and “dis” in Greek, meaning “twice,” “once and again.” To combine is to unite, to order two things. The number (arithmós), as such, is the harmonious ordering of the even and the odd, the unlimited and the limited, the infinite and the finite.↩

  14. Aristotle always tended to falsify other people’s thoughts. This can be seen in his treatment of Empedocles, Anaxagoras, and the Pythagoreans. However, this flaw does not exclude the great value he had as a first-rate philosopher.↩

  15. Aóristos is also used by the Pythagoreans in a limiting sense. Thus, determination is also sometimes limiting.↩

  16. Limitation is distinguished from determination because the former gives physical limits to the thing, while the latter gives only a formal profile. That is why we call limiting determination the one that constitutes a form in something limited, like the sculptor giving marble the form of Apollo (figure, here).↩

  17. It is essential, for the understanding of arithmós, to examine participation as a philosophical theme. In “Tratado de Simbólica” (Treatise on Symbolics), we present a synthesis of this theme, sufficient for the study of Pythagorean arithmós.↩

  18. Note that, for the Pythagoreans, limit and determination are often identified, which does not allow for greater clarity of thought.↩

  19. And it had no reason, because Christian thought is syncretic and brings together the positivities of human thought up to Christ. He did not come to replace, but to complete. His doctrine was not negative but affirmative of the positivities found until then, including the Pythagorean ones (for there is more Pythagoreanism in Christianity than one would think). Additionally, Christ maintained contact with Pythagorean sects in Palestine, such as the Essenes, whose leader, John the Baptist, was Pythagorean, as was the school of Melchizedek and its remnants from the time of Christ, as affirmed by categorized Pythagoreans of the modern era.↩

  20. This fragment attributed to Philolaus is congruent with the one we reproduced in paragraph 10 of his fragments.↩

  21. Analogy is the subject of our “Problematics of Analogy,” which is part of “Themes and Problematics of Concrete Philosophy.”↩

  22. It is said that Philosophy seeks the “whys,” and Science seeks the “hows” of facts. But within the realm of “whys” (reasons), there is a “how” that eludes us. The visualization of this “how” could only be obtained through scientific methods, which reveals that there is, within the realm of philosophy, a territory that is scientific, and that science and philosophy would complement each other. This aspect, due to its undeniable importance, suggests a topic for studies that will serve as material for our future work.↩

  23. We have been studying the Corpus Hermeticum for a long time and plan to publish it in Portuguese soon. On that occasion, we will discuss the various difficulties that have arisen for those who have devoted themselves to the study of Hermes Trismegistus.

    We will have the opportunity to demonstrate that hermetic thought has distant roots in Egyptian culture. Furthermore, we will show the baselessness of those who seek to reduce these ideas to a purely Greek origin.↩

  24. There is obviously an error, either by Aristotle, or by Alexandre, or by the scribe.

    The syllogism can be reduced as follows:

    The elements of the dyad are the One and the Great-Small; Now, the dyad is the first of the numbers, Therefore, the elements of numbers are the One and the Great-Small. This is what ultimately confirms the general conclusion of the entire passage.↩

  25. We will talk more about creation, in a Pythagorean sense, later.↩

  26. These verses are attributed to Lysis, a great Pythagorean, but he based them on the teachings of the master from Samos.↩

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