This is the first of five related discussions on the work of Pythagoras of Samos, generally regarded as one of the most universally enlightened men of the ancient world.
We should discuss his work, as it was in 1955 at the Pythagorean Congress in Athens. This philosopher's birthplace, the island of Samos, was renamed in his honor, and the Greek government expressed its remembrance of one of its most outstanding citizens in many ways.
Pythagoras is remembered scientifically for his contributions to astronomy, music, and mathematics. On the philosophical level, he is known for introducing the doctrine of transmigration among the Greeks. On a religious level, he emphasized the unity of world religions. His vast travel brought him into direct contact with the great religious systems of his time, and he practiced mystical disciplines at the university he established at Crotona.
Thus, we have a man whose achievements cover an extensive range. As a result, his memory has passed through many vicissitudes in the descent of time. Those who admire his scientific achievements are inclined to question his philosophical attainments or doubt his religious opinions. Yet, all in all, a grateful world has honored him. Through a better understanding of his philosophy, some doubts and uncertainties concerning his various contributions can be clarified, and something can be added to the luster surrounding his illustrious name.
Our discussions will primarily focus on the mathematical philosophy of Pythagoras, which is also divided into two essential parts: theoretical and practical. On the practical level, Pythagoras made the most significant contributions of the ancient world to advancing the science of mathematics. He is responsible for the later achievements of Euclid, who in turn has influenced all mathematical thinkers down to the present day. If it had not been for Pythagoras and Euclid, there might not have been a Steinmetz or an Einstein. The significant development of mathematics rested upon the practical findings of Pythagoras and the philosophy of numbers upon his philosophical speculations.
Pythagoras was one of the first, if not the first, to emphasize the importance of numbers and mathematics in advancing the total state of man's integration. It was extended to assume that so noble a science, so magnificent a system, was devised simply for the convenience of the banker and the money changer. The purpose of mathematics was to unfold, through orderly instruments, man's internal consciousness of the universe around him and the psychic life within him. Thus, to Pythagoras, mathematics was part of religion, philosophy, and all those great ideals and systems of human thought that have enriched, cultured, cultivated, and civilized the conduct of human beings.
This evening, we will discuss the Pythagorean concept of numeration and number. In this, we come to the theoretical phase of the master's work. We are interested in this phase because it is possible to progress in the science of mathematics for a practical purpose by attending many schools or by moderate study of mathematical textbooks. For the most part, however, progression of this kind does not include the philosophy of numbers or the great imponderables relating to the universal mystery, which Pythagoras regarded as the essence and substance of all mathematical speculation. Thus, we must understand the numbers taught by Pythagoras and the philosophy he interpreted through numbers, which has become identical to numerical speculation for most of those studying his system.
To begin with, Pythagoras recognized the existence of a primary and fundamental concept in mathematics, which he called archetypal number. As the term indicates, the archetypal number is a concept of numeration in the divine mind. As archetypes are patterns or designs which, when impressed upon substances, result in the gradual unfoldment of orderly sequences, the archetypal number represents the key to the grand design of being. It defines and unfolds the entire theory of existence and enables the individual, through the study of archetypal numbers, to apprehend or perceive the primary function of the divine mind. Thus, to Pythagoras, numeration was a kind of mathematics existing only in the consciousness of God, and mathematics themselves, the various branches of arithmetical science, represent these archetypal or divine numbers shadowed forth into the mundane world and becoming the guiding principle, the controlling design affecting the unfoldment, growth, evolution, and progress of every living thing.
Thus, he distinguished what he called numeration, which is a descent of numbers in principle, and he also recognized numbers themselves, which in turn were a descent of numeration on the objective level of numerical division.
Thus, if we approach the concept of numeration, we must approach the basic definition of deity as understood in the Pythagorean system. What is enumeration, and how does it differ from number? Numeration is a number in principle. Numeration is a concept of number but is not a number itself. When we say God is one, we are thinking about enumeration. In this case, one is a concept of totality, not a concept of unit first or an isolated numerical one. When we say the end of human experience is union with the divine, we think of a concept of union based upon the idea of unity, and unity is again a numerational concept of oneness. Yet, unity is not one. You wouldn't say two plus unity equals three. You wouldn't think that way. Unity has a different connotation. Unity is one, but it is a totality. Unity is one in terms of the destruction of interval because things in a state of unity are in a state of identity, sameness, or assimilation, where all division or separateness between them can no longer be conceived or calculated. Therefore, unity is the restoration of all division, the re-establishment of totality as a concept.
Man has never seen totality because, regardless of what he may see, the unseen must also be conceived as existing, and totality must be the complete unity of the seen and the unseen. It must be the entire identity of the known and the unknown. Therefore, man may not perceive it, but he may conceive it. Yet, even in his conception thereof, there is a limitation, and his conception is empiric. That is, it is an assumption based on the acceptance of the concept or the statement that the known plus the unknown equals one. This concept carries us into one of the most profound abstractions of thought, and we find it restored to us in a great school of Indian thought, the school of Yoga. Yoga means union. It means the state or consciousness of oneness. Oneness, in search of man for the nature of truth, assumes this condition at one moment. It is the ultimate state achieved by the progressive destruction of diversity or interval by which the human being gradually achieves a state of identity with total existence.
All these things are words, but words are symbols of ideas. Ideas are, in man, the shadows of archetypes. According to these concepts, the universe, creation itself, existence, and manifested being are all suspended from eternal ideas in the divine mind. Man, attempting to understand God and himself, engenders shadowy ideas within his intellect that are archetypal but not complete yet indicate to a limited degree the probable direction of the divine mind.
Numeration differs from number in that it represents either unity itself or aggregations of the parts of unity, no longer arranged side by side but conceived as a new unity composed of divided parts. Let us assume we have several straws in our hand, and we count them and find that we have twelve. This creates the possibility of two concepts of the nature of twelve. One idea is that we are holding twelve separate straws, thus representing a dozen of anything—twelve matches, twelve dollars, twelve straws. Twelve things may not all be alike; we may hold twelve objects of different categories and call them twelve unless we distinguish further by saying that four are lemons, two are oranges, and six are apples.
Another way of conceiving this is when twelve objects are brought together; they produce a grouping for which a single term is applicable. When we hold twelve objects, we don't repeatedly point at them and say "one"; we say "twelve." The moment we say "twelve," a numerical concept arises in the consciousness of the person we address, transferring the idea of this group as a collective. We know that twelve can stimulate an idea, not primarily twelve separate parts but a unit called twelve. This unit has a nature of its own, dimensions, boundaries, and proportions, and applies only to a particular order or numerical group. If we remove one straw, twelve is destroyed, and a new number, eleven, is created. Eleven is also a unit, a totality composed of eleven separate parts. All groups may be formed of individual parts or as units or unities consisting of a certain number of related elements or members. The twelve is a numeration; twelve ones considered separately constitute numbers. Twelve is an idea conveying oneness composed of twelve parts, while the parts themselves do not convey oneness nor necessarily imply it, as they may be separate when considered individually. When grouped, they engender a new idea: the sum of themselves.
Pythagoras emphasized this on a philosophical level, though it might not seem important initially but gains importance as we proceed. We begin with the first Pythagorean mathematical concept, unity or the monad. Unity is a term applicable to God; therefore, the Pythagoreans did not correctly regard it as a number. They considered it the first motion of being and the first expression of consciousness. The first conceivable or perceivable quality of the divine nature is unity.
Unity is a single term covering the phenomenon of ultimate and absolute diversity. We think of deity as the total of all its parts, the complete essence of creation. We think of God as that immense and incalculable being in whom we live, move, and have our existence. Therefore, the entire unit, the monad, is the one that is all. The one which is all may be conceived geometrically as a sphere, as a sphere is a geometrical solid with infinite surfaces. The assumption is that there is an inconceivable number of flat planes in the structure of a circle, making the circle unlimited or eternal in its surfacing. The sphere rests upon the most minor hypothetical surface of its nature. It is the most immediately subjected to motion, moving wherever the surface upon which it is placed is unbalanced. This is a Pythagorean theorem because, to the Pythagoreans, the first motion of being is due to the uneven nature of the eternal surface upon which the sphere operates or moves. These different principles become involved in a variety of ideas.
Pythagoras, beginning with the enumeration of the monad or the unit, began to philosophize: What is there in nature outside or beyond the nature of infinite being? Pythagoras defined God as an infinite being whose body was composed of the substance of light and whose soul was composed of the substance of truth. This definition is one of the noblest and most splendid in philosophical history. Conceiving deity as infinite and with no interval in its nature, Pythagoras concluded that the monad, the unit, the totality, is the only numeration that can exist per se without qualification, limitation, or restriction. It is also the only immortal number, the only eternal numeration, as it is the only thing that cannot be destroyed. Totality is indestructible, as all destruction merely reduces parts within the totality. Any creature can die, and its body can return to the elements from which it came, the soul returning to the sidereal powers. However, this destruction or disintegration is merely the dissolving of a compound. Nothing is lost; everything remains in the universe. All forms are subject to change, but nothing is destructible in the sense of being susceptible to destruction.
The only thing not subject to change, and therefore to what we call destruction, is totality, as there can never be less than all. There can never be more than all. Nothing can be added to all, and nothing can be taken from all. All cannot be multiplied nor divided or in any way qualified. Thus, Alness or the state of totality is a term always to be given to the sum of all conceivable and inconceivable parts. Regardless of their number, these parts are summarized by the concept of part or a partial existence, all about the number two.
Two represents any possible conceivable appearance of disunity in space. When we see two ones side by side and unite them, calling them two, the name two pertains to our illusional or materialistic lack of internal understanding. Two, composed of parts brought together, is not two, but one. So, what are two? We are holding two objects in our hands. Two is one in terms of halves. Regardless of the number of parts, the triad or three is one in thirds, and the tetrad or four is one in fourths. The moment we say four, we give the number of parts. When we say tetrad, we provide the name for the unity of those parts. Therefore, all numerations are names for unity when conceived as consisting of parts. This consciousness leads to recognizing the restoration of unities by re-integrating their parts. All division lies in the acceptance of the classification of numbers; all unity is in the acceptance of the classification of enumeration.
Advancing politically, sociologically, and culturally or overcoming religious prejudices or cultural intervals, we constantly seek to re-establish unity. This unity is the internal comprehension of the idea behind the concept of separation. Unless the idea is there and active, we are unaware that all divided parts, when brought together, create unity. Unities of various sizes, numbers, and orders, by their further gatherings and mingling, create still greater unities, and these unities unite to form more magnificent unities until, finally, all converge in total unity. Thus, the universe is an ascending order of unities composed of lesser parts, the lesser parts of greater unities. When regarded or contemplated from levels below themselves, all unities have one nature or appearance, and when conceived from levels above, they have another nature or appearance. A unity conceived from a level below is an aggregate of separate parts; seen from a superior position, it comprises individual parts. The superior always conceives the unity, discovers it, or experiences it, whereas the inferior sees the parts but cannot conceive the unity. Looking at twelve from below, we see twelve separate parts; from a superior position, we see unity manifesting through the numeral twelve.
This is a matter of perspective, opening a vast pageant of speculation. Man looking upward to the aggregates of unities above himself calls them gods. These gods, in turn, look toward unity transcending themselves and call this sovereign unity being or the supreme and one God above all division. Like the original cell of the human body, the universe is never divided by nature or substance. Cell multiplication occurs within the first denominated cell until the entire body develops within the original cell and never departs.
Similarly, the infinite diversity of creation occurs within the original cell of unity, being the inseparable nature of the deity itself. All division consists of a hypothetical division within eternal and absolute unity. The division exists only in the consciousness of creatures on a lower level than the phenomena involved. That which stands above diversity annihilates it, and that which stands below diversity may be intellectually or spiritually destroyed by the sense of diversity and, therefore, be unaware of the eternal unity.
Unity or the monad, an individual eternal unconditioned unlimited existence, properly signified by the sphere, represents the totality of all parts, all things equidistant from the eternal central zone of consciousness. Deity is also susceptible to another interpretation: if God is all or eternity, God is also one in the sense of the first, the superior, the most magnificent, excelling all other things. Its excellence or primordial place gives us another concept of unity. Therefore, we may worship God equally as all or the one. Like the one, what is different from totality? The nature of totality is complete inclusiveness; the nature of one represents complete exclusiveness. The one is the superior, the greatest, the first, representing not only total being but the supreme excellence of being.
In the concept of the human mind, there is this inevitable alternation between conceiving the supreme being as all or one. As one, as the first, as the primordial, it may properly be placed in the center of the circle or the hypothetical sphere of existence in the form of a dot, creating a simple universal mathematical symbol, often applied by primitive people to the sun and still used in astronomical symbolism: a dot in a circle. This dot in the circle tells us one in all and, by reverse, all in one. The dot tells us that the inconceivable, immeasurable, incalculable totality of being does not change. Still, man, unable to assume total consciousness, attempts to define all by bestowing upon it specific singular attributes that achieve uniqueness of concept. This uniqueness of concept is the first departure into relative thinking.
When we think of God as unique, we think of God as set apart from all things. If we think of God as a unit, we think of God as all things, the total and entire substance of all parts. In the concept of uniqueness, we transform or deform the idea of unity, recognizing God no longer as being but as a being, no longer as a principle, but as a principle, no longer as consciousness. This division is a reduction by which totality is destroyed, establishing a being or a total unit surrounded by an abyss of a non-identical nature. When we conceive of a superior or unique being, we create this concept by differentiating it from all other things, creating polarity. We have a being and a not-being, the not-being being the dark background or the background of the concept of a being created within ourselves.
The moment we create a God or a God concept, we inevitably bring into existence the not-God concept. The primary idea of this is the hypothetical division of spirit and matter. Duality or the two represented to the Pythagoreans spirit and matter, light and darkness, male and female, life and death, representing all polarized existences. The positive pole of these existences is conceived as a reality, and the negative pole as a shadow or non-substance, deprived of the essential nature of substance. Some ancient theologists and mythologists conceived that, initially, the universe was an infinite expanse of being. Gradually, this limitless expanse retired or restricted itself toward its center, creating a glowing spiritual existence and leaving the area from which it had retired deprived of its nature, leaving darkness behind it or the not-self in the great space left by the withdrawal of being to form a being or a center of consciousness.
If we have self-consciousness, we must also have an interval of some nature, for the self can only be unique because it is separate from or different from other-selves. Man, in the gradual development of his innate egoism, has come to this curious psychological situation in which uniqueness is the experience within each psychic entity, and each individual feels that he is unique, that the divine or essential spark within himself is a separate and distinct spark with a separate and distinct destiny. The whole conception of Western philosophy and theology has been built upon this concept. This concept makes this unique self capable of separation in quality from being, thus capable of ignorance, violation of the law of being, sin, and crime. Because it has a separate existence and no separate existence is eternal, the complex of selfhood brings the inevitable fear of death. Death is inescapable where a condition of uniqueness or separateness exists. Uniqueness implies a constant conflict between itself and the not-self, which surrounds it, ending in the exhaustion of selfhood because the valuable area of not-self infinitely outnumbers the spark of self in each creature self is but a spark in an infinite expanse of space, closing in upon that sense of selfhood. The self cannot have a victory over totality but must finally be exhausted in its psychological conflict with totality and be returned to the state of totality, as all buildings built by man upon the earth ultimately return to the earth. All worlds and planets return to the space from which they came. All conditioned beings, by being conditioned, are mortal and must return to an unconditioned state, which to our unenlightened intellect means extinction.
This conditioned space then becomes the hypothetical Pythagorean duad in terms of halves. Wherever we have this one in terms of halves, the positive pole assumes the dynamic position, becoming the agent, while the negative pole assumes the static position, becoming the patient. Agent and patient are in contrast through eternity. Agent and patient may be regarded as spirit, matter, intellect, and form. The Pythagoreans insisted that the acceptance of the concept of duality and the projection of categories based upon duality, the unfoldment of syllogisms based upon duality, all result in the mind descending into a state of illusion, as the existence of selfhood itself is a primary illusion. From this one fundamental illusion, all others naturally follow. This is essentially the teaching of Buddha, who was convinced that accepting the illusion of personal divinity or personal self ultimately leads to the complete corruption of the internal consciousness of man.
If we have the monad as all consciousness, the duad gives us self-consciousness and unconsciousness. Self-consciousness is the term we bestow upon our field of conscious awareness. Unconsciousness is the mysterious sea in which we try to swim, as it is the consciousness of all things except ourselves, regarded as enigmatic, unknowable, vague, dark, and beyond comprehension. In psychology, the term unconscious refers to what is unconscious in space or that we are unconscious, which may be superlatively conscious. Nothing is more mysterious to any individual than his neighbor's thoughts; we cannot know with certainty what anyone else thinks or believes. We have a spark of light called our thinking, surrounded by a strange darkness, the thinking of every other person. This thinking of every other person gathered is not thinking because we cannot share it or participate in it. It has no mental or visual imaging power within us.
The illusion rests in the simple point that every other person is unreal to each person. We are part of everyone else's unreality if we are honest with ourselves. This concept leads to the inevitable conclusion that we are not living in a world where we as a self can be unique, but this uniqueness is a delusion. Every other living thing is equally exceptional, and the totality of this uniqueness is unity, which is substantially indivisible. Uniqueness is only our point of view, which is incorrect upon extended consideration. Uniqueness is our egotistic assumption, and when the rest of the world has the same attitude, everyone is unique to himself and non-existent to everyone else. Here, we have the center of another kind of universe, the universe of illusion, in which each center is everywhere and its circumference nowhere, the reverse of the great circumference of reality.
The duad represents a certain inharmonious maladjustment or lack of understanding. According to the Pythagoreans, it is the first perceptible number, as the monad is too sacred to be a number. When you put the number one on it, you create uniqueness and destroy it. Therefore, neither the one nor the monad should be considered numbers but principles abiding forever in eternity. Two, as diversity, is the first valid number because diversity is the first aspect of being of which man is aware and able to have rational cognition. Pythagoras said the monad is intellect, the duality is science, the triad is opinion, and the tetrad is sense. We have this descent of powers.
The number three was significant in Pythagorean thinking as it relates to an instinct within the mystery of enumeration. The triad symbolizes equilibrium in space, consisting of three monads and two intervals. If you make three dots in a row, you have three dots and two intervals between them. Pythagoras emphasized the significance of intervals, as interval equals instruction. By interval or the concept of interval, conditioned existence can estimate realities, known by the spaces between them rather than their natures.
Man cannot define monads but learns their nature through their interactions within intervals. We observe their attributes through their functions, purposes, motions, or immutabilities over periods. These relationships constitute the Pythagorean concept of interval, perceivable through reactions operating within spaces. This motion within intervals is a gradual signature by which each thing writes a name for itself by its actions.
Similarly, man attempts to associate with other human beings through intervals, becoming aware of their existences not by what they are but by what they do. The nature of the deity in substance is unknowable. Still, the deity in motion within its interval causes the emergence of the detractors, the pyramid of dots consisting of ten monads. This internal motion of the deity results in the generation of existence, suggesting that the creative motion of consciousness within the deity corresponds to meditation, realization, concentration, or internal disciplines.
Greeks and Pythagoreans assumed creation was an internal experience in the divine consciousness. Brahmanas of India described the deity as extending to the outer boundaries of absolute space, implying that its motion is only within itself. Plato and other Greek philosophers assumed that all motion of universal consciousness is from the circumference to the center, creating duality, the dot, and the circle, a polarized experience of consciousness within the total being.
Pythagoras questioned whether totality lost awareness of its nature in polarization. He suggested the possibility that self-knowing, as recognition of separate selfness, exists only in conditioned beings. Eastern Buddhists refrained from defining the state of the mahaparanirvanic consciousness, the consciousness identical to reality, as a universally aware or submerged existence.
The Pythagorean concept that duality is the first of numbers implies that the state of consciousness rising from duality is the first illusion. From this illusion, all others naturally follow. Natural mathematics of the universe experienced the struggle to preserve equilibrium, as imbalance leads to motion. Dynamic symmetry, achieved by imbalance, causes sensory reactions in observers, leading to actions such as pain, war, and crime. Balance, or equilibrium, symbolizes unity, and imbalance symbolizes duality.
Pythagoras emphasized that achieving unity involves internal immovability, a positive state of restraint. Pythagoreans believed the triad-imposed equilibrium between the polarities of the duad, symbolizing the beautiful art as the great moderator of excesses. As an experience of equilibrium, the gorgeous represents the universal redeemer, the suspension of excess in all things.
The Pythagoreans completed their picture of numbers and numeration with the tetrad, the symbol of justice. The tetrahedron, or the four-faced symmetrical solid, encloses an area, symbolizing the material world, the arena of law. The physical universe is where all things learn obedience to universal law. Deity, extending from one through diversity to equilibrium, generates the power of the soul and, finally, forms in the four-square world of matter.
The numbers four, three, two, and one equal ten, the restoration of the monad. This numerical system reveals the descent of principles, moving from unity to diversity in appearance. Assuming the totality of the universe's parts, what is their relation to the primary consciousness that engendered them? Growth and unfoldment cannot affect totality, as nothing can outgrow it. Growth is a motion toward unity, not infinite extension. Things grow together, reducing in multitude, and evolution ends in resolving numbers.
Growth reduces numbers, returning to the primordial one. As parts grow, they unite into the next inevitable step, forming a larger unity. Evolution does not produce infinite numbers, as growth ascends like a pyramid to an apex, where divided parts return to unity.
Pythagorean theory of creation and the principle behind numeration and numbers suggest growth is a continual restatement of monads or numerations over numbers, revealing unity. Any act establishing unity is sacred; any act promoting division is profane. The will of the infinite is for all things to be one, restoring the infinite to its fullness. Unity's restoration ceases time and eternity, re-establishing the infinite's total recognition of its reality.
This substance is the Pythagorean theory of creation and numeration. We will proceed with this study in the following lecture.
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Manly P. Hall Seminar: Pythagorean Theory of Number 1: Basic Philosophy ...
