Whenever we may think of arithmetic and other subjects dealing with figures, we tend to remember most our early struggles in school with geometry based on Euclid's Elements. But among the eminent mathematicians in Greece, the Pythagoreans took an approach very different from his. For them, mathematics had a connection with philosophy; there was a mystical tone in their work similar to that in Oriental thought. The followers of Pythagoras were indeed meticulous in their thinking with figures and their empirical work with them, but their precise mathematical speculation was not severed from the depth of metaphysical concepts. The profound difference between these two points of view may be summed up by comparing Euclid's propositions, which are logical but display no philosophical basis, with the texts of Theon of Smyrna which are in a direct line from the older Pythagoreans. The latter proceeded from principles or "a priori established laws accompanied with simple demonstrations for verification and contemplation, but always using these as a means, for philosophizing about the nature of the Universe." [See p. xi, Theon of Smyrna: Mathematics Useful for Understanding Plato, by Theon of Smyrna, translated by Robert and Deborah Lawlor from the 1892 Greek/French edition of J. Dupuis (Secret Doctrine Reference Series, Wizards Bookshelf); the notes by J. Dupuis are invaluable, as also the translators' introduction and the publisher's glossary and note. The index and end-papers are of assistance, the latter showing a) a map of the ancient sites in Greece and Magna Graecia, and b) a table of five of the chief foreign alphabets.]
Among the books dealing with such aspects of mathematics, philosophy, and spiritual knowledge is Theon of Smyrna's study for understanding Plato. This work, dismissed by Sir Thomas Heath as lacking in intrinsic value but contributing something to the history of mathematics, should rather be regarded as a classic text in its own right. Its recent translation into English provides us with a valuable tool to unlock the meaning not only of Plato's mathematics but also of his concepts about the cosmos and, as a bonus, gives us insight into the objective of the Mystery Schools of Greece. This key appears in one paragraph in Theon's introduction, where he compares philosophy
to the initiation into things truly holy, and to the revelation of the authentic mysteries. There are five parts in initiation: the first is the preliminary purification, because participation in the mysteries must not be given indiscriminately to all who desire it, but there are some aspirants whom the harbinger of the path separates out, such as those of impure hands, or whose speech lacks prudence; but even those who are not rejected must be subjected to certain purifications. (Ibid., pp. 8-9).
Four further stages are described. After the purification, the tradition of sacred things ("which is initiation proper") is presented. The third is the opening up of "the full vision (the highest degree of the initiation)." This stage has also been called the epopteia or reception. The fourth part which "is the end and design of the revelation,"
is [the investiture] the binding of the head and fixing of the crowns. The initiated person is, by this means, authorized to communicate to others the sacred rites in which he has been instructed; whether after this he becomes a torch-bearer, or an hierophant of the Mysteries, or sustains some other part of the sacerdotal office. But the fifth, which is produced from all these, is friendship and interior communion with God, and the enjoyment of that felicity which arises from intimate converse with divine beings. — Cf. Thomas Taylor's translation in Eleusinian and Bacchic Mysteries, 4th ed., 1891, p. 85; the reference to "crowns" is symbolic, see also H. P. Blavatsky, Isis Unveiled, 2:101-2
Theon sheds light over a wide area, touching arithmetic, the numerical laws of music (including the various meanings of the word logos), and astronomy. In Gnosticism and early Christianity, this term was used for the Divine Mind. The Latin equivalent is verbum (see John, 1:1). His suggestions as to Plato's meaning facilitates also a new understanding of the Neoplatonists whose school was born long after Theon's time: Plotinus and Proclus at the two ends of the line show the influence. The translators comment that the tradition reaching us through Theon integrates "numerical or scientific modes of knowledge with metaphysical and mystic cosmologies. History indicates that this form was brought into Greece, the threshold of the Occident, from Egyptian, Indian, Babylonian and other Mid-Eastern and Eastern cultures" (p. x).
In addition to the understanding that Theon gives us of certain aspects of Neoplatonism, the modern Egyptologist and mathematician R. Schwaller de Lubicz found in this work the clues that enabled him to uncover the ancient Egyptian system of philosophy. This idea starts an exciting train of thought: Theon derived his views from the Pythagorean school, and hundreds of years earlier Pythagoras had spent some time in Egypt. The circle seems completed in our own time with de Lubicz using the Pythagorean element in Theon's text to discover the so-far-untranslated Egyptian wisdom-tradition hidden from public view within the stream of hieroglyphics. The alphabetic and syllabic language used for routine, daily affairs is one thing, but the symbolic expressions that charted the road to man's perfection as a human being and his relation to the universe is another. De Lubicz spent some fifteen full years in Egypt, doing on-site research among the remains of the old civilization, fruitful because of Theon's eye-opening effect upon him.
One of the most interesting themes among the many in Theon of Smyrna is that of the "One," when the One and the Monad are alluded to in more than a strictly numerical sense. "One" is a term used in religious philosophy for the godhead, and even in fundamentalist texts it occurs for a personalized God. A well-known problem exercising the minds of scholars is: how does the "One" (whether divinity per se or a personal God) produce or become the "many" — the material universe we know through our senses. Unfortunately, the English language does not possess an extensive or profound metaphysical vocabulary, and as Richard Robb, publisher of the Wizards Bookshelf edition, states, it is mainly concerned with things or possessions. He contrasts this with the old Greek which seems better equipped to deal with philosophical thinking, i.e. with ideas, "and is perhaps a more spiritual language" (p. viii). The "One" being intrinsic in the "many," there really is no problem of how the former can produce the latter, although logically the two terms contrast with each other.
The Monad is viewed as the foundation of all existences because it is in everything, it cannot be divided into parts. In other words, everything flows from and depends upon the One. The Pythagorean and other Greek philosophies used the plural form of Monad to stand for real entities regarded as the first principles of existence, their spiritual aspect. The Monads, then, are the indivisible centers of consciousness, the enduring part of the inherent nature or essence of entities that grow through experience. Henry More, 17th century Cambridge Platonist, expressed the idea of the Monad as: "One Steddy Good, centre of essences, Unmoved Monad, that Apollo hight." In The Song of the Soul (II, iii; III, xii), "hight" is an old expression for "named" or "termed," so the phrase means "was called" Apollo. His contemporary and fellow Platonist Ralph Cudworth said, "That which was called by [the Platonists and Pythagoreans] the Unity itself or a Monad — that is, the one most simple Deity." (The True Intellectual System of the Universe, I, iv, 225.)
The Pythagorean influence upon Theon being so strong, it is worth while to look briefly at the main teachings of that "school." First, the universe as a whole is considered to be one vast system of mathematically correct combinations, and the vibrations resonated in space by the different planets resemble the tones of notes emitted by various kinds of musical instruments, a concept known as "the music of the spheres." Pythagoras was also said to have taught that the sun is but an agent, or better a transmitter, of energies, and that therefore neither the sun nor the stars are actually the sources of light and heat. Another of their teachings is that Universal Mind is diffused through all things.
Pythagoras has been quoted as saying: "Know divinity, that is number and harmony. The human soul is number moving itself." It is obvious he does not mean mathematical figures per se, for his secret of numbers was based on the symbolic emergence of the Three and the Four from the One and solitary Monad. What does this mean? First there was the One, the consciousness inherent in cosmos, with inert substance or matter represented by Two. Both together reacted in manifestation to make the Three, a trinity, the "child" of the two previous numbers/entities, and also called the "soul of the world" or cosmos. Out of these three came the Four or material range of substance that we perceive as our habitat, the universe generally — cf. Pythagoras, Teacher of the Law, by Jeremiah Taylor. This pamphlet is very suggestive when considered with The Theoretic Arithmetic of the Pythagoreans by Thomas Taylor (no relation). Plato was using Pythagorean language when in the Timaeus he summarized the profound concepts within this system of numbers as One, plus two, plus three, plus four, equals ten, this last figure standing for the cosmos in its entirety — the subjective and objective (or apparent) halves of the whole.
The Pythagoreans symbolized all of this by a triangle of ten dots, called the tetraktys. This figure is capable of considerable extension, yet all of the dots stream from the single one at the apex. The school did not limit its use of the number system to the ten, but also stressed the number seven; for example, man considered as a duality of three higher, intangible elements, on the one hand, and four more material components on the other. They alluded also to seven planets — designated in some traditions as "sacred" — referring to those with a special "harmonic" relation with or magnetic influence upon our planet earth. Tied in with these concepts were views about light, color, and sound as correlates of the "One" interpenetrating or manifesting as the All.
These are the aspects of Pythagorean mathematics that not only provide a background for Theon's work but also reveal his implications in many areas. They likewise indicate what Euclid cut out of his Elements of geometry. The commentators in the new English translation point out the cleavage between the former system and Euclid's, for Theon does not deal exclusively with the arithmetical side of numbers, but also with the philosophy, and he enters into a discussion of the numerical laws of music and harmony, for instance, and very seminal passages they are.
Theon refers to his section on music as "a dissertation on the harmony of the world," and he uses the word symphony in the old sense of "agreement or concordance." Because symphony is now almost wholly applied to a musical work played by a large orchestra, the translators say, the original symphonia has been rendered as "consonance." As Theon states: "This consonance has the greatest power, being truth in reason, felicity in life, and harmony in nature; and this harmony, which is diffused throughout the world, will not be found unless it is revealed first through numbers" (p. 32). The view that harmony arises out of the relationships of sounds, colors, and numbers leads us to conclude that in this context, number means relations. Theon then considers the quaternary (or tetraktys) and the decad (10) in this connection.
He stresses that the importance of the quaternary in music is great "because all the consonances are found in it" (p. 62). And this was meant not only for its intrinsic sake, but because the Pythagoreans felt it represented or outlined the "entire nature of the universe." Theon adds that this was the reason their most sacred oath was:
I swear by the one who has bestowed the tetraktys to the coming generations, source of eternal nature, into our souls.
"The one who bestowed it" was, of course, Pythagoras; but whether he discovered it or transmitted it from ancient sources with which he had become acquainted, is a moot point. Theon holds that the two quaternaries of odd and even numbers: 1, 2, 4, 8 and 1, 3, 9, 27 are just those numbers that Plato, in the Timaeus (??35 b, c) applies to the soul.
Several other quaternaries are considered as well (see pp. 64-6). The fourth, for example, "is that of the simple bodies, fire, air, water and earth," originating out of the one, in this case fire, with air corresponding to 2, water to 3, and earth to 4. The sixth quaternary "is that of the created things, the seed being analogous to unity and to the point." The first seven quaternaries are designated as "material and perceptible." The eighth "contains faculties by which we are able to form judgment on the preceding, and which are its intellectual part, namely: thought, science, opinion and feeling." Thought is indicated as the unity in this set of four.
The ninth quaternary is that which composes the living things, body and soul, the soul having three parts, the rational, the emotional and the willful; the fourth part is the body in which the soul resides.
The tenth quaternary is that of the seasons of the year, through the succession of which all things take birth, that is, spring, summer, autumn and winter.
This supplies a Pythagorean background for Plato's famous image of the decussated cross (X) in space — here using one key of interpretation to link it with the relationships of each of the pairs of solstices and equinoxes to each other and as pairs per se also (Plato, op. cit., ??36 b, c). "The eleventh [quaternary] is that of the ages: childhood, adolescence, maturity and old age." And Theon goes on to say that these quaternaries result in the "perfect world" — perfect because "everything is part of it, and it is itself a part of nothing else." But this last expression needs to be qualified, because a world and its inhabitants are part of a solar system which in turn is a component of a galaxy, thereby of a larger assemblage of galaxies, and so on. Perhaps Theon was alluding to the unique individual quality of each grouping. In any case, he claims that the system he delineated above was the reason why the Pythagoreans used the oath.
Rounding out the theme of Theon's work discussed here, we return to its foundation in the original Unity, as summarized by Robert Lawlor:
there is the implication that the universal outpouring of the monad into the tangibility of number is of a cyclic nature which ultimately results in a return of the all-containing synthesis of the original Unity, but through its transformation into the formal, embodied expression of this Unity in the form of the decad. — p. xii
That is to say, the One emanates the ten — or the universe as we perceive it — and is not separate from it but dwells in it as it dwells in the One. Or as Theon expressed it:
Unity is the principle of all things and the most dominant of all that is: all things emanate from it and it emanates from nothing. It is indivisible and it is everything in power. It is immutable and never departs from its own nature through multiplication ( 1 x 1 = 1 ). All that is intelligible and cannot be engendered exists in it: the nature of ideas, God himself, the soul, the beautiful and the good, and every intelligible essence, such as beauty itself, justice itself, equality itself, for we conceive of each of these things as being one and as existing in itself. — pp. xi-xii
An old Greek word for spirit is pneuma, which means breath, and it is very interesting that in many ancient languages a word for spirit or divinity was derived from a root-term for breath. This brings before our vision the image of a continually pulsating universe, rhythmically expanding and contracting. The Hindu philosophers pictured the process as the Outbreathing and Inbreathing of a cosmic Intelligence or Consciousness vast beyond the range of human language to define, of quality inconceivable within the reach of the finite mind.
This universal motion carries us all onwards to a future beyond our imagination; the first breath of babies pulsates because of the universal rhythm; so does the earth in its seasonal changes and successions. The sun also expands and contracts, i.e. "breathes," according to the latest scientific discovery reports, and no doubt our home galaxy or Milky Way flows outward and returns inward to a center in so large a sweep of time that it may seem motionless to us. But the tiniest subparticle making up an atom of our physical body also must move to the beat of the cosmic drummer, as do all of us together.
(From Sunrise magazine, January, 1980. Copyright © 1980 by Theosophical University Press)