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1,2,3,4: PYTHAGORAS AND THE COSMOLOGY OF NUMBER
At the heart of vedic mathematics lies a principle that underscores most, if
not all, of the ancient wisdom traditions, the conveying of knowledge through
cryptic, highly compressed expressions, open to multiple levels of interpretation.
A prime example of this is the teaching of the Greek mathematician and sage
Pythagoras. According to his ancient biographers:
"In the Pythagorean school, knowledge was transmitted symbolically, through
the use of cryptic statements and riddles, in which a small number of words
was pregnant with multiple levels of interpretation. Students were required
to find meaning in these enigmatic lessons, sometimes through questioning
and dialogue, sometimes by meditating upon their many possible meanings."
(1)
If this was true of Pythagorean teachings, it was even
more significant in more ancient schools of knowledge; it was, after all, at
these schools, in Egypt, Babylon, and elsewhere, that Pythagoras gained his
knowledge. In the case of the Indian tradition, both in Vedic times and later
in the Hindu and Buddhist periods, the term most commonly encountered for this
kind of cryptic literature was the sutra or collection of sutras.
While this is often translated as "aphorism," or "formula," the word comes from
the Sanskrit root for "thread," a usage that persists in the modern word "suture."
As doctors use sutures to sow us up after surgery, the ancient sutras
tie together our knowledge and integrate our awareness. There is no better example
than the teachings contained in Patanjali's Yoga Sutras whose terse expressions
contain instructions for the development of higher states of consciousness.
Similarly, all the principles of vedic mathematics are encapsulated in sixteen
sutras, which, along with thirteen sub-sutras, provide the basis
for all the operations described in "The Cosmic Computer" (2).
If vedic mathematics can be counted as part of vedic literature,
its ultimate source is the Rg Veda. This is certainly not concise, consisting
of over 10,000 verses, but, as His Holiness Maharishi Mahesh Yogi has explained,
it has a unique structure in which the essence of the whole text is essentially
contained in one highly compressed expression--its first word. "It is the purpose
of all ciphers to invest a few signs with much meaning," Carlo Suarès tells
us. "In the severity of its beginning, in its first chapter, in its first sequence
of letter numbers, is the seed, and in the seed is the whole." (3)
Suarès is referring to the beginning of Genesis, in
which the process of creation is described, using the symbolism of gematria,
in which each letter is given a numerical value. (4) According to Maharishi,
the Rg Veda also sets forth a cosmogony in its first word-- Agni, but using
a purely linguistic symbolism based on the physiology of speech. The first letter,
or sound, AAAAAA…, pronounced with the mouth and throat fully open, and thus
with a fully open sound, represents the fullness of the unmanifest, unbounded
Brahman. But the letter G, a full glottal stop, introduces the first boundary
on the full openness of the sound AAAAAA…. As the wave value of a sub-atomic
particle collapses onto a point value when observed, so the unity, or samhita,
value of Brahman collapses onto a point and becomes the triadic value of rishi,
devata and chhandas, observer, process of observing, and object
of observation. From here the process of manifestation begins. As the full stream
of manifestation emerges, it leads on to the fullness of creation, and this
is represented by the syllable NI, the same name given to the leading tone in
Indian music (Sa, Re, Ga, Me Pa, Dha, Ni.....). The details of the process,
and the content of manifestation and evolution, are unfolded through the rest
of the verses of Rg Veda and commented upon by the rest of Vedic literature,
including vedic mathematics.
Unity, duality, diversity, wholeness. These are the
mechanics of creation described in different symbolic formulations in different
knowledge traditions. To find it in purely mathematical or numerical form we
return to the Pythagorean tradition, and its most concise expression comes from
his successor Plato. Considered the most Pythagorean of Platonic dialogues,
the Timaeus begins with a question by Socrates: "One, two, three ? but
where, my dear Timaeus, is the fourth of my guests of yesterday who were to
entertain me today?" (5) Commentators usually ignore this statement, but, as
we have seen, in ancient literature every expression is "pregnant with multiple
levels of meaning." This is particularly true when dealing with numbers.
"He [Pythagoras] held that the ultimate substances of all things, material
and immaterial, were numbers, which had two distinct and complimentary aspects.
On the one hand, they had a spatial and dynamic existence, and, on the other,
they were fundamental formulating principles which were purely abstract. Thus,
for example, the monad was understood by the Pythagoreans both as the number
one, which had physical properties that could be manipulated in nature, and
as an idea, which embodied the original unity at the source of all creation."
(6)
The fundamental formulating principles in the universe
are those values of unity, duality, diversity and wholeness we have already
encountered. In Pythagorean thought these principles are clearly expressed in
the first four numbers. Furthermore, this symbolism can be interpreted in terms
of the Quadrivium, the four Pythagorean mathematical disciplines: arithmetic,
music, geometry and astronomy.
Arithmetic was seen as the study of the abstract essence
of things. Thus each number had a cosmological, as well as mathematical, significance.
The monad, manifest as the number one, denotes the primordial unity at the basis
of creation. The transition from one to two, from the monad to the dyad, represents
the first step in the process of creation--unity polarizing within itself becomes
duality. Three, the triad, is the first true number. One contains the seed,
and two introduces potential. Three brings number into being, causing the potential
contained within the monad to manifest into its true expression, the world of
plurality and multitude.
If one and two initiate creation, three and four complete
the process. Therefore, the tetrad, four, represents completion. Everything
in the universe, both natural and numerical, is completed in the progression
from one to four as 1 + 2 + 3 + 4 = 10, which brings us to the decad, also known
to the Pythagoreans as the tetraktys, and representing their most sacred
symbol. The same sequence, from unity to multiplicity via duality and trinity,
is expressed even more graphically in the simplest and most basic musical relationships,
those expressed through the numbers 1,2,3,4. The simplest and most fundamental
musical relationship is the octave, discovered by Pythagoras to be the 1:2 relationship,
and by Joseph Saveur (1653-1716) many centuries later, to be the first relationship
in the harmonic overtone series. The experience of the octave is of two notes
that are the same and yet different, and these values, sameness and difference
are the fundamental substances used by the Demiurge to create the World-Soul
in the Timaeus. Further, the octave provides the boundary conditions
within which the musical universes contained within scales are formed, the values
of Do in Do, Re, Mi, Fa, So, La, Ti, Do. Of these intervals, the central ones
are those found to be next in the overtone series, 2:3, known as the fifth and
3:4 known as the fourth. These values are found in the first four harmonics
of the overtone series, first 1:2 (octave), then 2:3 (fifth) then 3:4 (fourth)
recapitulating the octave at the next power of two. In four simple sounds the
whole process of unity, duality, multiplicity and wholeness is presented to
the awareness.
In subsequent centuries, the science of geometry was developed
into a sacred form in which the same process is represented by the circle (unity),
contrasted with the square (diversity), and reconciled in the squaring of the
circle, in alchemical practice, and the development of the mandala in Eastern
art and architecture. "The object of sacred geometry being to depict that fusion
of opposites, the squared circle is therefore its first symbol. Temples and
cosmological cities throughout antiquity were founded on its proportions." (7)
For Pythagoras, the symbolism of wholeness (kosmos) and order (harmonia)
extended beyond mathematical to astrological phenomena. A theoretical planet
called the counter-earth was posited to bring the number of heavenly bodies
in the Pythagorean firmament to ten, the perfect number, the number of the tetraktys.
And over time, an association between planets and musical notes was developed
and elaborated into the famous "music of the spheres," a beautiful image of
the kosmos as a divine harmony.
Having seen its range of implications, it could almost be
stated that the sequence 1,2,3,4 sums up, in a compressed symbolism, the whole
range of Pythagoreanism. But if we delve deeper into Platonic thought, a further
dimension is revealed. In one of his most potent allegories, known as the "Divided
Line," Plato sets out his theories of ontology and epistemology, and again
it is done in terms of the number four. In this analogy, Plato makes a distinction
between the outer realm of the world, illuminated by the sun and the inner realm
of the mind, illuminated by the Good. The Divided Line passage divides each
of these realms into two further sections. Plato also deals with the state of
mind in which the resultant four realms are apprehended, resulting in the following
scheme:
Level Object Faculty
Type of Knowledge
IV
Forms
dialectic transcendental
cognition } internal
III mathematics
thinking, scientific
understanding } world
Etc.
reasoning
II
physical sense
common-sense
belief } external
objects perception
}
world
I shadows
illusory illusion
(8)
perception
It can be seen from this scheme that within the subjective
realm of the mind, Plato posits a level of knowledge higher than that which
deals with mathematical objects through the processes of thinking and reasoning.
This is the level of the forms and it is reached, Plato tells us, through the
use of the "second phase" of the dialectic, a technique that, according to Jonathan
Shear is similar to the practice of jñana yoga. (9) This again reflects
the Pythagorean approach to mathematics, one that must, on some level at least,
apply to vedic maths also:
"For Pythagoras, mathematics served as a bridge between the visible and invisible
worlds. He pursued the discipline of mathematics not only as a way of understanding
and manipulating nature, but also as a means of turning the mind away from
the physical world, which he held to be transitory and unreal, and leading
it to the contemplation of eternal and truly existing things that never vary.
He taught his students that by focusing on the elements of mathematics, they
could calm and purify the mind, and ultimately, through disciplined effort,
experience true happiness." (10)
Notes:
(1) John Strohmeier and Peter Westbrook. Divine Harmony: The Life and Teachings
of Pythagoras. (Berkeley, CA: Berkeley Hills Books, 1999), p. 54.
(2) Williams and Gaskell, The Cosmic Computer (Inspiration Books, 1997.
(3) Carlo Suarès The Cipher of Genesis (York Beach, Maine: Samuel Weiser,
Inc., 1992), p. 72.
(4) For more information on gematria see John Michell, The New View
Over Atlantis (London: Thames & Hudson Ltd., 1983) and Gordon Strachan,
Jesus the Master Builder: Druid Mysteries and the Dawn of Christianity.
(Edinburgh: Floris Books, 1998).
(5) Plato, Timaeus, 17a.
(6) Strohmeier & Westbrook (1999), p. 66.
(7) John Michell, The Dimensions of Paradise: The Proportions and Symbolic
Numbers of Ancient Cosmology (San Francisco: Harper & Row, 1988), pp. 66-67.
(8) Jonathan Shear, The Inner Dimension: Philosophy and the Experience of Consciousness
(New York: Peter Lang, 1990), p. 12, n2.
(9) It is interesting note that the Greek word harmonia has a similar
etymology to the Sanskrit yoga, viz. a joining together of opposite values.
(10) Strohmeier & Westbrook (1999), p. 66
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