Mathematics and
the Spiritual Dimension
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Archimedes and Pythagoras
A common belief among ancient cultures was that the
laws of numbers have not only a practical meaning, but also a mystical or religious one.
This belief was prevalent amongst the Pythagoreans. Prior to 500 B.C. E., Pythagoras, the
great Greek pioneer in the teaching of mathematics, formed an exclusive club of young men
to whom he imparted his superior mathematical knowledge. Each member was required to take
an oath never to reveal this knowledge to an outsider. Pythagoras acquired many faithful
disciples to whom he preached about the immortality of the soul and insisted on a life of
renunciation. At the heart of the Pythagorean world view was a unity of religious
principles and mathematical propositions.
In the third century B.C. E. another great Greek mathematician,
Archimedes, contributed considerably to the field of mathematics. A quote attributed to
Archimedes reads, "There are things which seem incredible to most men who have not
studied mathematics." Yet according to Plutarch, Archimedes considered
"mechanical work and every art concerned with the necessities of life an ignoble and
inferior form of labor, and therefore exerted his best efforts only in seeking knowledge
of those things in which the good and the beautiful were not mixed with the
necessary." As did Plato, Archimedes scorned practical mathematics, although he
became very expert at it.
The Abacus: A mechanical counting device
The Greeks, however, encountered a major problem. The Greek
alphabet, which had proved so useful in so many ways, proved to be a great hindrance in
the art of calculating. Although Greek astronomers and astrologers used a sexagesimal
place notation and a zero, the advantages of this usage were not fully appreciated and did
not spread beyond their calculations. The Egyptians had no difficulty in representing
large numbers, but the absence of any place value for their symbols so complicated their
system that, for example, 23 symbols were needed to represent the number 986. Even the
Romans, who succeeded the Greeks as masters of the Mediterranean world, and who are known
as a nation of conquerors, could not conquer the art of calculating. This was a chore left
to an abacus worked by a slave. No real progress in the art of calculating nor in science
was made until help came from the East.
Shulba Sutra
In the valley of the Indus River of India, the
world's oldest civilization had developed its own system of mathematics. The Vedic Shulba
Sutras (fifth to eighth century B.C.E.), meaning "codes of the rope," show that
the earliest geometrical and mathematical investigations among the Indians arose from
certain requirements of their religious rituals. When the poetic vision of the Vedic seers
was externalized in symbols, rituals requiring altars and precise measurement became
manifest, providing a means to the attainment of the unmanifest world of consciousness.
"Shulba Sutras" is the name given to those portions or supplements of the
Kalpasutras, which deal with the measurement and construction of the different altars or
arenas for religious rites. The word shulba refers to the ropes used to make these
measurements.
Math cannot take the mystery out of life without doing away with
life itself, for it is life's mystery, its unpredictabilitythe fact that it is dynamic,
not staticthat makes it alive and worth living.
Although Vedic mathematicians are known primarily for their
computational genius in arithmetic and algebra, the basis and inspiration for the whole of
Indian mathematics is geometry. Evidence of geometrical drawing instruments from as early
as 2500 B.C.E. has been found in the Indus Valley. [1] The beginnings of algebra can be
traced to the constructional geometry of the Vedic priests, which are preserved in the
Shulba Sutras. Exact measurements, orientations, and different geometrical shapes for the
altars and arenas used for the religious functions (yajnas), which occupy an important
part of the Vedic religious culture, are described in the Shulba Sutras. Many of these
calculations employ the geometrical formula known as the Pythagorean theorem.
This theorem (c. 540 B.C.E.), equating
the square of the hypotenuse of a right angle triangle with the sum of the squares of the
other two sides, was utilized in the earliest Shulba Sutra (the Baudhayana) prior to the
eighth century B.C.E. Thus, widespread use of this famous mathematical theorem in India
several centuries before its being popularized by Pythagoras has been documented. The
exact wording of the theorem as presented in the Sulba Sutras is: "The diagonal chord
of the rectangle makes both the squares that the horizontal and vertical sides make
separately." [2] The proof of this fundamentally important theorem is well known from
Euclid's time until the present for its excessively tedious and cumbersome nature; yet the
Vedas present five different extremely simple proofs for this theorem. One historian,
Needham, has stated, "Future research on the history of science and technology in
Asia will in fact reveal that the achievements of these peoples contribute far more in all
preRenaissance periods to the development of world science than has yet been
realized." [3]
The Shulba Sutras have preserved only that part of Vedic
mathematics which was used for constructing the altars and for computing the calendar to
regulate the performance of religious rituals. After the Shulba Sutra period, the main
developments in Vedic mathematics arose from needs in the field of astronomy. The
Jyotisha, science of the luminaries, utilizes all branches of mathematics.
The need to determine the right time for their religious rituals
gave the first impetus for astronomical observations. With this desire in mind, the
priests would spend night after night watching the advance of the moon through the circle
of the nakshatras (lunar mansions), and day after day the alternate progress of the sun
towards the north and the south. However, the priests were interested in mathematical
rules only as far as they were of practical use. These truths were therefore expressed in
the simplest and most practical manner. Elaborate proofs were not presented, nor were they
desired.
Evolution of Arabic (Roman) Numerals from India:
A close investigation of the Vedic system
of mathematics shows that it was much more advanced than the mathematical systems of the
civilizations of the Nile or the Euphrates. The Vedic mathematicians had developed the
decimal system of tens, hundreds, thousands, etc. where the remainder from one column of
numbers is carried over to the next. The advantage of this system of nine number signs and
a zero is that it allows for calculations to be easily made. Further, it has been said
that the introduction of zero, or sunya as the Indians called it, in an operational sense
as a definite part of a number system, marks one of the most important developments in the
entire history of mathematics. The earliest preserved examples of the number system which
is still in use today are found on several stone columns erected in India by King
Ashoka in about 250 B.C.E. [4 ] Similar inscriptions are found in caves near Poona
(100 B.C.E.) and Nasik (200 C.E.). [5] These earliest Indian numerals appear in a script
called brahmi.
After 700 C.E. another notation, called by the name "Indian
numerals," which is said to have evolved from the brahmi numerals, assumed common
usage, spreading to Arabia and from there around the world. When Arabic numerals (the name
they had then become known by) came into common use throughout the Arabian empire, which
extended from India to Spain, Europeans called them "Arabic notations," because
they received them from the Arabians. However, the Arabians themselves called them
"Indian figures" (AlArqanAlHindu) and mathematics itself was called "the
Indian art" (hindisat)
Evolution of "Arabic numerals" from Brahmi (250
B.C.E.)
to the 16th century.
Mastery of this new mathematics allowed the Muslim mathematicians
of Baghdad to fully utilize the geometrical treatises of Euclid and Archimedes.
Trigonometry flourished there along with astronomy and geography. Later in history, Carl
Friedrich Gauss, the "prince of mathematics," was said to have lamented
that Archimedes in the third century B.C.E. had failed to foresee the Indian system
of numeration; how much more advanced science would have been.
Prior to these revolutionary discoveries, other world civilizationsthe Egyptians, the
Babylonians, the Romans, and the Chineseall used independent symbols for each row of
counting beads on the abacus, each requiring its own set of multiplication or addition
tables. So cumbersome were these systems that mathematics was virtually at a standstill.
The new number system from the Indus Valley led a revolution in mathematics by setting it
free. By 500 C.E. mathematicians of India had solved problems that baffled the world's
greatest scholars of all time. Aryabhatta, an astronomer mathematician who
flourished at the beginning of the 6th century, introduced sines and versed
sinesa great
improvement over the clumsy halfcords of Ptolemy.
A. L. Basham, foremost authority on ancient India, writes in The
Wonder That Was India,
Medieval Indian mathematicians, such as Brahmagupta
(seventh century), Mahavira (ninth century), and Bhaskara (twelfth century), made several
discoveries which in Europe were not known until the Renaissance or later. They understood
the import of positive and negative quantities, evolved sound systems of extracting square
and cube roots, and could solve quadratic and certain types of indeterminate
equations." Mahavira's most noteworthy contribution is his treatment of fractions for
the first time and his rule for dividing one fraction by another, which did not appear in
Europe until the 16th century.
Equations and Symbols:
B.B. Dutta writes: "The use of symbolsletters of the
alphabet to denote unknowns, and equations are the foundations of the science of algebra.
The Hindus were the first to make systematic use of the letters of the alphabet to denote
unknowns. They were also the first to classify and make a detailed study of equations.
Thus they may be said to have given birth to the modern science of algebra." [7] The
great Indian mathematician Bhaskaracharya (1150 C.E.) produced extensive treatises
on both plane and spherical trigonometry and algebra, and his works contain remarkable
solutions of problems which were not discovered in Europe until the seventeenth and
eighteenth centuries. He preceded Newton by over 500 years in the discovery of the
principles of differential calculus. A.L. Basham writes further, "The
mathematical implications of zero (sunya) and infinity, never more than vaguely realized
by classical authorities, were fully understood in medieval India. Earlier mathematicians
had taught that X/0 = X, but Bhaskara proved the contrary. He also established
mathematically what had been recognized in Indian theology at least a millennium earlier:
that infinity, however divided, remains infinite, represented by the equation oo /X =
oo." In the 14th century, Madhava, isolated in South India, developed a power
series for the arc tangent function, apparently without the use of calculus, allowing the
calculation of pi to any number of decimal places (since arctan 1 = pi/4). Whether he
accomplished this by inventing a system as good as calculus or without the aid of
calculus; either way it is astonishing.
Spiritually advanced cultures were not ignorant of the principles
of mathematics, but they saw no necessity to explore those principles beyond that which
was helpful in the advancement of God realization.
By the fifteenth century C.E. use of the new mathematical
concepts from India had spread all over Europe to Britain, France, Germany, and Italy,
among others. A.L. Basham states also that
The debt of the Western world to India in this respect [the field
of mathematics] cannot be overestimated. Most of the great discoveries and inventions of
which Europe is so proud would have been impossible without a developed system of
mathematics, and this in turn would have been impossible if Europe had been shackled by
the unwieldy system of Roman numerals. The unknown man who devised the new system was,
from the world's point of view, after the Buddha, the most important son of India. His
achievement, though easily taken for granted, was the work of an analytical mind of the
first order, and he deserves much more honor than he has so far received.
Unfortunately, Eurocentrism has effectively concealed
from the common man the fact that we owe much in the way of mathematics to ancient India.
Reflection on this may cause modern man to consider more seriously the spiritual
preoccupation of ancient India. The rishis (seers) were not men lacking in
practical knowledge of the world, dwelling only in the realm of imagination. They were
well developed in secular knowledge, yet only insofar as they felt it was necessary within
a world view in which consciousness was held as primary.
In ancient India, mathematics served as a bridge between
understanding material reality and the spiritual conception. Vedic mathematics differs
profoundly from Greek mathematics in that knowledge for its own sake (for its aesthetic
satisfaction) did not appeal to the Indian mind. The mathematics of the Vedas lacks the
cold, clear, geometric precision of the West; rather, it is cloaked in the poetic language
which so distinguishes the East. Vedic mathematicians strongly felt that every discipline
must have a purpose, and believed that the ultimate goal of life was to achieve
selfrealization and love of God and thereby be released from the cycle of birth and
death. Those practices which furthered this end either directly or indirectly were
practiced most rigorously. Outside of the religioastronomical sphere, only the problems
of day to day life (such as purchasing and bartering) interested the Indian
mathematicians.
Poetry in Math:
One of the foremost exponents of Vedic math, the late Bharati Krishna Tirtha Maharaja,
author of Vedic Mathematics, has offered a glimpse into the sophistication of Vedic
math. Drawing from the Atharvaveda, Tirtha Maharaja points to many sutras
(codes) or aphorisms which appear to apply to every branch of mathematics: arithmetic,
algebra, geometry (plane and solid), trigonometry (plane and spherical), conics
(geometrical and analytical), astronomy, calculus (differential and integral), etc.
Utilizing the techniques derived from these sutras, calculations
can be done with incredible ease and simplicity in one's head in a fraction of the time
required by modern means. Calculations normally requiring as many as a hundred steps can
be done by the Vedic method in one single simple step. For instance the conversion of the
fraction 1/29 to its equivalent recurring decimal notation normally involves 28 steps.
Utilizing the Vedic method it can be calculated in one simple step. (see the next section
for examples of how to utilize Vedic sutras)
In order to illustrate how secular and spiritual life
were intertwined in Vedic India, Tirtha Maharaja has demonstrated that mathematical
formulas and laws were often taught within the context of spiritual expression (mantra).
Thus while learning spiritual lessons, one could also learn mathematical rules.
Tirtha Maharaja has pointed out that Vedic mathematicians prefer
to use the devanagari letters of Sanskrit to represent the various numbers in their
numerical notations rather than the numbers themselves, especially where large numbers are
concerned. This made it much easier for the students of this math in their recording of
the arguments and the appropriate conclusions.
Tirtha Maharaja states, "In order to help the pupil to
memorize the material studied and assimilated, they made it a general rule of practice to
write even the most technical and abstruse textbooks in sutras or in verse (which is so
much easiereven for the childrento memorize). And this is why we find not only
theological, philosophical, medical, astronomical, and other such treatises, but even huge
dictionaries in Sanskrit verse! So from this standpoint, they used verse, sutras and codes
for lightening the burden and facilitating the work (by versifying scientific and even
mathematical material in a readily assimilable form)!" [8] The code used is as
follows:
The Sanskrit consonants
ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.
Vowels make no difference and it is left to the author to select
a particular consonant or vowel at each step. This great latitude allows one to bring
about additional meanings of his own choice. For example kapa, tapa, papa, and yapa all
mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn
with double or triple meanings. Here is an actual sutra of spiritual content, as well as
secular mathematical significance
gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara
While this verse is a type of petition to Krishna,
when learning it one can also learn the value of pi/10 (i.e. the ratio of the
circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a
selfcontained masterkey for extending the evaluation to any number of decimal places.
The translation is as follows:
O Lord anointed with the yogurt of the milkmaids' worship
(Krishna), O savior of the fallen, O master of Shiva, please protect me.
At the same time, by application of the consonant code given
above, this verse directly yields the decimal equivalent of pi divided by 10: pi/10 =
0.31415926535897932384626433832792. Thus, while offering mantric praise to Godhead in
devotion, by this method one can also add to memory significant secular truths.
This is the real gist of the Vedic world view regarding the
culture of knowledge: while culturing transcendental knowledge, one can also come to
understand the intricacies of the phenomenal world. By the process of knowing the absolute
truth, all relative truths also become known. In modern society today it is often
contended that never the twain shall meet: science and religion are at odds. This
erroneous conclusion is based on little understanding of either discipline. Science is the
smaller circle within the larger circle of religion.
We should never lose sight of our spiritual goals. We should
never succumb to the shortsightedness of attempting to exploit the inherent power in the
principles of mathematics or any of the natural sciences for ungodly purposes. Our
reasoning faculty is but a gracious gift of Godhead intended for divine purposes, and not
those of our own design.
