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**Mathematics and the Spiritual Dimension
**

I remember the time my father pulled me aside and said, "Son,
you can explain everything with math." He was a rationalist, and for him
God existed only in the sentiments of the uneducated. At the time I believed
him, and I think his advice had a lot to do with my decision to pursue a degree
in physics. Somewhere along the way, however, in 1969, something happened
(something many people are still trying to figure out) which drew me away from
the spirit of that fatherly advice and subsequently my once promising career.

Unfortunately, I think I went too far to the other side. I threw
reason to the wind, so to speak, and unceremoniously became a self-ordained
"spiritual person." Science, the foundation of which is mathematics,
as I saw it, had nothing to offer. It was only years later, when the cloud of
my sentimentalism was dissipated by the sun of my soul's integrity, that I was
able to separate myself from yet another delusion-the first being the advice of
my father, and the second being the idea that I could wish myself into a more
profound understanding of the nature of reality.

Math cannot take the mystery out of life without doing away with
life itself, for it is life's mystery, its unpredictability-the fact that it is
dynamic, not static-that makes it alive and worth living. We may theoretically
explain away God, but in so doing we only choose to delude ourselves; I =
everything is just bad arithmetic.

However, before we can connect with our heart of hearts, our real
spiritual essence, we cannot cast reason aside. With the help of the
discriminating faculty we can know at least what transcendence is not.
Withdrawing our heart from that is a good beginning for a spiritual life.

Mathematics has only recently risen to attempt to usurp the throne
of Godhead. Ironically, it originally came into use in human society within the
context of spiritual pursuit. 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.
Intoxicated by the gross power inherent in mathematical principles, later
civilizations, succumbing to the all-inviting arms of illusion, employed these
principles and further explored them in an attempt to conquer nature. The folly
of this, as demonstrated in modern society today, points to the fact that
"wisdom" is more than the exercise of intelligence. Modern man's
worship of intelligence blinds him from the obvious: the superiority of love
over reason.

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 unpredictability-the fact that it is
dynamic, not static-that 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]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 pre-Renaissance 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" (Al-Arqan-Al-Hindu) 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 civilizations-the
Egyptians, the Babylonians, the Romans, and the Chinese-all 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 sines-a great
improvement over the clumsy half-cords 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."6 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 symbols-letters 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
self-realization 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 religio-astronomical
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 Atharva-veda, 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 easier-even for the children-to 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
self-contained master-key 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.

Vedic Mathematical Sutras

Consider the following three sutras:

1. "All from 9 and the last from 10," and its corollary:
"Whatever the extent of its deficiency, lessen it still further to that
very extent; and also set up the square (of that deficiency)."

2. "By one more than the previous one," and its
corollary: "Proportionately."

3. "Vertically and crosswise," and its corollary:
"The first by the first and the last by the last."

The first rather cryptic formula is best understood by way of a
simple example: let us multiply 6 by 8.

1. First, assign as the base for our calculations the power of 10
nearest to the numbers which are to be multiplied. For this example our base is
10.

2. Write the two numbers to be multiplied on a paper one above the
other, and to the right of each write the remainder when each number is
subtracted from the base 10. The remainders are then connected to the original
numbers with minus signs, signifying that they are less than the base 10.

6-4

8-2

3. The answer to the multiplication is given in two parts. The
first digit on the left is in multiples of 10 (i.e. the 4 of the answer 48).
Although the answer can be arrived at by four different ways, only one is presented
here. Subtract the sum of the two deficiencies (4 + 2 = 6) from the base (10)
and obtain 10 - 6 = 4 for the left digit (which in multiples of the base 10 is
40).

6-4

8-2

4

4. Now multiply the two remainder numbers 4 and 2 to obtain the
product 8. This is the right hand portion of the answer which when added to the
left hand portion 4 (multiples of 10) produces 48.

6-4

8-2

----

4/8

Another method employs cross subtraction. In the current example
the 2 is subtracted from 6 (or 4 from 8) to obtain the first digit of the
answer and the digits 2 and 4 are multiplied together to give the second digit
of the answer. This process has been noted by historians as responsible for the
general acceptance of the X mark as the sign of multiplication. The algebraical
explanation for the first process is

(x-a)(x-b)=x(x-a-b) + ab

where x is the base 10, a is the remainder 4 and b is the
remainder 2 so that

6 = (x-a) = (10-4)

8 = (x-b) = (10-2)

The equivalent process of multiplying 6 by 8 is then

x(x-a-b) + ab or

10(10-4-2) + 2x4 = 40 + 8 = 48

These simple examples can be extended without limitation. Consider
the following cases where 100 has been chosen as the base:

97 - 3 93 -
7 25 - 75

78 - 22 92 - 8 98 - 2

______ ______ ______

75/66 85/56 23/150 = 24/50

In the last example we carry the 100 of the 150 to the left and 23
(signifying 23 hundred) becomes 24 (hundred). Herein the sutra's words
"all from 9 and the last from 10" are shown. The rule is that all the
digits of the given original numbers are subtracted from 9, except for the last
(the righthand-most one) which should be deducted from 10.

Consider the case when the multiplicand and the multiplier are
just above a power of 10. In this case we must cross-add instead of cross
subtract. The algebraic formula for the process is: (x+a)(x+b) = x(x+a+b) + ab.
Further, if one number is above and the other below a power of 10, we have a
combination of subtraction and addition: viz:

108 + 8 and 13 + 3

97 - 3 8 -
2

_______ ______

105/-24 = 104/(100-24) =
104/76 11/-6 =
10/(10-6) = 10/4

The Sub-Sutra: "Proportionately" Provides for those
cases where we wish to use as our base multiples of the normal base of powers
of ten. That is, whenever neither the multiplicand nor the multiplier is
sufficiently near a convenient power of 10, which could serve as our base we
simply use a multiple of a power of ten as our working base, perform our
calculations with this working base and then multiply or divide the result
proportionately.

To multiply 48 by 32, for example, we use as our base 50 = 100/2,
so we have

Base 50 48 - 2

32 - 18

______

2/ 30/36 or
(30/2) / 36 = 15/36

Note that only the left decimals corresponding to the powers of
ten digits (here 100) are to be effected by the proportional division of 2.
These examples show how much easier it is to subtract a few numbers,
(especially for more complex calculations) rather than memorize long
mathematical tables and perform cumbersome calculations the long way.

Squaring Numbers

The algebraic equivalent of the sutra for squaring a number is:
(a+-b)2 = a2 +- 2ab + b2 . To square 103 we could write it as (100 + 3 )2 =
10,000 + 600 + 9 = 10,609. This calculation can easily be done mentally.
Similarly, to divide 38,982 by 73 we can write the numerator as 38x3 + 9x2 +8x
+ 2, where x is equal to 10, and the denominator is 7x + 3. It doesn't take
much to figure out that the numerator can also be written as 35x3 +36x2 + 37x +
12. Therefore,

38,982/73 = (35x3 + 36x2 +37x + 12)/(7x + 3) = 5x2 + 3x +4 = 534

This is just the algebraic equivalent of the actual method used.
The algebraic principle involved in the third sutra, "vertically and
crosswise," can be expressed, in one of it's applications, as the
multiplication of the two numbers represented by (ax + b) and (cx + d), with
the answer acx2 + x(ad + bc) + bd. Differential calculus also is utilized in
the Vedic sutras for breaking down a quadratic equation on sight into two
simple equations of the first degree. Many additional sutras are given which
provide simple mental one or two line methods for division, squaring of
numbers, determining square and cube roots, compound additions and
subtractions, integrations, differentiations, and integration by partial
fractions, factorisation of quadratic equations, solution of simultaneous
equations, and many more. For demonstrational purposes, we have only presented
simple examples.

Bibliography

1. E.J.H. Mackay, Further Excavations at Mohenjo-daro, 1938, p.
222.

2. Saraswati Amma, Geometry in Ancient and Medieval India, Motilal
Banarsidas, 1979, p. 18.

3. Dr. V. Raghavan, Presidential Address, Technical Sciences and
Fine Arts Section, XXIst AIOC, New Delhi, 1961.

4. Herbert Meschkowski, Ways of Thought of Great Mathematicians,
Holden-Day Inc., San Francisco, 1964.

5. Howard Eves, An Introduction to the History of Mathematics,
Rinehart and Company Inc., New York, 1953, p. 19.

6. A.L. Basham, The Wonder That Was India, Rupa & Co.,
Calcutta, 1967.

7. B.B. Dutta, History of Hindu Mathematics, Preface.

8. Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaja, Vedic
Mathematics, Motilal Banarsidass, Delhi, 1988.