PARALLEL DEVELOPMENTS IN PHILOSOPHY AND MATHEMATICS IN INDIA

In South Asia, sophisticated mathematics grew from the attempt to predict
celestial phenomena. It was the priests who had the time and the intellectual
training that enabled them to pursue this study. And the priests had an incentive:
To understand the workings of the heavens is to come closer to understanding the
nature of transcendent reality.
The mathematics in this project will focus on the development of trigonometry.
Trigonometry arose from and for over ¯fteen hundred years was used exclusively
for the study of astronomy and astrology. Basic concepts of trigonometry were
developed in classical Greece. These were imported to India where they °ourished,
creating the modern subject we know today. Hindu and Jain priests went so far as
to discover and use the basic calculus of trigonometric functions. Always, the purpose
was to further investigations of astronomical phenomena. Again, this is not
peculiar to South Asia. In Western Europe throughout the Middle Ages and Renaissance,
the terms Mathematician, Astronomer, and Astrologer were considered
synonomous.


 Trigonometry ¯rst appeared in the eastern Mediterranean. Hipparchus of Nic½a
in what is now Turkey (c. 161{126 bce) is considered the originator of trigonometry.
He developed it to determine and predict positions of planets. The ¯rst problem
to which he applied this new tool was the analysis of a very disturbing discovery:
The universe is lop-sided.
To explain his work, we need to begin with the Greek understanding of the
universe. This starts with the assumption that the earth is stationary. While
this was debated in early Greek science|does the earth go around the sun or
the sun around the earth?|the simple fact that we perceive no sense of motion
is a powerful indication that the earth does not move. In fact, when in the early
seventeenth century it became clear that the earth revolves about the sun, it created
a tremendous problem for scientists: How to explain how this was possible? How
could it be that we were spinning at thousands of miles per hour and hurtling
through space at even greater speeds without experiencing any of this? Surely
if the earth did move, we would have been °ung o® long ago. Newton's great
accomplishment in his Mathematical Principles of Natural Philosophy was to solve
this problem.
The astronomy of Hipparchus begins with a ¯xed and immovable earth. Above it
is the great dome of the night sky, rotating once in every 24 hours. In far antiquity
it was realized that the stars do not actually disappear during the day. They are
present, but impossible to see against the glare of the sun. The position of the sun
in this dome is not ¯xed. During the year, it travels in its own circle, called the
ecliptic, through the constellations. One can tell the season by locating the position
of the sun in its annual journey around this great circle. This is what the zodiac
does. The sign of the zodiac describes the location of the sun by pinpointing the
constellation in which it is located. 

Most stars are ¯xed in the rotating dome of the sky, but a few, called the
wanderers or, in Greek, the planetes (hence our word planets), also move across the
dome following this same ecliptic circle. If the position of the sun is so important in
determining seasons of heat and cold, rain and drought, it appears self-evident thatthe positions of the wanderers should have important|if more subtle|in°uences
on our lives. Astronomy/astrology was born.
The philosophers of ancient Greece had developed an orderly world-view that put
the earth in the center of the universe with the moon, sun, and planets embedded
in concentric, ethereal spheres that rotated with perfect regularity around it. This
model became the basis for a comprehensive understanding of the universe that
was tight and consistent and would last for almost two millennia. Hipparchus was
one of the ¯rst to ¯nd a serious °aw in this system.
The four cardinal points of the great circle traveled by the sun mark the boundaries
of the seasons: winter solstice, spring equinox, summer solstice, and autumnal
equinox. If the sun travels the ecliptic at constant speed, then this model would
imply that the four seasons should be of equal length. They are not (see Figure 1).
Winter solstice to spring equinox is a short 89 days. Spring is almost 90 days.
Summer, the longest season, is over 93 1=2 days. And fall comes close to 93 days.
If, in fact, the sun moves at a constant speed, this can only mean that the earth is
o®-center. Hipparchus tackled the problem of calculating the position of the earth.
Question 1. Given the lengths of the seasons, how far o® center is the earth?
Hint 1a The ¯rst thing that Hipparchus did was to determine the arclength traveled
by the sun during each season. Hipparchus measured this arclength in degrees.
Today we use degrees to measure angles, but in Greece and in India and even in
western Europe until the eighteenth century, degrees were used to measure distance
along the circumference of a circle. The total circumference is 360±. Winter lasts
89 days which is 89/365.25 year. How far, in degrees, does the sun travel during
winter? How far does the sun travel in each of the other seasons?
Hint 1b If the circumference of this circle is 360, what is its radius? This will be
the value of R.
Hint 1c We will need to ¯nd the length of the chord that connects the winter and
summer solstice and the length of the chord that connects the spring and autumnal
equinox. Before we do that, consider how we will use this information. Show that
if a chord has length L, measured in degrees, and the radius of the circle is R, then

Hint 1d The crux of the problem is to ¯nd the lengths of the chords. Given
arclength ® on a circle of radius R, we denote the length of the chord that connects
the endpoints of the arc by crd ®. Using Figure 2, show that crd ® = 2Rsin(®=2)
where we measure the argument of the sine in degrees. Set R = 360=2¼ and use
your calculator (remember to put it in degree mode) to ¯nd the lengths of the
two chords. Now ¯nd the distance from the earth to the center of the universe
(measured in degrees).

3.1. Philosophy in the Vedas. Throughout the middle of the second millennium
bce, a nomadic people migrated into northern India from neighboring Iran. Initially
settling along the upper reaches of the Indus Valley, they soon came to dominate
much of northern India. When these nomads entered India it was populated by
ethnically and linguistically diverse groups of people, but the Indus Valley itself
was home to a sophisticated (but probably already declining) civilization that has
come to be known as the Indus Valley Civilization.
The entering nomads, known as ¹Aryans (derived from ¹arya which means noble),
left behind a vast body of literature composed between the end of the second
millennium bce and the middle of the ¯rst millennium bce. Known as the Vedas,
these texts are still recognized by contemporary Hindus as having the highest scriptural
authority. The language of the Vedas, a close precursor of classical Sanskrit,
belongs to the same linguistic family as Greek, Latin and many modern European
languages. The ¹Aryans who ¯ltered into India must therefore have originated from
the same population that had migrated westwards into Europe. Once in India,
the ¹Aryans were in°uenced by the indigenous cultures that they encountered. One
of the enduring and most elusive questions about Vedic civilization is to ascertain
the extent of this in°uence and to separate it from ideas brought by the ¹Aryans
themselves into India.

Vedic practice and thought centered on the performance of a complex set of
ritual sacri¯ces, performed on behalf of wealthy and powerful patrons by hereditary
groups of ritual specialists, the br¹ahmanas (henceforth written as \brahmin").
Vedic sacri¯ces were performed on a daily basis, at crucial times during the year such
as the juncture between seasons, and on exceptional occasions such as the assertion
of sovereignty by a conquering king. Although the gods were believed to partake of
these ritual o®erings, the e±cacy of the ritual depended on its correct performance,
not on the capricious will of the gods. The underlying rationale of these sacri¯cial
rituals suggests a belief in hidden correspondences between ritual and cosmic realities.
As bearers of this ritual knowledge, the brahmins held a powerful position
in society alongside the ruling elites, or ks. atriyas, on whose behalf they performed
the sacri¯ces. The brahmins preserved this knowledge from one generation to the
next through the establishment of priestly lineages responsible for memorizing and
transmitting some portion of this sacri¯cial knowledge. Consisting of hymns to the
Vedic gods and sacri¯cial formul½, the earliest sections of the four books of the
Vedas (the four books are the R.
g, S¹ama, Yajur and Atharva Veda) began as orally
transmitted teachings and did not become written manuscripts until the beginning
of the Common Era.
The belief in a knowable, orderly cosmos amenable to human manipulation was
an underlying theme of Vedic thought. It remains an important theme for subsequent
Indian philosophical thought. Already in the Vedas this idea of order had
been extended to human conduct, and moral virtue was understood as acting in
accordance with the cosmic laws. Society itself was ordered into four hereditary
groups known as varn. as, each one with its appropriate and specialized sphere of
action. One Vedic hymn describes creation in terms of the sacri¯ce of a great primal
being. From the sacri¯ce of his thousand headed body came the whole of the created
order, including the four varn. as: the brahmins (priests) came from his mouth,
the ks. atriyas (warriors) from his arms, the vai¶syas (artisans) from his thighs, and
the ¶s¹udras (servants) from his feet. What this hymn tells us is that these social
divisions were viewed as much a part of the natural order of things as the division
between the sun and the moon, or the sheep and the goat.
The Vedic sacri¯cial rituals were the rituals of a nomadic people and involved the
construction of often elaborate but temporary altars. Like the rituals of the ancient
Greeks, the construction of these altars necessitated precise changes to be made in
the shape of the bricks used in their construction. A knowledge of geometry and
astronomy was required to ensure the correct construction of these altars, in the
correct location, at the correct time. This knowledge was recorded in appendices
to the Vedas where we can ¯nd, for example, a statement of what has come to be
known as the theorem of Pythagoras. Scholars continue to debate the origins of
this knowledge. Some speculate that knowledge of the Pythagorean theorem spread
into Greece and India from common Indo-European origins.
3.2. Philosophy in the Upanis.ads. In 800 bce India there was no Buddhism, no
Jainism, and little of what we would now characterize as being typically Hindu.
Many of the ideas associated with the Indian religious and philosophical traditions
were formed during this period (c.700{200 bce), a time of both great social change
and intellectual speculation. Almost a thousand years after the demise of the cities
of the Indus Valley Civilization, new urban centers appeared further eastward in the
Ganges valley. In this part of India, small tribal societies were being consolidated

into larger kingdoms and new cities appeared that were to become the administrative
and military centers for these larger political entities. This more cosmopolitan
environment facilitated the exchange of intellectual ideas as well as material goods,
and undoubtedly this social upheaval resulted in the growth of intellectual speculation
that marked this period. Such speculations are recorded in the Upanis.ads
(pronounced \Upanishads"), texts that constitute the ¯nal portion of the Vedas.
In the Upanis.ads, we see the formation of a worldview that became the framework
for almost all subsequent Indian philosophical and religious thought.
This worldview can be grasped by understanding three key concepts: sam. s¹ara,
karma and moks.a. Sam. s¹ara refers to the idea that each person lives not only once
but through a series of lives. It involves a cyclical view of existence inasmuch as
each death is followed by rebirth, either in this world or elsewhere. As belief in
sam. s¹ara became more clearly articulated, the Vedic belief in an orderly cosmos
persisted in the form of causal regularities that link one life to the next. Karma,
originally referring only to ritual actions, became a more generic term, referring to
all actions of moral consequence. An individual's actions, whether good or bad,
generate the energy that drives the process of rebirth, and the actions of a past life
have predictable causal repercussions in a subsequent life. Simply understood, good
actions generate a good rebirth, and bad actions a bad. \Good" and \bad" must
be viewed as relative terms, however, because the idea of any kind of continued
worldly existence came to be seen as problematic. Although the robust worldliness
of the earlier Vedic hymns was present in the Upanis.ads, temporal existence was
increasingly seen as unsatisfactory and inevitably involved su®ering. The search
for a way to transcend or be liberated (moks.a) from an existence determined by
the workings of karma becomes one of the central questions of the Upanis.ads.
One of the most prominent of teachers depicted in the Br.had¹aran. yaka Upanis.ad,
Y¹aj~navalkya, under repeated questioning at a great public debate, reduced the number
of gods from \three and three hundred, and three and three thousand" (3,306)
gods to one and a half, and ¯nally to one god. Apparently, Y¹aj~navalkya taught
that beneath the multiplicity of the Vedic pantheon is a simpler underlying transcendental
reality. After much speculation, this reality was identi¯ed as brahman.
Brahman had originally referred to the sacred power associated with the chants or
mantras used in the Vedic sacri¯ces. In the Upanis.ads, the concept of brahman
grew until it was ¯rmly established as a fundamental principle pervading and sustaining
all of the created worlds. The Vedas had presupposed a linkage between
cosmic and ritual reality. In the Upanis.ads, this logic was extended to a belief in a
correspondence between human and cosmic reality.
The Upanis.ads taught that brahman is present in each individual as the Self
(¹atman) and liberation from the world of transmigration is e®ected through an
experiential awareness of one's essential unity with the supra-mundane reality of
brahman. When brahman was considered only in ritualistic terms, then ritual
knowledge gave a su±cient knowledge of its reality. In the Upanis.ads, however,
there was a movement toward the authority of experience. Brahman became conceptualized
in such a way that it was to be known on the basis of experience.
There was a shift toward what we today would characterize as psychology.
The identi¯cation of brahman and ¹atman led to the investigation of human experience
in order to describe this brahman/¹atman in empirical terms. For many
Upanis.adic thinkers, the relationship of the created individual to brahman was arrived
at through an investigation of human states of consciousness waking, dreaming.


and deep sleep. Although impressed by the freedom and creative possibilities of
dreaming, and the lack of ego and su®ering of deep sleep, the Upanis.adic thinkers
eventually reached the conclusion that nothing in the domain of ordinary human
experience quali¯es as that state that is equivalent to brahman. They postulated
the existence of an entirely di®erent and transcendent state, a fourth (tur¹³ya) state
of consciousness. This mystical fourth state goes beyond anything a normal individual
would ordinarily experience. Its existence established the rationale for the
development of ascetic and yogic practices. In order to achieve this liberating state
of consciousness, the individual would need to engage in a set of practices developed
speci¯cally for that end. The elitism of the Vedic sacri¯cial tradition was replaced
by a new elite of skilled yogic practitioners.
Thus, although Upanis.adic thought was wedded to the logic of sacri¯cial thought,
there was a change in the character of Upanis.adic teachings. The shift towards a
new kind of experiential knowledge was paralleled by a change in recognized sources
of authority. Brahmins were still prominent in the Upanis.ads as teachers, but their
authority rested as much on their acquisition of this new knowledge as it did on
their °uency in sacri¯cial ritual. These texts captured a mood of existential urgency,
and we see women and non-brahmins engaged in this quest for moks.a alongside the
brahmins. The Upanis.ads re°ect a society in transition.
The life of the Buddha embodied all of these changes. He was born in Kapilavastu
situated at the northern edge of the Ganges plain that was by then the center of
North Indian civilization. He was born among the ¶S¹akya people and was probably
part of the ruling elite. He was not, however, a brahmin. Traditional accounts of his
life depict him as renouncing his comfortable existence as a young man in order to
seek liberation. As a renunciant he must have studied with teachers much like those
depicted in the Upanis.ads. He became well versed in the teachings and techniques
of these new teachers but ultimately challenged and rejected their solution. He
rejected the metaphysical idea of brahman and hence its human equivalent, ¹atman.
His rejection of the existence of ¹atman became a central tenet of Indian Buddhism.
In his rejection of ¹atman, the Buddha was rejecting the idea that human su®ering
could be transcended on the basis of grasping some one thing, some aspect of human
existence that wasn't subject to decay or su®ering. Rather, his teachings embrace
impermanence as a necessary quality of human existence. The Buddha diagnosed
desire as the cause of both human su®ering and rebirth. Buddhism is essentially a
set of views and practices that aim to eliminate desire, and Buddhist enlightenment
(nirv¹an. a) is simply viewed as the extinguishing of all such desires. The Buddha
rejected the belief in ¹atman precisely because understanding our impermanence was
seen as a crucial step in the elimination of desire. Much more than the Upanis.ads,
the teachings of the Buddha represented a real break from Vedic sacri¯cial thought.
3.3. The Development of Logic in India. According to Indian tradition, a fundamental
purpose of philosophy is to guide the individual toward moks.a, or release
from the su®erings of the world. Certainly this goal was one of the principle concerns
of the Upanis.ads. Subsequent philosophers in India continued to develop their
ideas under this rubric. But a preoccupation with moks.a did not exclude the development
of a more analytical philosophy. For many thinkers, holding correct views
about the world became a necessary component of moks.a, and the development of
such views required the critical skills of logical thinking.