Pakistan is a paradigm example of a failed state that has undergone an extremely dangerous form of radical Islamisation.
Science and Mathematics in India
In all early civilizations, the first expression of mathematical
understanding appears in the form of counting systems. Numbers in very early
societies were typically represented by groups of lines, though later different
numbers came to be assigned specific numeral names and symbols (as in India) or
were designated by alphabetic letters (such as in Rome). Although today, we take
our decimal system for granted, not all ancient civilizations based their
numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system
was in use.
The Decimal System in Harappa
In India a decimal system was
already in place during the Harappan period, as indicated by an analysis of
Harappan weights and measures. Weights corresponding to ratios of 0.05, 0.1,
0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been identified, as have
scales with decimal divisions. A particularly notable characteristic of Harappan
weights and measures is their remarkable accuracy. A bronze rod marked in units
of 0.367 inches points to the degree of precision demanded in those times. Such
scales were particularly important in ensuring proper implementation of town
planning rules that required roads of fixed widths to run at right angles to
each other, for drains to be constructed of precise measurements, and for homes
to be constructed according to specified guidelines. The existence of a gradated
system of accurately marked weights points to the development of trade and
commerce in Harappan society.
Mathematical Activity in the Vedic Period
In the Vedic period, records of mathematical activity
are mostly to be found in Vedic texts associated with ritual activities.
However, as in many other early agricultural civilizations, the study of
arithmetic and geometry was also impelled by secular considerations. Thus, to
some extent early mathematical developments in India mirrored the developments
in Egypt, Babylon and China . The system of land grants and agricultural tax
assessments required accurate measurement of cultivated areas. As land was
redistributed or consolidated, problems of mensuration came up that required
solutions. In order to ensure that all cultivators had equivalent amounts of
irrigated and non-irrigated lands and tracts of equivalent fertility -
individual farmers in a village often had their holdings broken up in several
parcels to ensure fairness. Since plots could not all be of the same shape -
local administrators were required to convert rectangular plots or triangular
plots to squares of equivalent sizes and so on. Tax assessments were based on
fixed proportions of annual or seasonal crop incomes, but could be adjusted
upwards or downwards based on a variety of factors. This meant that an
understanding of geometry and arithmetic was virtually essential for revenue
administrators. Mathematics was thus brought into the service of both the
secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication,
fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana
attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge (rekha-ganit)
are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600
BC) which describe techniques for the construction of ritual altars in use
during the Vedic era. It is likely that these texts tapped geometric knowledge
that may have been acquired much earlier, possibly in the Harappan period.
Baudhayana's Sutra displays an understanding of basic geometric shapes and
techniques of converting one geometric shape (such as a rectangle) to another of
equivalent (or multiple, or fractional) area (such as a square). While some of
the formulations are approximations, others are accurate and reveal a certain
degree of practical ingenuity as well as some theoretical understanding of basic
geometric principles. Modern methods of multiplication and addition probably
emerged from the techniques described in the Sulva-Sutras.
Pythagoras - the Greek mathematician and philosopher who lived in the 6th C
B.C was familiar with the Upanishads and learnt his basic geometry from the
Sulva Sutras. An early statement of what is commonly known as the Pythagoras
theorem is to be found in Baudhayana's Sutra: The chord which is stretched
across the diagonal of a square produces an area of double the size. A similar
observation pertaining to oblongs is also noted. His Sutra also contains
geometric solutions of a linear equation in a single unknown. Examples of
quadratic equations also appear. Apasthamba's sutra (an expansion of
Baudhayana's with several original contributions) provides a value for the
square root of 2 that is accurate to the fifth decimal place. Apasthamba also
looked at the problems of squaring a circle, dividing a segment into seven equal
parts, and a solution to the general linear equation. Jain texts from the 6th C
BC such as the Surya Pragyapti describe ellipses.
Modern-day commentators are divided on how some of the results were
generated. Some believe that these results came about through hit and trial - as
rules of thumb, or as generalizations of observed examples. Others believe that
once the scientific method came to be formalized in the Nyaya-Sutras - proofs
for such results must have been provided, but these have either been lost or
destroyed, or else were transmitted orally through the Gurukul system, and only
the final results were tabulated in the texts. In any case, the study of Ganit
i.e mathematics was given considerable importance in the Vedic period. The
Vedang Jyotish (1000 BC) includes the statement: "Just as the feathers of a
peacock and the jewel-stone of a snake are placed at the highest point of the
body (at the forehead), similarly, the position of Ganit is the highest amongst
all branches of the Vedas and the Shastras."
(Many centuries later, Jain mathematician from Mysore, Mahaviracharya further
emphasized the importance of mathematics: "Whatever object exists in this moving
and non-moving world, cannot be understood without the base of Ganit (i.e.
mathematics)".)
Panini and Formal Scientific Notation
A particularly important development in the history
of Indian science that was to have a profound impact on all mathematical
treatises that followed was the pioneering work by Panini (6th C BC) in the
field of Sanskrit grammar and linguistics. Besides expounding a comprehensive
and scientific theory of phonetics, phonology and morphology, Panini provided
formal production rules and definitions describing Sanskrit grammar in his
treatise called Asthadhyayi. Basic elements such as vowels and consonants, parts
of speech such as nouns and verbs were placed in classes. The construction of
compound words and sentences was elaborated through ordered rules operating on
underlying structures in a manner similar to formal language theory.
Today, Panini's constructions can also be seen as
comparable to modern definitions of a mathematical function. G G Joseph, in The
crest of the peacock argues that the algebraic nature of Indian mathematics
arises as a consequence of the structure of the Sanskrit language. Ingerman in
his paper titled Panini-Backus form finds Panini's notation to be equivalent in
its power to that of Backus - inventor of the Backus Normal Form used to
describe the syntax of modern computer languages. Thus Panini's work provided an
example of a scientific notational model that could have propelled later
mathematicians to use abstract notations in characterizing algebraic equations
and presenting algebraic theorems and results in a scientific format.
Philosophy and Mathematics
Philosophical doctrines also had a profound influence
on the development of mathematical concepts and formulations. Like the
Upanishadic world view, space and time were considered limitless in Jain
cosmology. This led to a deep interest in very large numbers and definitions of
infinite numbers. Infinite numbers were created through recursive formulae, as
in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different types
of infinities: infinite in one direction, in two directions, in area, infinite
everywhere and perpetually infinite. Permutations and combinations are listed in
the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).
Jain set theory probably arose in parallel with the Syadvada system of Jain
epistemology in which reality was described in terms of pairs of truth
conditions and state changes. The Anuyoga Dwara Sutra demonstrates an
understanding of the law of indeces and uses it to develop the notion of
logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to
denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama
various sets are operated upon by logarithmic functions to base two, by squaring
and extracting square roots, and by raising to finite or infinite powers. The
operations are repeated to produce new sets. In other works the relation of the
number of combinations to the coefficients occurring in the binomial expansion
is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing
reality, it probably helped in grappling with indeterminate equations and
finding numerical approximations to irrational numbers.
Buddhist literature also
demonstrates an awareness of indeterminate and infinite numbers. Buddhist
mathematics was classified either as Garna (Simple Mathematics) or Sankhyan
(Higher Mathematics). Numbers were deemed to be of three types: Sankheya
(countable), Asankheya (uncountable) and Anant (infinite).
Philosophical formulations
concerning Shunya - i.e. emptiness or the void may have facilitated in the
introduction of the concept of zero. While the zero (bindu) as an empty place
holder in the place-value numeral system appears much earlier, algebraic
definitions of the zero and it's relationship to mathematical functions appear
in the mathematical treatises of Brahmagupta in the 7th C AD. Although scholars
are divided about how early the symbol for zero came to be used in numeric
notation in India, (Ifrah arguing that the use of zero is already implied in
Aryabhatta) tangible evidence for the use of the zero begins to proliferate
towards the end of the Gupta period. Between the 7th C and the 11th C, Indian
numerals developed into their modern form, and along with the symbols denoting
various mathematical functions (such as plus, minus, square root etc) eventually
became the foundation stones of modern mathematical notation.
The Indian Numeral System
Although the Chinese were also
using a decimal based counting system, the Chinese lacked a formal notational
system that had the abstraction and elegance of the Indian notational system,
and it was the Indian notational system that reached the Western world through
the Arabs and has now been accepted as universal. Several factors contributed to
this development whose significance is perhaps best stated by French
mathematician, Laplace: "The ingenious method of expressing every possible
number using a set of ten symbols (each symbol having a place value and an
absolute value) emerged in India. The idea seems so simple nowadays that its
significance and profound importance is no longer appreciated. It's simplicity
lies in the way it facilitated calculation and placed arithmetic foremost
amongst useful inventions."
Brilliant as it was, this
invention was no accident. In the Western world, the cumbersome roman numeral
system posed as a major obstacle, and in China the pictorial script posed as a
hindrance. But in India, almost everything was in place to favor such a
development. There was already a long and established history in the use of
decimal numbers, and philosophical and cosmological constructs encouraged a
creative and expansive approach to number theory. Panini's studies in linguistic
theory and formal language and the powerful role of symbolism and
representational abstraction in art and architecture may have also provided an
impetus, as might have the rationalist doctrines and the exacting epistemology
of the Nyaya Sutras, and the innovative abstractions of the Syadavada and
Buddhist schools of learning.
Influence of Trade and Commerce, Importance of
Astronomy
The growth of trade and commerce, particularly
lending and borrowing demanded an understanding of both simple and compound
interest which probably stimulated the interest in arithmetic and geometric
series. Brahmagupta's description of negative numbers as debts and positive
numbers as fortunes points to a link between trade and mathematical study.
Knowledge of astronomy - particularly knowledge of the tides and the stars was
of great import to trading communities who crossed oceans or deserts at night.
This is borne out by numerous references in the Jataka tales and several other
folk-tales. The young person who wished to embark on a commercial venture was
inevitably required to first gain some grounding in astronomy. This led to a
proliferation of teachers of astronomy, who in turn received training at
universities such as at Kusumpura (Bihar) or Ujjain (Central India) or at
smaller local colleges or Gurukuls. This also led to the exchange of texts on
astronomy and mathematics amongst scholars and the transmission of knowledge
from one part of India to another. Virtually every Indian state produced great
mathematicians who wrote commentaries on the works of other mathematicians (who
may have lived and worked in a different part of India many centuries earlier).
Sanskrit served as the common medium of scientific communication.
The science of astronomy was also spurred by the need
to have accurate calendars and a better understanding of climate and rainfall
patterns for timely sowing and choice of crops. At the same time, religion and
astrology also played a role in creating an interest in astronomy and a negative
fallout of this irrational influence was the rejection of scientific theories
that were far ahead of their time. One of the greatest scientists of the Gupta
period - Aryabhatta (born in 476 AD, Kusumpura, Bihar) provided a systematic
treatment of the position of the planets in space. He correctly posited the
axial rotation of the earth, and inferred correctly that the orbits of the
planets were ellipses. He also correctly deduced that the moon and the planets
shined by reflected sunlight and provided a valid explanation for the solar and
lunar eclipses rejecting the superstitions and mythical belief systems
surrounding the phenomenon. Although Bhaskar I (born Saurashtra, 6th C, and
follower of the Asmaka school of science, Nizamabad, Andhra ) recognized his
genius and the tremendous value of his scientific contributions, some later
astronomers continued to believe in a static earth and rejected his rational
explanations of the eclipses. But in spite of such setbacks, Aryabhatta had a
profound influence on the astronomers and mathematicians who followed him,
particularly on those from the Asmaka school.
Mathematics played a vital role in Aryabhatta's
revolutionary understanding of the solar system. His calculations on pi, the
circumferance of the earth (62832 miles) and the length of the solar year
(within about 13 minutes of the modern calculation) were remarkably close
approximations. In making such calculations, Aryabhatta had to solve several
mathematical problems that had not been addressed before including problems in
algebra (beej-ganit) and trigonometry (trikonmiti).
Bhaskar I continued where Aryabhatta left off, and
discussed in further detail topics such as the longitudes of the planets;
conjunctions of the planets with each other and with bright stars; risings and
settings of the planets; and the lunar crescent. Again, these studies required
still more advanced mathematics and Bhaskar I expanded on the trigonometric
equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to
be an irrational number. Amongst his most important contributions was his
formula for calculating the sine function which was 99% accurate. He also did
pioneering work on indeterminate equations and considered for the first time
quadrilaterals with all the four sides unequal and none of the opposite sides
parallel.
Another important astronomer/mathematician was
Varahamira (6th C, Ujjain) who compiled previously written texts on astronomy
and made important additions to Aryabhatta's trigonometric formulas. His works
on permutations and combinations complemented what had been previously achieved
by Jain mathematicians and provided a method of calculation of nCr that closely
resembles the much more recent Pascal's Triangle. In the 7th century,
Brahmagupta did important work in enumerating the basic principles of algebra.
In addition to listing the algebraic properties of zero, he also listed the
algebraic properties of negative numbers. His work on solutions to quadratic
indeterminate equations anticipated the work of Euler and Lagrange.
Emergence of Calculus
In the course of developing a
precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the
concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal,
or near instantaneous motion of the moon, and express it in the form of a basic
differential equation. Aryabhatta's equations were elaborated on by Manjula
(10th C) and Bhaskaracharya (12th C) who derived the differential of the sine
function. Later mathematicians used their intuitive understanding of integration
in deriving the areas of curved surfaces and the volumes enclosed by them.
Applied Mathematics, Solutions to Practical
Problems
Developments also took place in applied mathematics
such as in creation of trigonometric tables and measurement units. Yativrsabha's
work Tiloyapannatti (6th C) gives various units for measuring distances and time
and also describes the system of infinite time measures.
In the 9th C, Mahaviracharya ( Mysore) wrote Ganit
Saar Sangraha where he described the currently used method of calculating the
Least Common Multiple (LCM) of given numbers. He also derived formulae to
calculate the area of an ellipse and a quadrilateral inscribed within a circle
(something that had also been looked at by Brahmagupta) The solution of
indeterminate equations also drew considerable interest in the 9th century, and
several mathematicians contributed approximations and solutions to different
types of indeterminate equations.
In the late 9th C, Sridhara (probably Bengal)
provided mathematical formulae for a variety of practical problems involving
ratios, barter, simple interest, mixtures, purchase and sale, rates of travel,
wages, and filling of cisterns. Some of these examples involved fairly
complicated solutions and his Patiganita is considered an advanced mathematical
work. Sections of the book were also devoted to arithmetic and geometric
progressions, including progressions with fractional numbers or terms, and
formulas for the sum of certain finite series are provided. Mathematical
investigation continued into the 10th C. Vijayanandi (of Benares, whose
Karanatilaka was translated by Al-Beruni into Arabic) and Sripati of Maharashtra
are amongst the prominent mathematicians of the century.
The leading light of 12th C Indian mathematics was
Bhaskaracharya who came from a long-line of mathematicians and was head of the
astronomical observatory at Ujjain. He left several important mathematical texts
including the Lilavati and Bijaganita and the Siddhanta Shiromani, an
astronomical text. He was the first to recognize that certain types of quadratic
equations could have two solutions. His Chakrawaat method of solving
indeterminate solutions preceded European solutions by several centuries, and in
his Siddhanta Shiromani he postulated that the earth had a gravitational force,
and broached the fields of infinitesimal calculation and integration. In the
second part of this treatise, there are several chapters relating to the study
of the sphere and it's properties and applications to geography, planetary mean
motion, eccentric epicyclical model of the planets, first visibilities of the
planets, the seasons, the lunar crescent etc. He also discussed astronomical
instruments and spherical trigonometry. Of particular interest are his
trigonometric equations: sin(a + b) = sin a cos b + cos a sin b; sin(a - b) =
sin a cos b - cos a sin b;
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of
the Islamic invasions and the conversion of colleges and universities to
madrasahs. But this was also the time when Indian mathematical texts were
increasingly being translated into Arabic and Persian. Although Arab scholars
relied on a variety of sources including Babylonian, Syriac, Greek and some
Chinese texts, Indian mathematical texts played a particularly important role.
Scholars such as Ibn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C,
Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th C, Maghreb, author of
Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C, Damascus, author of The book
of Chapters in Indian Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada,
11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born
Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba)
were amongst the many who based their own scientific texts on translations of
Indian treatises. Records of the Indian origin of many proofs, concepts and
formulations were obscured in the later centuries, but the enormous
contributions of Indian mathematics was generously acknowledged by several
important Arabic and Persian scholars, especially in Spain. Abbasid scholar Al-Gaheth
wrote: "India is the source of knowledge, thought and insight." Al-Maoudi (956
AD) who travelled in Western India also wrote about the greatness of Indian
science. Said Al-Andalusi, an 11th C Spanish scholar and court historian was
amongst the most enthusiastic in his praise of Indian civilization, and
specially remarked on Indian achievements in the sciences and in mathematics. Of
course, eventually, Indian algebra and trigonometry reached Europe through a
cycle of translations, traveling from the Arab world to Spain and Sicily, and
eventually penetrating all of Europe. At the same time, Arabic and Persian
translations of Greek and Egyptian scientific texts become more readily
available in India.
The Kerala School
Although it appears that original
work in mathematics ceased in much of Northern India after the Islamic
conquests, Benaras survived as a center for mathematical study, and an important
school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made
important mathematical discoveries that would not be identified by European
mathematicians till at least two centuries later. His series expansion of the
cos and sine functions anticipated Newton by almost three centuries. Historians
of mathematics, Rajagopal, Rangachari and Joseph considered his contributions
instrumental in taking mathematics to the next stage, that of modern classical
analysis. Nilkantha (15th C, Tirur, Kerala) extended and elaborated upon the
results of Madhava while Jyesthadeva (16th C, Kerala) provided detailed proofs
of the theorems and derivations of the rules contained in the works of Madhava
and Nilkantha. It is also notable that Jyesthadeva's Yuktibhasa which contained
commentaries on Nilkantha's Tantrasamgraha included elaborations on planetary
theory later adopted by Tycho Brahe, and mathematics that anticipated work by
later Europeans. Chitrabhanu (16th C, Kerala) gave integer solutions to
twenty-one types of systems of two algebraic equations, using both algebraic and
geometric methods in developing his results. Important discoveries by the Kerala
mathematicians included the Newton-Gauss interpolation formula, the formula for
the sum of an infinite series, and a series notation for pi. Charles Whish
(1835, published in the Transactions of the Royal Asiatic Society of Great
Britain and Ireland) was one of the first Westerners to recognize that the
Kerala school had anticipated by almost 300 years many European developments in
the field.
Yet, few modern compendiums on
the history of mathematics have paid adequate attention to the often pioneering
and revolutionary contributions of Indian mathematicians. A significant body of mathematical works were produced in
the Indian subcontinent. The science of mathematics played a pivotal role not
only in the industrial revolution but in the scientific developments that have
occurred since. No other branch of science is complete without mathematics. Not
only did India provide the financial capital for the industrial revolution India also provided vital elements of the scientific
foundation without which humanity could not have entered this modern age of
science and high technology.
Facts:
Mathematics and Music: Pingala (3rd C AD), author of Chandasutra
explored the relationship between combinatorics and musical theory anticipating
Mersenne (1588-1648) author of a classic on musical theory.
Mathematics and Architecture: Interest in arithmetic and
geometric series may have also been stimulated by (and influenced) Indian
architectural designs - (as in temple shikaras, gopurams and corbelled temple
ceilings). Of course, the relationship between geometry and architectural
decoration was developed to it's greatest heights by Central Asian, Persian,
Turkish, Arab and Indian architects in a variety of monuments commissioned by
the Islamic rulers.
Transmission of the Indian Numeral System: Evidence for the
transmission of the Indian Numeral System to the West is provided by Joseph
(Crest of the Peacock):-
-
Quotes Severus Sebokht (662) in a Syriac text describing the
"subtle discoveries" of Indian astronomers as being "more ingenious than those
of the Greeks and the Babylonians" and "their valuable methods of computation
which surpass description" and then goes on to mention the use of nine
numerals.
-
Quotes from Liber abaci (Book of the Abacus) by Fibonacci
(1170-1250): The nine Indian numerals are ...with these nine and with the sign
0 which in Arabic is sifr, any desired number can be written. (Fibonaci learnt
about Indian numerals from his Arab teachers in North Africa)
Influence of the Kerala School: Joseph (Crest of the Peacock)
suggests that Indian mathematical manuscripts may have been brought to Europe by
Jesuit priests such as Matteo Ricci who spent two years in Kochi (Cochin) after
being ordained in Goa in 1580. Kochi is only 70km from Thrissur (Trichur) which
was then the largest repository of astronomical documents. Whish and Hyne - two
European mathematicians obtained their copies of works by the Kerala
mathematicians from Thrissur, and it is not inconceivable that Jesuit monks may
have also taken copies to Pisa (where Galileo, Cavalieri and Wallis spent time),
or Padau (where James Gregory studied) or Paris (where Mersenne who was in touch
with Fermat and Pascal, acted as an agent for the transmission of mathematical
ideas).
Upanishadic philosophy: preparing
the ground for rationalism
Although the Upanishadic texts (like
some of the earlier Vedic texts) are primarily concerned with acquiring
knowledge of the "soul", "spirit" and "god" - there are aspects of Vedic and
Upanishadic literature that also point to an intuitive understanding of nature
and natural processes. In addition, many of the ideas are presented in a
philosophical and exploratory manner - rather than as strict definitions of
inviolable truth.
Although the Upanishadic texts goaded
the Upanishadic student to concentrate on comprehending the inner spirit,
rational investigation of the world by other scholars was not entirely
squelched, and eventually, the Upanishadic period gave way to an era which was
not inimical to the development of rational ideas, even encouraging scientific
observation and advanced study in the fields of logic, mathematics and the
physical sciences.
Following an era when rituals and
superstitions had begun to proliferate, in some ways the Upanishadic texts
helped to clear the ground for greater rationalism in society. Brahmin orthodoxy
and ideas of ritual purity were superseded by a spiritual perspective that
eschewed sectarianism and could be practised universally, unfettered by an
individual's social standing. Much of the emphasis was on discovering "spiritual
truths" for oneself as opposed to mechanically accepting the testimony of
established religious leaders. Although there is a thematic commonality to the
Upanishadic discourses, different commentators offered subtly varying
perspectives and insights.
The concept of god in Upanishadic (and
even earlier Vedic) thinking was quite different from the more common definition
of god as creator and dispenser of reward and punishment. The Upanishadic
concept of god was more abstract and philosophical. Different texts postulated
the doctrine of a universal soul that embraced all physical beings. All life
emanated from this universal soul and death simply caused individual
manifestations of the soul to merge or mingle back with the universal soul. The
concept of a universal soul was illustrated through analogies from natural
phenomenon.
"As the bees make honey by collecting the juices of distant
trees, and reduce the juice into one form. And as these juices have no
discrimination, so that they might say, I am the juice of this tree or that, in
the same manner, all these creatures, when they have become merged in the True,
know not that they are merged in the True. . . ."
"These rivers run, the eastern (like the Ganges) towards the
east, the western (like the Indus) towards the west. They go from sea to sea
(i.e., the clouds lift up the water from the sea to the sky and send it back as
rain to the sea). They become indeed sea. And as those rivers, when they are in
the sea, do not know, I am this or that river, in the same manner, all these
creatures, proceeding from the True, know not that they have proceeded from the
True. . . ."
In another story, the "wise" father, expounder of the
Upanishadic concept of god, asks his son to dissolve salt in water, and asked
him to taste it from the surface, from the middle and from the bottom. In each
case, the son finds the taste to be salty. To this his father replies that the
'universal being' though invisible resides in all of us, just as the salt,
though invisible is completely dissolved in the water.
(Chanddogya, VI)
As a corollary to this theory emerged
the notion that even as individual beings might refer to this universal soul -
i.e. god in varied ways - by using different names and different methods of
worship - all living beings were nevertheless related to each other and to the
universal god, and capable of merging with the universal god. This approach thus
laid the foundation for egalitarian and non-discriminatory philosophies such as
Buddhism and Jainism (as well as non-sectarian streams of Hinduism) that
followed the Upanishadic period. As is evident, such an approach was not
incompatible with secular society, and permitted different faiths and sub-faiths
to coexist in relative peace and harmony.
In the course of defining their
philosophy, the scholars of the Upanishad period raised several questions that
challenged mechanical theism (as
was also done in some hymns from the Rig Veda and Atharva Veda).
If god existed as the unique creator of the world, they wondered who created
this unique creator. The logical pursuit of such a line of questioning could
either lead to an infinite series of creators, or to the rejection or
abandonment of this line of questioning. The common theist solution to this
philosophical dilemma was to simply reject logic and demand unquestioning faith
on the part of the believer. A few theists attempted to use this contradiction
to their own advantage by positing that god existed precisely because "He" was
indescribable by mere mortals. But, by and large, this contradiction was taken
very seriously by the philosophers of the Upanishadic period. The Upanishadic
philosophers attempted to resolve this contradiction by defining god as an
entity that extended infinitely in all dimensions covering both space and time.
This was a philosophical advance in that it attempted to come to terms with at
least the most obvious challenges to the notion of god as a human-like creator
and did not require the complete rejection of logic.
Another philosophical advance of the
Upanishadic period was that religion was transformed from the realm of bookish
parroting of scriptures to the realm of advanced intellectual debate and
polemics. The Upanishadic philosophers did not lay down their conclusions as
rigid doctrines or inviolable laws but as seductive parables - sometimes
displaying remarkable worldly insight and analytical skill. By attempting to win
over their followers through analogies from nature, and by employing the methods
of abstract reasoning and debate, they created an environment where dialectical
thinking and intellectual exchanges could later flourish.
In the very process of their
questioning, (and albeit speculative reasoning about god), they had opened the
door for rationalists and even outright atheists who took their tentative
questioning about the role and the character of god as "creator" to conclusions
that rejected theism entirely. But in either case, many rationalist and/or
naturalist philosophical streams emerged from this initial foundation. Some were
nominally theistic (but in the abstract Upanishadic vein), others were agnostic
(as the early Jains), while the early Buddhists and the Lokayatas were atheists.
Thus even though the Upanishads contained much that should rightly be dismissed
as abstruse intellectual jugglery and philosophical mumbo-jumbo, the Upanishadic
philosophers had levelled the ground for the seeds of rationalism to flourish in
Indian soil.
The Vaisheshika School
The Vaisheshika school
(considered to be founded by Kanada,
author of the Vaisesika Sutra) was
an early realistic school whose main achievement lay in it's attempt at
classifying nature into like and unlike groups. It also posited that all matter
was made up of tiny and indestructible particles - i.e. atoms that aggregated in
different ways to form new compounds that formed the variety of matter that
existed on the earth.
Their philosophy was described through
the enumeration of the following concepts: Dravya (Substance), Guna
(Quality), Karma (Action), Samanya (Generality),
Visesa (Particularity), Samavaya (Inherence) and abhava
(non-existence).
Dravya (or substance) was
understood as the specific result of a particular aggregate effect - i.e. the
combination of atoms in a unique way. Substances were repositories for qualities
and actions. Guna or quality was that which resided in a dravya.
Qualities did not however contain qualities themselves. 24 qualities were
enumerated, such as - color, form, smell, touch, sound, number, magnitude,
distinctions, conjunction, disjunction, nearness, remoteness, heaviness,
fluidity and viscosity. (As was typical of the times, psychological attributes
such as pleasure, pain, desire, aversion, effort, tendency, cognition,
impression, and ethical attributes such as merit and demerit were also included
in the list, i.e. - qualities that were inapplicable to inanimate objects were
not treated separately)
Action or Karma represented
physical movement. Unlike quality which was passive, Karma was dynamic.
Action was the determinant of conjuction and disjunction. Five types of action
were noted: throwing upwards or downwards, contraction, expansion and
locomotion.
Satta or physical existence
was viewed as being the common attribute of substance, quality and action - i.e.
only existing (as opposed to imaginary) entities could have substance, qualities
and be capable of action.
Samanyata or 'generality' was
seen as a mental construct to create common classes of substances, qualities or
actions while Visesata (particularity) was used to identify and
separate individual items from their general classes. Samavaya
or inherence was a relation that
existed in those things that could not be separated without destroying them.
Four categories of Abhava as
negation or non-existance were listed: pragabhava or prior non-existance,
referring to the absence of an object before it's creation; dhvamsabhava
or posterior negation, as the absence of an object after it had been destroyed;
anyonyabhava or mutual non-existance, refering to an object being
distinct and different from the other; atyantabhava or absolute
non-existence, indicating non-existence in the past, present and future, citing
the example of air as permanently lacking in smell - (which was presumably true
in a period where air pollution must have been uncommon!).
An important contribution of the
Vaisheshika school was a careful study of the time-relation in a chain of
causes and effects. In a very rudimentary way, the school (along with other such
schools) anticipated the theory of time calculus which could also be extended to
space calculus.
The Vaisheshika school thus
served as an important step in the study of science by enumerating concepts that
could further the study of physics and chemistry. In addition, the the study of
medical science (including veterinary science) received considerable impetus
from such attempts at methodical observation and classification.
The Nyaya and related schools
The Nyaya schools complemented and
built on the Vaisheshika school by elaborating on the process of accumulating
valid scientific knowledge through accurate perception and generating valid
inferences.
The school articulated four means of
acquiring valid knowledge: pratyaksha or perception through one of the
senses; anumana or inference; upamana or comparison with a
well-known object; or shabda - verbal testimony.
The conditions of perception, and it's
range and limits were carefully studied. Trasarenu - the minima
sensibile (i.e. the minimum visible), anubhuta-rupa - the
infra-sensible, abhibhuta - the obscured perception , and
anubhuta-vriti - potential perception, were recognized as different types
of perception.
A general methodology of ascertaining
the truth (tattva) was described which consisted of describing a
proposition (uddesa), the ascertainment of essential facts obtained
through perception, inference or induction (laksan or uppa-laksana),
and finally examination and verification (pariksa and nirnaya).
This process could involve examples (drishtanta), logical arguments
(avayava), reasoning (tarka) and discussion (vada)
- , intellectual exchange, or interplay of two opposing sides in the process of
arriving at a decisive conclusion. A successful application of this method could
result in a siddhanta - i.e. established principle - (or in the case of
mathematics - a theorem or theory) elucidated through proofs (pramana).
Alternatively, it could lead to a rejection of the initial proposition.
The Nyaya school identified various
types of arguments that hindered or obstructed the path of genuine scientific
pursuit, suggesting perhaps, that there may have been considerable practical
resistance to their unstinting devotion to truth-seeking and scientific
accuracy. They list the term jalpa - an argument not for the sake of
arriving at the truth but for the sake of seeking victory (this term was coined
perhaps to distinguish exaggerated and rhetorical arguments, or hyperbole from
genuine arguments); vitanda (or cavil) to identify arguments that were
specious or frivolous, or intended to divert attention from the substance of the
debate, that were put-downs intended to lower the dignity or credibility of the
opponent; and chal - equivocation or ruse to confuse the argument.
Three types of chal are listed: vakchala - or verbal
equivocation where the words of the opponent are deliberately misused to mean or
suggest something different than what was intended; samanyachala or
false generalization, where the opponents arguments are deliberately and
incorrectly generalized in a way to suggest that the original arguments were
ridiculous or absurd; uparachala - misinterpreting a word which is used
figuratively by taking it literally. Also mentioned is jati, a type of
fallacious argument where an inapplicable similiarity is cited to reject an
argument, or conversely an irrelevant dissimiliarity is cited to reject an
argument.
The Nyaya school also recognized that
intelligent and meaningful debates were not possible if certain fundamental
principles and basic definitions and concepts were not mutually accepted.
Nigrahasthana was the term used to identify disagreements based on absence
of mutually acceptable first principles. An example might be a debate between a
theist who rejected logic, and a non-theist who rejected faith.
The Nyaya school also listed five
classes of logical fallacies (hetvabhasa) : savyabhichara or
the inconclusive type which employed reasoning from which more than one
conclusion could be drawn but was used to insist on a single specific
conclusion; viruddha or contradictory, where the reasoning
used actually contradicted the proposition to be established; kalatita -
where the elapse of time had made the argument invalid; sadhyasama,
the unproven type, where the reasoning employed rested on arguments or
principles that had not been proven and require proofs themselves - i.e. this
was the type of fallacy where one unproven result was merely converted into
another unproven result.; and finally prakaranasama - where the
reasoning employed provoked the very question it was designed to answer - i.e. a
recursive fallacy.
In this manner, the Nyaya school
defined a very sophisticated school of rational philosophy where the process of
scientific epistemology was analyzed threadbare and all the dangers of
unscientific reasoning and propaganda ploys were skillfully exposed.
Causality
Buddhist and Jain scholars, as well as
later Hindu scholars offered their own approaches to scientific reasoning.
Virtually all the rational schools were concerned with describing causality and
causal relationships, and recognized that effects may not have single causes but
may require a group or conjunction of causes to occur. Buddhist scholars
emphasized that cause and effect need not have a linear effect but that desired
effects may also require the right conditions for their fruition. (That is to
say that for a plant to grow successfully, it would not only need the right
seed, but that it would also need the right type of soil, fertilization,
sunlight and water.)
Both the Jains and the Buddhists
correctly speculated that a potential for the desired effect must also be
present in the cause or causal agent. (For instance, only a mango seed could
produce a mango tree because only the mango seed incorporated the potential of
developing into a mango tree.) As another example, one could note that
something with brittle properties such as glass might break upon impact whereas
something strong such as steel would survive. Thus a physical impact on
substances of different properties would have different results.
The Nyaya school also recognized
co-effects - i.e a series of antecedants could cause a series of effects -
either successive and staggered in time, or near simultaneous. Nyaya texts on
causality indicate that there was an awareness that light travelled at a very
high speed but the transmission of light was not instantaneous.
Buddhist and Jain Atomic Theories
The Buddhist and Jain philosophers
also proposed their own variations of the atomic theory. Like the Vaisheshikas,
atoms were perceived as infinitely small by the Jainas. But the Jainas went a
step further by positing that the union of atoms required opposite qualities in
the combining atoms - as is true in the case of electrovalent bonding. However,
they erred in thinking that covalent bonding (which does not require opposite
polarities in the combining atoms) could not occur. But their intuition that
opposite polarities created mutual attraction and facilitated chemical reactions
was correct. In the Buddhist view, matter was in fact an aggregate of rapidly
recurring forces or energy waves. Their theory was illustrated with examples
drawn from natural phenomenon involved with light emission. An atom was
perceived as a momentary flash of light combining and separating from other
atoms according to strict and definite laws of causality. Physical matter was
thus seen as a denser and more concentrated form of light. Although at odds with
other atomic theories of the time, their approach fit in with their general view
that all things in nature were temporal, that there was constant change in
nature - that degradation and renewal were continuous processes.
The Syadvada system of Jain Logic
Jain philosophers also made certain
important contributions to the science of epistemology by proposing that the
truth of a concept or observation could not only be true or false but
indeterminate - and combinations of the above - such as true under some
conditions (or true at a particular time or place - or true based on the
validity of certain inferences) and false under other conditions, or true under
some conditions but indeterminate under others, and so on. This led to a matrix
of seven possible states of the truth
(true, false, true or false, indeterminate, true or indeterminate, false or
indeterminate, true or false or indeterminate).
Jaina rationalists also studied the
relationship between the universal and the particular and made important points
concerning generalities and individual peculiarities. They also noted that
objects in the real world exist in a network of relationships with each other -
and have specific attributes that mark them temporally and spatially: "Every
real is thus hedged round by a network of relations and attributes, which we
propose to call its system or context or universe of discourse, which demarcates
it from others." Jaina philosophers also successfully synthesized earlier
debates on change and permanence by positing that all objects (or parts of
objects) passed through phases of "existence, persistence, and cessation" and
that reality was therefore a complex combination of things relatively permanent
yet also relatively changing.
These ideas thus formed the
foundations of Indian science and contributed to the gradual elaboration of
mathematics and astronomy, as well as agricultural and meteorological sciences.
Developments in metallurgy and civil engineering also followed. Medicine and
surgery perhaps received the greatest and the earliest impetus from these
developments. Developments in philosophy also led to concomitant developments in
the realm of art and culture.
Yet. to a considerable extent,
knowledge about the progress of science and reason in Indian history is often
scarce. These (and other such) historical contributions were either denied or
demeaned during the process of colonization, and are only now beginning to be
re-acknowledged within India and abroad. But in A. D 1068, Indian contributions
to the mainstream of science were held in great esteem and readily acknowledged
in some parts of the world:
Here is what Said Al-Andalusi,
an 11th C Spanish scholar, court
historian and chronicler wrote then:
"Among the nations, during the course of
centuries and throughout the passage of time, India was known as the mine of
wisdom and the fountainhead of justice and good government and the Indians were
credited with excellent intellects, exalted ideas, universal maxims, rare
inventions and wonderful talents ... They have studied arithmetic and geometry.
They have also acquired copious and abundant knowledge of the movements of the
stars, the secrets of the celestial sphere and all other kinds of mathematical
sciences. Moreover, of all the peoples they are the most learned in the science
of medicine and thoroughly informed about the properties of drugs, the nature of
composite elements and peculiarities of the existing things."
