URL: http://www.cerc.utexas.edu/~jay/india_science.html
Astronomy
* Earliest known precise celestial
calculations:
As argued by James Q. Jacobs, Aryabhata,
an Indian Mathematician (c. 500AD) accurately calculated celestial constants
like earth's rotation per solar orbit, days per solar orbit, days per lunar
orbit. In fact, to the best of my knowledge, no source from prior to the
18th century had more accurate results on the values of these constants!
Click here for details. Aryabhata's 499 AD computation of pi as 3.1416
(real value 3.1415926...) and the length of a solar year as 365.358 days
were also extremely accurate by the standards of the next thousand years.
* Astronomical time spans:
The notion of of time spans that
are truly gigantic by modern standards are rarely found in ancient civilizations
as the notion of large number is rare commodity. Apart from the peoples
of the Mayan civilization, the ancient Hindus appear to be the only people
who even thought beyond a few thousand years. In the famed book Cosmos,
physicist-astronomer-teacher Carl Sagan writes "... The dates on Mayan
inscriptions also range deep into the past and occasionally far into the
future. One inscription refers to a time more than a million years ago
and another perhaps refers to events of 400 million years ago, ... The
events memorialized may be mythical, but the time scales are pridigious".
Hindu scriptures refer to time scales that vary from ordinary earth day
and night to the day and night of the Brahma that are a few billion earth
years long. Sagan continues, "A millennium before Europeans were wiling
to divest themselves of the Biblical idea that the world was a few thousand
years old, the Mayans were thinking of millions and the Hindus billions"
[See 5].
* Theory of creation of the universe:
A 9th century Hindu scripture,
The Mahapurana by Jinasena claims the something as modern as the following:
(translation from [5])
Some foolish men declare that a
Creator made the world. The doctrine that the world was created is ill-advised,
and should be rejected. If God created the world, where was he before creation?...
How could God have made the world without any raw material? If you say
He made this first, and then the world, you are faced with an endless regression...
Know that the world is uncreated, as time itself is, without beginning
and end. And it is based on principles.
Theories of the creation of universe
are present in almost every culture. Mostly they represent some story portraying
creation from mating of Gods or humans, or from some divine egg, essentially
all of them reflecting the human endeavour to provide explanations to a
grave scientific question using common human experience.
Hinduism is the only religion that
propounds the idea of life-cycles of the universe. It suggests that the
universe undergoes an infinite number of deaths and rebirths. Hinduism,
according to Sagan, "... is the only religion in which the time scales
correspond... to those of modern scientific cosmology. Its cycles run from
our ordinary day and night to a day and night of the Brahma, 8.64 billion
years long, longer than the age of the Earth or the Sun and about half
the time since the Big Bang" [See 5]. Long before Aryabhata (6th century)
came up with this awesome achievement, apparently there was a mythological
angle to this as well -- it becomes clear when one looks at the following
translation of Bhagavad Gita (part VIII, lines 16 and 17), "All the planets
of the universe, from the most evolved to the most base, are places of
suffering, where birth and death takes place. But for the soul that reaches
my Kingdom, O son of Kunti, there is no more reincarnation. One day of
Brahma is worth a thousand of the ages [yuga] known to humankind; as is
each night." Thus each kalpa is worth one day in the life of Brahma, the
God of creation. In other words, the four ages of the mahayuga must be
repeated a thousand times to make a "day ot Brahma", a unit of time that
is the equivalent of 4.32 billion human years, doubling which one gets
8.64 billion years for a Brahma day and night. This was later theorized
(possibly independently) by Aryabhata in the 6th century. The cyclic nature
of this analysis suggests a universe that is expanding to be followed by
contraction... a cosmos without end. This, according to modern physicists
is not an impossibility.
And here is how -- a few billion
years ago, something known as the Big Bang happened and it is believed
that the universe, as we "know" it, came into existence; one that is continually
expanding after the Big Bang. That the galaxies are receding from us can
be proved by showing Dopler shifts of far off galaxies. Common belief is
that it happened from a mathematical point with no dimension at all. All
the matter in our universe was concentrated in that miniscule volume. Although
we know that we are living in an expanding universe, physicists are not
sure whether it will always be expanding. This is because it is not known
whether there is enough matter in the universe such that there is enough
gravitational cohesion in it that the expansion will gradually slow down,
stop and reverse itself resulting into a contracting universe. If we live
in such an oscillating universe, then the Big Bang is not the beginning
or creation of the universe, but merely the end of the previous cycle,
the destruction of the last incarnation of the universe in the very way
suggested by Hindu philosophers thousands of years ago!
A brand new theory -- that of a
"CYCLIC MODEL", developed by Princeton University's Paul Steinhardt and
Cambridge University's Neil Turok, made its highest-profile appearance
yet in April 2002, on Science Express, the Web site for the journal Science.
But past incarnations of the idea have been hotly debated within the cosmological
community from 2001. A jist of the claims can be found here. The PDF preprint
of the entire paper can be downloaded from here. The Hindu belief that
the Universe has no beginning or end, but follows a cosmic creation and
dissolution can be found here.
* Earth goes round the sun:
Aryabhata, it so happens, was apparently
quite sceptical of the widely held doctrines about eclipses and also about
the belief that the Sun goes round the Earth. He didn't think that eclipses
were caused by Rahu but by the Earth's shadow over the Moon and the Moon
obscuring the Sun. As early as the sixth century, he talked of the diurnal
motion of the earth and the appearance of the Sun going round it.
Mathematics/Computer Science
* Binary System of number representation:
A Mathematician named Pingala (c.
100BC) developed a system of binary enumeration convertible to decimal
numerals [See 3]. He described the system in his book called Chandahshaastra.
The system he described is quite similar to that of Leibnitz, who was born
in the 17th century.
* Earliest and only known Modern
Language:
Panini (c 400BC), in his Astadhyayi,
gave formal production rules and definitions to describe Sanskrit grammar.
Starting with about 1700 fundamental elements, like nouns, verbs, vowels
and consonents, he put them into classes. The construction of sentences,
compound nouns etc. was explained as ordered rules operating on underlying
fundamental structures. This is exactly in congruence with the fundamental
notion of using terminals, non-terminals and production rules of moderm
day Computer Science. On the basis of just under 4,000 sutras (rules expressed
as aphorisms), he built virtually the whole structure of the Sanskrit language.
He used a notation precisely as powerful as the Backus normal form, an
algabraic notation used in Computer Science to represent numerical and
other patterns by letters.
It is my contention that because
of the scientific nature of the method of pronunciation of the vowels and
consonents in the Indian languages (specially those coming directly from
Pali, Prakit and Sanskrit), every part of the mouth is exercised during
speaking. This results into speakers of Indian languages being able to
pronounce words from any language. This is unlike the case with say native
English speakers, as their tongue becomes unused to being able to touch
certain portions of the mouth during pronunciation, thus giving the speakers
a hard time to speak certain words from a language not sharing a common
ancestry with English. I am not aware of any theory in these lines, but
I would like to know if there is one.
* Invention of Zero:
Although ancient Babylonians were
known to have used what is often called "place holders" to distinguish
between numbers like 809 and 89, they were nothing more than blank spaces
or at times two wedge shapes like ". More importantly, they lacked the
realization that zero has a place in the number system as well as it comes
with a baggage of abstract interpretations. Hence, while they can be credited
with intelligently solving a practical problem of avoiding misinterpreting
two numbers like 809 and 89, they can hardly be credited with the invention
of the complex notion of zero and the even more complex notion of the abstract
idea of "nothingness".
The ancient Greeks were beginning
their contributions to mathematics around the time zero as an empty place
holder was being popularized by Babylonian mathematicians. The Greeks did
not adopt what is called a positional number system, a system that gave
a value to a number because of its relative position in the set of numerals.
This is because the Greeks' achievements were based on geometry. This resulted
into firstly, Greeks relating numbers with lengths of line segments, and
secondly, decoupling numbers from any potential abstract interpretations.
It is commonly thought that in Greek society numbers that required to be
"named" were not used by mathematician- philosophers, but by merchants
and hence no clever notation was needed. Thus even the eminent mathematician
like Ptolemy used the then recent place holder "zero" more as a punctuation
mark than any serious numeral. Although a few Greek astronomers began using
the symbol "O", the symbol more familiar to us now, to denote place holders,
zero was not thought of as a number by the Greeks.
The first notions of zero as a number
and its uses have been found in ancient Mathematical treatise from India
and thus India is correctly related to the immensely important mathematical
discovery of the numeral zero. This concept, combined with the place-value
system of enumeration, became the basis for a classical era renaissance
in Indian mathematics. Indians began using zero both as a number in the
place-value system of numerals as well as to denote an empty place (place
holder). Obviously, the use as a number came later. Aryabhata devised a
number system what has no zero yet a positional number system. There is
however, evidence that z dot has been used in earlier Indian manuscripts
to denote an empty position. Also contemporary Indian scriptures also tend
to use zero in places where unknown values are registered, where we would
use x. Later Indian mathematicians had names for zero, but no symbol for
it. Aryabhata used the word "kha" for position and it was also used later
as the name for zero.
The oldest known text to use zero
is an Indian (Jaina) text entitled the Lokavibhaaga ("The Parts of the
Universe"), which has been definitely dated to 25 August 458 BC [See 4]
An inscription, created in 876AD, found in Gwalior, acts as the first use
of zero as a number. Zero is not a "natural" candidate for being a number.
It is a great leap from physical to abstract that one needs to bridge when
dealing with zero. With zero also comes the notion of negative numbers
and along with all these comes a series of related questions about arithmetic
operations on natural numbers, both positive and negative and zero.
The development of the notion of
zero began, in my opinion, when mathematicians tried to answer these questions.
Three Hindu mathematicians, Brahmagupta, Mahavira and Bhaskara tried to
answer these in their treatise. In the 7th century Brahmagupta attempted
to provide rules for addition and subtraction involving zero.
The sum of zero and a negative number
is negative, the sume of a positive number and zero if positive, the sum
of zero and zero is zero. A negative number subtracted from zero is positive,
a positive number subtracted from zero is negative, zero subtracted from
a negative number is nagative, zero subtracted from a positive number is
positive, zero subtracted from zero is zero.
Brahmagupta also says that any number
multiplied by zero is zero. But problems arise when he tries to explain
division. While he is unsure about what division of a number by zero means,
he wrongly gives zero divided by zero to be zero. Brahmagupta's is the
first attempt from any mathematician to explain the arithmetic operations
on natural numbers and zero.
In the 9th century, Mahavira updated
Brahmagupta's attempts at defining operations using zero. Although he correctly
finds out that a number multiplied by zero is zero, but wrongly says that
a number remains unchanged when divided by zero.
The next valiant attempt came from
Bhaskara in the 11th century. Division of zero still remained an illusive
mystery.
A quantity divided by zero becomes
a fraction the denominator of which is zero. This fraction is termed an
infinite quantity. In this quantity consisting of that which has zero for
its divisor, there is no alteration, though many may be inserted or extracted;
as no change takes place in the infinite and immutable God when worlds
are created or destroyed, though numerous orders of beings are absorbed
or put forth.
This, in its face value seems correct,
by suggesting that any number when divided by zero is infinity, Bhaskara
suggeted that zero multiplied by infinity is any number, and hence all
numbers are equal, which is not correct. But Bhaskara did correctly find
out that the square of zero is zero, as is the square root.
The Indian numeral system and its
place value, decimal system of enumeration came to the attention of the
Arabs in the seventh or eighth century, and served as the basis for the
well known advancement in Arab mathematics, represented by figures such
as al-Khwarizmi. Al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning
that described the Indian place-value system of numerals based on numerals
1 through 9 and 0. Scholars like ibn Ezra and al-Samawal used the notion
of zero from al-Khwarizmi's work. In the 12th century al- Samawal extended
arithmetic operations using zero as follows.
If we subtract a positive number
from zero the same negative number remains, ... if we subtract a negative
number from zero the same positive number remains.
Zero also reached eastwards from
India to China, where Chinese scholars Chin Chiu-Shao and Chu Shih-Chieh
made use of the same symbol O for a places-based system in the 12th and
13th centuries respectively. From the time of Han (206 to 220 BC), Chinese
scholars used a place-value system called the suan zi ("calculation using
rods") that was a regular system that used horizontal and vertical lines
that used to denote the nine numerals. Ifrah says that "Thus one could
be forgiven for assuming that following the links established between India
and China at the beginning of the beginning of the first millennium BC,
Indian scholars were influenced by Chinese mathematicians to create their
own system in an imitation of the Chinese counting method." [See 4] He
goes on to argue that in suan zi, the zero appeared at a much later date.
Thus the notion of zero helps one to recognize the originality of the Indian
mathematicians vis-a-vis their Chinese counterparts. Ifra also establishes
that the Chinese scholars overcame the difficulties the absence of zeros
caused in trying to represent numbers like 1,270,000 often either using
characters of their ordinary counting system (a non-positional system that
did not require the use of a zero) or simply by empty spaces. After providing
a sequence of clues, [in 4], Ifrah continues "It was only after the eighth
century BC, and doubtless due to the influence of the Indian Buddhist missionaries,
that Chinese mathematicians introduced the use of zero in the form of a
little circle or dot (signs that originated in India),...".
Zero reached Europe in the twelfth
century when Adelard of Bath translated al-Khwarizmi's works into Latin
[See 1]. Fibonacci was one of the main mathematicians who accepted the
concepts of zero in Europe. He was an important link between the Hindu-Arabic
number system. In his treatise Liber Abaci ("a tract about the abacus"),
published in 1202, he described the nine Indian symbols together with the
symbol O for zero, but it was not widely accepted until much later. Significantly,
Fibonacci spoke of numbers 1 through 9, but a "sign" O. Although he brought
the notion of zero to Europe, it is clear that he was not able to reach
the sophistication of Indians like Brahamagupta, Mahavira and Bhaskara,
nor of the Arabic mathematicians like al-Samawal. The Europeans were at
first resistant to this system, being attached to the far less logical
Roman numeral system (notably the Romans never propounded the idea of zero),
but their eventual adoption of this system arguably led to the scientific
revolution that began to sweep Europe beginning by the middle of the second
millennium. However, it was not until the 17th century that zero found
widespread acceptance through a lot of resistance.
* The word "Algorithm":
Al-Khwarizmi, an eminent 9th century
Arab scholar, played important roles in importing knowledge on arithematic
and algebra from India to the Arabs. In his work, De numero indorum (Concerning
the Hindu Art of Reckoning), it was based presumably on an Arabic translation
of Brahmagupta where he gave a full account of the Hindu numerals which
was the first to expound the system with its digits 0,1,2,3,...,9 and decimal
place value which was a fairly recent arrival from India. Because of this
book with the Latin translations made a false inquiry that our system of
numeration is arabic in origin. The new notation came to be known as that
of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of
numeration making use of the Hindu numerals came to be called simply algorism
or algorithm, a word that, originally derived from the name al-Khwarizmi,
now means, more generally, any peculiar rule of procedure or operation.
The Hindu numerals like much new mathematics were not welcomed by all.
Click here for details.
* Representing Large numbers:
Mathematicians in India invented
the base ten system in ancient times. But research did not stop there.
The practice of representing large numbers also evolved in ancient India.
The base ten system of calculation that uses nine numerals and the zero
stood as an efficient way to represent numbers ranging from a very small
decimal to an inconceivably large number. The biggest number known to Greeks
was the myriad (10,000) whereas the Chinese, until recent times, had 10,000
as the largest unit of enumeration and the ancient Arabs knew only until
1,000. The notion of representing large numbers as powers of 10, one that
was invented in India, turned out to be extremely handy. The Yajur Veda
Samhitaa, one of the Vedic texts written at least 1,000 years before Euclid
lists names for each of the units of ten upto the twelfth power [See 1].
Later other Indian texts (from Buddhist and Jaina authors) extended this
list as high as the 53rd power, far exceeding their Greek contmporaries,
mainly because of the latter's handicap of not being able to accept the
fundamental Mathematical notion of abstract numerals. The place value system
is built into the Sanskrit language and so whereas in English we only use
thousand, million, billion etc, in Sanskrit there are specific nomenclature
for the powers of 10, most used in modern times are dasa (10), sata (100),
sahasra (1,000=1K), ayuta (10K), laksha (100K), niyuta (106=1M), koti (10M),
vyarbuda (100M), paraardha (1012) etc. Results of such a practice were
two-folds. Firstly, the removal of special imporatance of numbers. Instead
of naming numbers in grops of three, four or eight orders of units one
could use the necessary name for the power of 10. Secondly, the notion
of the term "of the order of". To express the order of a particular number,
one simply needs to use the nearest two powers of 10 to express its enormity.
Evidences of using very large numbers
have been found in the Vedas which are ancient Hindu scriptures. Vedas
are the most ancient written texts written in any Indo-European language.
They were written in Sanskrit from around 500BC, although traces go back
to 2000BC [See 4]. In the Taittiriya Upanishad, which is a part of the
third Veda, Yajur Veda, there is a section (anuvaka), that extols the "Beatific
Calculus" or a quasi-mathematical relationship between bliss of a young
man, who has everything in the world to the bliss of the Brahman, or "realization".
Translated roughly as follows, summarized from one done by Max Muller,
firstly it says that fear is all-pervasive. It continues by assuming that
a young, good man who is fit, healthy and strong, and has all the wealth
in the world, is one unit of human bliss. The anuvaka provides a precise
calculation of a series of multiplications by 100 to give number 10010
units of human bliss that can be had when one attains Brahman. The previous
anuvaka exhorts the aspirants to be fearless and strong, as only such a
person may realize the absolute within.
* "... true birthplace of our numerals":
Georges Ifrah:
Famed French scholar Georges Ifrah
spent years travelling and studying the mystery of the evolution of numbers.
While it is hard to prove that India is truly the birthplace of our modern
numerals, in my brief survey of the topic, it seems that there is no better
authority in the field other than Ifrah. I would refer the interested reader
to his authoritative book [See 4] to get a crisp, yet convincing account
supporting his claims. Ifrah provides a total of 45 pieces of evidences,
supported by numerous research work from contemporary scholars. Of the
45, 17 are from scholarly work from Europe that includes work of scholars
like Laplace, Fibonacci, and Adelard of Bath, and 28 are from work from
Arabic sources that includes work of scholars like al Biruni. He refers
to 24 evidences from scriptures from India, whose dates range from 1150
BC until 458 BC, when the Jaina text Lokavibhaaga dates back to. Of particular
interest was the work by Bhaskaracharya (1150 BC) where he makes a reference
to zero and the Indian place-value system as being creations of Brahma,
indicating that by that time they were considered "to have always been
used by humans, and thus to have constituted a "revelation" of the divinities",
[See 4]. Ifrah goes on to explain, with furious objectivity aided by a
plethora of evidences that are not isolated pieces of information, but
"a huge collection of proofs from all disciplines, dating from the most
significant eras", to establish his claim. He also shows how the numerals
evolved to look as they look today. His suggested pathway to the modern
numerals is:
* Brahmi (often called the "mother"
of all Indian writing) numerals
* Shaka, Kushana inscriptions
* Gupta style
* Nagari style
* Arabic from the "Gubar" style
* European late middle ages (cursive
forms of the Algorisms)
* modern.
Ifrah salutes the Indian researchers
saying that the "...real inventors of this fundamental discovery, which
is no less important than such feats as the mastery of fire, the development
of agriculture, or the invention of the wheel, writing or the steam engine,
were the mathematicians and astronomers of the Indian civilisation: scholars
who, unlike the Greeks, were concerned with practical applications and
who were motivated by a kind of passion for both numbers and numerical
calculations."
References
[1] B. V. Subbarayappa. "India's
Contributions to the History of Science" in Lokesh Chandra, et al., eds.
India's Contribution
to World Thought and Culture. Madras:
Vivekananda Rock Memorial Committee, pp47-66, 1970.
[2] G. G. Joseph. The crest of the
peacock: Non-European Roots of Mathematics. Princeton University Press,
1991.
[3] B. van Nooten. "Binary Numbers
in Indian Antiquity" in T. R. N. Rao and Subhash Kak, editors. Computing
Science in Ancient India, pp. 21-39.
[4] G. Ifrah. The Universal History
of Numbers: From Prehistory to the Invention of the Computer. Translated
from French to English by David Bellos, E. F. Harding, Sophie Wood and
Ian Monk. The Harvill Press, London, 1998.
[5] C. Sagan. Cosmos. Ballantine
Books, New York, 1980.
[6] R. Kaplan. The Nothing that
Is: A Natural History of Zero. Oxford University Press, 2000. Bibliography
and notes: click here.
[7] C. Seife. Zero: The Biography
of a Dangerous Idea. Viking, 2000.
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