Born: about 520 BC in Shalatula (near Attock), now Pakistan
Died: about 460 BC in India
Panini was born in Shalatula, a town near to Attock on the Indus
river in present day Pakistan. The dates given for Panini are pure guesses.
Experts give dates in the 4th, 5th, 6th and 7th century BC and there is also no
agreement among historians about the extent of the work which he undertook.
What is in little doubt is that, given the period in which he worked, he is one
of the most innovative people in the
whole development of knowledge. We will say a little more below
about how historians have gone about trying to pinpoint the date when Panini
lived.
Panini was a Sanskrit grammarian who gave a comprehensive and
scientific theory of phonetics, phonology, and morphology. Sanskrit was the
classical literary language of the Indian Hindus and Panini is considered the
founder of the language and literature. It is interesting to note that the word
"Sanskrit" means "complete" or
"perfect" and it was thought of as the divine language, or language
of the gods.
A treatise called Astadhyayi (or Astaka ) is Panini's major work.
It consists of eight chapters, each subdivided into quarter chapters. In this
work Panini distinguishes between the language of sacred texts and the usual
language of communication. Panini gives formal production rules and definitions
to describe Sanskrit grammar. Starting with about 1700 basic elements like
nouns, verbs, vowels, consonants he put them into classes. The construction of
sentences, compound nouns etc. is explained as ordered rules operating on
underlying structures in a manner similar to modern theory. In many ways
Panini's constructions are similar to the way that a mathematical function is
defined today.
Joseph writes in [2]:-
[Sanskrit's]
potential for scientific use was greatly enhanced as a result of the
thorough systemisation of its grammar
by Panini. ... On the basis of just under 4000 sutras [rules expressed as aphorisms], he built virtually the
whole structure of the Sanskrit
language, whose general 'shape' hardly changed for the next two
thousand years. ...
An indirect consequence of Panini's efforts to increase the
linguistic facility
of Sanskrit soon became apparent in the character of scientific
and mathematical literature. This may
be brought out by comparing the grammar of Sanskrit with the geometry of Euclid - a particularly apposite
comparison since, whereas
mathematics grew out of philosophy in ancient Greece, it was ... partly an outcome of linguistic developments in India.
Joseph goes on to make a convincing argument for the algebraic
nature of Indian mathematics arising as a consequence of the structure of the
Sanskrit language. In particular he suggests that algebraic reasoning, the
Indian way of representing numbers by words, and ultimately the development of modern
number systems in India, are linked through the structure of language.
Panini should be thought of as the forerunner of the modern
formal language theory used to specify computer languages. The Backus Normal
Form was discovered independently by John Backus in 1959, but Panini's notation
is equivalent in its power to that of Backus and has many similar properties.
It is remarkable to think that concepts which are fundamental to today's
theoretical computer science
should have their origin with an Indian genius around 2500 years
ago.
At the beginning of this article we mentioned that certain
concepts had been attributed to Panini by certain historians which others
dispute. One such theory was put forward by B Indraji in 1876. He claimed that
the Brahmi numerals developed out of using letters or syllables as numerals.
Then he put the finishing touches to the theory by suggesting that Panini in
the eighth century BC (earlier than most
historians place Panini) was the first to come up with the idea
of using letters of the alphabet to represent numbers.
There are a number of pieces of evidence to support Indraji's
theory that the Brahmi numerals developed from letters or syllables. However it
is not totally convincing since, to quote one example, the symbols for 1, 2 and
3 clearly don't come from letters but from one, two and three lines
respectively.
Even if one accepts the link between the numerals and the
letters, making Panini the originator of this idea would seem to have no more
behind it than knowing that Panini was one of the most innovative geniuses that
world has known so it is not unreasonable to believe that he might have made
this step
too.
We also promised to return to a discussion of Panini's dates.
There has been no lack of work on this topic so the fact that there are
theories which span several hundreds of years is not the result of lack of effort,
rather an indication of the difficulty of the topic. The usual way to date such
texts would be to
examine which authors are referred to and which authors refer to
the work. One can use this technique and see who Panini mentions.
There are ten scholars mentioned by Panini and we must assume
from the context that these ten have all contributed to the study of Sanskrit
grammar. This in itself, of course, indicates that Panini was not a solitary
genius but, like Newton, had "stood on the shoulders of giants". Now
Panini must have lived later than these ten but this is absolutely no help in
providing dates since we have absolutely no
knowledge of when any of these ten lived.
Panini's grammar has
been evaluated from various points of view. After all these
different
evaluations, I think that the grammar merits asserting ... that it is one of
the greatest
monuments of human intelligence.
Article by: J J O'Connor and E F Robertson
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Panini.html
Panini's grammar (6th century
BCE or earlier) provides 4,000 rules that describe the Sanskrit of
his day completely. This grammar is acknowledged to be one of
the greatest intellectualachievements of all time. The great variety of
language mirrors, in many ways, the complexity of nature and, therefore,
success in describing a language is as impressive as a complete theory ofphysics.
It is remarkable that Panini set out to describe the
entire grammar in terms of a finite number of rules.
Scholars
have shown that the grammar of Panini represents a universal
grammatical and computing system. From this perspective it
anticipates the logical framework of modern computers. One
may speak of a Panini machine as a model for the most
powerful computing system.
Source: Staal, F. 1988. Universals. Chicago: University of
Chicago Press.
In a treatise called Astadhyayi Panini distinguishes between the
language of sacred
texts and the usual
language of communication. Panini gives formal production rules and definitions
to describe Sanskrit grammar. The construction of sentences,
compound nouns etc. is
explained as ordered rules operating on underlying structures in a manner
similar to modern theory.
Panini should be thought
of as the forerunner of the modern formal language theory used to specify
computer languages. The Backus Normal Form was discovered independently by John
Backus in 1959, but Panini's notation is equivalent in its power to that of
Backus and has many similar properties.
http://history.math.csusb.edu/Mathematicians/Panini.html
Before the end of the period of the Sulbasutras, around the
middle of the third century BC, the Brahmi
numerals had begun to appear.
Here is one style
of the Brahmi numerals..
These are the earliest numerals which,
after a multitude of changes, eventually developed into the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 used
today. The development of numerals and place-valued number systems are studied in the article Indian numerals.
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html#s23
Vedic religion gave rise to a study of mathematics for
constructing sacrificial altars, then it was Jaina cosmology which led to ideas
of the infinite in Jaina mathematics. Later mathematical advances were often
driven by the study of astronomy. Well perhaps it would be more accurate to say
that astrology formed the driving force since it was that "science"
which required accurate information about the planets and other heavenly bodies
and so encouraged the development of mathematics.
Religion too played a major role in astronomical investigations
in India for accurate calendars had to be prepared to allow religious
observances to occur at the correct times. Mathematics then was still an applied
science in India for many centuries with mathematicians developing methods to
solve practical problems.
Yavanesvara, in the second century AD, played an important role
in popularising astrology when he translated a Greek astrology text dating from
120 BC. If he had made a literal translation it is doubtful whether it would
have been of interest to more than a few academically minded people. He popularized
the text, however, by resetting the whole work into Indian culture using Hindu
images with the Indian
caste system integrated into his text.
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html#s23
The Indian methods of computing horoscopes all date back to the
translation of a Greek astrology text into Sanskrit prose by Yavanesvara in 149
AD. Yavanesvara (or Yavanaraja) literally means "Lord of the Greeks"
and it was a name given to many officials in western India during the period
130 AD – 390 AD. During this period the Ksatrapas ruled Gujarat (or Madhya
Pradesh) and these "Lord of the
Greeks" officials acted for the Greek merchants living in the
area.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Yavanesvara.html
By about 500 AD the classical era of Indian mathematics began
with the work of Aryabhata. His work was both a summary of Jaina mathematics
and the beginning of new era for astronomy and mathematics. His ideas of
astronomy were truly remarkable. He replaced the two demons Rahu, the Dhruva
Rahu which causes the phases of the Moon and the Parva Rahu which causes an
eclipse by covering the Moon or Sun or their light, with a modern theory of
eclipses. He introduced trigonometry in order to make his astronomical
calculations, based on the Greek epicycle theory, and he solved with integer
solutions indeterminate equations which arose in astronomical theories.
Aryabhata headed a research centre for mathematics and astronomy
at Kusumapura in the northeast of the Indian subcontinent. There a school
studying his ideas grew up there but more than that, Aryabhata set the agenda
for mathematical and astronomical research in India for many centuries to come.
Another mathematical and astronomical centre was at Ujjain, also in the north
of the Indian subcontinent, which grew up around the same time as Kusumapura.
The most important of the
mathematicians at this second centre was Varahamihira who also
made important contributions to astronomy and trigonometry.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Yavanesvara.html
The main ideas of Jaina mathematics, particularly those relating
to its cosmology with its passion for large finite numbers and infinite
numbers, continued to flourish with scholars such as Yativrsabha. He was a
contemporary of Varahamihira and of the slightly older Aryabhata. We should
also note that the two schools at Kusumapura and Ujjain were involved in the
continuing developments of the numerals and of place-valued number systems. The
next figure of major importance at the Ujjain school was Brahmagupta near the
beginning of the seventh century AD and he would make one of the most major contributions
to the development of the numbers systems with his remarkable contributions on
negative numbers and zero. It is a sobering thought that eight hundred years
later European mathematics would
be struggling to cope without the use of negative numbers and of
zero.
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html#s23
