Author: NS Rajaram

Publication: The Pioneer

Date: September 23, 2007

The Universal History of Numbers, 3 volumes,
Georges Ifrah, Penguin, $25

Finally it all came to pass as though across
the ages and the civilisations, the human mind had tried all the possible
solutions to the problem of writing numbers, before universally adopting the
one which seemed the most abstract, the most perfected and the most effective
of all." In these memorable words, the French-Moroccan scholar Georges
Ifrah, the author of the monumental but somewhat flawed book, The Universal
History of Numbers, sums up the many false starts by many civilisations until
the Indians hit upon a method of doing arithmetic, which surpassed and supplanted
all others -- one without which science, technology and everything else that
we take for granted would be impossible. This was the positional or the place
value number system. It is without a doubt the greatest mathematical discovery
ever made, and arguably India's greatest secular contribution to civilisation.

The recent publicity over the use of infinite
series by Kerala mathematicians several centuries before Newton and Gregory
has failed to note that it is not a new discovery. CT Rajagopal and K Mukunda
Mura wrote about it in 1944. While the rediscovery of India's contribution
to calculus is certainly welcome, it should not obscure other important contributions
to mathematics coming from India. Of these none is more important than the
modern number system.

This brings us back to George Ifrah's book
mentioned at the beginning. It tells the story of humanity's 3,000-year struggle
to solve the most basic and yet the most important mathematical problem of
all -- counting. The first two volumes recount the tortuous history of the
long search that culminated in the discovery in India of the 'modern' system
and its westward diffusion through the Arabs. From our viewpoint, the second
volume is the most interesting. The third volume, on the evolution of modern
computers, is not on the same level as the first two.

While not without limitations, especially
with regard to alphabetical writing, The Universal History is fascinating
to read. It shows that the term 'Arabic numerals' is a misnomer; the Arabs
always called them 'Hindi' numerals. What is remarkable is the relatively
unimportant role played by the Greeks. They were poor at arithmetic and came
nowhere near matching the Indians. Babylonians, a thousand years before them,
were more creative, and the Maya of pre-Colombian America far surpassed them
in both computation and astronomy. So, the Greek Miracle is a modern European
fantasy.

The discovery of the positional number system
is a defining event in history, like man's discovery of fire. It changed the
terms of human existence. While the invention of writing by several civilisations
was also of momentous consequence, no writing system ever attained the universality
and the perfection of the positional number system. Today, in the age of computers
and the information revolution, computer code has all but replaced writing
and even pictures. This would have been impossible without the Indian number
system, which is virtually the universal alphabet as well.

What makes the positional system perfect is
the synthesis of three simple yet profound ideas: Zero as a numerical symbol;
zero having 'nothing' as its value; and, zero as a position in a number string.
Other civilisations, including the Babylonian and the Maya, discovered one
or other feature but failed to achieve the grand synthesis that gave us the
modern system.

The synthesis was possible because of the
Indians' capacity for abstract thought: They saw numbers not as visual aids
to counting, but as abstract symbols. While other number systems, like the
Roman numerals, expressed numbers visually, Indians early broke free of this
shackle and saw numbers as pure symbols with values.

The economy and precision of the positional
system has made all others obsolete. Some systems could be marvels of ingenuity,
but led to incredible complexities. The Egyptian hieroglyphic system needed
27 symbols to write a number like 7659. Another indispensable feature of the
Indian system is its uniqueness. Once written, it has a single value no matter
who reads it. This was not always the case with other systems. In one Maya
example, the same signs can be read as either 4399 or 4879. It was even worse
in the Babylonian system, where a particular number string can have a value
ranging from 1538 to a fraction less than one! So, a team of scribes had to
be on hand to crosscheck numbers for accuracy as well as interpretation.

The zero was usually indicated by a blank
space until first a dot and then the modern symbol came to be used. It was
more than 500 years before the Indian system made it to Europe. Leonardo of
Pisa, better known as Fibonacci, is credited with being the first to use it
in Europe. It may be said that until the 15th century India was ahead of Europe
in mathematics, but it began to fall behind in the 16th and the 17th century.

A question for historians of science is why
this decline came about. Arab scholar Alberuni claims that the Islamic invasions
drove Indian science away from great centres like Ujjain to places further
south where the hands of invaders did not reach until much later. It is probably
no coincidence that the last great school of mathematics to flourish happened
to be in Kerala, the southern-most State.

Modern India has not produced historians of
science of the first rank. Following an ideological rather than a scientific
approach, Indian historical writings generally tend to be imitative and derivative.
No wonder the most significant work on the Sulbasutras -- 'Vedic mathematics'
-- was done by American mathematician A Seidenberg.

One expects the younger generation of Indian
historians to study India's scientific heritage as earnestly as Ifrah has
done.

The reviewer is currently working on a cultural
history based on the evolution of writing and mathematics