I
have always been at a loss of words about what to say when Japanese people tell
me that all Indians are mathematical geniuses because they memorize more
multiplication tables from a young age, and hence, are born for the IT
industry.
(Needless
to say, the IT industry in India employs less than 0.5% of the Indian
population.)
First,
let us take a look at how both Japan and India have performed at the
International Mathematical Olympiad over the years.
Performance
at the IMO
First participation: 1989.
Number of participations: 24.
Gold medals: 11. Silver medals:
58. Bronze medals: 51. Honourable mentions: 18.
Performance
at the IMO
First participation: 1990.
Number of participations: 23.
Gold medals: 32. Silver medals:
61. Bronze medals: 34. Honourable mentions: 5.
Japan
stands out with almost thrice the number of gold medals than India.
So,
why all this fascination about Indian mathematics?
Surely, the Japanese who memorize
almost 2,000 alphabets to read their own language, can memorize more
multiplication tables if they actually wanted to? Also, despite several drawbacks, the Japanese education system is definitely one of the better ones on this planet. Japan has produced several nobel laureates and has top ranked universities. (None of India's famous universities rank in the top 200.)
However,
there are several books sold in Japan, under the title of Indoshiki or Indian mathematics, which mention tricks of how to
calculate faster. TV programs often show Indian classrooms where students multiply
figures like 333 x 333. Why not more complex numbers, I wonder!
(This apparent "admiration" could soon turn into disappointment when they realize that this so-called ability neither builds problem solving skills nor makes efficient personnel.)
But,
is the ability to calculate faster an essential talent required by
mathematicians?
Jayant
Narlikar, one of India’s top astrophysicists educated at Cambridge University
and whose academic advisor was none other than Sir Fred Hoyle, has lucidly
written on this subject in his wonderful book, The Scientific Edge.
He
further recommends using a pocket calculator to provide fast answers to
numerical problems.
This
should be an ample reply to all foreigners who are inordinately fascinated by the mental
gymnastics of calculating faster or being able to recite multiplication tables
from memory. This is also a probable reason why some foreigners are so quick to glibly and randomly criticize
India at the drop of a hat when things don't always turn out to be as per their expectations, because their understanding is often of a very
elementary level to start with.
The
following is an extract from Professor Narlikar’s highly readable book:
What Is Vedic
Mathematics?
These
words should normally mean mathematics from the Vedas, of the kind we
encountered in the first chapter. However, in the popular mind, the phrase has
been hijacked to mean a particular piece of work, which is described in Vedic Mathematics by Jagadguru
Shankaracharya Swami Shri Bharati Krishna Tirthaji Maharaj, one of the foremost
leaders of the Hindus. Numerous books and articles have since appeared, most of
them written by amateur mathematicians, interpreting the contents, commenting
on them and generally extolling its high mathematical content. Here is a
typical extract from one such publication:
It
contains sixteen simple mathematical Sutras from the ‘Vedas’ and forms a class
by itself, not pragmatically conceived and worked out as in the case of other
scientific works, but the result of the intuitional visualization of
fundamental mathematical truths and principles.
First
of all, can one say that what Swamiji has described is of Vedic origin? The
book was published after the author died, and so we do not have his account of
how he came across the information in the Vedas. The foreword to the book is
given by Manjula Trivedi, a disciple of Swamiji. Neither of these accounts
gives any evidence that the work highlighted in the book is of Vedic origin.
The book contains sixteen Sutras,
(rules, results or formulae) and thirteen Upa-Sutras. In the preface, the
author claims that these Sutras are contained in the parishishta (an appendix) of the fourth of the four Vedas, the Atharva Veda. Unfortunately, no
authorized edition of the Atharva Veda
contains these Sutras.
In this context, S. G. Dani, a
mathematician at the Tata Institute of Fundamental Research in Mumbai who has
written authoritatively on Vedic mathematics, cites an episode narrated by K.
S. Shukla, a renowned scholar of ancient Indian mathematics. He recalled
meeting Swamiji, showing him an authorized edition of Atharva Veda and pointing
out that the sixteen Sutras were not in any of its appendices. Swamiji is said
to have replied that they occurred in his own parishishta and in no other! In short, Swamiji claimed that these
Sutras were Vedic on his own authority, without any independent evidence to
support his assertion. Alas no one, however exalted, has the right or privilege
to add anything supplementary to the Vedas and claim that it is as authentic as
the Vedas themselves, or else there is no authenticity left in any claimed part
of the Vedas as being original to those works.
The Type of
Mathematics
Leaving
aside the question of authenticity, we now consider the book itself. Tf the
contents were remarkable in themselves as judged by modern mathematical
standards, then we could rejoice in the fact that something of Indian origin
(Vedic or otherwise) has turned out to be so advanced.
As an aside, one may compare the
notebook left behind by the Indian mathematician Srinivasa Ramanujan and
discovered in the Trinity College Library. It contained many important and
hitherto unknown results of number theory. Characteristically, many of the
results were simply stated as facts by the illustrious author, and the later
generations of mathematicians spent considerable time and effort proving them.
The results were new and made significant additions to the present literature
in mathematics. So here we have a compendium of results that has passed the
test of being relevant to higher mathematics.
Judged by this standard, the
sixteen Sutras stand nowhere, and it does not require a professional
mathematician to tell you so. One needs, however, to get over the popular
misconception that tricks and short cuts to arithmetical operations like
multiplication, division and finding the square root constitute higher mathematics.
There are gifted persons who can do mental calculations with large numbers very
fast. Public performances (often pitting them against calculators) are very
popular and leave their audiences impressed. There is no question that people
who have these computational abilities can boast of a very remarkable skill,
but they are not mathematicians.
Real mathematics is not number
crunching but the interplay of logical reasoning, starting from relatively
innocuous-looking postulates that leads to profound conclusions. Take, for
example, Euclid’s theorem that the number of prime numbers is infinite. (A
prime number is one which has no other divisors except the number itself and
one.) For example, the first few primes are 2, 3, 5, 7, but a number like 12 is
not a prime since it is also divisible by 2, 3, 4 and 6, besides 12 and 1. So
the question is, does the sequence of primes end. In other words, is there a
last a last prime so that any number greater than it is composite, , i.e.
divisible by a factor other than 1 and the number itself? If such a last number
does not exist, then the number of primes must be infinite.
Euclid proved that this is in fact
the case by the following argument. Suppose we multiply 2 and 3 and add 1 to
the sum. We get 7, and if we try to divide it by 2 or 3, the remainder is 1.
The number 7 is a prime of course. But we cannot guarantee that the product of
all primes starting from 2 and going up to any level, plus 1 will necessarily
be a prime. So Euclid gave this ingenuous argument. Suppose, the number of
primes is not infinite. Then there must exist a prime that is the largest of
them all. Call it P. So, by definition, there is no prime greater than P. Now
consider the number obtained by multiplying all primes up to P and adding one
to the total. This number is
2 x 3 x 5 x 7 x . . . x P + 1.
This
number is clearly not divisible by any of the primes up to P. for, if we divide
it by any of these numbers, we will get the remainder equal to 1. Now let us
examine whether this number is itself a prime or not. If it is a prime, then we
have a contradiction, because this new prime is greater than P, which was
claimed to be a largest prime in existence. If the number is not a prime, then
it has to be divisible by some number. Such a number will either itself be a
prime or prime factor belong to our set of primes going all the way up to P,
because none of these primes divide this number exactly. (recall that they all
leave a remainder 1.) So this prime will have to be greater than P. again we
have a contradiction. So the assumption that the number of primes is finite is
wrong, and their number is indeed infinite.
This argument demonstrates the
power and beauty of logical reasoning that leads to a profound conclusion
without number crunching. Notice that although we used the notion of
multiplication of a large number of prime factors, nowhere did we actually
multiply them out to look at the result!
Thus we can say that although
mathematics started out as an exercise involving numbers, it eventually
expanded far beyond the original concept, and it is the results that command
the widest generality and applicability that get recognition as being profound.
Mere number crunching is not considered a significant part of higher
mathematics.
