Saturday, April 20, 2013

Reality Check: The Power Of Real Mathematics.

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?[1]

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.

[1] Narlikar, Jayant V., The Scientific Edge – The Indian Scientist from Vedic to Modern Times, PENGUIN Books 2003, (p. 26~29).

Tuesday, April 9, 2013

Reaching Out!

"Being dim-witted but aspiring for the success of the learned, I will end up in ridicule, like a dwarf raising his hands towards a fruit that can be reached only by a tall person."

- Kalidasa   (Raghuvansha).
Kalidasa went on to produce a masterpiece after writing this.