Wonderful world of mathematics

There is an apocryphal story about Leonhard Euler, a brilliant mathematician of the 19th century, in the court of Catherine the Great, the empress of Russia. Denis Diderot, the French philosopher, known for his atheistic views, visited the royal court in St. Petersburg engaging in philosophical arguments. The bottomline of Diderot’s arguments was that God does not exist. Since none of the Russian philosophers could match Diderot in the depth of their arguments, the empress called upon Euler to engage in disputations with the visiting scholar. Euler had found out that Diderot’s weak point was mathematics. So when they began their disputation, Euler wrote a simple mathematical formula and said that in view of the equation written it follows that God exists. When called upon to reply, Diderot was completely flummoxed by this line of argument, as he had no clue to the mathematical content of what Euler had written. He asked for time to reply and then quietly made his way to Paris.

This story indicates that mathematics is beheld as a difficult subject, not a cup of tea for the common man or woman. This has resulted in pressures, social as well as intellectual, that its teaching should be reduced to the bare minimum so that the typical student has no difficulty in passing the school leaving examination. Any action by the educational authorities to ease the pressure of mathematics is welcome to the typical students and their guardians. This type of development, however, goes in opposite direction to the role mathematics is playing in the modern times. Although mathematicians like to claim that their subject is abstract with no visible connection to any tangible effect on human existence, it is well known that many abstract results of mathematical origin have tangible links with real life. Indeed, the rapid growth of science and technology that is influencing our existence today is intimately related to the growth and progress of mathematics. An example will illustrate how important is the role of mathematics in technological development.

In 1957, the Soviet Union launched its first satellite, the Sputnik, which marked the advent of space technology. However, the launch caused a great deal of shock and consternation in the US, mainly because against the Cold War background the Soviet feat appeared sinister. It represented the advance of space technology in that country compared to that in the US. For strategic reasons, the US felt that its own space programme must be boosted. Several steps were taken to reduce the implied gap with respect to the USSR. One of the most serious steps was to boost the teaching of mathematics all the way from school to the research level. It was felt that technology cannot flourish without the backing of science and science cannot flourish without the backing of mathematics. An excellent introduction to the multifaceted applications of mathematics is seen from the four-volume compilation of mathematical ideas in the book, The World of Mathematics by James R. Newman.

It shows how mathematics has spread in so many unexpected areas. An interesting chapter is on the Battle of Trafalgar, which was won by Admiral Nelson in the year 1805. Nelson had 40 ships while the attacking French and Spanish armada had 46. In a head-on confrontation, Nelson figured that his smaller fleet would be annihilated. So he followed the following strategy. He divided the enemy armada into two equal parts and sent a diversionary force of eight ships to fight one half of the enemy armada. The other half of the enemy fleet of 23 he attacked with 32 of his remaining ships. This latter encounter wiped out the enemy fleet and left Nelson with 22 ships. In the first encounter he lost all ships but reduced the enemy ships to 21. Thus in the second part of the battle his 22 boats won by destroying all 21 of the enemy boats. Of his own fleet five survived.

This divide and attack policy came to Nelson intuitively as a result of his long experience. But in modern times one can understand the process by mathematical modelling. This has been discussed by Newman and the model leads to the following simple result. Suppose two forces are fighting with similar weaponry but with unequal starting numbers of fighting units. To fix ideas suppose force X has a starting strength of 40 and the opposing force Y has strength 32. The mathematical solution then tells us that as the fighting proceeds the forces lose strength but in such a way that the difference of their squares stays constant. So in the above example the difference of squares is 576, which stays fixed. So when force Y is annihilated the strength left with force X is the square root of 576, that is, 24. By the same law if Nelson had opted for a direct encounter, he would have lost all ships while the enemy would have been left with around 21-22 ships. As it turns out in this example, the application of the mathematical model leads us to a result very similar to what Nelson obtained intuitively.

The use of mathematics in daily life has come in an unexpected way, so much so that even the creators of a mathematical result are taken by surprise. For example, the theory of numbers has results about prime numbers — that is numbers that are not divisible by any number (except one or the number itself). The eminent mathematician G.H. Hardy confidently asserted that one of his highly abstract results would have no application at all. He was proved wrong! Today very large primes are used as keys to codes that transmit very confidential messages. Thus to avoid interaction with mathematics on the grounds that it is a difficult subject is an altogether wrong approach. That way we would be missing a key subject that is indispensable to the development of any nation. Rather one should improve the technique of teaching it. With a good teaching method and an imaginative textbook one can open out a vista of delights in the world of mathematics.