I begin by sharing an experience which, apart from being funny is deeply worrying. It pertains to the strange spectacle of an encounter with someone who upon knowing that you are involved in teaching/ conducting research in mathematics waxes lyrical in taking inexplicable pride in saying that he was always weak in mathematics during his school days, notwithstanding whether he has or has not achieved success in his chosen profession. What is uniquely strange about this experience is that it is peculiar to mathematics and one doesn't normally get to hear such things being spoken about other sciences like physics, chemistry, life sciences or about literature, history etc.! Part of the reason for this willingness to gloat over what has come to be recognized as a serious handicap in an individual's educational background may perhaps be pinned down to their understanding of mathematics as an endeavor which is:
(a) Not terribly important for meaningful education and hence could be dispensed with.
(b) Too difficult for people with normal intelligence and ought to be the preserve of the rare crop of geeks having a natural flair for mathematics.
The fact, however, remains that neither of these assumptions serves to explain such perceptions about mathematics being in vogue in the public. A more reasonable explanation for such perceptions to exist in a large segment of the educated public is surely rooted in the poor quality of teachers having taught them mathematics at the school level. The attendant malady involving math phobia follows as a natural corollary which again is a condition that is typically encountered in mathematics and not in other academic disciplines. Rather than take measures to identify and address the reasons for the preponderance of math phobia among the students, there are reports that the J&K Board of School Education is contemplating an 'alternative' to mathematics at the high school level in order to arrest the dropout rate among girl students in government schools. This amounts to throwing the baby with the bath water as it also underscores the abysmally low level of understanding of those who seem to be completely clueless about the whole thing. Much as one may argue that the primitive methods of teaching as they are in vogue in our schools at the moment being among the many reasons for poor performance of students in maths, perhaps one of the most effective ways to address the scourge of math phobia would be to appoint qualified but highly motivated individuals as teachers in our schools. That would necessitate a paradigm shift in the current recruitment system as it prevails in our educational institutions which is deeply flawed and faulty to the core.
Apart from this twin phenomenon involving the inexplicable pride in being bad in mathematics and the phenomenon of math phobia, there is yet another feature which is unique to the enterprise of mathematics. This pertains to the perception of beauty which is generally believed to be the chief motivation for studying and pursuing mathematics – or science, for that matter – as a career option. As opposed to the physical beauty which appeals to the physical eye, the beauty that one talks about in mathematics comes from the abstract nature of the subject and is quintessentially intellectual, and also demanding. Among other things, it comes from the harmonising order of its parts and is earned and experienced through active engagement, unlike a game in sports where a spectator enjoys the game as much as the players, even without the need to participate as a player. The question, therefore, arises: what is it that entails 'active engagement' in mathematics.
A true appreciation of mathematics doesn't come from the mere understanding of the statement of a theorem, but in the highly demanding requirement of knowing what is actually going on which comes only through the grind of making sense of its proof in all its fine details. In that sense, one may get a fair amount of idea underlying a mathematical statement/theorem without the tedium of getting to know its proof in much the same way as, say in the case of a complicated gadget or a machine which one can use without having to understand the intricate technology that has gone into the making of that device. The statements involving the well-known theorem of Pythagoras on the one hand and that of Fermat's last theorem (FLT) on the other are prototypical mathematical facts which can be explained to the 'first man in the street', as it were. And whereas the proof of Pythagoras can be made intelligible to anyone with an exposure to pre-high school maths, the proof of FLT which was achieved in 1995 and hailed as the crowning glory of the mathematics of the last century is beyond the ken of some of the most noted mathematicians including number theorists who have not been actively involved in that part of number theory where FLT truly belongs! In that sense, mathematics comes across as a great leveller. To drive home the latter point, I can't do better than bring up an innocuous looking problem dealing with as familiar an object as the positive integers which has defied the efforts of some of the best mathematicians who have tried their hand at this problem. The problem is based on the following simple observation: pick up a positive integer n at random. Divide it by 2 if it is even, otherwise consider the number 3n+1 and repeat the process. One would observe that regardless of whichever number you start with, the process ends at 1 (as the last quotient)! As an example take n=6. The above process leads to the following chain of numbers which ends at 1: 6, 3, 10, 5, 16, 8, 4, 2, 1. This simple observation has led mathematicians to suspect that the statement is valid for all positive integers and has come to be known as Collatz conjecture in the literature. The point is that in mathematics an empirical observation which may be seen to hold for a huge sample set of individual objects will not qualify to be considered part of the mathematical truth, until a rigorous and flawless proof has been devised and verified to be valid within the logical framework of mathematics.
This takes us back to the question involving the pursuit of mathematics as an endeavor essentially devoted to the study and understanding of abstract ideas and structures. What use is the act of doing mathematics for its own sake, without any care for its use in real life situations? There is a part of mathematics, the so-called applied mathematics which seeks to address itself to such situations and which has been such a spectacular success both in diverse areas of science and technology, primarily on account of the human understanding of mathematics as an abstract science. One can't do better than quote what famous theoretical physicist Victor Weisskopf had said during a talk he had given in Brussels way back in April 1964:
"The value of fundamental research does not lie only in the ideas it produces. If science is highly regarded and the importance of being concerned with the most up-to-date problems of fundamental research is recognized, then a spiritual climate is created which influences all other activities. An atmosphere of creativity is established which penetrates every cultural frontier. Applied sciences and technology are forced to adjust themselves to the highest intellectual standards, which are determined in pure research; that is what attracts productive people and brings productive scientists to those countries where science is at its highest level. Fundamental research creates the intellectual climate in which our modern civilization flourishes. It pumps the lifeblood of ideas and inventiveness not only into the technological laboratories and factories, but into every cultural activity of our time".
Sadly, there is a universal, albeit a pathological trend across countries in the world to reduce funding in fundamental research. There already are signs of deleterious effects of this trend being visible in certain research centers of repute devoted to fundamental research in basic sciences which have suffered on account of the funds having dried up in recent years. The effect has certainly been far more pronounced in India where the lopsided emphasis on research that would directly impact the society and the life of its citizens has driven hordes of even talented scientists towards market driven research. That betrays a shallow and a very poor understanding of things by our policy makers. Here again, Weisskopf comes up with yet another plea where he so forcefully argues the case for generous state funding in fundamental research:
"A small part only of a nation's total income is needed to keep fundamental research in full swing. It would be wrong to try to save a fraction of this small part if such savings weakened the most vital and active part of our intellectual life, the part which we all should regard with pride as one of the highest achievements of our century".
I hope and pray that these lessons are not lost on the policy pundits in our own state of Jammu and Kashmir.
Prof M A Sofi teaches at the Department of Mathematics, Kashmir University, Srinagar.