A UTILITARIAN APPROACH TO MATHEMATICS

As also in every other subject, planners and educators in mathematics are faced with significant changes and rising expectations in preparing the kind of work force a country will need in the future. The world of work is now less manual but more mental and less static but more varied. Now working smarter is more important than just working harder. We need workers who match this requirement.

There is a general misconception that mathematics is about calculations and symbols. It is due to the fact that the technical trappings of the subject, its symbolism and baffling terminology, tends to obscure its real nature. Hence a common view paves in that, by mastering the skills of mathematical manipulations, one can solve everyday problems. The public gets this wrong impression of mathematics that by the use of an equation or a formula one can solve all the social, economic and environmental problems of a country. It is a matter of historical record that this "utilitarian approach" towards mathematics has ruined a genuine scientific culture which is an important part of the infrastructure for the development of a country.

Let us be sure that mathematics is not about how a few equations or numbers can be manipulated for everyday use. Mathematics, since antiquity, is the result of sophisticated intellectual endeavours.

Had motivational force of mathematics been only its applications, creation of matrices, linear operators or differential equations connected with waves would not have taken place. Mathematics would not have 61 main branches and more than 3,525 sub-branches encompassing theorems concerning characteristics about numbers, shapes, arrangements, movements and chances. Today there are 1,500 professional journals in mathematics publishing about 25,000 research papers a year in more than a hundred languages.

The example of just one single theorem is enough evidence to the fact that mathematics is not about calculations and routine manipulations. The "classification of all finite simple groups" is one such a theorem where its complete proof, developed over a 30-year period by about 100 group theorists, is the union of some 500 journal articles covering approximately 10,000 printed pages. Even these 500 papers required careful selection from among some 2,000 research papers on simple group theory.

Mathematics is about ideas and the way different ideas relate to each other. Abstraction is the real strength of mathematics. In mathematics, the effort should not be just to get the right answer in the shortest possible time. It should be more of a matter of understanding why an answer is possible at all and why it takes the form that it is. For instance, for a mathematician it is not so important that 1 + 4 = 5. Rather it is more important to deduce from here that every prime number of the form 4k + 1 can be expressed as a sum of two squares.

One can indeed put in use Pythagoros theorem to calculate the length of the hypotenuse of a right-angled triangle but more rewarding is to conceive the concept of irrational numbers from the theorem. The first one is straightforward but the second objective requires mathematical insight.

In learning mathematics, intuition is as important as the logic. Intuition allows one to understand why a mathematical deduction must be correct, whereas the logic provides firm bases to show that it is true. Therefore while teaching mathematics, it is important, besides other things, to emphasize on developing mathematical intuition and logical mind. On the other hand, cultivating skills in students to do laborious routine calculations for sake of finding solutions of  "practical problems" will have negative psychological repercussions. Inadvertently we would instil in them a myopic and defective approach towards mathematics, that is, avoid abstraction and always look for so-called "useful mathematics".

It is a historical fact that the spread of utilitarian approach towards mathematics has brought more harm than good. Too much emphasis on this aspect of mathematics can deprive students from having a sound foundation of mathematics. Novel mathematics is not amenable to an industrial approach. We should not let this idea prevail that every result of mathematics should be usable in everyday life. Unrealistic expectations from a subject can eventually create disappointment about the subject. One can imagine the devastating psychological effects on students when eventually they will find that they have learnt only uses of a few tools of the trade at the cost of not learning about the trade itself.

It is therefore imperative that proper and balanced emphasis is laid on the applications of a mathematical concept. One should not inadvertently generate the idea that a mathematical concept or an equation is a panacea. Some very beautiful and significant mathematics also turns out to be useless in practice, because the real world just does not work that way.