MATHEMATICS: AN ALLY OF ANALYTICAL THOUGHT

Published in Dawn, 24th March 2002

It is doubtful whether today within the community of Pakistani mathematicians a consensus can be achieved as to what precisely is the role of a university lecturer in mathematics. They, by and large, are usually indifferent on this issue. Perhaps because they are pre-occupied with matters much more immediate and far less abstract and academic. However, though very small to the extent of being negligible, there are a few who believe that a university lecturer of mathematics should be an instrument of social change. But the rest are unlikely to indulge in a dispassionate explanation of the long-term effects of what is now being done in mathematics.

Before one makes an effort to understand the role of a university lecturer in mathematics, one needs to understand what mathematics is about. As a well-known British mathematician says: “Mathematics is not about symbols and calculations. They are just tools of the trade. Mathematics is about ideas. In particular it is about the way that different ideas relate to each other. If certain information is known, for example, what else must necessarily follow? The aim of mathematics is to understand such questions by stripping away the inessentials and penetrating to the core of the problem. It is not just a question of getting the right answer; it is more a matter of understanding why an answer is possible at all, and why it takes the form that it does.”

The spirit of mathematics is one of free inquiry. It is an expression of the whole man. It breads rationalism, which, in Iqbal’s words, is an ally of analytical thought. Mathematics is a mixture of analytical description and synthetic interpretations. It opens up new dimensions of wisdom.                        

Mathematicians are sensitive to the similarity because pattern and structure are what the subject is all about, and that is the point. Mathematics is not a matter of remembering formulae to do long multiplication, solve quadratic equations and find areas of triangles. Those may be among its raw materials, but mathematics is about identifying patterns, recognising structures, investigating the logical consequences of hypotheses. These skills are necessary before anything else when making a decision, passing a judgement, using a computer or reading the news.

One can argue that there is not enough imagination in the way mathematics is lectured at the tertiary level. If that is the view, then there is clearly something wrong. Every day, many students sit in mathematics lectures which are dull, or which they are not properly equipped to understand. Not unnaturally, they are bored, and the result is that not only do they learn nothing, but they form a negative impression of mathematics as being nothing but a lot of useless formulae.

There is a need to ponder seriously about how to inculcate mathematical sense amongst university students. There is a need to exchange ideas about methods and materials. And there is also a need for a wider public understanding of mathematics and the way it should be taught at all levels. Perhaps the problem is the lack of perception of the importance of mathematics and mathematics-related subjects in society. Attempting to explain that to understand the simple scientific principles one must also explain the esoteric, usually earns one a blank stare.

Mathematics is not mundane and thus should appeal to the imaginative and creative students. The students who lack these qualities, do not continue with it the way it should be learnt. Unfortunately, many of the students lack the open mind to cope with the esoteric parts of the subject.

Mathematicians tend to ignore many specific features of the object in question. There are three modes of thinking that highlight the difference between mathematics and other disciplines: abstraction, deduction, and induction. They are listed in decreasing order of importance for mathematicians, but this would be in increasing order of importance for scientists generally. The process of abstraction involves the sense of pattern recognition. The patterns are not usually the visual ones of everyday experience. The drive to find common themes from disparate areas seems to be part of a mathematician’s temperament.

The process of abstraction is a vital characteristic of mathematical thought probably more distinctive than the method of deduction. Most scientists practice deduction but not to the extent of mathematicians. However, other disciplines are comparatively restricted in the amount of abstraction that they allow themselves. The desire of abstraction seems to be an essential part of a mathematician’s psyche. It is not just a matter of abstracting from the physical world to the mathematical. Mathematicians, in fact, strive to find themselves just the right level of abstraction for a given setting, seeking the perfect balance of the twin goals of utility and generality. Another feature of scientific method is induction, the attempt to generalize conclusions from a number of particular instances. Mathematicians practise this more often than usually realized, however, in a special way.

The very fact is that time spent on teaching has become more rewarding money-wise than perhaps learning and acquiring the ability to develop the three modes of thinking, namely, abstraction, deduction, and induction. In an effort to generate real mathematical culture with intellectualism as one of the most essential parts of it, lecturers and teachers need to emphasis on inculcating the mathematical sense and intellect amongst student rather than the habits of memorization and reproduction of routine mathematical formulae, techniques, and theorems.

The pursuit of mathematical knowledge is a rewarding intellectual exercise in its own right. As the renowned British mathematician, Professor Graham Higman FRS, said in a PTV interview with the author during the former’s visit to Pakistan in 1987: “We do fundamental research, not only to acquire results solely but because the process is an ennobling one – it is one that makes you more worthwhile than before; it is something that if you cut yourself off, you are making yourself less human than you ought to be.”

Only if one takes up this responsibility can one avoid that contempt and disrespect for oneself of those who create knowledge today.