MATHEMATICS LIKE THE TAJ MAHAL

BORNEO BULLETIN, 30th June 1994

            Meeting the new challenges of the modern scientific age depends in part on the quality and quantity of its mathematical research. Mathematics holds the key to the mastery of not only the natural sciences but the social sciences as well. Consequently it is the key to the understanding of the laws of nature and society. It has a deep impact on social development.

Mathematical knowledge has been growing with enormous speed. The expansion has been the greatest during the last two centuries. By one estimate, the additions to mathematics in the 19th and 20th centuries, both in quality and quantity, far outweighed the total combined productivity of all preceding ages. These centuries probably are also the most revolutionary in the history of mathematics.

The following statistical data can perhaps give us some idea as to what extent mathematics has expanded. The ancient Pythagoreans had classified mathematics into four branches, namely, arithmetic, geometry, astronomy and music. Since then, research in mathematics has grown so much that according to the American Mathematical Society's Subject Classification, there are over 61 main branches of mathematics and more than 3,525 sub-branches. Today, there are some 1,500 professional journals in mathematics publishing about 25,000 research papers a year in more than a hundred languages.

Mathematics is a difficult subject. Its technical trappings, its symbolism and terminology tend to obscure its real nature. As a British mathematician Dr. Ian Stewart has said: "Mathematics is not about symbols and calculations. These are just tools of the trade. Mathematics is about ideas; in particular it is about the way different ideas relate to each other".

Mathematics of the 20th century is the result of a sophisticated intellectual activity. Much of the subject that today is known as mathematics is an outgrowth of thought that originally centered in the concepts of number, shape, arrangement, movement and chance.

           Because of its complexity, technical language, symbolism and rigour of logic, teaching mathematics is not a job which can be easily accomplished. As the Pakistani Professor of Mathematics Khwaja Masud has said: "In mathematics, a bad teacher, can very easily reduce the Taj Mahal into a heap of stones."

The German Professor of Mathematics, Felix Klein, who was the most outstanding teacher of his day, made a strong plea for a humanistic along with a rigorous approach to the teaching of mathematics. Such a teaching, he said, should take full cognizance of the present state of the subject, of its long history, of its applications, of its plastic beauty, of its relations with philosophy and of its deep impact on social development.

The secret of mathematical power lies in the process of abstraction which means the process of stripping an idea of its concrete accompaniments. By doing so, we free our minds from burdensome and irrelevant details. Therefore, we are able to accomplish more than if we have to keep the whole physical picture before us. Thus objects become less important than the relations between these objects. The paradox is now fully established that the utmost abstraction and logic are true weapons with which to control our thought of concrete fact.

The British mathematician Professor Alfred N.Whitehead puts it: "Nothing is more impressive that the fact that as mathematics withdrew increasingly into the upper regions of ever greater extremes of abstract thought, it returned to earth with a corresponding growth of importance for the analysis of concrete fact."

Abstraction is the real strength of mathematics. For students of mathematics, it becomes a bugbear. It is the duty of a teacher of mathematics to rid his students of this baseless fear by taking them into the realm of abstraction step by step, first taking them to the concrete reality which gave birth to abstraction. In mathematics an effort should not be just to get the right answer; for a teacher it should be more of a matter to make students understand why an answer is possible at all, and why it takes the form that it is.

The trend of greater generality has been accompanied by an increased standard of logical rigour. Logical rigour provides a restraining influence which is of great value when dealing with subtleties which are an important part of mathematics. The more complicated and extensive the subject becomes, the more important it is to adopt a critical attitude.

Mathematics and its implications can have full impact on our scientific thinking only if it is properly taught and understood. If a theorem is proved by enormous calculations, the result is not properly understood until the reasons why the calculations work can be isolated.

In learning mathematics intuition is as important as the logical. An intuitive proof of a theorem or a solution of a problem allows one to understand why the theorem must be true or the solution must be correct, whereas the logic provides firm grounds to show that it is true.

          Therefore while teaching mathematics, it is important, besides other things, to emphasis on developing mathematical intuition and logical mind rather than cultivating skills in students to do laborious routine calculations. In this way mathematics will look like the Taj Mahal and not a heap of stones.