HARMFUL TRENDS IN MATHEMATICS
Published in THE MUSLIM, 15th May 1992
No country can develop scientific infrastructure for economic development without relying on mathematics. Meeting of the new challenges of the modern scientific age depends in part on the quality and quantity of its mathematical research.
The ancient Pythagoreans had classified mathematics into ten branches/sub-branches (see table 1). Since the, research in mathematics has grown so much that according to the American Mathematical Society (AMS) Subject Classification 1990, there are over 61 main branches of mathematics and more than 3,525 sub-branches. Today there are some 1500 professional journals in mathematics publishing about 25,000 research papers a year in more than a hundred languages. These figures give us an insight into the horizontal and vertical expansion of mathematics today.
This expansion has been the greatest during the last two centuries. 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 are also the most revolutionary in the history of mathematics.
One major problem of mathematics is to explain to the layman what is it all about. The technical trappings of mathematics, its symbolism and terminology tend to obscure its real nature.
As 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.
It is not so important for a mathematician that 1 + 4 = 5. A mathematician’s aim is to make use of particular examples and come out with a useful such as with reference to the above example, that every prime number of the form 4k + 1 is a sum of two squares.
One can consider another example of the famous Pythagorean theorem, which had tremendous effect on later ages of mathematicians. When particular values were substituted in the formula of Pythagoras, a new generation of numbers came into existence known as irrational numbers.
Similarly the solution of a particular equation x2 + 1 = 0 yielded a new type of numbers now known as complex numbers. This throws light on the fact that particular cases are important for the mathematicians to the extent that they form bases for the development of mathematical ideas, which are deep and useful.
The driving force in mathematics is problems. A good problem is one whose solution opens up entirely new vistas, as was the case in the few examples mentioned earlier. Most good problems are difficult but not all difficult problems are good.
Mathematical ideas have a long, lifetime. The Babylonian solution of quadratic equations is as fresh and useful now, as it was 4000 years ago.
The calculus of variations first bore fruit in classical mechanics yet survived the quantum revolution unscathed. The way it was used changed but the mathematical ideas have remained at the forefront of mathematical research since his death in the 19th century. Thus mathematical ideas have permanence that perhaps the physical theories lack.
In Pakistan, research activity in mathematics has been growing since independence. The number of research papers produced per year has gone up. Considering the fact that the adequately educated mathematical manpower in Pakistan is far less than what it should have been, the production and quality of research papers by international standard is fairly good.
But this production is primarily due to sporadic efforts by a few mathematicians. Not all researchers in mathematics in the country are producing good mathematics. Research papers produced are generally published in obscure professional journals.
Very often, papers are published for the sake of increasing the quantity of papers, or for the sake of having more papers to one’s credit. Moreover, mathematical research in Pakistan is basically repetitive (repeating what others in other countries have already done) and contains what is generally known as routine mathematics.
Over the years, several trends have developed in mathematics in Pakistan, which are damaging to the development of mathematics in the country. One of these is the utilitarian approach towards mathematics. There is too much emphasis on the development of mathematics which can be used for research and development of our industry and science and technology. Really good mathematical ideas are hard to come by. They result from the combined work of many people over long periods of time. Such results cannot be produced at will.
Novel mathematics is not amenable to an industrial research and development approach. But the results pay for all the effort by their durability and versatility. As Professor Sharp has said: There is nothing deadlier for a mathematician that to be placed in a beautiful office and be instructed to lay golden eggs. Creativity is never directly sought after. It comes indirectly.
Mathematical laws are almost in conformity with the laws of nature. For instance in the giant sunflower (helianthus maximus), the florets naturally arrange themselves into logarithmic spirals. Another example is seen in the human body where various measurements of the human body exist in certain proportionality.
This notwithstanding, it is dangerous to have an industrial approach towards mathematics or any other science for that matter. We should not expect every result of mathematics to be usable in everyday life. Such an attitude is harmful for creativity, which demands the full mental activity of a mathematician without any social, economic, or psychological constraints. The flower of creativity can only stem from an environment which is not polluted with these constrains.
The utilitarian approach in mathematics should at best be only expected of the applied mathematicians. But the irony is that on the one hand the so-called applied mathematicians themselves have no idea what pure mathematics is about. They think that 'pure mathematics is bad mathematics' and that 'number theory is no theory'. Whereas they themselves are producing research which is abstract and theoretical to the extent that the only use of it is that it is intellectually stimulating. Yet they expect pure mathematicians to be application oriented.
Most people, even some cultivated scientists, think that mathematics applies because you learn theorem 'A' and the theorem A some how mysteriously explains the laws of nature. That does not happen even in science fiction novels. The results of mathematics are seldom directly applied; it is the definitions that are really useful. What applies is the cultural background one gets from a course, not the specific theorems taught in the subject.
There should not be such a sharp distinction between pure and applied mathematics. At least keeping in mind the kind of mathematical research that is being done in Pakistan. One needs extraordinary wisdom and competence to draw the line of 'usefulness'. Such a division or controversy in mathematics is decreasing its utility and research activity.
The other harmful trend in mathematics in Pakistan is the tendency to use 'flexible' yardsticks to judge the worth of one's research work. The worth of research work of a mathematician cannot be judged by counting the number of his papers nor can one say that those who have less number of papers will certainly have more worthwhile papers.
We have examples in the past of great mathematicians like Euler, Cayley and Cauchy, who had around 1000 paper each, yet more or less every paper of theirs turn out to be seminal. Then we have examples of mathematicians like Burnside, Gian-Carlo Rota and Sir Michael Atiyah who have comparatively less number of papers, and more or less, all of them are significant. In Pakistan different yardsticks are being used for different people depending on whether one is in favour or against that person whose work is being judged. Professor Gian-Carlo Rota says: 'There is a ratio by which one can measure how good a mathematician is and that is how many crackpot ideas he must have in order to have one good idea. If it is ten to one then he is a genius. For the average mathematician, it may be one hundred to one'.
The fact is that because of the dearth of mathematicians in Pakistan working in any one same field, mathematicians in general are neither qualified nor competent enough to judge another specialty. Yet irresponsible sweeping statements are often pass on one another's research work and when a mathematician passes judgment on another mathematician's work, non-mathematicians will believe it as a divine verdict.
The effect and impact of this kind of behaviour on the growth of mathematical research is harmful to say the least. The deserving mathematicians do not get due encouragement while the undeserving flourish in the country. Is this an age of mediocrity which ruined 19th century Norwegian mathematician Abel during his lifetime? Or are our mathematicians facing what the 19th century French mathematician Galois faced?