The Beauty of Abstraction

Baalu Kharel

(Freelance jourlist

and short-story writer)

On Lighter Vein

It is sad to see students nurse some strange phobias regarding mathematics. When a child is initiated into the world of mathematics, beginning with numbers, and then with multiplication tables following which division comes, he finds it interesting to a certain extent. But beyond a certain point, he gets bored. Boredom starts when the business of cramming takes deep roots. He has to ‘learn by heart’, as they say – cramming, that is – a whole array of multiplication tables. Until the multiplication table of 10, it is perhaps alright. But beyond that, all hell breaks loose. Then begins fear. All kinds of phobia at their worst.

The unfortute part of the story when it comes to mathematics education in our school system is that children are not inspired to appreciate the complex yet beautiful universe of mathematics. Seldom does a teacher inspire his young learners to go beyond mere memorization of certain facts and figures; that mathematics ultimately becomes synonymous with memorization, and eventually turns out to be a mechanical engagement with numbers playing with themselves through the strange – as it appears to be – routes of addition, subtraction, multiplication and division, is nothing but a mathematical catastrophe.

The other aspect pertains to altertives. The beauty of mathematics lies in altertives. A problem can be solved in more ways than one. There can be multiple approaches to solve a riddle in mathematics. Of these approaches, some may be rigorous than the others; one of them may be the best one – the most rigorous yet the most beautiful one, inspiring new ideas, leading to new insights, new knowledge horizons, and eventually new realities. These realities may even lead to a different stream of mathematics altogether, with huge implications in the real world, especially physics.

But the above relates to mathematics – pure mathematics, to be precise – at higher levels, say at the Master’s or PhD level. When it comes to lower-level mathematics, or school mathematics, things are simpler; yet higher-level ideas can be altered and brought down to jell with lower-level needs. Take a very simple case when it comes to altertives. When a child is asked to multiply 10 with 10, he instantly comes up with the answer, 100 – not because he knows what has really been done when 10 is multiplied by 10, as is the case generally, but because he has memorized it all. If this seems trivial, take the case of 12 being multiplied by 13. If the child knows what it is to multiply, and how, he will take a paper and pencil, and do the work as taught to him by his teacher – a typical methodology we all know. The answer comes out to be 156. The child is correct. And he gets full mark. So, he has knowledge in that case. But is there any altertive way of telling the child what has really happened?

Well, what has happened is that when 12 is multiplied by 13, you are adding 12 to itself as many times! It essentially boils down to addition. In other words, in the case of tural numbers like 1, 2, 3 etc, multiplication is just a manifestation of addition! How many teachers in our schools, including the expensive ones charging hefty fees, teach the young, impressioble lot that way? It is simple, on a lighter vein. If this is done, I believe the child will be greatly fascited. He will realize that there is something more to what he has learnt typically; he will begin to think innovatively; and then creativity will happen – the flowering of which is crucial to the generation of new knowledge, and eventually a better world to live in.

At the higher level, mathematics is total abstraction. Here we do not deal with numbers. We deal with abstract symbols and operations. For instance, the operation of addition takes a different meaning altogether. In highly abstract streams of mathematics such as topology – the study of properties of a geometrical figure which do not change even when the figure is deformed by stretching, bending, twisting etc – all that remains of mathematics is total abstraction. One might wonder what on earth such study will do, how mankind will be benefited, how it can help alleviate poverty, and all such kinds of thing. But the fact of the matter is that topology, at its abstract best, has led to incredible advances in the field of modern physics, and hence modern technology, and without the modern technology we have today, the comforts we are living in would have been mere fantasies of the wildest kind. In fact, quantum mechanics – arguably the most revolutiory world in modern physics – crumbles to pieces without the use of topology. We would not have quantum electronics then, too. And without quantum electronics, how would we have all these sophisticated e-gadgets? At the end of the day, it is the beauty of abstraction. And, as John Keats said so rightly, a thing of beauty is a joy for ever.