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Sunday July 21, 2024

The beauty of mathematics

Mathematician and philosopher Alfred Whitehead explains this best

By Sarmad Khawaja
January 27, 2024
This representational image shows a person solving a maths problem. — Pexels
This representational image shows a person solving a maths problem. — Pexels 

Public discourse is mind-boggling these days: economists peddle political manifestos; politicians debate arcane economic issues; lawyers don’t make sense whatsoever; and rational persuasion gives no return.

So, it is mighty encouraging to see our mathematicians talking about mathematics and promoting best teaching practices. Last Friday, they gathered in Islamabad to award the prestigious Medal of the Pakistan Mathematical Society posthumously to Prof Khwaja Masud (1922-2010) for his “exceptional contributions to various branches of knowledge and his profound impact on the field of mathematics”.

Dr Qaiser Mushtaq, the internationally honoured mathematician and the society’s president, said that Prof Khwaja Masud inspired thousands of students since 1944, instilled in them a deep passion for mathematics, fostered a vibrant and dynamic educational atmosphere, and left an enduring legacy through his profound impact on mathematics, history of science, philosophy and politics.

His speech is a student’s heartwarming gratitude for his teacher who encouraged his students to embrace the excitement of mathematics. That is one precious gift an inspired teacher gives to an eager student. Through his passion, knowledge and high expectations, the teacher stimulates the critical abilities of the student, encourages her/him to produce excellence; and sets her/him off on a life-long journey in search of the beauty of mathematics. As St Augustine wrote, “One kindly spirit sets another on fire”.

But what is this excitement of mathematics? It is the ability to appreciate its beauty, which Bertrand Russell says is supreme, cold and austere – like that of sculpture – yet sublimely pure and capable of perfection such as only the greatest art can possess. Mathematics is beautiful because it precisely reflects the harmony, elegance and beauty of the natural world.

But to be able to appreciate this beauty and grasp its excitement you need much background and effort. It is a skill taught skillfully.

Here’s the difficulty: mathematics is all about abstraction; about stripping ideas of detail, that is, removing the fluff we comprehend with our senses, and reducing them to numbers, which we cannot easily comprehend; about manipulating these numbers, seeking patterns and teasing out the underlying structures. The primary aim being to make everything as simple as possible.

Mathematician and philosopher Alfred Whitehead explains this best. He says that abstraction gives mathematics beauty and immense power: as it withdraws into the upper regions of ever-greater extremes of abstract thought, it returns to earth with a corresponding growth of importance for the analysis of concrete facts. These utmost abstractions are true weapons with which to control our thoughts of concrete fact. And they give mathematics the key to mastering not only natural but social science, and to paving the way for scientific advancement.

So then, the proper way to teach mathematics is by taking the students into the realm of abstraction step by step starting with the concrete reality where it arose.

The bad news is this does not happen in our schools, colleges and universities. And so, our children do not see the beauty that is mathematics. My heart sinks when I find they do not even know that ‘zero’ was discovered in our region; that this region is at the heart of the history of numbers; that Al Beruni measured the Earth’s circumference near Pind Dadan Khan using trigonometry now taught in schools.

Such ignorance has serious implications. It means losing grip on almost anything that matters, from understanding the grand book of the universe, which Galileo said, is written in the language of mathematics to everyday commerce and industry which deal with facts and quantities, with theories and ideas. In the modern world you need to count and calculate, says Will Durant, measure and design, with competitive accuracy and speed, which is impossible without mathematics.

Our indifference to mathematics is costly. Our faulty curriculum is costly. And so is our dreadful way of teaching the sublime subject of mathematics. As a result, we lose too much talent.

How did we end up like this? The answer is another story. But I do not see any way to regard all this as anything but a disastrous failure of our mathematics education.

We disregard the importance of mathematics at our peril. Now, to the other gift of a good mathematics teacher: it is educating for liberation. That is, making the students aware of both the socio-cultural reality which shapes their lives and of their capacity to change that reality.

It means education is a double-edged sword. There is education for pacification, and there is education for liberation. So, there is no neutrality in education. For 200 years, our education has pacified us. It subordinates us to an unjust status quo. It numbs our minds to injustice all around us, to widespread poverty, rampant illiteracy and child malnutrition, and to the ill fates of countless men, women and children who toil and perish in obscurity.

Our economic, political and legal systems have deep marks of pacification. And our schools, colleges and universities continue this unholy undertaking.

Education for liberation means throwing out the yoke of subordination. It means embracing rationality, objectivity and humanity. It means living for these principles and making your life and conduct an example to follow.

These two gifts – the excitement of mathematics and educating for liberation – like intertwining threads make up the fine mosaic of good teaching. That is how you teach mathematics. And popularize it. History is evidence that no country advances economically or otherwise without a mastery of mathematics. A nation strong in mathematics is bound to be strong in all realms of knowledge.

The good news is this is a wonderful moment for the Pakistan Mathematical Society. Kudos to its members for spotlighting these two precious gifts of its finest teacher, and pulling the country’s centre of gravity more towards rationalism, objectivity and humanism when all around us is ‘sound and fury,’ ‘signifying nothing.’


The writer is a freelancecontributor. He can be reached at: Khwaja.Sarmad@gmail.com